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About this book

The volume includes papers from the WSCMO conference in Braunschweig 2017 presenting research of all aspects of the optimal design of structures as well as multidisciplinary design optimization where the involved disciplines deal with the analysis of solids, fluids or other field problems. Also presented are practical applications of optimization methods and the corresponding software development in all branches of technology.

Table of Contents


Part I: General Approaches and Strategies: Multi-Disciplinary Optimization


Multidisciplinary System Optimisation on the Design of Cost Effective Space Launch Vehicle

This paper presents the methodology and the optimization strategy applied by Bertin Technologies for over 10 years to perform space launch vehicle design and implemented by using the property software platform HADES V15.0. The problem formulation consists in finding the best launch vehicle concept i.e. the one maximising performances (payload mass on final orbit) and minimising launch cost while satisfying technical, mission and architecture constraints. The strategy is based on a Multidisciplinary Design Feasible (MDF) approach coupled with the use of Genetic Algorithms (GA) for global optimization, and Gradient-Based Algorithms for final tuning and results refining. HADES V15.0 platform provides the associated software environment integrating a number of technical and economic modules consistently interconnected within a system optimization loop. The main disciplines taken into account in the platform are related to the launcher’s propulsion, structure, aerodynamics, trajectory optimization and cost. The use of an integrated platform for multi-objective and multi-disciplinary optimization enables an efficient process and quick optimization. This methodology is particularly well fitted to the design of a small space launch vehicle, allowing to take into account the multidisciplinary nature of such a complex system and to manage the inherent sensitivity for this kind of vehicle. The application case presented was used to design Bertin Technologies’ cost-effective expandable Space Launch Vehicle (SLV) for Microsatellites, ROXANE.

Cédric Dupont, Andrea Tromba, Sophie Missonnier

Multidisciplinary Design Optimization of Body Exterior Structures

Multidisciplinary Design Optimization (MDO) uses optimization methods to solve design problems by incorporating all relevant disciplines simultaneously. In vehicle engineering the objective typically is to optimize weight while maintaining performance involving various load cases from multiple CAE attributes. The MDO process can be divided into three successive steps: (1) data generation entailing creation, submission and post-processing of various DoE’s, (2) meta-modeling and optimization, and (3) validation of optimization proposals. Recent efforts have been targeted at improving the efficiency of the “meta-modeling and optimization” phase in terms of throughput time and quality of solutions by the development of:Automated preparation and submission of CAE models to the HPC cluster.Automated meta-modeling producing reproducible, high-quality meta-models based on Gaussian Processes with Automatic Relevance Determination at reduced HPC work load.An enhanced NSGA-II algorithm for constrained multi-objective optimization.These developments resulted in 20% reduction in throughput times in conjunction with further weight saving potential. The viability of these improvements is illustrated by findings in recent optimization projects.

Michel H. J. W. Paas, Hessel C. van Dijk

An Augmented Sequential Optimization and Reliability Assessment for Reliability-Based Design Optimization

Involving inevitable uncertainties in an engineering system design optimization is a necessity and requirements to achieve a reliable design which reliable design optimization approaches are expressed as Reliability-Based Design Optimization (RBDO) methods. Despite tremendous efforts have been made in the field of RBDO, computational efficiency improvement is still a significant challenge. Sequential Optimization and Reliability Assessment (SORA) has made great efforts to improve computational efficiency by decoupling RBDO problem into sequential deterministic optimization and reliability assessment as a single-loop method. In this paper, in order to further improve computational efficiency and extend the application of the current SORA method, an Augmented SORA (ASORA) is proposed by refraining from reliability assessment for satisfied probabilistic constraints in each cycle until all probabilistic constraints are be satisfied. The accuracy and efficiency of the proposed method compare to the SORA method is illustrated through some single and multi-disciplinary mathematical examples and a multi objective example.

Jafar Roshanian, Ali A. Bataleblu, Benyamin Ebrahimi, Ali A. Amini

Metamodel-Based Multidisciplinary Design Optimization of a General Aviation Aircraft

Computational burden is still a significant challenge in the in multidisciplinary design optimization (MDO) of complex engineering systems. This challenge can be arising from the curse of dimensionality of the design space and the multiplicity of disciplines involved in the design problem. Tremendous efforts have been made to improve the computational efficiency, especially in the field of MDO. Meta-modeling is one of the powerful tools to facilitate this problem and has been received increasing attention in the past decades. Meta-models are used to provide simpler models instead of the complex original models and by admitting a small percentage of error reduces computing time of the problem. Kriging meta-model, due to its high efficiency in medium dimension problems has been attracted the attention of many researchers. Due to lack of continuity in the complex design problems, creating a comprehensive and appropriate meta-model with acceptable accuracy to cover the entire design space is difficult and almost impossible. This paper proposed a strategy to improve the accuracy of the created meta-models using the elimination of outlier data from sampled points and re-designing the effective Kriging meta-model parameters. The proposed strategy is applied to the conceptual design of a General Aviation Aircraft (GAA) using MDO methodology and appropriate Kriging meta-model. Meta-models of the design disciplines including propulsion, aerodynamics, weight and sizing, performance criteria and stability disciplines are created and integrated based on Multidisciplinary Design Feasibility (MDF) structure to improve the aircraft performance. The gross weight of the aircraft and cruise phase range are considered as the objective functions. The NSGA-II multi-objective evolutionary optimization algorithm is utilized to demonstrate a set of possible answers in the form of the Pareto front.

Jafar Roshanian, Ali A. Bataleblu, Mohammad H. Farghadani, Benyamin Ebrahimi

How to Deal with Mixed-Variable Optimization Problems: An Overview of Algorithms and Formulations

Real world engineering optimization problems often involve discrete variables (e.g., categorical variables) characterizing choices such as the type of material to be used or the presence of certain system components. From an analytical perspective, these particular variables determine the definition of the objective and constraint functions, as well as the number and type of parameters that characterize the problem. Furthermore, due to the inherent discrete and potentially non-numerical nature of these variables, the concept of metrics is usually not definable within their domain, thus resulting in an unordered set of possible choices. Most modern optimization algorithms were developed with the purpose of solving design problems essentially characterized by integer and continuous variables and by consequence the introduction of these discrete variables raises a number of new challenges. For instance, in case an order can not be defined within the variables domain, it is unfeasible to use optimization algorithms relying on measures of distances, such as Particle Swarm Optimization. Furthermore, their presence results in non-differentiable objective and constraint functions, thus limiting the use of gradient-based optimization techniques. Finally, as previously mentioned, the search space of the problem and the definition of the objective and constraint functions vary dynamically during the optimization process as a function of the discrete variables values.This paper presents a comprehensive survey of the scientific work on the optimization of mixed-variable problems characterized by continuous and discrete variables. The strengths and limitations of the presented methodologies are analyzed and their adequacy for mixed-variable problems with regards to the particular needs of complex system design is discussed, allowing to identify several ways of improvements to be further investigated.

Julien Pelamatti, Loïc Brevault, Mathieu Balesdent, El-Ghazali Talbi, Yannick Guerin

Comprehensive PHEV Powertrain Co-design Performance Studies Using MDSDO

Plug-in hybrid electric vehicle (PHEV) powertrain design is a complex process that requires an approach which enables the simultaneous integration of component design and powertrain control strategy decisions. Combined optimal design and control (co-design) methods are generally used to support this design paradigm. Although several PHEV powertrain co-design problems have been explored in the past, there have been no studies that have simultaneously addressed the impact of performance criteria such as acceleration performance and all-electric range (AER) along with predefined duty cycles on component design and hybrid mode (non-AER) supervisory control strategies. This is problematic as these performance criteria tend to strongly affect component sizing, which in turn can affect the supervisory control strategy in such a way that a non-performance-based co-design solution may become suboptimal. Therefore, this research addresses these issues by solving a comprehensive PHEV powertrain co-design performance study using a co-design method known as multidisciplinary dynamic system design optimization (MDSDO). In particular, MDSDO is implemented to simultaneously identify the optimal component designs, powertrain supervisory control strategies, and AER performance of a mid-size PHEV during a predefined vehicle duty cycle and a 0–60 mph acceleration maneuver such that the vehicle operating cost is minimized. A family of optimal solutions is generated by performing a parametric study for three distinct values of AER. The results from this study indicate that the formal inclusion of the performance criteria in a co-design problem has a significant impact on both component design and hybrid-mode supervisory control strategies.

Saeed Azad, Mohammad Behtash, Arian Houshmand, Michael Alexander-Ramos

Benchmarking Approaches for the Multidisciplinary Analysis of Complex Systems Using a Taylor Series-Based Scalable Problem

In the practical use of multidisciplinary design optimization, the prevalent approach for the multidisciplinary analyses (MDA) is nonlinear block Gauss–Seidel iteration, which consists in solving each discipline in a sequential manner, and repeating this sequence until convergence. This approach is easy to implement but often exhibits slow convergence rates or does not converge at all. An alternative is to use approaches based on Newton’s method to solve the coupled system, also known as tightly coupled or monolithic approaches. Past work, especially in the field of fluid-structure interaction, shows that Newton-based tightly coupled approaches can be more efficient and robust than loosely coupled approaches for the analyses of coupled systems. With the computing power and methods currently available, it is expected that the application of MDA and MDO to systems of greater complexity in terms of coupling and number of disciplines will increase. This makes it important to compare loosely and tightly coupled approaches for complex systems. To address the lack of literature providing such comparisons, we use a novel and highly flexible Taylor series-based analytical scalable problem with OpenMDAO—an open-source framework for MDA and MDO—to compare coupled Newton and nonlinear block Gauss–Seidel approaches for complex systems. We find that assembly time of the linear systems involved, linear solver efficiency, and strength of coupling in the problem play a major role in determining which approach is more efficient for a given problem. We also observe that the coupled Newton approaches are more robust and scale better than the nonlinear block Gauss–Seidel approaches as the strength of coupling between components increases.

Shamsheer S. Chauhan, John T. Hwang, Joaquim R. R. A. Martins

Convergence Strategy for Parallel Solving of Analytical Target Cascading with Augmented Lagrangian Coordination

Analytical Target Cascading (ATC) is a decomposition-based optimization methodology that partitions a system into subsystems and then coordinates targets and responses among subsystems. Augmented Lagrangian relaxation with Alternating Direction method (AL-AD) has been widely used for the coordination process of both hierarchical ATC and non-hierarchical ATC, and theoretically guarantees convergence under the assumption that all subsystem problems are convex and continuous. One of the main advantages of ATC is that it can solve subsystem problems in parallel, thus allowing it to reduce computational cost by parallel computing. Previous studies have proposed AL coordination strategies for parallelization by eliminating interactions among subproblems. This is done by introducing a master problem and support variables or by approximating a quadratic penalty term to make subproblems separable. However, conventional AL-AD does not guarantee convergence in the case of parallel solving. Our study found that, in parallel solving using targets and responses of the current iteration, conventional AL-AD causes mismatch of information in updating the Lagrange multiplier (LM). Therefore, the LM may not reach the optimal point, and as a result, increasing penalty weight causes numerical difficulty in the AL penalty function approach. To solve this problem, we propose a modified AL-AD for parallel solving in non-hierarchical ATC. The proposed algorithm uses the subgradient method with adaptive step size in updating the LM, which is independent of quadratic penalty terms and keeps quadratic penalty terms at the initial value. Without approximation or introduction of an artificial master problem, the modified AL-AD for parallel solving can achieve similar accuracy and convergence with much less computational cost, compared with conventional methods.

Yongsu Jung, Namwoo Kang, Ikjin Lee

Efficient Global Optimization Strategy Considering Expensive Constraints

This paper proposes a novel augmented Lagrange multiplier based efficient global optimization strategy (denoted as L-EGO) to solve black-box design optimization problems involving computationally expensive constraints. The original objective function, constraint functions, multipliers and exterior penalty function are integrated to construct an augmented objective function. By optimizing the expected improvement of the augmented objective function, the sample points are sequentially generated to refine the Kriging metamodel, and the Lagrange multiplier and penalty factor are updated during the iteration, which leads the optimization process converging to the feasible optimum efficiently. Two benchmark problems are used to test the proposed method via comparing with another metamodel-based optimization algorithm (i.e., CiMPS). The comparison results show that L-EGO outperforms CiMPS in terms of global convergence, efficiency, and robustness. Finally, L-EGO is applied to solve a practical all-electric GEO satellite multidisciplinary design optimization (MDO) problem, which involves seven expensive constraints. Compared with the initial design, the optimized solution reduces the total mass of the satellite by 7.1% and satisfies all the constraints, which demonstrates the effectiveness and practicality of the proposed L-EGO method.

Bin Yuan, Li Liu, Teng Long, Renhe Shi

Part II: General Approaches and Strategies: Multi-Objective Optimization


Producing Smart Pareto Sets for Multi-objective Topology Optimisation Problems

To date the design of structures via topology optimisation methods has mainly focused on single-objective problems. However, real-world design problems usually involve several different objectives, most of which counteract each other. Therefore, designers typically seek a set of Pareto optimal solutions, a solution for which no other solution is better in all objectives, which capture the trade-off between these objectives. This set is known as a smart Pareto set. Currently, only the weighted sums method has been used for generating Pareto fronts with topology optimisation methods. However, the weighted sums method is unable to produce evenly distributed smart Pareto sets. Furthermore, evenly distributed weights have been shown to not produce evenly spaced solutions. Therefore, the weighted sums method is not suitable for generating smart Pareto sets. Recently, the smart normal constraints method has been shown to be capable of directly generating smart Pareto sets. This work presents an updated smart normal constraint method, which is combined with a bi-directional evolutionary structural optimisation algorithm for multi-objective topology optimisation. The smart normal constraints method has been modified by further restricting the feasible design space for each optimisation run such that dominant and redundant points are not found. The algorithm is tested on several different structural optimisation problems. A number of different structural objectives are analysed, namely compliance, dynamic and buckling objectives. Therefore, the method is shown to be capable of solving various types of multi-objective structural optimisation problems. The goal of this work is to show that smart Pareto sets can be produced for complex topology optimisation problems. Furthermore, this research hopes to highlight the gap in the literature of topology optimisation for multi-objective problems.

David J. Munk, Gareth A. Vio, Grant P. Steven, Timoleon Kipouros

Multicriterial Optimization of Geometrical and Structural Properties of the Basic Module of a Single-Branch Truss-Z Structure

Truss-Z (TZ) is an Extremely Modular System (EMS). Such systems allow for creation of structurally sound free-form structures, are comprised of as few types of modules as possible, and are not constrained by a regular tessellation of space. Their objective is to create spatial structures in given environments connecting given terminals without self-intersections and obstacle-intersections. In an EMS, the assembly, reconfiguration and deployment difficulty is moved towards the module, which is relatively complex and whose assembly is not intuitive. As a result, an EMS requires intensive computation for assembling its desired free-form geometrical configuration, while its advantage is the economization of construction and reconfiguration by extreme modularization and mass prefabrication.TZ is a skeletal modular system for creating free-form pedestrian ramps and ramp networks among any number of terminals in space. TZ structures are composed of four variations of a single basic module (Truss-Z module, TZM) subjected to affine transformations (mirror reflection and rotation). The previous research on TZ focused on global discrete optimization of the spatial configuration of modules. This contribution is the first attempt at structural optimization of the TZM for a single-branch TZ. The result is a multicriterial optimization, where the Pareto front provides the means to strike the optimal balance between geometric and structural assessment criteria.

Machi Zawidzki, Łukasz Jankowski

Pseudo Expected Improvement Matrix Criteria for Parallel Expensive Multi-objective Optimization

Many engineering optimization problems involve multiple objectives which are sometimes computationally expensive. The multi-objective efficient global optimization (EGO) algorithm which uses a multi-objective expected improvement (EI) function as the infill criterion, is an efficient approach to solve these expensive multi-objective optimization problems. However, the state-of-the-art multi-objective EI criteria are very expensive to compute when the number of objectives is higher than two, thus are not practical to use in real-world problems. In the early work, the authors have proposed three cheap-to-calculate and yet efficient multi-objective EI matrix (EIM) criteria for the expensive multi-objective optimization. In this work, the three EIM criteria are extended for parallel computing to further accelerate the search process of the multi-objective EGO algorithm. The approach selects the first candidate at the maximum of an EIM criterion, and then multiplies the EIs in the EI matrix by the influence function of the first candidate to approximate the updated EIM function. The influence function is designed to simulate the effect that the first candidate will have on the landscape of each EI function. Then the second candidate can be selected at the maximum of the approximated EIM criterion. As the process goes on, a desired number of candidates can be generated in a single optimization iteration. The proposed parallel EIM (called pseudo EIM in this work) criteria have shown significant improvements over the single-point EIM criteria in terms of number of iterations on the selected test instances. The results indicate that the proposed pseudo EIM criteria can speed up the search process of the multi-objective EGO algorithm when parallel computing is available.

Dawei Zhan, Jiachang Qian, Jun Liu, Yuansheng Cheng

Optimal Near Sun Synchronous Orbital Design of a Nadir-Pointing Cubic Satellite with the Purpose of Thermal Load Control

Orbital parameter assessment as an integral part of a mission design has a parental effect on other subsystems of a satellite. The design and performance requirements of subsystems are intensely coupled with the orbital parameters. Additionally, mission requirements impose some constraints on the orbit design. For instance, in Earth Observation missions, the orbit should be designed in a way that keeps its local time in an allowable range during the mission lifetime. This allowable range of the local time depends on the imaging requirements and the quality of the received light from the region.Some parameters such as eclipse time, sun incidence angle, albedo, and earth IR depend on satellite orbital parameters. In this study, by doing a dynamic simulation of every aforementioned parameter over the satellite lifetime, investigation of an objective function and some constraints have been provided for every second of the mission. There are two subsystems of a satellite which are mostly under the influence of the orbital characteristics: Thermal Control Subsystem and Electrical Power Subsystem. Moreover, these two subsystems are strongly in interaction with each other so that considering only one of them without the other one is not analytically and practically possible. This paper mainly aims at seeking for an optimal near sun-synchronous orbit for a nadir-pointing satellite using Evolutionary Algorithms. Mission and Power requirements are considered as constraints to be satisfied as well as controlling thermal load applied on the satellite. A computational code has been developed to simulate the performance of the satellite mission characterized by high accuracy.

Asad Saghari, Shima Rahmani, Amir-reza Kosari

Part III: General Approaches and Strategies: Design of Experiments and Surrogate Models (Meta-Models)


Simple Intuitive Multi-objective ParalLElization of Efficient Global Optimization: SIMPLE-EGO

This paper describes how to parallelize the Efficient Global Optimization (EGO) algorithm, by making use of a simple multi-objective formulation. EGO constructs a Kriging approximation of the cost function, and then improves this approximation by adding additional designs to the initial designs. These aditional designs are added sequentially, one by one. In a specific iteration, the design is added where the Expected Improvement (EI) acquisition function is maximized. The EI function is typically maximized in the vicinity of the current best sampled point, or in a region that has large uncertainty. We demonstrate that instead of using the EI function, a multi-objective formulation can be used to decide where to add points. The two objectives that dictate where new designs should be sampled are function value, and uncertainty. The resulting Pareto front contains multiple designs that can be analyzed in parallel in the next iteration. Our study concludes with numerical examples.

Carla Grobler, Schalk Kok, Daniel N. Wilke

Gaussian Process for Aerodynamic Pressures Prediction in Fast Fluid Structure Interaction Simulations

The interaction between inertial, elastic and aerodynamic forces for structures subjected to a fluid flow may cause unstable coupled vibrations that can endanger the structure itself. Predicting these interactions is a time consuming but crucial task in an aircraft design process. In order to reduce the computational time surrogate reduced order models can be used in both structural and aerodynamic models. More over it is possible to avoid launching CFD computations at every time step. A database of aerodynamic pressure distribution on the structural component can be created conveniently sampling the space of the structural model DoF. Starting from the knowledge of the pre-computed data-set a Gaussian Process can be applied to predict the pressure distribution on an unexplored point of the space of DoF. The knowledge of the standard deviation can be used to give indications on where to launch further CFD computations to enrich the database. This technique will be first applied to a database of pressures obtained using the software Xfoil®, later it will be applied to CFD simulations of type RANS launched with elsA® on one Flap track Fairing of an Airbus aircraft.

Ankit Chiplunkar, Elisa Bosco, Joseph Morlier

Efficient Metamodeling Strategy Using Multivariate Linear Interpolation for High Dimensional Problems

Metamodeling method has been widely used to solve engineering problems which require significant computation, and there have been a large number of studies to propose efficient metamodeling methods in the last two decades. However, researches on practical metamodeling method dealing with high dimensional design space are insufficient. There sometimes exist high dimensional, expensive and black-box (HEB) problems, and handling these HEB problems with metamodeling method is challenging because of computational burden and complexity. In this paper, the efficient metamodeling strategy is proposed to handle HEB problems. The proposed strategy decomposes high dimensional design space into multiple less dimensional design spaces based on the coefficient of the multivariate linear interpolation equation. After the decomposition, sub-metamodels are generated using a sequential sampling method in each design space, and the final metamodel is constructed using these sub-metamodels. Engineering example verifies that the proposed strategy reduces required number of samples to satisfy the specified target accuracy compared to existing metamodeling methods.

Kyeonghwan Kang, Ikjin Lee, Donghyun Kim

Surrogate Modeling in the Design Optimization of Structures with Discontinuous Responses with Respect to the Design Variables – A New Approach for Crashworthiness Design

Advances to computational technology have resulted in the reduction of computational effort for crashworthiness analysis, hence enabling structural design optimization. Surrogate modeling has been shown to further reduce computational effort as well as to smooth noisy responses. Crashworthiness optimization problems are, though, ill posed as they include nonlinear, noncontinuous and noisy responses. This violates the Hadamard conditions for well-posed problems and therefore the applicability of gradient-based algorithms is limited.Here, discontinuities in the responses with respect to the design variables will be handled that result in large changes in the system functions with only small changes in the design variables using a novel surrogate modeling technique. The applicability of typical global surrogate models is limited when critical discontinuities are present. An efficient method has been developed here to identify the number of discontinuities and their position in the design domain. Previous works assume a said number of discontinuities; here though, the number of discontinuities is not given a priori. The discontinuities are identified by examining the relative difference in the response value of samples in immediate proximity of each other. Samples in the same continuous subdomain are clustered and a support vector machine for classification is exploited to locate discontinuities. Local approximations are then used for the continuous subspaces between the discontinuities. Lastly, a surrogate-based design optimization is carried out.Starting with a two-bar truss, demonstrating a snap-through discontinuity, this method is shown to account for such discontinuities. This is then integrated into an optimization framework. Further, a crash-absorbing tube is optimized that is impacted with an angle resulting in a noncontinuous design space: desired axial crushing and undesirable global buckling. After summarizing the results, advantages and possible limitations are discussed.

C. Boursier Niutta, E. J. Wehrle, F. Duddeck, G. Belingardi

RBF-Based High Dimensional Model Representation Method Using Proportional Sampling Strategy

To effectively tackling high dimensional, expensive, black-box (HEB) problems, this paper proposes a modified radial basis function based high dimensional model representation method using proportional sampling strategy (denoted as RBF-HDMR-PS). Different from the standard RBF-HDMR, the proposed RBF-HDMR-PS sequentially adds first order sample points with a predetermined proportion coefficient to effectively construct each component RBF, which avoids the stochastic influence of random sampling process in RBF-HDMR. The proposed RBF-HDMR-PS using different proportion coefficients is tested through two benchmark numerical problems with highly nonlinear first order components for comparing with RBF-HDMR. A best proportion coefficient is chosen and integrated into RBF-HDMR-PS. The comparison results show that RBF-HDMR-PS outperforms RBF-HDMR in terms of approximation accuracy.

Xin Li, Teng Long, G. Gary Wang, Kambiz Haji Hajikolaei, Renhe Shi

A Surrogate-Based Optimization Using Polynomial Response Surface in Collaboration with Population-Based Evolutionary Algorithm

The evaluation of system design is undoubtedly a time-consuming process with limited computational budget especially when some criteria such as reliability maximization or cost minimization are introduced as main objectives. This attracts many attentions to utilize the effectiveness of meta-models (surrogate-based methods) in the context of optimization. In this study, a collaboration between the population of Evolutionary Algorithms (population-based) and a polynomial surrogate model leads to reach global optimal points. As a population is formed to search the design space for the best solution, a response surface formation is intended in light of the fitness evaluation of population simultaneously. The accuracy of the response surface then can be increased by making beneficial use of original function evaluation of the population in the next iteration. To be more precise, construction of the surrogate model occurs from the first optimization iteration by means of population values (using original fitness function) and updating of this surrogate model is possible using the population cost of the next iterations. Meanwhile, the best solution of the surrogate model has to be injected into the population as a new member to empower the optimization search engine. The proposed creativity brings about promising results of global optimal solution with fewer function evaluations.

Shima Rahmani, Masoud Ebrahimi, Ayat Honaramooz

Using Gaussian Process to Enhance Support Vector Regression

Support vector regression (SVR) is a common surrogate model for computationally expensive simulation. It is able to balance the model complexity and the error tolerance. Whether SVR interpolates the training samples is dependent on its parameters. For the nonlinear function approximation without noise, when SVR is not an interpolator, it is advisable to model the errors and use them to compensate the prediction response. In this paper, the errors of SVR are modeled by using Gaussian process, and the final model response is obtained by the combination of SVR and the Gaussian process of the errors. The numerical experiments show the proposed method is able to further improve the prediction accuracy of SVR.

Yi Zhang, Wen Yao, Xiaoqian Chen, Fred van Keulen

Part IV: General Approaches and Strategies: Uncertainty and Robust Design


Improved Sequential Optimization and Reliability Assessment for Reliability-Based Design Optimization

The sequential optimization and reliability assessment (SORA), which is a decoupled reliability-based design optimization (RBDO) method, is generally more efficient than conventional double-loop RBDO methods but less efficient than single-loop approaches because of the most probable point (MPP) search in the reliability assessment loop of SORA. This study presents improved SORA (ISORA) which is a single-loop version of SORA to further improve its efficiency. Simplified reliability assessment is performed at each iteration instead of the reliability assessment loop of SORA. The optimum with higher reliability than the deterministic optimum is searched by moving the limit state function toward the safe region utilizing a design shift vector in SORA which is approximated using the concept of the mean value method in ISORA. ISORA is divided into ISORAx and ISORAu in this study, according to the space used to define the approximated design shift vector. ISORAu is more effective than ISORAx because it does not need to approximate MPP and design point whereas ISORAx needs to approximate them for non-normal random variables. Numerical study shows that ISORAu is the most effective among RBDO methods tested in this study in terms of efficiency, accuracy and robustness.

Sang-Hyeon Choi, Ikjin Lee

Improved Adaptive-Loop Method for Non-probabilistic Reliability-Based Design Optimization

Taking the unique superiority to handle the uncertain-but-bound problem, convex model is widely used in non-probabilistic reliability-based design optimization (NRBDO). Typically, parallel-loop method, serial-loop method and single-loop method are three basic approaches for non-deterministic design optimization. It is an essential issue to combine them together and exploit their own advantages to the full. To improve the optimization efficiency, an improved adaptive-loop method (IALM) for NRBDO is proposed in this paper. In the first phase, the reconstruction constraint function is formulated to enhance the reliability analysis with single-loop method. Due to the efficiency of single-loop method and robustness of parallel-loop method, the IALM can adaptively select a proper one based on judgment criteria. To further improve the robustness and efficiency of parallel-loop method, the enhanced chaos control (ECC) method is introduced in this paper. Numerical example is utilized to demonstrate the effectiveness of the proposed method by comparison with other existing methods in terms of numerical efficiency and stability.

Yutian Wang, Peng Hao, Chen Liu, Wu Fangzhou, Bo Wang

Multi-objective Reliability-Based Design Optimization for Energy Absorption Components Considering Manufacturing Effects

Light weight and crashworthiness design have been two main challenges in the vehicle industry, which often conflict with each other. To achieve light weight while improving the crashworthiness, design optimization techniques have been widely used. However, traditional crashworthiness design and process optimization are always performed respectively. That is, few process requirements are considered in the crashworthiness design which may lead to non-optimal or even impractical process scheme, and vice versa. Meanwhile, most of the energy absorbing members in vehicle body are fabricated by stamping process which will cause non-uniform thickness, residual strains and stresses, especially for high strength steel or advanced high strength steels. Furthermore, the uncertainties of the material properties, process and geometry generally propagate from process to crashworthiness responses, which will lead the uncontrollable fluctuations of crashworthiness. In other words, a deterministic optimization could lead to unreliable or unstable designs. To address these issues, a multi-objective reliability-based design optimization coupled process-performance (MORBDOCP) was proposed to optimize the double-hat thin-walled structure (DHTS). First, a finite element-based sequential coupled process-performance approach was developed to simulate the forming and crashworthiness of the DHTS, in which the material properties, process parameters and component geometry can be coupled and propagated from forming simulations to crashworthiness simulations efficiently. Then, the metamodel technologies were adopted to approximate the forming and crashworthiness responses. Finally, the multi-objective particle swarm optimization (MOPSO) approach, coupled with Monte Carlo Simulation (MCS), was employed to seek optimal reliability solutions. The optimal results show that the proposed method not only significantly improved the formability and crashworthiness, but also was capable of enhancing the reliability of Pareto solutions.

Huile Zhang, Guangyong Sun, Guangyao Li, Qing Li

Robust Design Optimization of Vehicle and Adaptive Cruise Control Parameters Considering Fuel Efficiency

In the past, the development of an adaptive cruise control (ACC) algorithm considering fuel efficiency and the development of an ACC system considering performances such as fuel efficiency, ride comfort and trackability have been carried out. In addition, research on vehicle and ACC parameters optimization considering fuel efficiency, ride comfort, trackability and safe distance have been carried out. However, in real world, vehicle sprung mass and center of gravity are changed due to the number of vehicle occupants, and there are uncertainties in vehicle parameters such as tire radius, tire spring constant and so on. Therefore, ACC should be designed considering uncertainties due to the variations of vehicle parameters.In this paper, robust design optimization of vehicle and ACC parameters considering uncertainties is carried out to make the robustness of performances such as fuel efficiency, ride comfort, trackability, and safe distance. Before performing the robust design optimization, vehicle parameters which have significantly influence on the performances are analyzed through analysis of variance (ANOVA) and uncertainty quantification is performed by analyzing information of vehicle occupants. Since numerous function calls of high fidelity model analysis are needed to perform design optimization that kriging surrogate model which is a mathematical model that can replace the high-fidelity model is employed and performed robust design optimization by using constructed kriging surrogate model.

Hansu Kim, Tae Hee Lee, Yuho Song, Kunsoo Huh

Bootstrap Guided Information Criterion for Reliability Analysis Using Small Sample Size Information

Several methods for reliability analysis have been established and applied to engineering fields bearing in mind uncertainties as a major contributing factor. Small sample size based reliability analysis can be very beneficial when rising uncertainty from statistics of interest such as mean and standard deviation are considered. Model selection and evaluation methods like Akaike Information Criteria (AIC) have demonstrated efficient output for reliability analysis. However, information criterion based on maximum likelihood can provide better model selection and evaluation in small sample size scenario by considering the well-known measure of bootstrapping for curtailing uncertainty with resampling. Our purpose is to utilize the capabilities of bootstrap resampling in information criterion based reliability analysis to check for uncertainty arising from statistics of interest for small sample size problems. In this study, therefore, a unique and efficient simulation scheme is proposed which contemplates the best model selection frequency devised from information criterion to be combined with reliability analysis. It is also beneficial to compute the spread of reliability values as against solitary fixed values with desirable statistics of interest under replication based approach. The proposed simulation scheme is verified using a number of small and moderate sample size focused mathematical example with AIC based reliability analysis for comparison and Monte Carlo simulation (MCS) for accuracy. The results show that the proposed simulation scheme favors the statistics of interest by reducing the spread and hence the uncertainty in small sample size based reliability analysis when compared with conventional methods whereas moderate sample size based reliability analysis did not show any considerable favor.

Eshan Amalnerkar, Tae Hee Lee, Woochul Lim

Stochastic Sensitivity Analysis for Robust Topology Optimization

Topology optimization under uncertainty poses extreme difficulty to the already challenging topology optimization problem. This paper presents a new computational method for calculating topological sensitivities of statistical moments of high-dimensional complex systems subject to random inputs. The proposed method, capable of evaluating stochastic sensitivities for large-scale, robust topology optimization (RTO) problems, integrates a polynomial dimensional decomposition (PDD) of multivariate stochastic response functions and deterministic topology derivatives. In addition, the statistical moments and their topology sensitivities are both determined concurrently from a single stochastic analysis. When applied in collaboration with the gradient based optimization algorithm, the proposed method affords the ability of solving industrial-scale RTO design problems. Numerical examples indicate that the new method developed provides computationally efficient solutions.

Xuchun Ren, Xiaodong Zhang

An Improved MPP-Based Importance Sampling Method for Reliability Analysis

Importance sampling (IS) as an efficient technique in Monte Carlo probability simulation has been widely applied for high reliability system analysis, which can greatly reduce the simulation numbers and improve the efficiency. Among the IS methods, Most Probable point (MPP)-based Importance Sampling Method (MISM) has gained wide attention because of its effectiveness and easy implementation. However, the traditional MISM uses the Acceptance-Rejection (A-R) technique to sample points from the important regions. The uniform distribution is often set as the proposal distribution and its value is equal to the value of the point in the Probability Density Function (PDF) where the first-order derivative is zero. This brings about efficiency-scarified problems when the derivative exceeds the sampling interval in different dimensional cases. Moreover, in order to increase the efficiency by use of the A-R process, it often performs the variable transformation (e.g. log-transformation). Unfortunately, it doesn’t work while the PDF of the random variables is not a log-concave function. In this paper, two feasible strategies were proposed to solve these drawbacks. The first strategy is that we set the maximal value of the PDF as the value of the uniform function, which is obtained by calculating all samples point by point in the target interval. It can be applied to any dimensions as an instructional strategy. The other one is that a well-designed normal distribution function, instead of the uniform distribution function, is used as the proposal distribution to avoid transformation of the variables. Finally, two numerical examples are given to illustrate the effectiveness and accuracy of the proposed method.

Guijian Tang, Wen Yao, Xiaoqian Chen, Yong Zhao

Characterization of Geometric Uncertainty in Gas Turbine Engine Components Using CMM Data

Measurements of component geometry are routinely made for inspection during manufacturing. Typically this results in ‘clouds’ of points or pixels depending upon the measuring system. Examples include points form laser-based or touch-probe co-ordinate measuring machines (CMMs). The point density may vary as will the cost and time taken to make measurements. There can also be gaps and occlusions in data, and sometimes it is only practical to collect sparse sets or points in a single dimension.This data often provides an untapped source of quantitative uncertainty information pertaining to manufacturing methods. It is proposed that state-of-the-art uncertainty propagation and robust design optimization approaches, often demonstrated using assumed normal input distributions in existing parameters, can be improved by incorporating these data. Inclusion of this information requires, however, that the point cloud be converted to an appropriate parametric form.Although the design intent of a component may be described using simple geometric primitives joined with tangency or at vertices, manufactured geometry may not exhibit the same simple form, and line and surface segment end locations are notoriously difficult to locate where there is tangency or shallow angles. In this paper we present an approach to first characterise point cloud measurements as curves or surfaces using Kriging, allowing for gaps in data by extension to universal Kriging. We then propose a novel method for the reduction of variables to parameterize curves and surfaces again using Kriging models in order to facilitate practical analysis of performance uncertainty. The techniques are demonstrated by application to a gas turbine engine blade to disc joint where the contact surface shape is measured and the notch stresses are critical to component performance.

Jennifer Forrester, Andy Keane

An Optimal Configuration of an Aircraft with High Lift Configuration Using Surrogate Models and Optimisation Under Uncertainties

Nowadays many simulations are computationally expensive, which is disadvantageous if one is interested in the quantification of uncertainties, parameter studies or in finding an optimal or robust design. Therefore often so-called surrogate models are designed, which are a good approximation of the original model but computationally less expensive.In this paper we first look for an approximation method to design a surrogate model for the simulation of a civil aircraft with active high lift configuration. Such aircrafts have the advantage that only small runways for take-off and landing are necessary. A first result, presented in this paper, is a configuration of the aircraft, where the direct operating costs (DOCs) are minimised. For the optimisation process seven parameters are chosen, for example the Mach number in the cruise flight and the area of the wing. In a second step we define 28 uncertain parameters and repeat the optimisation process including these uncertain parameters to derive a robust configuration.

Joachim Rang, Wolfgang Heinze

Reliability-Based Topology Optimization for Continuum Structures with Non-probabilistic Uncertainty

A non-probabilistic reliability-based topology optimization (NRBTO) method for continuum structures is proposed for structures with correlated interval parameters based on the multidimensional parallelepiped (MP) model. A topology optimization model is formulated to minimize volume of structure under displacement constraints. An equivalent optimization model is given and solved based on the efficient performance measurement approach (PMA). A numerical example is used to demonstrate the effectiveness of the proposed method.

Jing Zheng, Zhen Luo

Big-Data Based Rule-Finding for Analysis of Crash Simulations

Efficiently and systematically extracting knowledge from design of experiments studies in the field of vehicle crash simulation is increasingly challenging due to the number and size of the simulation models and the complexity of the occurring mechanisms. As an alternative to common statistical postprocessing, this paper proposes the usage of data-mining methods from the field of knowledge discovery in databases to condensate important effects and give insight into their origins.Before application of mining techniques, one has to separate the data into two groups regarding a criterion. This criterion may be in general a boolean condition, such as a response with specified bounds or the affiliation to a specific deformation mode. For the latter case, the categorization by deformation behavior is used by clustering of simulation results with similar distributions of plastic strain.Once the simulation runs have been separated into two groups, our proprietary rule mining technique can extract rules. A rule is a classical IF-THEN condition, where the IF-part are humanly readable boolean conditions of variables fulfilling the conclusion. All discovered rules need to be filtered regarding certain rule mining criteria, so that an engineer is only confronted with a few reasonable, interesting and important rules. The filtered rules usually contain only few variables, which is also a benefit from decision tree learning in the background.In contrast to optimization the result of this technique is not a design point but multiple design subspaces. An engineer may now either choose a design point within these spaces directly or use the variable ranges as constraints for an additional optimization.

C. Diez, P. Kunze, D. Toewe, C. Wieser, L. Harzheim, A. Schumacher

Mathematical Models and Methods of Effective Estimation in Multi-objective Optimization Problems Under Uncertainties

The selection of the optimal criteria for solutions and sought values of the objective functions (multi-objective decision making) is a complex task. It becomes a real challenge when prior data are uncertain. In this paper you will find a new approach to solve this task. The new method uses the updated method of getting the scalar convolutions of the criteria for the described task. The updated method references Ashby`s law of Requisite Variety, Kolmogorov’s concept of power averages and Tikhonov`s ideas of regularization.The obtained set of the scalar convolutions was evaluated from the maximum likelihood principle. The set can be used for synthesis of robust meta-models, mathematical model identification, and robust optimal engineering.The scalar convolutions for the criteria were obtained from Student`s statistics. It served as a criterion to check the equality of distribution centers for representative samples from two multidimensional general t populations. Student`s statistics also played a role of multidimensional analogue of Romanovsky criterion Ro to check the hypothesis about the equality of covariance matrices Ro or statistics H (H is the mutual information) instead of statistics Ro.The paper contains the mathematical formulations and computational methods for the synthesis of quasi-solutions of stochastic optimization problems with mixed conditions. The implementation of the research results will provide the developers with the robust estimations of the sought values even if the prior data are uncertain.The paper also deals with a new probability-based method to solve the direct problem of dimensional engineering networks. In accordance with the probability-based method the mathematical expectations and confidence intervals of the control variables of functional elements are obtained from the mathematical expectations and confidence intervals of the decision selection criteria or from the phase variables of considered systems or processes.To make the processing time several times less one proposed the memetic algorithm with the consistent application of advanced real coded evolutionary method, decremental neighborhoods method, randomized path relinking method.At the end of the paper you can observe the interactive decision support system «Concept_Pro_St®» that focuses on a wide range of users in the fields of: engineering, project management, data-monitoring oriented management and production supervision to ensure product quality (Design for Six Sigma), industrial safety, environmental, pharmaceuticals, medicine, etc., working on issues of construction of robust meta-models (formal mathematical models in the form of regression equations), robust optimal design and diagnostics of systems and processes.To validate the method regarding to some particular object one solved the problem of robust optimal designing of centrifugal impeller fitted with backward curved blades in the conditions of stochastic nature of the input data.

Meniailov Ievgen, Khustochka Olexandr, Ugryumova Kateryna, Chernysh Sergey, Yepifanov Sergiy, Ugryumov Mykhaylo

A Shifted-Constraint RBDO Framework Using Monte Carlo Simulations

In this paper, a sampling-based framework for reliability-based design optimization problems has been developed to provide a comprehensive structure adaptable to MCS methods such as importance sampling strategy. This framework has the capability of adopting any reliability analysis by decoupling two integral parts of a RBDO algorithm: reliability analysis and optimization problem. A shifting vector strategy has been proposed what imposes the constraints to immediately satisfy the reliability requirements. The main advantage of this framework is its independency on the approximation of the constraints since it aims at finding an optimum shifting vector based on the probability density function of the constraints themselves.

Shima Rahmani, Asad Saghari, Masoud Ebrahimi

Optimization of Manufacturing Tolerances on Sheet Metal Components in the Development Process

In an increasingly challenging environment of growing competition and stricter legal regulations, the concept of lightweight design gets more and more important for the automotive industry. Usually, the goals of lightweight tend to conflict with the robustness of the designs. The scattering of the geometry within the manufacturing tolerance leads to a divergence between the test bench results and the prediction from simulation. For an optimized lightweight design, this deviation will often lead to a violation of the requirements. In this case, the part has to be re-designed to improve its robustness, causing an additional time expense, higher costs and a higher weight.This paper presents a new way to answer the conflict of objectives between lightweight and robustness. By the integration of manufacturing tolerances in an early stage of the product design, the robustness of a part can be improved without increasing its weight. A parametric CAE model is used to represent all deviations of the geometry and material properties within the tolerance range. Additionally, an analytic expression of the relevant simulation results is built. These tools are not only used to predict exactly the scattering of the test bench result, but also allow optimizing the tolerance ranges in order to obtain a 100% satisfaction of the requirements in every tolerated configuration.The optimized tolerance ranges make it possible transferring the weight advantage generated in the structural optimization into the design of prototypes. Through a better prediction of the test bench results in the simulation, unnecessary development loops can be avoided. The presented procedure is the first one that integrates manufacturing tolerances within the simulation with the objective of optimizing the tolerance ranges beyond a simple parameter optimization. Thereby, new weight potentials are allowed, and the development process can be substantially shortened.

C. Hayer, S. Fiebig, T. Vietor, J. Sellschopp

Part V: General Approaches and Strategies: Sensitivity Analysis and Parameter Identification


A Gradient-Based Topology Optimisation for Radar Cross Sections in Two-Dimensional Acoustics

Radar cross section (RCS) is a measure of ability of object to reflect radar signals in the direction of radar receiver. The concept is often used in electro-magnetics, however, the RCS can be extended in acoustics to evaluate the sound performance of object. By manipulating the RCS of acoustic material, we can design sophisticated acoustic devices. In this study, to design an acoustic material which has a desired RCS pattern, a gradient-based topology optimisation is proposed. The topological derivative, which characterises the sensitivity of the RCS for a pre-fixed direction to an appearance of an infinitesimal circular object is rigorously derived. We show that the topological derivative consists of two terms, one of which is evaluated with the adjoint variable method and the other is with direct estimation. The derived topological derivative is implemented in a topology optimisation. We use the level set method to express the sound object and the boundary element method to solve the acoustic problems in the process of the optimisation. These choices are reasonable since the RCS is evaluated with (a square of) a boundary integral. It is important to express the boundary of objects and evaluate the boundary integral in an accurate manner. We show several numerical examples of optimal designs for acoustic object with desired RCS patterns and some applications for inverse problems.

Hiroshi Isakari, Toru Takahashi, Toshiro Matsumoto

A Topology Optimisation of Wave Absorbers in Two-Dimentional Electro-Magnetic Field with an Accelerated BEM by the -Matrix Method

In this study, we propose a new topology optimisation for electro-magnetic wave absorbers. We define an objective function as the amount of time-averaged inward Poynting’s vector on boundary of absorbing material, and consider to maximise the objective function. The sensitivity of the objective function is strictly derived as a variation of the objective function when an infinitesimal hole is created in the design domain by using the adjoint variable method. For the efficient and accurate computation of the topological derivative, we employ the boundary element method and the $${\mathscr {H}}$$-matrix method. In the boundary element method, radiation condition is automatically satisfied with the help of the Green function, which enables us to treat problems defined in the infinite domain strictly. Also, by using the accelerated LU decomposition with the $${\mathscr {H}}$$-matrix method to solve the algebraic equations derived by the BEM, we achieve the fast computation of the sensitivity. We apply the sensitivity analysis to the level-set based topology optimisation.

Kenta Nakamoto, Hiroshi Isakari, Toru Takahashi, Toshiro Matsumoto

High-Fidelity Aero-Structure Gradient Computation Techniques. Application to the Onera M6 Wing

This paper presents some recent achievements towards high-fidelity aero-structure gradient computation with respect to structural design variables. A first part details the new module developed and implemented in the Onera elsA software capable of computing aero-structure gradients using intrusive direct and adjoint methods. In a second part an alternative improved uncoupled non-intrusive approach is proposed. Both approaches are evaluated and compared on the Onera M6 wing considering criteria such as accuracy, efficiency, and applicability on practical industrial problem.

Timothée Achard, Christophe Blondeau, Roger Ohayon

Identification for Input Sound Pressure Level in Hammering Test Based on Adjoint Variable and Finite Element Methods

In this study, we present the inverse analysis for identification of hammering signal in non-destructive hammering test. The performance function is defined by square sum of residual between the obtained and the computed sound pressure. Here, the problem is to find the input sound pressure so as to minimize the performance function. The formulation for this problem is carried out by the adjoint variable method, and the numerical simulation of the sound pressure propagation is carried out based on the wave equation and the finite element method. As a result of numerical experiments of forward analysis sets to correcting wave of sound pressure in input point of inverse analysis.

Eiki Matsuoka, Takahiko Kurahashi, Yuki Murakami, Shigehiro Toyama, Fujio Ikeda, Tetsuro Itama, Yoshihiro Tawara

Application of Digital Image Correlation to Material Parameter Identification

This paper expounds the parameter identification of material models using simulation-based optimization and experimental results obtained using Digital Image Correlation (DIC). DIC is an optical method which provides full-field displacement measurements for mechanical tests of materials and structures. It can be used to obtain strain field histories from an experimental coupon which can be combined with the corresponding fields obtained from a Finite Element Analysis to identify constitutive properties. The methodology, which involves the solution of an inverse problem, has been implemented in the optimization code LS-OPT®. A core feature is multi-point curves: response curves which are evaluated at multiple locations and extracted from simulations and experimental data. An interface to a commercial optical measurement package was created and an example of a tensile test was used to demonstrate the methodology based on the measurement of point-wise strains vs. tensile force. The Hockett-Sherby flow curve function, using two parameters, was used to model the material. The example validated the code but revealed potential problem areas requiring further investigation. A prominent issue is the method used for matching the experimental and computational curves. To identify sources of ill-posedness of the regression problem, several diagnostic tools are suggested. Some of these tools, notably sensitivity analysis will be demonstrated at the conference.

Nielen Stander, Katharina Witowski, Christian Ilg, Andre Haufe, Martin Helbig, David Koch

Part VI: General Approaches and Strategies: General Aspects of Single-Objective Optimization


A Novel Adaptive Region-Based Global Optimization Method for High Dimensional Problem

Surrogate models are widely used in simulation-based engineering design and optimization to save the computational cost. In this work, an adaptive region-based global optimization method is suggested. During the sampling process the approach uses hyperrectangles to partition the design space and construct local surrogates according to existing sample points. The sizes of hyperrectangles are adaptively generated by the maximum distance of the centered sample point between other points. The large size of the hyperrectangle indicates that the constructed local surrogate might be low accuracy, and the extended size of the hyperrectangles indicates that new sample points should be sampled in this sub-region. On the other hand, considering the exploration of the design space, an uncertainty predicted by using the Kriging model is integrated with the local surrogate strategy and applied to the global optimization method. Finally, comparative results with several global optimization methods demonstrates that the proposed approach is simple, robust, and efficient.

Fan Ye, Hu Wang

Coupling of Computer-Aided Methods: Supporting Product Developer During Embodiment Synthesis

The high complexity during the development of new product generations is nowadays one of the biggest challenges in the product development process. One solution to support the product developer is the use of computer-aided methods. These methods allow simulating various physical behaviors in order to predict the validity of the new product generation. Furthermore, the integration of those methods into optimization procedures allows generating optimal design proposals. One example for this is the topology optimization. However, those methods are limited because they do not consider the results of previous simulations or information from other domains in a fully automated way. This work proposes a strategic approach to overcome the current challenges by coupling different simulation and optimization methods. By this, newly coupled computer-aided methods are developed in order to support the product developer during embodiment synthesis. The presented coupled methods show the benefits of this strategy. For example, the linkage of the topology optimization with the form fill simulation was carried out. Hereby, the calculation of the fiber orientations is done in each iteration of the topology optimization process. This results in initial design proposals for parts made of fiber-reinforced polymers which consider the local anisotropies caused by the manufacturing process. Another example for the successful coupling of existing methods is the merging of topology optimization and multi-body simulation. This allows reducing the energy consumption of accelerated systems by minimizing the inertia of moving structures within the topology optimization process. The presented coupled computer-aided methods are based on established and commercially available tools. This ensures a short-term availability of the developed methods for product developer. Therefore, additional qualification measures and long training periods are not required.

Albert Albers, Markus Spadinger, Manuel Serf, Stefan Reichert, Steffen Heldmaier, Micha Schulz, Nikola Bursac

Optimization Design of Smart Reversible Diaphragms Using Shape Memory Polymer

As a new intelligent actuating material, shape memory polymer (SMP) attracts great interests because it deforms under high temperature conditions and can maintain the temporary deformation after unloading under low temperature condition. Currently the tank diaphragm is mainly using metal in the field of aerospace, but it never can be recycled. Instead of traditional diaphragm, a novel reversible diaphragm of shape memory polymer is developed in this work. The overturning and recovery behavior of the SMP diaphragm are analyzed by using nonlinear finite element method to exhibit the advantage of the shape memory material. The parametric effects of the thickness, height, radius of the diaphragm and temperature on overturning performance are investigated. Then an optimization model is established with the objective function of required minimum pressure for completing overturning deformation and the design variables are thickness, height and radius. Optimized SMP diaphragm is obtained under the critical pressure constraints. This research can contribute to the design and application of novel SMP diaphragm.

Qing-Sheng Yang, Ran Tao, Pin Wen

Experimental and Numerical Analysis of Mechanical Properties of Tape Spring Hinges and Optimal Design

A tape spring with outstanding self-driven, self-locking performances is exploited for deployable structure. A novel hinge using three tape springs is design, and this paper presents a complete set of optimization process for the hinge. The quasi static simulation techniques for deploying and folding processes are studied in ABAQUS, and verified by experiments using particularly designed apparatus. Impact ratio is defined to measure the overshoot of the hinge when deploying, then analysis of effects from geometric parameters on performances of the hinge is carried out. And then, optimal model of the hinge that aims to minimize impact ratio subjected to allowable stress is established based on the Response Surface Methodology (RSM) and the analysis of parameters’ effects. Large Scale Generalized Reduced Gradient algorithm (LSGRG) in Isight is selected to solve optimal equations. The optimal results show that impact ratio is decreased by 58.6% and Mises stress is within the safety limits. The proposed method is of great significance for designing novel deployable structure with higher stability and reliability.

Hong-ling Ye, Yang Zhang, Qing-sheng Yang, Ramana V. Grandhi

Multidisciplinary Structural Optimization Using of NSGA-II and ɛ-Constraint Method in Lightweight Application

In recent years, the automobile industry to produce lighter vehicles with higher levels of safety and competitive price was under more pressure. Light weight structure is engaged with different types of external loads such as static, cyclic, impact, and aerodynamics forces. These multi-disciplines are influenced by a wide range of discrete parameters, such as the number of layers of a composite part, material properties and joining technology beside the continuous parameters such as the size of cross-sections and thickness.Minimization of weight and cost and maximization of strength and durability of a multi-material structure made of metal and composite goes toward a multi-objective and multi-discipline problem. NSGA-II is implemented as a powerful algorithm in multi-objective problems for finding the best combination of discrete and continuous variables in structural optimization. Since no estimation about the minimum and maximum values of objectives is not present, the use of weighted sum method may incline to results away from the global optimum. While Epsilon method which provides much better organization on the objectives and constraints, seems a better method to select suitable members during the optimization process.To cut a long time of cyclic simulation for calculating the fatigue life of structure, Epsilon method in two stages is implemented in this study. In the first stage, weight, strength, cost and maximum of Mises and Tsai-Wu stress were assessed and evaluation of second stage is carried out only on structures which have satisfied the provisions of the first stage. The combination of Epsilon method and NSGA-II to solve the mixed variable and multi-objective and multi-discipline problem can be seen as a comprehensive method for structural optimization without any limitations on the number and type of variables and objectives.

Vahid Ghaffari Mejlej, Paul Falkenberg, Eiko Türck, Thomas Vietor

Fast Dynamic Analysis of Beam-Type Structures Based on Reduced-Order Model

Dynamic numerical simulation of large-scale complicated beam-type structures is unavoidable in modern engineering calculations; however, the inherent nature of the models often leads to unmanageable demands on the computational resources. The model reduction method aims to reduce this computational burden by generating reduced-order models (ROMs) that are faster and cheaper to simulate, yet accurately represent the original structures behavior. Therefore, a novel reduced order model is proposed to determine the natural frequencies of the beam-type structures in this study, which is established by using a reduction basis along with the polynomial interpolation function (PIF) depends on a set of parameters. The basic idea is to translate the displacements of FEM nodes in each cross section into a small amount of nodes with a few generalized DOFs. Moreover, the proposed ROMs have the ability to identify shell lobe-type modes and coupled modes. Then, a fast optimization framework for thin-walled cylinders is established based on the reduced order model. Finally, numerical examples demonstrate the effectiveness of proposed method.

Yuwei Li, Bo Wang, Peng Hao, Yan Zhou, Yang Zhao

Parametric Modeling and Optimal Design of Space Tubular Extendable Booms via a One-Dimensional Unified Formulation

The increasing demand for greater satellite mission capabilities has led to the need for lighter, stronger space deployable structures within the limited packaged volume. Tubular extendable booms, characterized by small stowed volume, light weight and large magnification ratios, have been widely used in small satellite missions, especially in cubesat missions. However, the deployment accuracy of these booms remains to be carefully analyzed and optimized, as the actual parameters are nonlinear. This paper presents a simple and general methodology for parametric modeling and optimal design of typical tubular extendable booms via a one-dimensional unified formulation. Using enhanced capabilities of refined beam elements, we obtained parametric models with remarkable reductions in computational costs are obtained to detect shell-like solutions for the booms. Then optimal design of the boom structure is easily achieved, as changes to the cross-section only alter the integration domain during analytical procedure. Finally, the optimized mechanical performance of these tubular extendable booms (including boom bending/torsional stiffness, relative tip displacements) is evaluated and compared under various loading conditions. Comparisons have shown that the structural stiffness of storable tubular extendable member (STEM) boom could be significantly improved by its proper configuration/material improvement. While the collapsible tube mast (CTM) boom and triangular rollable and collapsible (TRAC) boom have increased their anti-bending capabilities. All these improvements are of instructive significance in structural design of these tubular extendable booms, make the booms better at performing deployment tasks.

Yi Hu, Yong Zhao, Zhouhui Tuo, Jie Wang

Part VII: Optimization Algorithms: Local Mathematical Methods


Multi-Fidelity Optimization of Complex Physics Involved Engineering Systems

This work presents an optimal design technique which allows for the use of multiple simulation models of varying physics and complexity (fidelity levels). An advancement of traditional Surrogate Based Optimization (SBO) techniques is intended to alleviate the computational cost associated with structural and multidisciplinary design optimization while maintaining a high degree of accuracy typically associated with fully coupled, nonlinear, complex physics-based models. This methodology, termed Bayesian Influenced Low-Fidelity Correction Approach to Multi-Fidelity Optimization, utilizes a combination of surrogate modeling, Bayesian statistics, and Trust Region Model Management (TRMM) techniques. A novel Bayesian Hybrid Bridge Function (BHBF) was developed to serve as the low-fidelity correction technique. This BHBF is a Bayesian weighted average of two standard bridge functions, additive and multiplicative. The correction technique is implemented in parallel with a modified Trust Region Model Management (TRMM) optimization scheme. It is shown that optimization on the corrected low-fidelity model converges to the same local optimum as optimization on the high-fidelity model in fewer high-fidelity function evaluations and ultimately lower computational cost. This work also extends the low-fidelity correction optimization beyond the traditional bi-fidelity (limited to 2 fidelity levels) optimization to that of a novel approach to handling optimization with multiple (2 or more) fidelity objective and constraint functions with commercial optimization solvers. It is shown that implementation of this Bayesian low-fidelity correction optimization approach results in high-fidelity results at a reduced computational cost. This is demonstrated on computationally different engineering design problems. First, a 27% computational savings over traditional optimization techniques is observed in the unconstrained minimization of thermally induced stresses in a quarter symmetric panel represented via 4 differing fidelity levels. Finally, a 71% reduction in computational cost is observed in an airfoil shape optimization in which both the objective and constraints are represented by 2 fidelity levels.

C. Corey Fischer, Ramana V. Grandhi

Efficient Optimal Surface Texture Design Using Linearization

Surface textures reduce friction in lubricated sliding contact. This behavior can be modeled using the Reynolds equation, a single partial differential equation (PDE) that relates the hydrodynamic pressure to the gap height. In a previous study, a free-form texture design optimization problem was solved based on this model and two competing design objectives. A pseudo-spectral method was used for PDE solution, which was treated as a black box in the optimization problem. This optimization implementation did not exploit model structure to improve numerical efficiency, so design representation fidelity was limited. Here a new strategy is introduced where design representation resolution and computational efficiency are improved simultaneously. This is achieved by introducing a new optimization variable involving both pressure gradient and the cube of gap height at each mesh node location, and simultaneously solving the flow and texture design problems. This transformation supports linearization of the governing equations and design objectives. Sequential Linear Programming (SLP) is used with the epsilon-constraint method to generate Pareto-optimal texture designs with high resolution and low computational expense. An adaptive trust region is used, based on solution improvement, to manage linearization error. Comparing to the non-linear programming implementation, the solution converged to a set of slightly suboptimal points (maximum 25% objective function degradation when normalized apparent viscosity is 0.5, and moderately better when normalized apparent viscosity is 0.2), but results in significant improvement in computational speed (8.4 times faster).

Chendi Lin, Yong Hoon Lee, Jonathon K. Schuh, Randy H. Ewoldt, James T. Allison

Quadratic Multipoint Exponential Approximation: Surrogate Model for Large-Scale Optimization

Sequential Linear Programming (SLP) is a well-known first-order optimization method. Sequential quadratic programming (SQP) is generally preferred for smooth problems, because it is second-order accurate; however, it suffers from the curse of dimensionality for large numbers of design variables. For large-scale problems, SLP may be a good alternative, although its performance depends on the move-limit strategy, the efficiency of the LP solver, and obviously the nonlinearity of the functions. A robust implementation of SLP with a trust region strategy is implemented here in conjunction with a large-scale LP solver. The number of SLP outer-loop iterations to converge is demonstrated to be reduced by an intermediate variable transformation during the linearization. The well-known two-point exponential approximation (TPEA) is extended to take advantage of more than two previous points in determining intervening variables, which can be beneficial particularly for temporarily inactive variables. A single set of intermediate variables are selected for use with all constraint and objective functions, based on Lagrangian sensitivity, to maintain a linear sub-problem for SLP. Quadratic terms constructed in a reduced sub-space are explored for efficient large-scale SQP.

Robert A. Canfield

Topology Optimization of General-Joint Planar Linkage Mechanisms with an Application to Finger Rehabilitation Device Design

Although topology optimization methods can inspire designers, their use for the design of linkage mechanisms is not yet fully developed. However, there are a number of new mechanism design problems, as in robots or robotic rehabilitation devices. One of the critical limitations of earlier linkage mechanism synthesis methods by gradient-based topology optimization is that mechanisms only with one type of joint, a revolute joint, can be synthesized. The handling of revolute joints alone limits the capability of topology optimization considerably, and some mechanisms involving prismatic joints cannot therefore be synthesized. Here, we aim to develop a topology optimization method that can synthesize mechanisms with general joints including revolute and prismatic joints. The specific application in mind is the topology-optimization-based design of several robotic rehabilitation mechanisms. To this end, we propose a new ground model termed the joint element connected rigid-block model (JBM). The key idea of the proposed JBM is that the existence of prismatic joints and revolute joints are directly controlled by design variables. After testing the validity of the developed formulation with the JBM, we applied it to the design of a robotic finger rehabilitation device. An interesting design, not available in the existing literature, was synthesized, and the result may inspire designers in the field. It is expected that this method can be more useful as an inspirational design tool for the robotic industry in the future.

Seok Won Kang, Jeong Han Yu, Sang Min Han, Yoon Young Kim

Part VIII: Optimization Algorithms: Global Methods (e.g. Evolutionary Algorithms)


A Cross-Entropy Optimization Algorithm for Continuous Function Based on Improved Sampling

In this paper, the Cross-Entropy (CE) optimization algorithm for continuous functions is studied. Aiming at the problem of computational efficiency caused by the large number of samples in Monte Carlo sampling, the Dynamic Weight strategy (DW) and the Adaptive sample Size strategy (AS) are proposed. The former can get the weighting coefficient of each sample by measuring the difference between the elite sample and the best sample, which can speed up the convergence rate of the algorithm. The latter can greatly reduce the number of calls to the objective functions by building relationship between the sample size and the standard deviation of elite samples, which can effectively improve the optimization efficiency. Finally, the validity of the proposed algorithm is verified by four unconstrained multimodal standard test cases and two constrained practical project examples. The results show that with similar capability to obtain the global optimal solution, the number of calls to the objective functions is reduced by 73.44%, 69.44%, 47.33%, 77.88%, 40.21% and 25.95% respectively compared with the traditional CE algorithm.

Zhengyang Ma, Wen Yao, Yong Zhao, Yiyong Huang

Surrogate Based Global Optimization Using Adaptive Switching Infill Sampling Criterion

A novel infill sampling criterion is proposed for a surrogate based global optimization algorithm. Due to the extensive amount of calculations required for the meta-heuristic global optimization methods, a surrogate model was employed. In the surrogate based global optimization, SBGO, an iterative process of constructing a model and sampling new points are repeated until a stopping criterion is met. An infill sampling criterion, ISC, controls which data point should be sampled, however, because the characteristics of a design problem are prone to influence the performance of the algorithm, an adaptive ISC should be developed. Thus, in this study, an algorithm that adaptively searches globally and locally considering the current existing samples is proposed. The novel ISC is integrated with a global search measure weighted minimum distance, WD, which considers not only the most ambiguous regions but also accounts for the response values for higher efficiency. The algorithm was tested on unconstrained mathematical functions including the Dixon-Szego test functions and the results were compared with other SBGO algorithms. Additionally, the algorithm was further expanded and implemented to constrained optimization problems using penalizing coefficients.

Dohyun Park, In-Bum Chung, Dong-Hoon Choi

Enhanced Firefly Algorithm with Implicit Movement

The question for an optimal solution to a certain real-world problem often turns into a complex optimization problem. The sizing of the cross sections for bars of a truss structure is generally hampered by interdependencies. This prevents local search methods from finding a sufficient optimum. For those issues, there is a demand for fast and reliable global optimization algorithms. The Firefly Algorithm is a swarm-intelligence-based method frequently used for solving multi-modal optimization problems. The algorithm maintains a set of individuals, each corresponding to a point within the solution space. During the optimization process, the individuals move within the solution space under certain rules in order to find the global optimum. This paper presents an enhancement of the Firefly Algorithm by an implicit backward Euler movement. Therefore, in each iteration a linear system of equations must be solved to determine the new positions of the individuals. To evaluate the performance of the implicit movement, it is applied to continues benchmark optimization functions. The optimization process is compared to the basic Firefly Algorithm to specify the effect of implicit movement. Furthermore, a discrete parameter optimization of a ten-bar truss in the sense of a weight reduction is carried out. The optimization results are compared to the basic Firefly Algorithm as well as to the results of four state-of-the-art algorithms. The implicit movement provides an intuitive and an easy to implement modification of the Firefly Algorithm. Simulation results show that the implicit movement causes a significant improvement in the convergence behavior compared to the basic Firefly Algorithm and outperforms state-of-the-art algorithms in terms of the solution quality and convergence behavior. Due to its generality, the proposed implicit movement can be implemented to several swarm-intelligence-based algorithms and offers a promising universal approach for enhancement.

Ronald Bartz, Sierk Fiebig, Thilo Franke, Paul Falkenberg, Joachim Axmann

Application of Multilevel Optimization Algorithms

In our industry researches we often face very difficult problems where ordinary algorithms fail to find the global optimum. Mostly they have difficult, high-dimensional count, very large design space where even the concept of direction and distance are non-existent and have to be defined, the neighbourship in the feasible space also needs definition. In these cases, these terms are often defined and calculated by heuristic functions. On these problems the applied optimization methods often fail, they stuck in local optima, working very slowly and find suboptimal solution. So we decided to try to link optimization methods and create multi-level optimization methods to cope these problems. As a base concept in the first stage we use some simple, fast, rapidly converging algorithm, then some finer grade algorithm like population based swarm optimization method. In this paper, we will show and evaluate some multi-level optimization methods tested on several test functions, comparing the convergence and computational needs.

László Kota, Károly Jármai

Part IX: Structural Optimization: Sizing


Structure Sizing Optimization Capabilities at AIRBUS

Structural sizing optimization at Airbus leans on different approaches depending on the needs. Two main solutions have been defined, mainly for questions of performance and deployment. A rapid solution (PRESTO) has been developed based on a pure discrete algorithm to perform parametric trade-off studies in early design stages. The structure optimization process has been simplified in such a way that it is as quick as possible. This is the right solution to be used by non-expert users in early overall aircraft studies. It exploits large databases and surrogate models in a Big Data mindset. A more advanced solution (ACO-AMO) has been developed to support later design stages and more detailed optimization. It uses a standard continuous optimization approach based on gradient information and state of the art mathematical programming algorithms. A second step allows performing a discrete ply-by-ply optimization for composite covers using an optimization heuristic (ant colony paradigm). This is the right technology to address advanced structure optimization and to be included in MDO processes. Both methods are under integration in a single platform (SOFIA) to provide adequate sizing optimization services for the relevant components (e.g. wing, fuselage) and design stages. A status will be made in this paper on the technologies used in the different tools, the applications performed and the current and future research and development axes especially in the effort to merge solutions or to integrate them in multidisciplinary processes.

Stéphane Grihon

Mixed-Integer Linear Programming Reformulation Approach for Global Discrete Sizing Optimization of Trussed Steel Portal Frames

This paper presents a method to find the global solution of combined truss-frame size optimization problems. The approach is based on a reformulation of the optimization problem as a Mixed-Integer Linear Programming problem (MILP) which is solved by means of a branch-and-bound method. A portal frame that consists of both beam and truss elements is adopted as a test case. The optimal sections of the portal frame have to be selected from a square hollow sections catalog. The design of the portal frame has to meet the requirements prescribed by the Eurocodes. These requirements are adopted as constraints by reformulating them as or approximating them by a linear equation. Not only the Eurocode constraints related to member strength and stability but also all Eurocode constraints related to the joints of the structure are taken into account during the optimization. As a consequence, a post-processing step to account for other constraints is avoided, therefore optimality is retained and additional engineering time is reduced. The optimization results are presented and the influence of the joint constraints on the optimal design is discussed. In addition, the efficiency of the method is compared with the efficiency of a genetic algorithm.

Roxane Van Mellaert, Kristo Mela, Teemu Tiainen, Markku Heinisuo, Geert Lombaert, Mattias Schevenels

Optimal Design of Double-Pipe Heat Exchangers

Heat exchangers are used in industrial and household processes to recover heat between two process fluids. This paper shows numerical investigations on heat transfer in a double pipe heat exchanger. The working fluids are water, and the inner and outer tube was made from carbon steel. There are several constructions which able to transfer the requested heat, but there is only one geometry which has the lowest cost. This cost comes from the material cost, the fabrication cost and the operation cost. These costs depend on the material types and different geometric sizes, for example inner pipe diameter, outer pipe diameter, length of the tube. The performance of the heat exchanger and the pressure drop are in a close interaction with the geometry. Optimum sizes can be calculated from the initial conditions (when one of the process fluid inlet and outlet temperature and the flow rate is specified). The correlations to the Nusselt number and the friction data come from experimental studies.

Máté Petrik, Gábor Szepesi, Károly Jármai

Part X: Structural Optimization: Fiber and Composite Optimization


Optimization of Oriented and Parametric Cellular Structures by the Homogenization Method

We present here a topology optimization method based on a homogenization approach to design oriented and parametrized cellular structures. The present work deals with 2-D square cells featuring a rectangular hole, because their structure is close to that of rank-2 sequential laminates, which are optimal for compliance optimization. For several cells, the value and the parametric sensitivities of their effective elastic tensor can easily be computed, by the resolution of a cell problem. The obtained results can be used to build a surrogate model for the homogenized constitutive law. Moreover, we add the local orientation of the cells to our problem. Then, an optimal composite shape is computed thanks to an alternate directions algorithm. The crucial ingredient of the methodology is the extraction of a quasi-periodic and additive manufacturable structure from the previously obtained composite shape, based on the introduction of a space transformation.

Perle Geoffroy-Donders, Grégoire Allaire, Julien Cortial, Olivier Pantz

Generating the Best Stacking Sequence Table for the Design of Blended Composite Structures

In order to improve the ability of a large-scale light-weight composite structure to carry tensile or compressive loads, stiffeners are added to the structure. The stiffeners divide the structure into several smaller panels. For a composite structure to be manufacturable, it is necessary that plies are continuous in multiple adjacent panels. To be able to prescribe a manufacturable design, an optimization algorithm can be coupled with a reference table for the stacking sequences (SST). As long as the ply stacks are selected from the SST, it is guaranteed that the design is manufacturable and all strength related guidelines associated with the design of composite structures are satisfied. An SST is made only based on strength related guidelines. Therefore, there exist a large number of possibilities for SSTs. Minimized mass is a typical goal in the design of aircraft structures. Different SSTs result in different values for the minimized mass. Thus it is crucial to perform optimization based on the SST which results in the lowest mass. This paper aims to introduce an approach to generate a unique SST resulting in the lowest mass. The proposed method is applied to the optimization problem of a stiffened composite structure resembling the skin of an aircraft wing box.

F. Farzan Nasab, H. J. M. Geijselaers, I. Baran, A. de Boer

A Lean Method for Local Patch Reinforcement Using Principal Stress Lines

Composite materials offer the possibility to design tailored laminates for a broad spectrum of applications such as airplane wings or monocoques in the automotive industry. Especially when carbon fiber reinforced plastic is used, parts with enormous performance can be designed. The downside of designing a laminate is the complexity of the task itself, as well as the price of the carbon fiber. Placing the expensive carbon fiber only at the necessary positions can help to reduce the costs of the parts and to increase the performance, specifically the stiffness, compared to a glass fiber laminate itself. Designing the laminate gets even harder, an algorithm is needed to support the engineer in this task.This paper will introduce a lean method using the principal stress line to locate the best position for carbon fiber patches using a Michell structure. The Michell structure is optimized for tensile and pressure forces. Unidirectional tapes have outstanding properties in fiber direction, but weak performance orthogonal to the direction of the fiber. Placing the unidirectional tapes along the Michell structure will load the tapes in an optimal way. First of all the algorithm will determine the principal stress lines connecting load and bearing. These will be used as a starting point for a Michell structure. The generated Michell structure will be optimized to increase the performance. After the substructure is found, it will be mapped back into the original laminate. The new improved laminate will be compared with the original one, concerning weight, stiffness and pricing. A significant reduction in weight could be achieved at a constant price and improved stiffness.

Philipp Gebhardt, Eiko Türck, Thomas Vietor

Frequency Response Characteristics of 2D Wings in Uncertain Environments: A Random Matrix Theory Approach

Whilst structural design processes in engineering have been extensively developed, the additional consideration of uncertainty quantification (UQ) provides a more holistic forecast on its long term sustainability. UQ methods such as Polynomial Chaos Theory have received attention in numerous fields within engineering for its ability to approximate statistical moments with good accuracy and with low computational expense. This study explores a probabilistic approach to analyze frequency response in 2D wings facilitated by Random Matrix Theory (RMT). This UQ method has not been explored thoroughly within the aerospace sector. Uncertainties are enforced on the length, width and Young’s modulus of two wings varying in geometry and the natural vibration mode characteristics are determined via finite element modeling. Baseline characteristics are compared to an approximation derived from RMT and the worst case scenarios. It was found that RMT provided a good estimate of both the frequency and magnitude shifts of the vibrational modes under the enforced uncertainty. The more complex geometry was found to be more robust to the imposed variations, and RMT was able to capture this behavior effectively.

Aditya Vishwanathan, David Munk, Gareth Vio

Gradient Based Structural Optimization of a Stringer Stiffened Composite Wing Box with Variable Stringer Orientation

The structural optimization plays a key role in multidisciplinary optimization. The proof of structural integrity is a prerequisite for the performance assessment of a wing design. In addition, the modification of the structural design allows changing the bend-twist coupling properties in a beneficial way for overall cruise performance. Due to the high number of design variables affecting the mass and stiffness of a wing box, a gradient based process is established. It meets the needs of fast convergence and enables the coupling with aerodynamic analyses that provide gradients as well. A well suited parametrization of the design variables is necessary, especially for composite materials. Therefore, lamination parameters are used, which are proven to be suitable for gradient based optimization. In order to consider stiffening structures (i.e. stringer on the wing cover), a smeared stiffener approach is used. With this approach, it is not necessary to model the stringer explicitly in the Finite Element model. The influence of different stringer shapes and their orientation can be evaluated with the method suggested in this paper. In order to reduce calculation time, the numerical model is evaluated using analytic formulations for global and local stability as well as strength. The two approaches, smearing the stiffeners or explicitly modeling stiffeners, are validated by comparison of global deformations. The optimization process is applied to a representative wing box loaded with an eccentric load. The influence of different stringer orientations on the structural deformation is examined in conjunction with the optimization of lamination parameters.

Sascha Dähne, Christian Hühne

Optimization Approach for Free-Orientation of a Laminated Shell Structure with Orthotropic Material

In this study, we propose a non-parametric material orientation optimization method for optimum design of laminated composite shell structures consisting of orthotropic materials. We consider a single objective in terms of the compliance in the present work and minimize it under the state equation constraint. The material orientation in all layers are taken as design variables. The optimum design problem is formulated as a distributed-parameter optimization problem, and the sensitivity function which respect to the material orientation variation is theoretically derived. The optimum orientation variations are determined by using the H1 gradient method with the Poisson’s equation, where the sensitivity functions are applied as the fictitious internal heat generation with the Robin condition to vary the material orientation. With the propped method, we can obtain the arbitrary and smooth distribution of the material orientations of all layers to reduce the compliance without design parametrization. The optimum design examples show that the proposed optimization method can effectively obtain the optimum distribution of the material orientation.

Yoshiaki Muramatsu, Masatoshi Shimoda

Structural Optimization of Stiffened Composite Panels for Highly Flexible Aircraft Wings

Within this paper, the structural optimization of stiffened composite panels is considered. The application example is a wing of a long–range aircraft, whereby the flexibility has to be increased due to flight physical demands. To develop the new structural design concept, numerical optimization is used. The structural analysis is carried out utilizing detailed finite element models (DFEM) with linear static and linear eigenvalue simulations. Using this DFEM– and gradient–based optimization environment, designs are analyzed and optimized regarding their mass, considering damage tolerance and stability requirements.The advantages of the chosen simulation and optimization approach are presented. The new design concept is used to improve the aircraft wing and the differences to the state of the art wing design are depicted.

Tobias Bach, Christian Hühne

SIMP Based Topology Optimization for Injection Molding of SFRPs

Today there exists a huge demand for technologies which enable and facilitate the mass production of fiber reinforced composites. Injection molding of Short Fiber Reinforced Plastics (SFRP) is a quite popular method especially in the automotive industry, providing high stiffness levels on the one hand and complex moldable shapes on the other hand. Due to the high cost of mold production and injection molding machines, nowadays lots of research is done to improve models and to develop software for the simulation of this process. This allows to detect problems with the mold design and optimization of the part performance and quality at an early stage of the development.In the case of SFRP injection molding, the mechanical properties of the finished part are mainly influenced by the local fiber orientation, which itself depends on the shape/topology of the part. We investigate an approximate approach for compliance-based topology optimization for such parts, where we replace the costly filling and fiber orientation simulation by the solution of an eikonal equation which determines the principal fiber orientation approximately.Sensitivities for compliance minimization are derived for the classical Solid Isotropic Material with Penalization (SIMP) method in combination with a transversely isotropic material law. The optimization problem is then discretized using the Finite Element Method and solved using a projected gradient method.

Felix Ospald, Roland Herzog

Part XI: Structural Optimization: Shape Optimization


Optimization of Stepped Plates in the Elastic Plastic Range

An optimization technique is developed for circular plates of piecewise constant thickness. The plates under consideration have been manufactured of an ideal elastic plastic material obeying Tresca’s yield criterion. Necessary optimality conditions are derived with the aid of the theory of optimal control. Obtained system of differential-algebraic equations is solved numerically in the case of the plate with single step of the thickness. Effectiveness of the design is assessed numerically.

Jaan Lellep, Julia Polikarpus

Geometric Design of Tumbling Mill Lifter Bars Utilizing the Discrete Element Method

The energy consumption of bulk materials handling consumes around 10% of the annual energy utilization on the planet, often driven by inefficient processes. A primary process in the reduction of particle size to fine grind are ball mills. Ball mills utilize ball charge to aid the grinding and are usually used as a primary grinding solution. Ball mills usually operate with charge around 30% with ball mill units utilizing up to 20 MW while standing up to a height of around 9 m. Industrial ball mills draw around 0.0011% of the world’s total power. We consider the geometric design optimization of the lifter bars for a ball mill. By utilizing the discrete element method (DEM) the collision frequency, relative velocity between colliding particles and contact force to estimate the specific impact energy for the ball mill is available. In this study we investigate the effect of the geometry of the lifter bar, that is height and bar angle with respect to the side wall, has on the specific impact energy for various charge distributions. In addition to the specific impact energy the energy spectra of the ball mill is computed to assess the energy distribution in normal and shear interactions. As the discrete element method is computationally demanding we investigate the extent to which the computational cost can be mitigated by following a multi-fidelity approach in which the number of particles and particle sizes are scaled to reduce the initial computational cost and only refined as the design optimization converges. Towards this aim we consider a virtual problem in which we aim to design the lifter of a tumbling mill for a specified power draw of the mill. The problem under consideration is denoted virtual as the solution of the problem is known a priori. This allows us to quantify and assess the quality of the obtained designs using the various strategies. The associated computational cost when considering the full computational model against the multi scaled computational models are then quantified.

Daniel N. Wilke, Nicolin Govender, Raj K. Rajamani, P. Pizette

Shape Optimization of Shell Structure for Controlling Transient Response

In this study, we propose a new approach to control transient response by optimizing the shape of a shell structure. The free-form optimization method for shells, a parameter-free shape optimization method developed by one of the authors, is extended for the transient response problem of a shell structure. The design objective is to minimize the dynamic compliance or to control the displacement at arbitrary points and times or time periods to the desired values under volume constraint. We solve this problem directly without converting the discrete equivalent static loads, which is generally employed in time-dependent optimization problems. The optimum design problem is formulated as a distributed-parameter optimization problem and the sensitivity function for this problem is theoretically derived. Based on the gradient method in the function space, we can determine the optimal free-form of a shell structure. With the proposed method, we can obtain the optimal design of a shell structure to satisfy the design objective and constraint while maintaining the surface smoothness. Some optimum design examples are demonstrated and the results are discussed at last.

Mamoru Wakasa, Masatoshi Shimoda

Shape Optimization for Microstructure Design of Porous Materials Described by the Biot model in the Homogenization Framework

The paper deals with shape optimization of microstructures generating porous locally periodic materials saturated by viscous fluids. The porous material is described as the Biot continuum derived by the homogenization method. The effective medium properties are given by the drained skeleton elasticity, the Biot stress coupling, the Biot compressibility coefficients, and by the hydraulic permeability of the Darcy flow model. These are computed using characteristic responses - solutions of the state problems defined in the representative unit cell constituted by an elastic skeleton and by a fluid channel generating the porosity. The design of the channel is described by a B-spline box which embeds the whole representative cell. The sensitivity analysis for all the homogenized material coefficients is derived using the domain method of the design velocity approach. The optimality criterion is aimed to maximize stiffness of the drained porous material and allow for a sufficient permeability and vice versa. Issues of the spline box parametrization, the channel shape regularity and FE mesh updates are discussed. The maximization problems are solved using the sparse nonlinear optimizer SNOPT.

Eduard Rohan, Daniel Hübner, Vladimír Lukeš, Michael Stingl

Optimum Morphing Shape Design for Morphing Wing with Corrugated Structure Using RBF Network

The morphing wing changes the geometrical shape seamlessly and continuously to improve the aerodynamic performance. Several studies have been conducted with variety of approaches. This study focuses on obtaining the optimum airfoil shape to maximize the lift-to-drag ratio or the maximum lift coefficient for the morphing wing using the corrugated flexible structures arranged in the rear side of the airfoil developed by one of the authors. The corrugated structure is connected to the upper skin to bend the airfoil as a morphing flap. During the morphing, the upper skin length is unchanged, but, on the other hand, the lower skin length is shorten. This airfoil morphing shape is modeled by using non-uniform rational basis spline (NURBS) curves with considering the above curve length condition, where the NURBS passing points are adopted as design variables. For numerical efficient optimization, sequential approximate optimization method using radial basis function (RBF) network is adopted. During the optimization, a design candidate will have an irrational wavy shape that brings an ill condition to the aerodynamic analysis, where the two-dimensional panel method is adopted as the aerodynamic analysis. To avoid the aerodynamic analysis for the violated design, the concept of tabu search is introduced to the RBF network. That is, if a violated wavy design candidate is selected during the optimization, the design is added to the tabu list without performing aerodynamic analysis. That is, the RBF network learning makes the searching region enclosed into the feasible design space and then the optimum design found rationally. The idea of the tabu search is also adopted to find the feasible initial design candidates. Through numerical examples, the validity of the proposed optimization method is demonstrated. Then, the optimum morphing wing shape is discussed.

Gen Nakamura, Kengo Uehara, Nozomu Kogiso, Tomohiro Yokozeki

Part XII: Structural Optimization: Topology Optimization with Density Methods - Principal Approach


Comparison of Different Formulations of a Front Hood Free Sizing Optimization Problem Using the ESL-Method

Topology and free sizing optimization are very important tools in the design phase to compute optimized design proposals. However, they are only well established for optimization problems based on linear analysis. In contrast, in the nonlinear analysis area – in particular in crash analysis - such kind of optimization cannot be applied due to the unavailability of gradients. A workaround is to create linear auxiliary load cases, which approximate the nonlinear load case at different time steps and which can then be used in optimization based on linear statics analysis. The ESL (Equivalent Static Load) method provides a procedure to create such auxiliary load cases in a well-defined way. A general drawback of such an approach is that responses in the nonlinear system and the related requirements are not defined in the linear statics system. Hence, a formulation must be derived translating the nonlinear requirements to the linear statics system. This is sometimes very difficult and is also the reason why very often the strain energy is used as objective function even if this does not reflect the real objective in the optimization task.In this study different formulations for a front hood free sizing optimization problem are investigated to answer the question to what extent it is possible to translate the real objective of the nonlinear system – the minimization of the HIC score – to the linear statics system. It turns out that even though such formulations can be found, they show no advantage in comparison to simpler formulations. The best performance can be obtained by using the mass as objective function, whereas the strain energy formulation fails due to the tendency to favor huge mesh deformations in void areas. The reason for such behavior and the related issues appearing especially in topology and free sizing optimization are explained and discussed in detail. Finally, some recommendations are given how to improve the performance of ESL-based topology and free sizing optimization.

Artem Karev, Lothar Harzheim, Rainer Immel, Matthias Erzgräber

A Study on the Design of Large Displacement Compliant Mechanisms with a Strength Criteria Using Topology Optimization

In this work, large displacement compliant mechanisms are designed using a gradient-based topology optimization formulation. Since such devices are designed to fulfill a kinematic task, take into account a nonlinear approach is very important. Besides that, as the mechanism strain is used to achieve the kinematics, a strength criteria must be included in the optimization problem. The use as well as the need of geometrical and/or material (compressible hyperelasticity) nonlinearities are discussed as well as the effectiveness of the stress constraint. The method of moving asymptotes is applied for the design variables updating. The derivatives are calculated analytically, by the adjoint method. In order to ensure topologies as near as possible to 0/1 solution, projection filtering techniques are applied. The well-known hinge problem in compliant mechanisms is also discussed and the role of both nonlinear analysis and stress constraint are discussed by means of benchmark examples. Eventually, the results are compared with a linear formulation.

Daniel M. De Leon, Juliano F. Gonçalves, Carlos E. de Souza

Efficient Density Based Topology Optimization Using Dual-Layer Element and Variable Grouping Method for Large 3D Applications

In this paper, efficient density based topology optimization method is proposed. FEA (Finite Element Analysis) meshes are distinguished from topology density element in this method. FEA element stiffness can be integrated by summation of density element stiffness using the MTOP (Multi-resolution Topology Optimization) method, and this dual layer element approach can reduce FEA computations. In spite of FEA time reduction, numerous density element cause the increasing of calculation time during the optimization process. Therefore, variable grouping method is proposed. Variable reduction process is performed according to histogram of multiplication of sensitivity and density variable. The result of compliance minimization example is presented for the large 3D application. Due to Dual layer element approach and variable grouping method, computation time of the density based topology optimization can be extremely reduced.

Jaeeun Yoo, Ikjin Lee

Topology Optimization and Reinforcement Derivation Method (RDM®) of a Hybrid Material Sump

This project aims to demonstrate how the power of various optimization methods, when coupled with experienced design engineers, can deliver weight savings and performance improvements. Topology optimization methods were applied to develop a new lightweight hybrid material sump, whilst achieving the same stiffness and stress requirements as the baseline aluminium sump casting. The optimization study explored three design options with various levels of differential integration into the sump casing, each generating different package spaces. Topology optimization in Genesis was applied to three options while keeping the same objectives and constraints to determine the load paths and remove superfluous material. The design which gave the best compromise between mass saving and design risks was then chosen for development.Manufacturing constraints were added to the topology optimization region containing the load-bearing aluminium structure for the design interpretation phase. Using the raw optimization results, a further collaborative work program between GRM’s Finite Element (FE) analysts and Jaguar Land Rover’s (JLR) design engineers led from an interpretation of an organic-shape into a manufacturable aluminium sump. A lightweight non-structural plastic reservoir inside the sump ensured the design was oil-tight. The combination of the aluminium skeleton and the oil-retention plastic formed the hybrid material sump.The Reinforcement Derivation Method (RDM®) was used on more mature design iterations to reinforce the new sump by highlighting areas altered during the interpretation phase and guarantee all stiffness requirements were met.Topology optimization and RDM® with stiffness, strength and manufacturing constraints were used to develop a hybrid material sump from a blank page leading to a mass reduction of 20% against the baseline without losing any performance and gaining additional secondary benefits.

Marine Favre Decloux, Alex Desmond, Lucy Fusco, Martin Gambling, Markus Hose

Topology Optimization with Stress Constraints Using Isotropic Damage with Strain Softening

Considering stress constraints in continuum topology optimization is challenging because the optimization involves a large number of design variables as well as large number of constraints. The most common approach is to aggregate the constraints into a single or a few global approximate functions. A different approach was proposed recently in which elasto-plastic response is considered and plastic strains are minimized, so that the yield conditions can be enforced implicitly. The main drawback of this approach is the added computational cost due to the nonlinear analysis. In the current work we aim to reduce the computational burden by using a simpler, yet effective, nonlinear material law. In the proposed approach, the continuum domain is modeled with isotropic damage and strain softening. The optimization problem involves only compliance and volume that can play the roles of either the objective or the constraint. An attractive aspect of this formulation is that no actual stress constraint is necessary. Once strain softening is considered, material that is strained beyond the yield strain becomes uneconomical in terms of strength. Therefore, the design is driven toward the utilization of material up to the yield strain and not beyond it – meaning that the stress constraint is imposed implicitly by the model. Numerical experiments show promising results: the proposed procedure is capable of generating stress constrained topological layouts that match results achieved by various other approaches.

Yakov Zelickman, Oded Amir

Simultaneous Topology Optimization of Material Density and Anisotropy

This paper presents an optimization methodology in order to find simultaneously optimal density and anisotropy. The objective is to maximize the structure global stiffness measured by the compliance. The density and the elasticity tensor are defined as design variables. The numerical procedure is composed of finite element stress calculations and local minimization problems. Thanks to the polar method, these local minimization problems are solved analytically. The method proposed turns out to be straightforward. Two numerical results are illustrated in the 2D-case: optimal designs depending on anisotropy, and optimal design and anisotropy.

Narindra Ranaivomiarana, François-Xavier Irisarri, Dimitri Bettebghor, Boris Desmorat

A Simple Approach to Deal with Zero Densities in Topology Optimisation

Topology optimisation techniques typically use a very small positive density $$\varepsilon $$ to model voids. Despite its simplicity and generally acceptable results, this approach can impose a number of difficulties. The weak material should be weak enough to validate the approximation of void areas, but on the other hand, using a very weak material can result in ill-conditioning of the stiffness matrix. Further and more serious complications can arise, for example in non-linear problems where weak elements cause numerical instabilities in the solution procedure. By studying the mechanical responses of structures when $$\varepsilon \rightarrow 0$$, this paper presents a simple approach to use arbitrarily weak material properties in void areas. This approach would effectively allow us to actually remove the void areas from the mesh in a range of problems and avoid the above-mentioned complexities.

Kazem Ghabraie

Using Exact Particular Solutions and Modal Reduction in Topology Optimization of Transient Thermo-Mechanical Problems

Design of transient thermo-mechanical systems is a challenging task often encountered during the design of high precision machines and instrumentation. Topology optimization can provide valuable insight during the design process, however, for large scale problems the backward time integration required to obtain adjoint sensitivity information is undesired. Previous work has illustrated how the introduction of a reduced modal basis allows to eliminate the backward time integration to obtain the adjoint variables. In order to reduce computational effort further, additional reduction approaches are considered. The focus is specifically on design cases where the relevant heat loads can be expressed or approximated analytically by combinations of harmonic, polynomial or exponential functions of time. Using the method of undetermined coefficients, an exact particular solution is obtained using the full system. Then, the corresponding homogenous solution is expressed using a reduced modal basis, for which a relatively small set of modes is required to obtain an accurate approximation. For the cases where the time component of the heat loads are expressed by the considered analytical functions, the backward time integration is eliminated from the calculation of the design sensitivities, while the forward integration is handled by convolutions.

Max van der Kolk, Evert C. Hooijkamp, Matthijs Langelaar, Fred van Keulen

Optimal Tendon Layouts for Concrete Slabs in Buildings Derived Through Density-Based Topology Optimization Algorithms

Post-tensioned (PT) concrete flat-plates are highly efficient gravity systems, especially in high-rise building construction. Efficiency is particularly present in multi-span applications with orthogonal grids and regular support arrangements, reducing slab thickness and producing benefits in architecture, engineering and construction. Traditionally in the United States, following the requirements of ACI318, the system is resolved by a quasi-orthogonal tendon layout with banded strands in one direction and distributed strands in the perpendicular direction.A novel approach to determining PT tendon layouts informed by topology optimization results has been applied by Skidmore, Owings and Merrill, LLP (SOM) to several recent projects. The optimization utilizes density methods in order to find optimal load paths in a finite element continuum design space defined by the concrete floor plate. By interpreting the density distribution and orienting the PT tendons along the optimal load paths suggested by topology optimization, it has been shown that savings of 25% or more on tendon quantities can be achieved. The majority of the observed arrangements do not follow a traditional banded/distributed layout. As a result, the deflection performance is also significantly more uniform since the optimized tendon layout reduces the load path throughout the floor. This can help alleviate common issues with thin flat-plate gravity systems such as irregular floor flatness due to warping induced by PT systems and inconsistent deflection at the perimeter.Pioneering applications of this new design approach have been evaluated for three buildings in California, United States and coordinated with construction teams for efficiency. This paper will discuss the design procedure from initial concepts derived from topology optimization to complete construction documents as applied to the case studies, and will create a conversation that will be of interest to both academics and practicing engineers.

Mark Sarkisian, Eric Long, Alessandro Beghini, Rupa Garai, David Shook, Ricardo Henoch, Abel Diaz

Contributions to Handle Maximum Size Constraints in Density-Based Topology Optimization

The maximum size formulation in topology optimization restricts the amount of material within a test region in each point in the design domain, leading to a highly constrained problem. In this work the local constraints are aggregated into a single one by p-mean and p-norm functions, classically used for stress constraints. Moreover, a new test region is investigated which is a ring instead of the classical circle around the element. These developments were implemented for compliance minimization with the MBB beam test case. Results indicate that p-mean performs better in the maximum size field than p-norm, because it underestimates the most violated constraint. This gives some relaxation to the problem that allows stiffer connections. Similar effect has been observed for the ring-shaped region which reduces the amount of holes that are introduced in the structure, specially in the connection of solid members. In addition, it is shown that the maximum size formulation allows the definition of the minimum gap between solid members which gives designers more control over the geometry. The developments have been illustrated and validated with compliance minimization tests of 2D-domains.

Eduardo Fernández, Maxime Collet, Simon Bauduin, Etienne Lemaire, Pierre Duysinx

Multimaterial Topology Optimization of Contact Problems Using Allen-Cahn Approach

The paper deals with the numerical solution of the multimaterial topology optimization problems for bodies in contact. The unilateral contact phenomenon between the elastic body and the rigid foundation with Tresca friction is governed by the elliptic boundary value problem with inequality boundary conditions. The materials distribution function is chosen as the design variable. The structural optimization problem consists in finding such topology of the domain occupied by the body in terms of the design variable that the normal contact stress along the boundary of the body is minimized. The original cost functional is regularized using the multiphase volume constrained Ginzburg-Landau energy functional rather than the perimeter functional. The first order necessary optimality condition is provided. The derivative of the cost functional is used to formulate the generalized gradient flow equation of Allen-Cahn type. The optimal topology is obtained as the steady state of the phase transition governed by this equation. The optimization problem is solved numerically using the operator splitting approach combined with the gradient projection method. Numerical examples are provided and discussed.

Andrzej Myśliński

Conceptual Design of Aircraft Structure Based on Topology Optimization Method

Structure optimization including topology optimization, size optimization, et al. has been proved to be efficient in improving structure performance; this means that the same performance can be achieved by using less material or using the same material for better performance. And optimizations play a bigger role in field of automotive and aircraft design. In this paper, an efficient topology optimization method is proposed for aircraft structure conceptual design. This is achieved as follows: first, based on the load and design requirements, we design a series of optimization model with the same explicit constrains (EC) including maximum displacement, center of gravity in x direction (COGX), first order of nature frequency, which are explicitly required, and varies mass constrain. By topology optimization analysis, the mass requirement will be determined matching with the deterministic constraints. And the material distribution model with high bearing capacity is obtained. Second, according to the topology optimization results, considering the functional requirements, such as devices space, operating cap, a two-dimensional parameter optimization model will be setup. Parameter optimization is carried out with the matching mass and others EC as constrains, and the total compliance as objective. The parameter optimization results can be used to guiding structural design. At the same time, the structure conceptual sketch will be analyzed in the optimization processing. Finally, the conceptual model of parameterized aircraft structure is obtained. Numerical examples based on x-43 aircraft show that the proposed method can obtain good solutions to aircraft design problems. Furthermore, the problem of large computational efforts in the solution process can also be solved under the recent Moving Morphable component (MMC) solution framework.

Guanghui Shi, Yupeng Zhang, Dongliang Quan, Dongtao Wu, Chengqi Guan

Singular, Large-Scale Solutions in Local Stress-Constrained Topology Optimization

In this paper, the union of sequential approximate optimization (SAO) and the ‘simultaneous analysis and design’ (SAND) formulation of the local stress-constrained topology optimization problem is delineated. The method is a standard structural optimization approach based on strictly convex and separable approximate subproblems, except for a straight-forward extension to include nonlinear equality constraints. The reason is, unlike conventional ‘nested analysis and design’ (NAND) methods, the finite element equilibrium equations are retained in the optimization problem as a set of nonlinear equality constraints. This implies that the state variables—i.e., the displacements—are primal optimization variables alongside the material-density variables. Because all optimization variables are independent, the computational cost and the dense coupling which is often consequential to nested formulations and calculation of the associated sensitivity derivatives, reduces to the manipulation of simple and sparse partial derivatives. Moreover, the equilibrium constraints may be violated up to the point of convergence, and, because the global stiffness matrix is not inverted per se, the material-density variables may take on a value of zero (0) exactly on the lower bound. The decoupling of the design and state variables on the one hand, and the exact representation of void material on the other, permits convergence to the singular optima which are typically not available in nested formulations without resorting to constraint relaxation techniques. Finally, numerical experiments on a simple benchmark problem indicates that high levels of computational efficiency may be achieved.

Dirk Munro, Albert Groenwold

Robust Multi-material Topology Optimization for Lattice Structure Under Material Uncertainties

Recent advances in additive manufacturing have enabled the manufacture of hollow shapes with complex external geometry and multiple materials using networks of small periodic cells known as lattice structures. The porosity and high number of elements in these structures generate lightweight designs with improved performance and functionality. This paper proposes a design approach for lattice structures considering both multi-material design and uncertainty in material properties. A density-based multi-material topology optimization method is proposed to find the optimal geometry and material layout. In addition, a new interpolation scheme for using multiple and arbitrary materials is presented. For robustness of structural compliance against uncertainty in material properties, a weighted sum of the mean and standard deviation is chosen as an objective function. To estimate statistical moments by the material uncertainty, the univariate dimension-reduction (UDR) method combined with Gauss-type quadrature sampling is employed. Two numerical examples demonstrate the efficiency of the proposed approach and the effect of material uncertainty on the robust design results.

Kohei Shintani, Yu-Chin Chan, Wei Chen

Part XIII: Structural Optimization: Topology Optimization with Density Methods – Special Extensions


An Element Deactivation and Reactivation Scheme for the Topology Optimization Based on the Density Method

Topology optimization results based on the homogenization or density method highly depend on the discretization of the design space. The smallest possible dimension of the optimized structure is one element edge length. If the mesh can be refined, smaller design features can be represented and thereby the performance of the optimized structures can be improved. But the mesh refinement is limited by the computational cost of the Finite Element Analysis (FEA).The aim of this contribution is the reduction of the computation time due to a reduction of the FE-model. Therefore a scheme for the density method is introduced, which deactivates low dense elements and enables a reactivation of important elements. These low dense elements usually do not have a mechanical importance and they also have nearly zero sensitivity considering compliance, stress, nodal displacements or eigenfrequency using the SIMP-approach (Solid Isotropic Material with Penalization of intermediate densities). Nevertheless all elements of the design space are usually included in the FEA during the topology optimization. Our approach reduces the number of active elements continuously with the iterations as the design converges to a black-and-white-design.

Robert Dienemann, Axel Schumacher, Sierk Fiebig

Topology and Cost Optimization Applied to Develop New Designs for a Monorail Structure

In this study, topology optimization was performed using a design methodology to minimize both the cost and weight of new designs for the skirt support structure of a monorail. Finite element analysis, topology optimization, manufacturing analysis, and a comparative cost analysis were conducted to develop novel lightweight designs. Verification of the designs was done by comparing them against the original structure.An analysis of the original structure was undertaken: structural analysis was performed using finite element analysis, and a cost breakdown was done based upon provided information by the industrial partner and the cost of the original structure. Manufacturing techniques used to create the original structure were identified.Topology optimization was performed in several iterations to converge on a quality result for the different designs. The large scale of the design space necessitated a new approach for creating detailed results from a large coarse mesh. Multiple loading cases and constraints were accounted for. The topology optimization results were then used to generate manufacturable designs. Different manufacturing methods were analyzed for the best trade-off between representation of the raw optimization result, structural performance, and cost efficiency.The final step was a cost analysis. The cost analysis was done comparatively to ensure the most qualitative and accurate comparison. The cost of the original structure and its manufacturing methods were used to build and validate the cost model. Features such as the number of parts, joints, bends, and length of cuts were used as metrics for the cost model.Two new, optimum designs resulted from this study: the first was 55% lighter than the original structure and 54% cheaper to manufacture, and the second was 53% lighter than the original structure and 58% cheaper to manufacture. These results greatly exceeded the initial target of 30% weight savings and 25% cost savings.

Christopher Carrick, Il Yong Kim

Knowledge Discovery in Dataset Generated by Topology Optimization

In this paper, we propose a new framework to obtain knowledges for appropriate formulations in topology optimization. The basic idea of this framework is incorporating Knowledge Discovery in Databases (KDD) and topology optimization. That is, we construct a database that records various material distributions generated by topology optimization for various structural design problems, including the varieties of design domain shapes, boundary conditions, and performance indicator functions, and then, we find appropriate material distributions in the database for a given design domain, boundary condition, and performance indicator functions. Furthermore, the proposed framework is designed such that we can easily change the performance indicator functions used to evaluate the material distributions. By doing so, we can easily test various indicator functions and obtain knowledges for appropriate setting of indicator functions, i.e., formulation. In contrast to conventional topology optimization methods which find optimized material distributions under given formulations, the proposed framework aims to find appropriate formulations themselves. We present a simple prototype system based on the proposed framework and discuss the potential of the proposed framework.

Shintaro Yamasaki, Kentaro Yaji, Kikuo Fujita

Automatic Definition of Density-Driven Topology Optimization with Graph-Based Design Languages

Today, the product development process is characterized by increasing diversity. A trend towards customer-tailored products can be observed. This trend demands new processes for product development and manufacturing. Increasing product individuality up to lot size one can be faced with methods, which automate the design process and avoid redundant manual work. As the number of tools involved in the product development process is ever-increasing, one goal is to eliminate the distribution of knowledge into several software tools and begin with one central and consistent data model from where all the software tools used can be triggered automatically. This goal can be addressed by using graph-based design languages [1], which themselves are based on the Unified Modeling Language (UML). On the manufacturing side, one technology can be seen in the additive manufacturing process. For finding a good structure, a combination of additive manufacturing and topology optimization can be advantageous because of the ability of additive manufacturing to create almost arbitrary geometry. Using the example of a lightweight quadrocopter, we propose a graph-based design language which integrates topology optimization and can cover different aspects of the multi-disciplinary product development process: requirements, abstract product functions, design constraints (i.e. equations) and costs among others. The topology optimization triggered by the design language can take into account different product configurations (i.e. packaging configurations) and accordingly makes design proposals (i.e. structural proposals for the frame of the quadrocopter). The executable nature of the graph-based design language reduces the design time significantly.

Manuel Ramsaier, Ralf Stetter, Markus Till, Stephan Rudolph, Axel Schumacher

A PDE-Based Approach to Constrain the Minimum Overhang Angle in Topology Optimization for Additive Manufacturing

Additive Manufacturing allows for considerably more form freedom compared to existing manufacturing technologies but still faces the limitation of building overhanging parts. The overhang limitation in additive manufacturing prevents the direct production of topology optimized parts. We present an overhang constraint that incorporates this manufacturing limitation into topology optimization. The overhanging regions in a design iteration are detected using front propagation and a global constraint is formulated by aggregating the local constraints within the design domain. Since the constraint is formulated in a continuous manner, it can be discretized for any type of mesh, and with an arbitrary minimum allowable overhang angle. Furthermore, it is easily extensible to 3D. The Ordered Upwind Method is used to solve the constraint, and adjoint sensitivities are used for efficient evaluation. The newly developed constraint is demonstrated on 2D examples having an unstructured mesh. Overhang free designs are obtained with smooth convergence behaviour.

Emiel van de Ven, Can Ayas, Matthijs Langelaar, Robert Maas, Fred van Keulen

Optimal External Support Structure Design in Additive Manufacturing

A series of strategies for designing support structure to fabricate overhanging features in additive manufacturing (AM) is proposed. The focus of this study is to maximize the external support stiffness while requiring supports that can be easily removed, minimizing the time required to add the supports, and using the least amount of material. These design requirements become necessities not only for adding supports to improve processing and post-processing efficiency, but also for reducing the cost of supports while maintaining specimen geometry. A repulsion index (RI) is proposed for satisfying the easy removal requirement and minimizing the size of artifacts left on the specimen surface; a weighting function is applied to quantify the time consumption to build the supports. The proposed RI and cost due to additional material and time consumption in adding the support are formulated within a multi-objective topological optimization constructed by the simple isotropic material with penalization method, continuous approximation of material distribution, and method of moving asymptotes. Numerical simulations demonstrated that rational and cost effective support layouts can be determined by the proposed cost-based formulation. This allows designers to find design solutions with a compromise between the deformation and the cost of support structure.

Yu-Hsin Kuo, Chih-Chun Cheng

Topology Optimization of Large Scale Turbine Engine Bracket Assembly with Additive Manufacturing Considerations

The objective of this paper is to perform topology optimization of an assembly structure considering additive manufacturing and the associated expanded design spaces. The examined assembly is an existing turbine engine design, comprised of 44 components, which has undergone rigorous real world testing and verification.Two different topology optimization approaches were considered, using four distinct load cases, considering various mechanical forces and anticipated inertial loads, meant to replicate the most extreme load cases experienced in takeoff, landing, and operation at altitude. The first optimization considers cut-out topology optimization, in which the sheet metal profile of the original bracket assembly design is maintained, while the size and arrangement of cut-outs is optimized. The second optimization features an expanded design space, meant to be representative of the improved design flexibility afforded by recent advances in metal additive manufacturing technology.When the design space limited to conventional sheet metal arrangements was considered, a total weight savings of approximately 25% was obtained, while maintaining equivalent maximum displacement and compliance values. In comparison, the increased geometric flexibility associated with the additive manufacturing design space allowed for a weight reduction of 66%, while reducing maximum displacement within the assembly by approximately 50% in each load case.The expanded design space and the associated drastic volume reduction from the initial design vector present several complications. An incremental design space reduction and refinement is presented. All designs are re-interpreted for manufacturing, with manufacturable designs verified through finite element analysis for comparison with original design. A variety of recommended modelling considerations are presented.

Bradley Taylor, Jamal Zeinalov, Il Yong Kim

Solving 2D/3D Heat Conduction Problems by Combining Topology Optimization and Anisotropic Mesh Adaptation

Topology optimization was recently combined with anisotropic mesh adaptation to solve 3D minimum compliance problems in a fast and robust way. This paper demonstrates that the methodology is also applicable to 2D/3D heat conduction problems. Nodal design variables are used and the objective function is chosen such that the problem is self-adjoint. There is no way around the book keeping associated with mesh adaptation, so the whole 5527 line MATLAB code is published ( The design variables as well as the sensitivities have to be interpolated between meshes, but MATLAB does not support interpolation on simplex meshes and it is thus handled as part of the local operations in the mesh adaptation. This functionality is available for nodal as well as element-wise design variables, but we have found the former to be superior. Results are shown for various discretizations demonstrating that the objective function converges, but comparison to optimizations with fixed meshes indicate that the use of mesh adaptation results in worse objective functions, particularly in 3D. Out of the 5018 statements only 100 is used for the actual optimization loop, 100 for 2D/3D geometry/mesh setup and 50 for the forward problem. It is thus feasible to use the script as a platform for solving other problems or for investigating the details of the methodology itself.

Kristian Ejlebjerg Jensen

Part XIV: Structural Optimization: Topology Optimization with Level Set Methods


Integrated Topology Optimization of Multi-component System Considering Interface Behavior of Interconnection Based on Conforming Mesh and Interface Elements

In integrated topology optimization of multi-component structures, the behavior of connecting interfaces between the components and host structure is a crucial issue closely related to the structural integrity. We propose a topology optimization framework to consider these interface behaviors. We treat the connecting interfaces with the cohesive zone model, which can effectively describe adhesively bonded interface behavior. The conforming mesh is adopted to discretize the evolving structure, and the interface elements are inserted to discretize the connecting interfaces. To clearly describe the structural boundaries and connecting interfaces, we also suggest a multi-material level set model. This level set model can conveniently define the locations of the connecting interfaces, and describe multi-material distribution without redundant phases. The objective function is the sum of strain energy and the work dissipated on the connecting interfaces. We treat the evolving velocities of the level set and the embedded components as design variables. The mathematical programming method (MMA in this study) is employed to obtain these design variables on the basis of adjoint-variable sensitivity analysis. With the proposed optimization formulation, multiple constraints and mixed design variables (i.e. level set design variables and components’ design variables) in level set-based optimization can be easily handled. The Hamilton-Jacobi equation is used to advance the design with the design variables as inputs. The numerical example shows the applicability and effectiveness of the proposed method.

Pai Liu, Zhan Kang

Stress Topology Optimisation for Architected Material Using the Level Set Method

This paper presents a stress-based level set topology optimisation method applied to microstructural design of architected material. Microscopic architected material systems are of interest due to the rise of additive manufacturing. Multiscale topology optimization leads to small members that may be more prone to high stress. The main contribution herein will be the combination of microstructural optimization with a von Mises stress p-norm functional. This type of function has been used to address the local nature issue of stress in macroscale design, approximating the maximum stress in the structure with a single function. The p-norm stress shape sensitivities will be presented using the material derivative method. The proposed level set method formulates a sub-optimization problem in each iteration and uses mathematical programming to obtain the optimal boundary velocities. The Ersatz material approach is used to link the level set method with the finite element structural analysis. First, numerical results for the macroscale are presented to show that stress concentrations are removed. Finally, a microstructural stress analysis based on the homogenization method is devised to reduce stress levels in microstructural topology optimization.

Renato Picelli, Raghavendra Sivapuram, Scott Townsend, H. Alicia Kim

Part XV: Structural Optimization: Topology Optimization with Other Methods


Multi-objective Structural Optimization and Design of Microsatellite Supporting Legs

The supporting legs of TianTuo-3 microsatellite support the satellite and can protect the antennas and other devices, but original design relies on experience of engineers and experiment. The design period is long and the cost is expensive, and structure is always conservative. Optimization Driven Design Process (ODDP) is applied and a multi-step optimization strategy which is more efficient is proposed to find a better design of the supporting legs in TianTuo-3 Microsatellite. First, an optimal topology configuration of leg is obtained by utilizing Solid Isotropic Material Penalty (SIMP) approach, considering minimization of compliance. Then, a multi-objective shape and size optimization is built considering strength, stiffness and stability requirements based on Non-dominated Sorting Genetic Algorithm-II (NSGA-II). Finally, after several rounds of design iteration by engineer, a brand new design is given. The mass of one leg, axial fundamental frequency and maximum von Mises stress are decreased by 17.93%, 4.52% and 11.67% respectively compared with original design. Buckling factor and lateral fundamental frequency are increased by 74.24% and 6.52% respectively. It shows that the multi-step strategy has advantages than traditional design process.

Hao Xu, Yong Zhao, Wen Yao, Ning Wang, Bingxiao Du

Dynamic Behavior of Hanging Truss Having Shape Memory Alloys (From the Optimization Viewpoint of Vibration Isolation and Attenuation)

Attenuation and isolation are typical means to cope with the environmental vibration. In this research, we consider a kind of truss structural system which is expected to possess both of the capabilities. Vibration attenuation is achieved by utilizing the shape memory alloy (SMA) wires characterized by its hysteretic loop of the pseudo-elasticity in relatively high temperature condition. Vibration isolation is realized based on relatively low stiffness of the wire members as well as the hanging configuration of the truss structural system itself; that is, the truss structural system in hanging configuration is able to demonstrate the property of the pendulum. In the current study, we deal with a hanging truss structural system having SMA and ordinary wire members. Both types of wire members have different mechanical characteristics; by combining these two types of wires, the attenuation and isolation capabilities and the stability of the hanging truss against the environmental vibration can be simultaneously achieved. We examine the abilities of this truss structural system through the calculations of the dynamic equations. In order to tackle nonlinearity of such kind of dynamic problems, a time integration method coupled with nonlinear iterative method is introduced. A combinatorial optimization problem is formulated and is solved by multi-objective genetic algorithm. Optimal configurations of the truss structures and the corresponding dynamic behaviors are discussed.

Xuan Zhang, Kazuyuki Hanahara, Yukio Tada

A Novel Heuristic Generator of Structural Topologies Based on Sorted Compliances

Topology optimization, although well recognized and widely developed, remains still up-to-date challenge. It has gained recently more attention since large computational ability become available for designers. This process is stimulated simultaneously by variety of emerging, innovative optimization methods. It is observed that traditional gradient-based mathematical programming algorithms, in many cases, are replaced by novel and efficient heuristic methods inspired by biological, chemical or physical phenomena. These methods become useful tools for structural optimization because of their versatility and easy numerical implementation. In this paper a novel heuristic algorithm for minimum compliance topology optimization is proposed. Its effectiveness is illustrated by the results of numerical generation of optimal topologies for selected plane and spatial structures.

Monika Mazur, Katarzyna Tajs-Zielińska, Bogdan Bochenek

Modifications of Bidirectional Evolutionary Structural Optimization for Structure Compliance

The paper mainly presents a modification of evolutionary rate in bidirectional evolutionary structural optimization (BESO) for structural compliance minimization. In the approach, a self-adapting evolutionary rate based on variation of the structure compliances is proposed. It means the volume change in each iteration will be adjusted to balance its effect on the objective function. Additionally, golden section search is used in averaging the sensitivity of compliance instead of bi-section search in the referred BESO (Huang and Xie, 2010). Furthermore, initial evolutionary rate, filter radius and penalty exponent in material model are tuned to get the appropriate values. The proposed simple technique improves results of benchmark problems in the literature.

Vu Truong Vu

Constrained Versions of the Free Material Design Methods and Their Applications in 3D Printing

The paper deals with the free material design and its constrained versions constructed by imposing: (a) cubic symmetry (cubic material design, CMD), (b) isotropy with: (b1) independent bulk and shear moduli (isotropic material design, IMD), and (b2) fixed Poisson’s ratio (Young’s modulus design, YMD). In the latter case the Young modulus is the only design variable. The moduli are viewed as non-negative, thus allowing for the appearance of void domains within the design domain. The paper shows that all these methods (CMD, IMD, YMD) reduce to two mutually dual problems:− the stress-based minimization problem in which the integrand is equal to a norm of the test stress field. The norm $$\rho (\cdot )$$ involved reflects the type of the constraints imposed;− the displacement-based problem in which the virtual work is subject to maximization over the adjoint displacement fields associated with strains, the dual norm $$\rho ^*(\cdot )$$ of which is bounded almost everywhere.Upon solving the former problem and finding the minimizer one can determine the optimal moduli; they assume non-zero values within the material domain and they vanish outside this domain, thus allowing for cutting out the final shape from the initial design domain. Therefore, the material design methods discussed determine simultaneously: the topology, the optimal shape and elastic material characteristics. The holes appear there where the minimizer vanishes.The YMD method has been made suitable for 3D printing. By using the inverse homogenization method the isotropic and nonhomogeneous YMD designs are replaced by equivalent discrete structures of hexagonal microstructure of varying cross sections of the ligaments. The appropriate numerical codes are prepared to make it possible to produce these fibrous structures by additive manufacturing. The produced prototypes are characterized by high stiffness with respect to the given load.

Tomasz Lewiński, Sławomir Czarnecki, Radosław Czubacki, Tomasz Łukasiak, Paweł Wawruch

Macroscopically Isotropic and Cubic-Isotropic Two-Material Periodic Structures Constructed by the Inverse-Homogenization Method

The present paper deals with the inverse homogenization problem: to reconstruct the layout of two well-ordered elastic and isotropic materials characterized by bulk and shear moduli (κ, μ) and given volume fraction ρ, within a 2D periodicity cell, corresponding to the predefined values of the effective isotropic (κ*, μ*) or cubic symmetry (κ*, μ*, α*) periodic composites. The algorithm used follows from imposing the finite element approximations on the solutions to the basic cell problems of the homogenization theory. The isotropy or cubic symmetry conditions, usually explicitly introduced into the inverse homogenization formulation, do not appear explicitly, as being fulfilled by the special microstructure construction. The square or hexagonal basic cell with the proper internal symmetry is uniformly divided into finite elements each of different element-wise constant material properties. In order to recover the periodic structure of the assumed properties a variety of composites are constructed with different underlying microstructures i.e. the 1-parameter SIMP-like isotropic mixture and 2nd rank orthogonal laminates.

Tomasz Łukasiak

Pylon and Engine Mounts Performance Driven Structural Topology Optimization

Engine deformations during operation are an increasing concern for engine performances. The tip-clearance, defined as the radial gap between the blade tip and the engine casing, can show small variations induced by aircraft maneuvers. These variations can produce increased tip leakage flow, secondary flows and vortex losses that can sensibly increase the engine trust specific fuel consumption (TSFC). In this paper the impact of pylon and nacelle design parameters on in-operation engine performance and overall mass is investigated for a turbofan engine. The structural behavior of the integrated engine will be simulated thanks to a linear finite element model that models the mechanical behavior of the engine under different load cases. Both rotor and stator are modelled to evaluate tip clearance deformations at different engine stages. In a first order approach these engine deformations are evaluated as the difference of radial displacements of superposed points of rotor and stator relative to blade tip. Then an in house performance engine model evaluates thrust specific consumption as a function of the tip clearances.

Simone Coniglio, Christian Gogu, Rémi Amargier, Joseph Morlier

Human-in-the-Loop Layout and Geometry Optimization of Structures and Components

Numerical truss layout optimization employs a ground structure which comprises a large number of potential structural elements from which an optimal subset is sought. The basic problem formulation can be solved via linear programming, which means that very large problems can be solved quickly. However, although layout optimization has been found to provide a highly effective means of identifying (near-)optimal truss layouts, these can be overly complex in form and hence unsuitable for practical use. To address this, a range of practical considerations can potentially be incorporated in layout optimization formulations directly (e.g. via the inclusion of nonlinear and/or nonsmooth terms). However, this will often greatly increase computational cost. Also, some practically important constraints are difficult to specify mathematically (e.g. aesthetic considerations). Furthermore, rather than being presented with a single ‘optimal’ design, designers often seek more flexible tools that allow them to interactively modify a design for a wide range of reasons.The present work addresses these issues by using a so-called ‘human-in-the-loop’ optimization framework, in which a designer can step into the design loop to make modifications as required. Initially a standard layout optimization is undertaken, followed by refinement via a geometry optimization step. ‘Human-in-the-loop’ refinement is then undertaken by the designer, who can manually modify the structural layout and/or apply additional design constraints. The modified design can then be checked and, if necessary, optimized further, utilizing geometry optimization. Several practical design problems are presented to demonstrate the efficacy of the interactive design tool developed, ranging from the design of small-scale mechanical components, suitable for fabrication via additive manufacturing (3D printing) techniques to larger scale trusses, for use in building structures.

Linwei He, Matthew Gilbert, Thomas Johnson, Chris Smith

Young’s Modulus Control in Material and Topology Optimization

We discuss compliance minimization from the broad perspective of Free Material Design (FMD) with Young’s moduli in place of Hooke’s tensor. By this, we attempt to optimize these elastic moduli which tightly correspond to principal components of stress at each point in the design space. The optimization task under study takes a form of the following constrained variational problem: In a given design space, fix two variables for which the total stored energy functional is minimal: (i) design variable - the directional Young modulus field restricted by an isoperimetric condition, and (ii) state variable - the statically admissible stress field. The functional is parametrically dependent on the Poisson ratios; these moduli are not optimized in the proposed approach. Similarly to FMD, our problem also reduces to minimizing a functional of linear growth in stresses. A characteristic feature is that the functional to be minimized is Michell-like, i.e. it takes a form of an integral of absolute values of principal components of stress tensor if the Poisson ratios are set to zero.

Grzegorz Dzierżanowski, Tomasz Lewiński

Regularization Scheme for Controlling Length Scale in Topology Optimization Based on Bacterial Quemotaxis

In order to avoid numerical problems in topological optimization such as checkerboards or mesh dependence, different regularization schemes have been used including penalization, continuation and filtering [24]. This research shows a new regularization scheme to avoid these problems in the Bacterial Chemotaxis Based Topology Optimization Algorithm- BCBTOA [23]. This algorithm simulates the behavior of marine bacteria that forms honeycomb patterns to represent the distribution of material in a domain [23], The original algorithm suffered from a faulty tuning of parameters [23] And after making some modifications we have obtained two fundamental parameters that control the algorithm, these parameters are the radius of action of the chemotactic model of communication between bacteria, (parameter R) and the desired volume fraction (parameter f) [21]. When the parameter R changes it restricts the size of the cavities and solid zones in the structure, the solid zones being larger as R increases and retaining the desired volume fraction f. From the above we deduce that the parameter R functions as a regularization parameter to control the size of the structural members. As a result, we obtained several layouts for different topology optimization problems of continuous bi-dimensional structures.

J. X. Leon-Medina, J. F. Giraldo-Avila, M. A. Guzmán

Structural Optimization Under Buckling Constraints Using Frame Elements with Anisotropic Cross Sections

Structural optimization is of increasing interest in a wide variety of application fields. In this paper, we focus on structural optimization under stress and buckling constraints. We use frame elements and investigate the impact of considering anisotropic cross sections, i.e. whose dimensions in the principal directions of the cross section are different (e.g. rectangles instead of squares, ellipses instead of circles). This allows us a better tailoring of the cross sections to the applied loading, thus having the potential to lead to lighter structures. We propose a formulation as a mixed-integer nonlinear optimization problem with a tailored objective function and decision variables involving cross sectional dimensions, considering more degrees of freedom than what is generally done. Moreover, a new algorithm tailored to the considered problem is proposed. Numerical results show that the proposed approach provides interesting structural weight savings.

Florian Mitjana, Sonia Cafieri, Florian Bugarin, Christian Gogu, Fabien Castanie

On the Numerical Approximation of Michell Trusses and the Improved Ground Structure Method

The paper deals with an improved method of solving large-scale linear programming problems related to Michell trusses. The method is an extension of the adaptive ground structure method developed recently by the author. As before, both bars and nodes can be switched between active and inactive states, but now, the adjoint displacements of inactive nodes, i.e. nodes in empty regions, are adjusted by solving an auxiliary optimization problem. This procedure is relatively cheap and allows a more efficient cutting off the design domain and a significant reduction of the problem size. Thus, the numerical results can be attained for denser ground structures than before, giving better approximations of Michell structures. It is especially important for 3D problems with multiple load conditions. The preliminary results of such problems are reported in the paper and clearly indicate high efficiency of the proposed method.

Tomasz Sokół

Cost and Weight Optimization of Hybrid Parts Using a Multi-material Topology Optimization Approach

Topology optimization is widely used in the industry with the objective to minimize the weight of structures or to maximize their stiffness. Typically, there is no special focus on the costs of the resulting part, as there is a direct correlation between material costs and weight.When looking at hybrid designs, combining a lightweight material with high costs and a material with lower costs but worse specific properties, this simplified view on costs is no longer sufficient. The correlation between costs and weight of the part is no longer met, as the material costs highly depend on which material is primarily used. The optimization problem, therefore, extends to a multi-objective problem with the competing aims ‘minimization of weight’ and ‘reduction of costs’. Additionally, a more complex manufacturing process has a major share in the overall costs of the part.The presented approach uses an extended cost calculation model to estimate the manufacturing costs of the hybrid component based on its geometrical properties. The epsilon-constraint-method is used to transform the costs objective function into an additional constraint. To be able to satisfy this new cost constraint, a group of fuzzy rules, which influence the usage of each material and therefore the costs, were added to the step size controller used by the topology optimization approach. By varying the costs constraint, different Pareto-optimal solutions can be found.

Paul Falkenberg, Eiko Türck, Thomas Vietor

Part XVI: Optimization with Emphasis on Particular Physics Model: Considering Non-Linear Effects (e.g. Material, Geometric, Contact)


Topology Optimization of Orthotropic Elastic Design Domains with Mortar Contact Conditions

In this paper we perform topology optimization of orthotropic elastic design domains in unilateral contact with non-matching meshes by adopting the mortar approach. The motivation is 3D printing of assemblies of parts, where the build direction as well as the contact interfaces between the parts strongly will influence the optimal solutions. Thus, topology optimization of a standard isotropic formulation with tied interfaces might not be a proper approach for this kind of design problems. This is studied in this work by maximizing the potential energy of orthotropic linear elastic bodies in unilateral frictionless contact. In such manner, no extra adjoint equation is needed to be solved and non-zero prescribed displacements are also included in the formulation properly. This will not be true if the compliance is minimized. The contact conditions of the non-matching meshes are treated by deriving the mortar integral from the principle of virtual power by representing the normal contact pressure with the trace of the global displacement shape functions. The mortar integral is then approximated with the Lobatto rule using a high number of integration points. In such manner, we obtain a set of Signorini conditions for each non-matching interface which we solve using the augmented Lagrangian approach and Newtons method. The design domains are formulated with orthotropic linear elasticity where intermediate density values are penalized using SIMP or RAMP, and the regularization is obtained by applying sensitivity or density filters following the approaches of Sigmund and Bourdin. The implementation of the methodology is efficient and produces reliable solutions. This is demonstrated for assemblies of design domains in unilateral contact which is rotated with respect to the build direction of a 3D printer. In particular, compliance as function of build direction is generated for different problems. This kind of curves might be most useful when orienting the parts in order to minimize the volume of support structures.

Niclas Strömberg

Topology Optimization of Structures with Elasto-Plastic Strain Hardening Material Modeling

The objective of this paper is to investigate the influence of the plastic model and the hardening rules adopted for a multiphase material structure on the resulting topologies by incorporating the elastoplasticity material models into the density-based topology optimization. A topology optimization method associated with multiphase elastoplastic materials is developed to maximize the load capacity under a prescribed displacement. This method enables optimizing a design domain composed of two-phase material composites, in which each material may has a specific plastic model. Particularly, by applying the von Mises and the Drucker-Prager yielding criterion to each material and adopting kinematic hardening rules. The interpolations of the elastic and the plastic response are achieved by applying dependency on the design variable to both elastic modulus and yield function. The sensitivity of the stated optimization problem is derived using a path-dependent adjoint method. The capability of the proposed optimization framework is presented through two numerical examples. From the results, it can be concluded that opposite to the post yielding behavior of strain-hardening having negligible effect on the resulting topologies, the type of yield criterion chosen for the material modeling and the magnitude of the prescribed displacement applied to the structure have a significant influence on the optimized layout.

Mengxiao Li, Hexin Zhang

Investigation of Contact Settings on the Result of Topology Optimization to Avoid Contact Stiffness Supports

Considering adjacent parts in the finite element analysis (FEA) leads to a contact problem. There are several necessary configuration parameters in computational contact mechanics, like the contact stiffness or discretization. Due to this variety there is a multitude of papers regarding the correct range for contact parameters in terms of FEA.In topology optimization - especially if the optimization target is set to maximal stiffness - the configuration of the contact problem is much more sophisticated. An overlarge contact stiffness for example leads to a phenomena which is called contact stiffness supports within this paper. This behavior results in a design proposal, which causes major load flow through the contact zone, whereas the rest of the component is removed.The most difficult part in determining the appropriate range for configuration parameters is comparing different simulations. In topology optimization, it is not sufficient to compare certain stress or displacement values of selected degrees of freedom like it is common in structural mechanic simulations. Instead it is necessary to collate the topology of the resulting design proposals.For this purpose an evaluation algorithm was designed, which supersedes the necessity of a visual evaluation by describing the similarity of two design proposals by means of a histogram. The result of a simulation with the identical design space but with a matching mesh - thus without contact zone - is used as reference for this computer-aided evaluation. This automated investigation of arbitrary contact configurations shows, that the values of the parameters are usually within the standard values (recommended in literature), but in a notably smaller interval. In general, smaller contact stiffness is required and the examined parameters are more sensitive in topology optimization. The valid intervals for configuration parameters, which were determined by this method, enable a more realistic way of considering the elastic environment in stiffness optimization.

Daniel Billenstein, Christian Glenk, Pascal Diwisch, Frank Rieg

Optimal Design of Skeletal Structures Exhibiting Nonlinear Response

A unified approach that accounts for buckling and stress limitations in optimization of skeletal structures is presented. Global buckling, local buckling and exceptions from allowed stresses in frame members are considered by optimizing the geometric and material nonlinear response instead of by imposing a large number of constraints. In the proposed approach, each frame member is modeled as a sequence of co-rotational beam elements with hyperelastic material behavior. Design variables are cross-section properties so that both topology and sizes can be optimized, and to node coordinates so that shape optimization can be pursued. Sensitivity analysis follows the adjoint method and optimization is solved using well-established first-order methods. We show that the procedure leads to a buckling-resistant and stress-constrained design by maximizing the sustained load for a given prescribed displacement. A detailed discussion on key aspects of the proposed approach is presented.

Hazem Madah, Oded Amir

Evolutionary Topology Optimization for Designing Cellular Fluid Actuators

Bio-inspired shape adapting systems, based on plants that can reshape themselves by altering the pressure in their cells, can attain very desirable features, such as light weight, large actuation and simple input. It is proposed to design efficient fluid actuators as cellular compliant mechanisms, in which each cell is internally loaded by pressured fluid cavities, through topology optimization techniques. In this work, a model for such cells was proposed, the finite element analysis, sensitivity analysis and optimization method were described and implemented. The chosen optimization method was the soft-kill Bidirectional Evolutionary Structural Optimization (BESO) method with material penalization. The loads dependency on the topology together with two considered nonlinear effects (geometrical nonlinearity and load nonlinearity) act as hindering factors to the optimization process as a whole. Cells with maximized expansion were obtained for two cases: cells that actuate only horizontally; and cells that actuate both horizontally and vertically.

Daniel Candeloro Cunha, Renato Pavanello

Part XVII: Optimization with Emphasis on Particular Physics Model: Considering Dynamic and Accoustic Load-Cases


Topological Design of Vibro-Acoustic Structures Using a Generalized Incremental Frequency Method

Topology design optimization of vibro-acoustic structures are studied, where the structures are assumed to be subjected to time-harmonic mechanical loading with given amplitude and a prescribed low or high excitation frequency. One of the difficulties of such a problem is that the design often easily converges to a local optimum because the excitation frequency may be located in an interval between two less appropriate consecutive resonance frequencies. In this paper, we seek for a new design strategy of overcoming the previous methods and deal with the problem in a ‘global’ and efficient manner. An ‘incremental frequency technique’ (IF technique) is first presented to show that it is often important to consider different design paths in order to obtain a desirable solution of the vibro-acoustic topology optimization problem. Then, a ‘generalized incremental frequency method’ (GIF method) based on the IF technique is proposed for any high or low prescribed value of the external excitation frequency. The excitation frequency is defined to be ‘low’ for positive values up to and including the fundamental resonance frequency of the structure, and to be ‘high’ for values beyond that. The GIF method provides a way of searching for different local optimized solutions in a systematic manner in different disjointed resonance frequency sub-intervals, and hereby the optimum solution of the problem may be identified from among a small number of obtained candidate local optimum solutions. Numerical examples show that the optimum designs obtained in different resonance frequency intervals normally exhibit different periodicity of structural topology. Numerical tests on the design of minimization of dynamic compliance also show that the ‘best’ (i.e. global optimum) design subject to a prescribed value of the external excitation frequency may not be identified in an interval between two consecutive resonance frequencies that have the same orders as the two resonance frequencies embracing the excitation frequency of the initial design chosen for the topological optimization procedure. The method developed in this paper may be applied to general vibro-acoustic design problems (e.g. minimization of sound power/emission/pressure, minimization of dynamic compliance/global displacement amplitude, minimization of force amplitude transmitted from the vibrating structure to the foundation, etc.) with multiple disjointed design sub-spaces, and may also be extended further for optimization problems concerning multiple external excitation frequencies or excitation frequency bands.

Niels Olhoff, Jianbin Du

An Approach to Use the Structural Intensity for Acoustical Topology Optimization

The aim of vibroacoustic engineering is to find a design of a part which is optimal in strength, weight and acoustics. To find the optimal construction shape in early design stages, topology optimization is the most widely used tool. Based on numerically calculated local mechanical values, the optimization method considered in this contribution decides to delete or add material in the region concerned, based on one value, e.g. stress. The aim is to find the best possible utilization of the material’s mechanical strength, regarding the component weight. This approach works very well for static problems.It is desirable to reduce all mechanical information to a single mean value, comparable to von Mises stress in case of static problems, to make a decision on structural changes. However, in acoustics, a dynamic system has to be solved. It is important to take the mechanical behavior of the adjacent regions of the focused area into account. The whole system together provides the specific acoustic characteristics. In addition, a frequency dependency exists. A reliable value to assess local areas of a construction, regarding the relevance for the overall acoustical behavior, is still missing.The idea of this paper is to use the structural intensity (STI) as a basic value for an acoustic assessment of finite elements. It combines two essential mechanical properties: the stress-tensor and the acoustically important velocity-vector. The STI represents the structure-borne sound energy flow and its direction at each point. These information could be used to lead the optimization algorithm to build up a component topology with improved acoustical properties. The approach presented shows how the structural intensity could be used to assess and evaluate each voxel concerning its acoustical impact.

Sebastian Rothe, Sabine C. Langer

Three-Dimensional Topology Optimization of a Flexible Multibody System via Moving Morphable Components

In this work, a novel three-dimensional (3D) topology optimization methodology for a flexible MBS undergoing both large overall motion and large deformation is proposed. Firstly, the flexible MBS of concern is accurately modeled via the 3D solid brick element of the absolute nodal coordinate formulation (ANCF), which utilizes positions of nodes and slopes as sets of generalized nodal coordinates. Secondly, to deal with the dynamic characteristics in the optimization process, the equivalent static load method is employed to transform the dynamic topology optimization problem into a static one. Last but not least, in order to reduce the computation time, the newly-developed moving morphable components (MMC) based topology optimization method is reevaluated to optimize the 3D flexible MBS. The MMC based framework incorporates more geometrical and mechanical information into the topology optimization directly and can optimize large-scale flexible MBS with high efficiency. Two numerical examples are presented to validate the accuracy of the solid element of ANCF and the effectiveness of the proposed optimization methodology, respectively.

Jialiang Sun, Qiang Tian, Haiyan Hu

Part XVIII: Optimization with Emphasis on Particular Physics Model: Considering Crash Load-Cases


Metamodel-Based Global Optimization of Vehicle Structures for Crashworthiness Supported by Clustering Methods

This work introduces a metamodel-based global optimization method for crashworthiness with the ability to synthesize continuum structures with an optimal distribution of material phases or gauges. The proposed optimization method makes use of fully nonlinear, dynamic crash simulations and consists of three main elements: (1) the generation of a conceptual design from the structures crash response, (2) the optimal clustering of the conceptual design to define the location of the material phases or gauges, (3) the metamodel-based global optimization, which aims to find the optimal settings for each cluster. The conceptual design can be generated from extracting finite element analysis information or by using structural optimization. The conceptual design is then clustered using clustering analysis to reduce the dimension of the design space. The global optimization problem aims to find the optimal material distribution on the reduced design space using metamodels. The metamodels are built using sampling and cross-validation, and sequentially updated using an expected improvement function until convergence. The proposed methodology is demonstrated using examples from multi-objective crashworthiness design examples.

Kai Liu, Duane Detwiler, Andres Tovar

Automatic Generation, Validation and Correlation of the Submodels for the Use in the Optimization of Crashworthy Structures

The structural optimization of large crashworthy systems like a vehicle body in a crash loaded case is a time consuming and costly process. The computation time can be reduced by dividing the large system (main model) into small systems called submodels. These submodels can be effectively used in the optimization to shorten the response time of the simulation. The generation of submodels by hand is challenging and requires a lot of effort and knowledge to create and validate them. This paper presents a workflow to automatically generate and validate the submodels using various mathematical functions. A submodel is a region of interest cut out from the large system which is to be analyzed in detail [1]. This detailed analysis can be useful to enhance the structural performance of a large crash system. There are two important parameters to generate a submodel using the so called connecting island algorithm, the threshold ratio and the connecting island value. These parameters are based on an evaluation function which is a structural response with time averaging and space averaging. The size of the submodel depends on these two parameters. The validation of a submodel is a four step process i.e. a local, global, mean and a response validation. These four steps measure the deviation between the submodel and their counterparts in the large system. The validation process is used to identify the quality of a submodel. It is discussed how the validation criteria effects the submodel quality and its size. The aim of this research work is to create a method for automatic generation and validation of submodels which is universally applicable to different crash models. The method is demonstrated on two different examples. The first one is an academic crash model of a cantilever frame hit by a rigid sphere. The submodels are generated and validated for different crash scenarios, where the rigid sphere hits the cantilever frame at different positions. The second is an industrial crash example of Toyota YARIS in front and side crash.

Carlos J. Falconi D., Alexander F. Walser, Harman Singh, Axel Schumacher

Multidisciplinary Optimisation of an Automotive Body-in-White Structure Using Crushable Frame Springs and Sub Space Metamodels in Trust-Regions

A metamodel based multidisciplinary design optimisation of a reduced fidelity automotive body in white model is presented. The considered disciplines are global stiffness, local attachment point stiffness and crashworthiness.The crushable structure in the crash model consists of so called crushable frame springs which can be used to represent the non-linear crushing response of thin-walled tubes in reduced fidelity models. Such springs need characterisation requiring component level finite element analysis. This means that even though the reduced fidelity models are much less computationally expensive than a full shell model, it is still far more expensive to evaluate than the model used for static stiffness.The multidisciplinary design optimisation is therefore carried out using a method for taking into account disparity in design variable dependence of the disciplines. This design variable dependence is specified by the designer and used to build metamodels in the reduced space of the significant variables to each discipline. This decreases the required number of points for each metamodel, and hence the associated computational cost for their evaluation. The method is implemented within the optimisation framework known as the mid-range approximation method together with a recovery mechanism for erroneous identification of significant variables.It is shown that crushable frame springs can be used together with sub space metamodels in a trust-region framework in order to carry out reduced fidelity multidisciplinary optimisation of automotive structures including both NVH and crash load cases.

Charles Mortished, Jonathan Ollar, Peter Benzie, Royston Jones, Johann Sienz, Vassili Toropov

Topology Optimization of Thin-Walled Structures Under Static/Crash Loading Case in the Hybrid Cellular Automaton Framework

Crashworthiness design and optimization is of great importance in the automotive industry. However, due to the high computational cost and numerical noise, crashworthiness topology optimization is not studied so intensively. In this paper, a relatively new method, the Hybrid Cellular Automata for Thin-walled Structures (HCATWS) is used in its improved version. In particular, its applicability is extended from structures with an initially regular grid to structures with different size of cells (sets of a higher number of finite elements). The corresponding modifications of the algorithm are discussed here. This also affects the updating rules used in the improved version; hence, the theory is revised and modified where necessary. In the outer loop of the HCATWS, bi-section search within limited length is used to define the target mass. In the inner loop, HCATWS utilizes proportional updating to redistribute the mass for each cell. Then mass correction is conducted to make sure the real mass converges to the target mass. Here the different sizes of the cells need to be considered. As applications, one linear static case is studied to demonstrate efficiency of the approach. Then, additional crash cases using nonlinear dynamic FEM are considered. Finally, the potential of using this approach for identification of optimal cross-sections of structures originating from Additive Manufacturing (AM) is explored. Here, it is important that optimized topology results from HCATWS are more easily manufactured compared to those obtained by traditional element-based, i.e. voxel-based, topology optimization.

Duo Zeng, Fabian Duddeck

A Topology Optimization Scheme for Crash Loaded Structures Using Topological Derivatives

This paper deals with a topology optimization scheme for crash loaded structures. The presented approach combines the numerical calculation of Topological Derivatives with a level-set method. In a microscale investigation on a planar shell with a hole under perpendicular stresses a meta-model for the Topological Derivatives is derived. This meta-model provides the sensitivities for the optimization of the macroscale mechanical problem. The iterative volume reduction with a level-set method, a linear mapping scheme and filtering allows the crash simulation with smooth boundaries. An academic example illustrates the applicability of the optimization scheme.

Katrin Weider, Axel Schumacher

Finding Optimized Layouts for Ribs on Surfaces Using the Graph and Heuristic Based Topology Optimization

A promising approach in reducing weight is using thin shell structures with supporting ribs instead of thick shell structures, in order to only use the material where it is needed. Nevertheless, the optimization of these structures remains a big challenge, especially in crash load cases, because of material nonlinearities, contact and large displacements.The Graph and Heuristic Based Topology Optimization (GHT) presents a solution to generate better layouts for crashworthiness structures, but up to now it has mainly been used for the design of extrusion profiles with a uniform cross section. In order to apply this method to shell structures with ribs, the GHT is enhanced, so that the mathematical graph can represent ribs that are attached to a given surface, which has no undercuts in the area of the new ribs.During the optimization with the GHT competing layouts with new ribs are suggested by already existing heuristics, which change the topology based on expert knowledge regarding crash applications with extrusion profiles. In a final shape and sizing optimization the thickness, curvature and height of the ribs as well as the thickness of the surface that they are attached to can be optimized. The responses of the optimization can include any value that can be extracted from the simulation results (e.g. accelerations).This feature enables finding better layouts for thin walled crashworthiness structures with supporting ribs. Although causing a large simulation effort, it is one of the few methods that can provide an optimized solution for structures in crash applications in consideration of a wide range of manufacturing constraints and only using shell elements, which result in short calculation times during the simulation.

Dominik Schneider, Axel Schumacher

Part XIX: Optimization with Emphasis on Particular Physics Model: Considering Fatigue/Durability/Damage


Blend Repair Shape Optimization for Damaged Compressor Blisks

During service life of an aircraft engine, highly loaded compressor blades are subject to wear and may get damaged. In order to prevent further damage, blades are refurbished and partial defects are removed by blend repairs. Although the blending of the blade only leads to small shape modifications, it impacts the structural properties significantly. Even a slight variation of the geometry may cause a significant change in the modal behavior. Therefore, it has to be analyzed how blend repairs affect the vibrations of the refurbished blade and how the repair processes can be improved. In the present work, the optimal shape of blend repairs is determined. The first objective function is supposed to be the related quantity of material removed. The second objective of the optimization results from the tuning of the blade’s eigenfrequencies. The location, form, and orientation of the blend is determinable by a parameterizable repair model serving as input for a multi-objective Particle Swarm Optimization algorithm. Studies are carried out aiming at establishing an efficient optimization procedure for the blend repair of integral bladed compressor disks. In particular, the study assesses and presents the influence of the blend repair parameters on the decisive eigenfrequencies related to material removal.

Ricarda Berger, Jan Häfele, Benedikt Hofmeister, Raimund Rolfes

Optimization of Fail-Safe Lattice Structures

In the current work, a fail-safe optimization of lattice structures is carried out. For the optimization, unit cells are not homogenized, but their members are modeled as beam elements. This allows applying a commonly used engineering approach for obtaining a fail-safe design. It consist of removing one beam element at a time and optimizing the remaining structure. At the end, the maximum beam radii are used for the final design. This approach is computationally extremely expensive for lattice structures, as it requires one optimization per removed beam. In our contribution, we show that the design obtained from this approach actually does not fulfill the desired fail-safe behavior. We therefore apply an alternative approach in which the fail-safe requirement is an optimization constraint. This is still computationally demanding and therefore, criteria are discussion for reducing the number of beam elements to be considered for the fail-safe requirement within the optimization.

Benedikt Kriegesmann, Julian Lüdeker, Micah Kranz

Probability-Based Damage Detection of Structures Using Surrogate Model and Enhanced Ideal Gas Molecular Movement Algorithm

Generally, updating a finite element model can be considered as an optimization problem where its physical parameters may be adjusted such that analytically computed features, using the updated FE model, are consistent with those obtained from experimentally. The objective function can be defined as a sum of squared difference between analytically computed and experimentally measured data. To meet this goal in this paper therefore, for efficiently reducing the computational cost of the model during the optimization process of damage detection, the structural response is evaluated using properly a trained surrogate model. Surrogate models have received increasing attention for use in detecting damage of structures based on vibration modal parameters. However, uncertainties existing in the measured vibration data may lead to false or unreliable output results from such model. Here, an efficient approach based on Monte Carlo simulation is proposed to take into account the effect of uncertainties in developing a surrogate model. The probability of damage existence (PDE) is calculated based on the probability density function of the existence of undamaged and damaged states.The Cascade Feed Forward Neural Network (CFNN) is chosen as a metamodeling technique and enhanced version of the Ideal Gas Molecular Movement (EIGMM) algorithm is used as the main procedure for updating the model. The developed approach is applied to detect simulated damage in numerical models of a 72-bar space truss and a 120-bar dome truss structuresThe simulation results show that the proposed method can implement well in probability-based damage detection of structures with less computational efforts compared to direct finite element model.

Mohammad Reza Ghasemi, Ramin Ghiasi, Hesam Varaee

Optimization of Finite Element Mesh Division Considering Stress Singularity for Bonded Structures

In this study, we present optimization of finite element mesh division considering stress singularity for bonded structures. If tensile and bending are applied to the bonded structure, singular stress field occurs around singular point, i.e., interface edge of bonded structure. It is known that stress and strain distribution are expressed by the function of distance from the singular point. In addition, in case of the stress analysis based on the FEM, the value of the stress component at singular point increases with decreasing mesh size around singular point. Therefore, fracture of the bonded structure evaluates by the intensity of stress singularity obtained by the stress distribution. In this study, we propose the optimal mesh division technique for the evaluation of the intensity of stress singularity.

Kengo Yamagiwa, Takahiko Kurahashi

Part XX: Optimization with Emphasis on Particular Physics Model: Considering Piezoelectricity, Magnetic and Electrical Fields


Topology Optimization of Power Semiconductor Devices

In this paper, topology optimization is applied to the design of power semiconductor devices. The doping density distribution of power semiconductor devices is optimized using a density-based topology optimization method. The density method is suitable for the design of power semiconductor devices because doping density can take on continuous values and is intrinsically free from the gray-scale problem. To verify the effectiveness of topology optimization, optimization was conducted for two types of two-dimensional design problems. At first, optimization of a p-n diode was performed to improve the trade-off between the breakdown voltage and on-resistance. This is formulated as a single-objective optimization problem with the Kreisselmeier-Steinhauser (KS) objective function of the electric field, which indicates the breakdown voltage characteristics, under the constraint of the on-resistance. By optimization, a p-i-n diode, which is a well-known diode structure, is obtained and the trade-off is improved. Next, optimization of an edge termination structure was performed to improve the breakdown voltage characteristics with the consideration of ion implantation, which is one of the fabrication processes used for semiconductor devices, under the process variation. The optimized structure obtained is ensured to be manufacturable and more robust with respect to the dose amount variation of ion implantation than the initial structure. These results demonstrate the effectiveness of topology optimization for the design of power semiconductor devices.

Katsuya Nomura, Tsuguo Kondoh, Tsuyoshi Ishikawa, Shintaro Yamasaki, Kentaro Yaji, Kikuo Fujita

Conductor Layout Optimization for Reducing the Magnetic Coupling Noise of a Filter Circuit Board

The aim of this research is to reduce the amount of high frequency noise propagating through space in a low-pass filter. Generally, high frequency noise is generated from power electronics equipment that is responsible for converting direct current and alternating current in hybrid cars. A device consisting of capacitors and inductors called the low-pass filter is used for absorbing the high frequency noise. Ideally, the low-pass filter absorbs more noise as its frequency is higher, but in practice, the noise absorption rate decreases beyond a certain frequency. This is because of the following reasons. The magnetic flux is generated from a current loop in the circuit according to Ampere’s circuital law. Then, this magnetic flux causes high frequency noise at the output terminal according to Faraday’s law of induction. This phenomenon is called magnetic coupling. The amount of noise that appears at the output terminal depends on the layout of the conductor on the circuit board. Conventionally, designers reduced the area of the current loop to avoid the influence of magnetic coupling. However, it was difficult for them to quantitatively predict the influence and design the conductor pattern optimally. In contrast, a topology optimization method is valid for obtaining a mathematically evident optimal structure. Topology optimization method usually allows grayscale area, which holds intermediate density value. For the conductor layout problem, it is difficult to appropriately set the current conductivity to the intermediate material density, and this possibly causes a computational error. For these reasons, we propose a grayscale-free topology optimization method for the conductor layout that minimize the influence of the magnetic coupling. Several numerical examples are provided to confirm that the appropriate optimal structures are obtained.

Hiroki Bo, Shintaro Yamasaki, Kentaro Yaji, Katsuya Nomura, Atsuhiro Takahashi, Kikuo Fujita

Integrated Design of Permanent Magnet Synchronous Motor by Incorporating Magnet Layout and Yoke Topology Optimizations

This paper presents an optimization method for the design of permanent magnet synchronous motor (PMSM) rotor considering a yoke structure and a permanent magnet (PM) layout. PMSM rotor is composed of the yoke and PMs, whose designs greatly influence the PMSM performance. There was a problem in conventional PMSM optimal design method that an obtained PM shape becomes complex. Therefore, the layout of PM with a specific shape is optimized in this study. The topology optimization is applied to the yoke structural optimization. To incorporate the yoke topology and the PM layout optimization problems into the PMSM rotor design problem, it is necessary to correlate those solutions during the optimization process. However, it is difficult to solve those problems simultaneously or individually because of those problems being connected strongly. In order to address the issue, response surface methodology is employed to combine the yoke topology optimization problem and the PM layout optimization problem by focusing on a hierarchy of problems. The effectiveness of the proposed method is shown by a numerical example.

Shun Maruyama, Shintaro Yamasaki, Kentaro Yaji, Kikuo Fujita

Part XXI: Optimization with Emphasis on Particular Physics Model: Considering Other Specialty Disciplines


Shape and Structural Design Optimization of Graphene Sheets in Natural Vibration Problem

Because of their superior mechanical, structural, and electronic properties, graphene sheets (GSs) are supposed to be components of nanoelectromechanical systems (NEMS). In this study, we carry out shape and structural optimization of GSs with topological defects in natural vibration problem to enhance their fundamental frequencies. At first, we model GSs as continuum frame models based on the molecular mechanics method. Then, we carry out shape design optimization of GSs by using a developed free-form optimization method for frame structures. In the shape design optimization, we use the fundamental frequency as objective function and enhance it considering volume constraint and repeated eigenvalues. Next, we derive the optimal atom structures of GSs with topological defects using a combination of the Phase-Field-Crystal method, the Voronoi tessellation method, and molecular dynamics simulation. The numerical results show that the fundamental frequencies of GSs can be significantly increased according to the shape and structural design optimization, and we can get their optimal atom structures with defects, which is helpful for designing GSs as components in NEMS.

Jin-Xing Shi, Keiichiro Ohmura, Masatoshi Shimoda

Two-Scale Concurrent Topology Optimization with Multiple Micro Materials Based on Principal Stress Direction

This paper studies two-scale concurrent topology optimization with multiple micro heterogeneous materials subject to volume constraints. Unlike the existing work on concurrent two-scale optimization where only one material with optimal microstructure is used or that with multiple micro material where each material is distributed in a number of prescribed geometrical domains, selection of micro heterogeneous materials in this work is based on direction of principal stresses in macro structure. For a structure composed of m micro materials, the macro elements are classified into m categories according to their principal stress direction and each category is assigned with a uniform micro material. The interpolation scheme for macro elements is based on Discrete Material Optimization (DMO), where each element is assigned with m macro design variables. The categorization process of the macro structure is achieved by proper modification of volume constraints, where the macro design variables are multiplied by penalty functions. The penalty functions make it uneconomical for the usage of micro materials, which do not correspond to the principal stress direction of their macro element. The macro structure and micro material are connected through effective property, which is calculated through novel numerical implementation of asymptotic homogenization method (NIAH). Both macro structure and micro materials are optimized concurrently and analytical sensitivities are calculated with adjoint method. One minimum compliance numerical example of an L-bracket subject to volume constraints, where one micro material correspond to macro domain with principal stress angle near 0 or 90 degrees and another correspond to that with principal stress angle near 45 or 135 degrees, is presented to show the potential of the proposed method.

Liang Xu, Gengdong Cheng

Topology Optimization of Viscoelastic Materials for Maximizing Damping and Natural Frequency of Macrostructures

The topology optimization algorithm of viscoelastic material microstructure based on bi-directional evolutionary structural optimization (BESO) method is proposed for macroscopic damping characteristics of the structures. The optimization aims to obtain the optimal topologies of the material microstructures within given volume fraction so that the resulting structure has optimal damping characteristics. The design concept of this scheme is essentially a two-scale design which considers the effective properties of material microstructures and macroscopic performance. Viscoelastic material is used for the damping of the macrostructure and the frequency constraint is also applied so that the resulting macrostructure has the best damping performance with prescribed natural frequencies. The microstructures of the material are represented by periodic unit cells (PUCs) and the effective properties of the material microstructures are homogenized and integrated into the finite element analysis of the macroscopic structures. The sensitivity analysis is conducted for iteratively updating the topologies of material microstructures. Numerical examples are presented to demonstrate the effectiveness of the proposed optimization algorithm.

Qiming Liu, Xiaodong Huang

Design of Adsorbed Natural Gas Tanks with Metal Inclusions by Topology Optimisation

Adsorbed Natural Gas (ANG) tanks are constituted by porous materials able to store gas at high density and low pressure through adsorption. This phenomenon causes gas molecules to accumulate on the surface of micro-pores in a density near the one from liquid phase. Adsorption is exothermic and highly dependent on the temperature. Also, adsorbent materials present poor thermal conductivity. Thus, the efficiency of the thermal management determines the feasibility of an ANG tank. This work presents the implementation of the Topology Optimisation Method (TOM) for designing ANG tanks aiming to improve the heat exchange for more efficient adsorption and desorption cycles. This is achieved by the optimal placement of aluminium throughout the activated carbon adsorbent bed aiming the maximisation of the total mass of gas adsorbed by the end of the filling cycle and the minimisation of mass after the emptying cycle. The TOM is performed adopting the Solid Isotropic Material Model with Penalisation (SIMP) and a term for the regularisation of the design variables in the objective functions. Resulting topologies for different design domains are presented and compared to the respective homogeneous cases.

R. C. R. Amigo, R. W. Hewson, E. C. N. Silva

Part XXII: Optimization with Emphasis on Particular Physics Model: Considering Manufacturing Aspects


Topology Optimization for Unifying Deposit Thickness in Electroplating Process

Uniformity of deposited thickness in electroplating processes is vital to the realization of desirable surface qualities in many products. The thickness distribution of deposits is affected by numerous factors, such as the arrangement and shapes of auxiliary cathodes, anodes, and shields as well as the detailed configuration of the electroplating process. Deposit thickness reflects the amount of ions transported from anodes to cathodes, particularly to the object being plated, although auxiliary cathodes are sometimes placed to prevent excess plating in certain areas of the product, as are shields that impede current flow.This study presents a topology optimization method for achieving uniform deposition thickness, applied to the design of the shields, auxiliary cathodes and anodes placed in an electroplating bath. The proposed method is based on the level set method and the FEM is used to analyze the electrochemical field. The shapes and arrangement of shields in an electrolyte are expressed according to the distribution of electric conductivity via a level set function, and the shapes and arrangement of anodes and auxiliary cathodes are expressed with respect to ion sources using other level set functions. The Kreisselmeier-Steinhauser function for the current density distribution on a cathode is employed as an objective function, since current density is nearly proportional to the thickness of the resulting electroplating. The magnitude of the current density on the cathode is used as a constraint so that it does not fall below a certain value, to avoid lengthy plating times that would occur if the current density were too low. Numerical examples are presented to confirm the utility of the proposed method and the results demonstrate that the proposed method can obtain appropriate shapes and arrangements of shields and anodes.

Naoko Ishizuka, Takayuki Yamada, Kazuhiro Izui, Shinji Nishiwaki

Multiscale, Thermomechanical Topology Optimization of Cellular Structures for Porous Injection Molds

During the injection molding cycle, molten material is injected at high pressure inside the mold and cooled down to form a solid part. This creates thermomechanical stresses that are alleviated by the correct design of a cooling system. In conventional molds, the cooling system consists of straight-line cooling channels, which can be manufactured using machining processes; however, they are thermally inefficient and unable to cool the injected part uniformly. The emergence of metal-based additive manufacturing techniques such as direct metal laser sintering (DMLS) allows the fabrication of molds with conformal cooling channels. Conformal cooling molds cool down the part faster and more uniformly; however, they face limitations. First, their fabrication cost is 10 to 20 times higher than the one of a conventional mold. Second, the DMLS process, which is the most popular fabrication method of conformal cooling molds, produces internal thermal stresses that distort the mold. The development of structural optimization methods such as multiscale topology optimization offers the potential to create novel and complex cellular structures that alleviate these current limitations. The objective of this research is to establish a multiscale topology optimization method for the optimal design of non-periodic cellular structures subjected to thermomechanical loads. The result is a hierarchically complex design that is thermally efficient, mechanically stable, and suitable for additive manufacturing. The proposed method seeks to minimize the mold mass at the macroscale, while satisfying the thermomechanical constraints at the mesoscale. The thermomechanical properties of the mesoscale cellular unit cells are estimated using homogenization theory. A gradient-based optimization algorithm is used for which macroscale and mesoscale sensitivity coefficients are derived. The design and evaluation of a porous injection mold is presented to demonstrate the proposed optimization method.

Tong Wu, Kim Brand, Doyle Hewitt, Andres Tovar

Multidisciplinary Shape Optimization of Ductile Iron Castings by Considering Local Microstructure and Material Behaviour

During the casting process and solidification of ductile iron castings, a heterogeneous microstructure is formed throughout the casting. This distribution is strongly influenced by the item geometry and the process related factors, as chemical composition and local solidification conditions. Geometrical changes to the geometry of the casting thus alters the local mechanical behavior and properties, as well as the distribution of stresses and strains when the casting is subjected to load. In order to find an optimal geometry, e.g. with reduced weight and increased load-bearing capacity, this interdependency between geometry and local material behavior needs to be considered and integrated into the optimization method.In this contribution, recent developments in the multidisciplinary integration of casting process simulation, solidification and microstructure modelling, microstructure-based material characterization, Finite Element Analyses (FEA) with local material behavior and structural optimization techniques are presented and discussed. The effect and relevance of considering the local material behavior in shape optimization of ductile iron castings is discussed and evidenced by an industrial application. It is shown that by adopting a multidisciplinary optimization approach by integration of casting simulation and local material behavior into shape optimization, the potential of the casting process to obtain components with high performance and reliability can be enabled and utilized.

Jakob Olofsson, Riccardo Cenni, Matteo Cova, Giacomo Bertuzzi, Kent Salomonsson, Joel Johansson

Topology Optimization with Integrated Casting Simulation and Parallel Manufacturing Process Improvement

Affordable lightweight design is one of the key topics for the automotive future. Increasing worldwide competition, stricter legal guidelines and weight-intensive electronic components of new drive systems require lighter and more economic chassis components. Aluminum parts enable activation of lightweight design potentials in terms of good material properties and affordable costs. To ensure cost-effective production of cast parts, a lean and efficient manufacturing process is necessary. Furthermore, a lighter design lowers the material costs of the part directly. Topology optimization offers a lightweight design in a fast automatic procedure and is nowadays used at the beginning of many development processes for cast parts. Commercial optimization software considers the castability of the part only insufficiently on the basis of simple geometric rules. Thus, manual modifications for the manufacturing process are needed.In this work, an integration of a casting simulation into the topology optimization is presented. This leads to a considerably increased castability of the part. In parallel to the stress-based optimization in order to improve the mechanical properties, turbulence and vorticity during the filling process are reduced, cold run is prevented and a directional solidification is ensured. This procedure does not only result in a castable lightweight design but it also provides a basis for an economic casting process. For that reason, the casting process is adapted in parallel to the part’s optimization. The geometry is reduced to two-dimensional graphs and an innovative functional is deduced from them. Its maximization will identify the ideal pouring position as well as the positions and sizes of necessary feeders. This method is discussed with the help of a corner module’s optimization. The resulting design is characterized by a low weight, suitability for economic casting and an easy transformation into a CAD-part in a short development process.

Thilo Franke, Sierk Fiebig, Karsten Paul, Thomas Vietor, Jürgen Sellschopp

Part XXIII: Optimization with Efocusing on Particular Industrial Applications: Automotive


Parameterization Setup for Metamodel Based Optimizations of Tailor Rolled Blanks

Tailor Rolled Blanks (TRB) are an established lightweight application for highly stressed structural parts in automotive industry. By varying the rolling gap, parts with load adapted thickness profiles and continuous transitions are manufactured. The rolling process itself is subjected to several manufacturing constraints which have to be considered in the numerical optimization. It has been shown that the thickness run parametrization has significant impact on the optimization result and the associated computational cost.The goal of the presented methods is to systematically assess the number of areas of constant thickness (plateaus) and setup a thickness run parametrization for subsequent metamodel based TRB optimization.Starting from a baseline design, the rolling direction is determined based on an optimality criterion. Three approaches are then compared to create an initial thickness distribution along the axis of maximum inhomogeneity without the need of additional non-linear function evaluations:1.Single iteration ESL-optimization2.Single iteration HCA scheme3.Scaling of stress distributionThe created distributions have to be interpreted in order to generate a parametrization for metamodel based optimization which takes all TRB design constraints into account. The interpretation is performed by solving an optimization tasks, resulting in the thickness run with the least deviation to the input thickness distribution.Two application examples are presented. A crash beam subjected to three point bending is optimized using two load case setups. Also a submodel of the NCAC Toyota Yaris in full frontal crash is used to optimize the front rail assembly. The results of the subsequent metamodel based optimization are compared to the results of metamodel based optimizations with a general parametrization setup.

Niklas Klinke, Axel Schumacher

A Study of Topology Optimization for Joint Locations of Automotive Full Vehicle

The optimization method for searching the joint locations of spot-welding and adhesives in a automotive body made of steel sheets was studied [1]. The key point is that the topology optimization method was applied to the optimization of joint elements by using a full vehicle model in order to improve vehicle performance. In addition to static stiffness using constraints, stiffness while driving is required in the body stiffness of the full vehicle. Inertia relief is known as a method for expression of behavior while driving.In this study, stiffness optimizations were carried out for an automotive full vehicle model by using static stiffness and inertia relief for stiffness while driving. The optimal target is addition of spot-welding and addition of adhesives. The results show that the developed topology optimization method for joint locations is valuable in optimization of automotive bodies made of steel sheets. Optimization of the location of spot-welding shows the efficiency of the topology method compared with the conventional method, and optimization of the location of adhesives shows the minimum quantity of adhesive for stiffness by the topology method. The results of optimization of joint locations differed between the static stiffness using constraints and the stiffness using inertia relief.

Takanobu Saito, Yoshikiyo Tamai, Jiro Hiramoto

Part XXIV: Optimization with Efocusing on Particular Industrial Applications: Aircraft


On Fast Design of Innovative Hierarchical Stiffened Shells Against Imperfections

Inspired by the hierarchical features of some bionics structures like glass sponges, we propose the innovative hierarchical stiffened shell from the point-of-view of increasing the hierarchies of grid-patterns. Differing from the traditional hierarchical stiffened shells, the innovative ones are composed of major and minor stiffeners with diverse grid-patterns. However, its post-buckling analysis and imperfection sensitivity analysis are too time-consuming. Therefore, it is crucial to develop an efficient equivalent model and optimization framework. Since various buckling modes may occur in hierarchical stiffened shells, it is essential to establish a more flexible equivalent strategy than the traditional single equivalent strategy. Firstly, the Numerical-based Smeared Stiffener Method (NSSM) is derived. Based on the NSSM, a reasonable adaptive equivalent strategy is developed from the concept of hierarchy reduction. Its core idea is to self-adaptively decide which hierarchy of the structure should be equivalent according to the critical buckling mode rapidly predicted by NSSM. On basis of the proposed adaptive equivalent strategy, an effective grid-pattern optimization framework is proposed. Firstly, the RBF surrogate model is constructed based on the equivalent model. Then, the inner optimization is performed on the surrogate model using global optimization algorithm. Furthermore, the optimum result is validated with the corresponding exact detailed model simulation, and the relative percentage error is served as a convergence criterion, to decide whether the adding the exact detailed model result into original surrogate model to generate a new one. Until the relative percentage error converges, the entire optimization is finally completed. Finally, representative illustrative examples indicate that one hierarchical stiffened shell with closely-spaced major stiffeners and triangle gird sub-structure is competitive in resisting imperfections.

Kuo Tian, Bo Wang, Tianyu Zhu, Sijun Xiong, Ke Zhang, Peng Hao

Mixed Variable Structural Optimization: Toward an Efficient Hybrid Algorithm

Designing a structure implies a selection of optimal concept and sizing, with the aim of minimizing the weight and/or production cost. In general, a structural optimization problem involves both continuous variables (e.g., geometrical variables, …) and categorical ones (e.g., materials, stiffener types, …). Such a problem belongs to the class of mixed-integer nonlinear programming (MINLP) problems. In this paper, we specifically consider a subclass of structural optimization problems where the categorical variables set is non-ordered. To facilitate categorical variables handling, design catalogs are introduced as a generalization of the stacking guide used for composite optimization. From these catalogs, a decomposition of the MINLP problem is proposed, and solved through a branch and bound method. This methodology is tested on a 10 bars truss optimization inspired from an aircraft design problem, consistent with the level of complexity faced in the industry.

Pierre-Jean Barjhoux, Youssef Diouane, Stéphane Grihon, Dimitri Bettebghor, Joseph Morlier

Part XXV: Optimization with Efocusing on Particular Industrial Applications: Civil Engineering


Optimal Estimation of Tidal Flow Based on Kalman Filter FEM Using Time History of Water Elevation

In this study, we present optimal estimation of tidal flow based on Kalman filter FEM using time history of water elevation. The shallow water equation is employed as the governing equations. The Galerkin and the selective lumping methods are employed to discretise the governing equation in space and time, respectively. The Kalman filter FEM is applied to estimate the flow field in Tokyo Bay.

Takahiko Kurahashi, Taichi Yoshiara, Yasuhide Kobayashi, Noboru Yamada

Topology Optimization of Elastic Wave Barriers Using a Two-and-A-Half Dimensional Finite Element Methodology

In order to reduce environmental ground vibration due to railway traffic, mitigation measures on the transmission path can be applied to impede propagation of ground vibration from source to receiver. Accounting for the three-dimensional character of the wave field is essential, but three-dimensional models are computationally demanding. When the cross sections of the barrier is invariant, a computationally efficient two-and-a-half-dimensional finite element method can be applied. In this paper, a 2.5D methodology is used to model a wave barrier that reduces the vibration levels in a building close to a railway track. Topology optimization is used to find the optimal barrier design. A gradient based approach is used, where the sensitivities are calculated with the adjoint method. The resulting design outperforms classical rectangular wave barrier with the same volume.

Cédric Van hoorickx, Mattias Schevenels, Geert Lombaert

Buckling Length in Mixed-Integer Linear Frame Optimization

In structural optimization of trusses and frames, the member profiles have to be selected from material supplier’s selection. This means that the optimization problem becomes discrete. The discrete frame optimization problem can be formulated as mixed-integer linear program (MILP) and thus solved for global optimality using well-known deterministic methods such as branch-and-cut. Within the formulation it is possible to include member buckling constraints. When using design standards as basis for member buckling resistance evaluation, the critical forces or buckling lengths of the members are required. Buckling length can be determined using many methods, both numerical and analytical. Regardless of the method, buckling length of a single member is dependent on surrounding members’ stiffness which makes it practically impossible to include the correct buckling lengths in MILP formulation directly. In general, the question of buckling length in frame optimization has rarely been discussed in the structural optimization literature. Therefore, in this paper, an iterative approach to determine the correct buckling lengths is presented. In this approach, the MILP optimization is run several times. Linear stability analysis is performed between MILP runs to update buckling length data. The performance of the proposed method is illustrated in example calculations. The example structures are steel frames and Eurocode 3 is used as basis for member resistance constraints. In the examples, the method converges with a relatively low number of iterations.

Teemu Tiainen, Kristo Mela, Markku Heinisuo

Optimization of Extradosed Concrete Bridges

Extradosed bridges combine the main elements of both a prestressed box girder bridge (PBG) and a cable-stayed bridge. With their shallow cable-stays and stiff decks they represent an economic alternative to PBG bridges and cable-stayed bridges for main spans of 100 to 200 m. Their structural behavior combines the concepts of cable suspension and bending of the high stiffness box girder. The extradosed cable-stays introduce a highly effective prestress on the box girder which enhances its structural efficiency. The use of optimization techniques in the design of large and complex structures like extradosed bridges naturally arises as an efficient way to compute the two sets of prestressing forces (box girder and cable-stays) and the cross-sectional dimensions of the tower and girder, aiming at reducing the material costs and thus obtaining economical and structurally efficient solutions.The current research work is a development of previous research works by the authors concerning the optimization of concrete cable-stayed bridges. In this work a numerical model for the design of extradosed concrete bridges was developed. The structural analysis includes all the actions and relevant effects, namely, the construction stages, the time-dependent effects and the geometrical nonlinearities. The discrete direct method is used for sensitivity analysis. The design of extradosed concrete bridges is formulated as a multi-objective optimization problem with objectives of minimum cost, minimum deflections and stresses and a Pareto solution is sought. An entropy-based approach is used to find the minimax solution through the minimization of a convex scalar function. The design variables considered are the extradosed cable areas and prestressing forces, the deck prestressing forces and tendon areas and the towers and deck cross-sections.The features and applicability of the proposed method are demonstrated by a numerical example concerning the optimization of a real sized extradosed concrete bridge.

Alberto M. B. Martins, Luís M. C. Simões, João H. J. O. Negrão

Optimization of Concrete Cable-Stayed Bridges with Discrete Design Variables

This work presents a procedure for finding the discrete optimum design of concrete cable-stayed bridges. The behaviour of this type of structures is governed by the stiffness of the load-bearing elements (towers, deck and cable stays) and the cable force distribution. In concrete bridges the stresses and deformations are significantly influenced by the construction sequence and by the concrete time-dependent effects. Furthermore, the geometrical nonlinear behaviour that arises when dealing with cables, large and flexible structures should also be considered in the analysis. A finite-element approach is used for structural analysis. It includes a direct analytic sensitivity analysis module, which provides the structural behavior responses to changes in the design variables. The optimum design of cable-stayed bridges involves a significant amount of design variables and design objectives. It can be stated as the minimization of stresses, displacements and bridge cost. To solve the continuous design problem an equivalent multi-criteria approach was used transforming the original problem into the sequential minimization of unconstrained convex scalar functions, from which a Pareto optimum can be obtained. This is followed by the rounding up or down of the continuous cross section variables to the nearest available discrete sections to find a discrete solution. Once the discrete design variables are fixed the solution is then improved by optimizing the cable installation forces (continuous variables) to meet the stress and displacements criteria. If this solution is infeasible, the segmental method is used to find the sizing variables which need to be modified. A concrete cable-stayed bridge example is presented to illustrate the proposed procedure.

L. M. C. Simões, A. M. B. Martins, J. H. J. O. Negrão

A Discrete Particle Swarm Algorithm for Sizing Optimization of Steel Truss Structures

Traditionally, the algorithms to solve structural engineering optimization problems consider that the design variables are continuously valued. However, in steel structural design practice, some variables are chosen from discrete sets of steel profiles during the sizing process of structural elements to avoid extra costs associated with the use of non-standard section sizes. Despite of that, there is an extensive bibliography concerning continuous valued design variables and particle swarm optimization (PSO) algorithms. On the other hand, smaller attention has been paid to solve problems with discrete valued design variables with PSO algorithm support. In this paper, a discrete particle swarm algorithm is presented and applied in some benchmark problems. Displacement and tensions constraints are also considered in accordance with Eurocode 3.

Waldir N. Felippe, Luiza F. Carneiro

Design of Cellular Materials and Mesostructures with Improved Structural and Thermal Performances

Honeycomb mesostructures and other types of cellular material such as wood, coral, and cancellous bone have properties that make them suitable for use in many structural engineering applications. This includes not only superior mechanical behavior and lightweight high-strength characteristics, but also better thermal conductivity, electrical resistivity, etc. A good understanding of these structures and materials can help structural engineers to design lightweight building systems with, for example, improved stability combined with improved thermal and acoustic insulation. Moreover, advances in additive manufacturing methods capable of producing these cellular structures also add to the motivation. In this paper, the application of cellular materials for construction applications in the building sector is illustrated and FEM analysis is used to examine several types of mesostructures. Furthermore, we focus on optimizing the equivalent thermal conductivity and stiffness of structures with a relative density of 0.5. Special attention is given to situations for which the required properties are not necessarily equal in all directions. Results show that using multidisciplinary topology optimization methods, the structural and thermal performances of these structures can be efficiently optimized.

Gieljan Vantyghem, Marijke Steeman, Wouter De Corte, Veerle Boel

Modified Ideal Gas Molecular Movement Algorithm Based on Quantum Behavior

Recently, the ideal gas molecular movement (IGMM) algorithm was proposed by the authors as a new metaheuristic optimization technique for solving single and multi-objective optimization problems. Ideal gas molecules scatter throughout the confined environment quickly. This is embedded in the high speed of molecules, collisions between them and with the surrounding barriers. In IGMM algorithm, the initial population of gas molecules is randomly generated and the governing equations related to the velocity of gas molecules and collisions between those are utilized to accomplish the optimal solutions. In this paper a modified IGMM algorithm is proposed based on quantum theory. Quantum based IGMM (QIGMM) is intended for enhancing the ability of the local search and increasing the individual diversity in the population. QIGMM improve capability of IGMM in avoiding the premature convergence and eventually finding the function optimum. startlingly, all these are obtained without introducing additional operators to the basic IGMM algorithm. The effectiveness of these improvements is tested by some standard benchmark optimization problems. experimental results show that, QIGMM algorithm is more effective and efficient than the original IGMM and other approaches.

Mohammad Reza Ghasemi, Hesam Varaee

Part XXVI: Optimization with Efocusing on Particular Industrial Applications: Energy Systems


Development of a Multi-Objective Genetic Algorithm for the Design of Offshore Renewable Energy Systems

This paper describes the development of a framework using a genetic algorithm in order to aid in the design of a mooring system for offshore renewable energy devices. This framework couples numerical models of the mooring system and structural response to cost models in order for the genetic algorithm to effectively operate considering multiple objectives. The use of this multi-objective optimization approach allows multiple design objectives such as minimum breaking load and the material cost to be minimized simultaneously using an automated mathematical approach. Through the application of this automated approach, a wider set of designs will be considered allowing the system designers to select a design which appropriately balances the trade-off between the competing objectives. In this work, a set of mooring designs that represent efficient solutions for the stipulated constraints are found and presented. The developed framework will be applicable to other offshore technology subsystems allowing multi-objective optimization and reliability to be considered from the design stage in order to improve the design efficiency and aid the industry in using more systematic design approaches.

Ajit C. Pillai, Philipp R. Thies, Lars Johanning

Life Cycle Assessment of Welded Structures Using Cost Optimization

The paper deals with the Life Cycle Assessment (LCA) to evaluate the environmental burdens associated with products by identifying and quantifying energy and materials used and wastes released to the environment. The assessment includes the entire life cycle of the products, encompassing, extracting and processing raw materials; manufacturing, transportation and distribution; use, re-use, maintenance; recycling, and final disposal. The investigations show, that reducing the mass of the product, the structure, one can reduce the harmful effects. A cost calculation system is introduced. The examples show the effect of optimization to improve the LCA behaviour of the structure.

Károly Jármai

A New Optimisation Framework for Investigating Wind Turbine Blade Designs

We propose a new optimisation framework developed for the investigation of innovative wind turbine blade designs. The design of wind turbines has progressively evolved over recent decades as part of an ongoing effort to provide economically competitive solutions for wind energy production. In particular, rotors have increased in size so as to capture more wind energy while limiting installation costs. At the same time blade designers have had to continually improve the structural efficiency of blades in order to accommodate higher extreme and fatigue loads resulting from growing rotor diameters. Modern wind turbine designs are the result of these incremental improvements, limiting financial risks but also confining the design space and effectively reducing opportunities for more radical innovation. In this paper, we enable the wider exploration of the wind turbine blade design space by means of a new optimisation framework. For that purpose we develop and combine state-of-the-art tools for the aero-servo-elastic analysis and optimisation of wind turbines aiming to explore the uncharted design space resulting from decades of incremental changes. Our framework relies on the use of B-spline surfaces and lamination parameters to provide a compact and continuous means of describing blade structures, also enabling the use of gradient-based optimisers. This structural parameterisation is further combined with beam and shell finite element models to provide further confidence in preliminary structural designs. The proposed framework is presented and verified herein. Validation results show good agreement with the modern large scale DTU 10 MW blade design. Additionally, the coupled bend-twist behaviour of the beam model is found to agree well with higher fidelity finite element model predictions.

T. Macquart, V. Maes, D. Langston, A. Pirrera, P. M. Weaver

Part XXVII: Optimization with Efocusing on Particular Industrial Applications: Others


Optimum Design on Neck Embossing Decoration of Aluminum Beverage Bottles

This paper proposes a new forming method to emboss the aluminum bottle neck with various relatively fine embossing patterns. In the proposed method, a directional pressure is applied to the inside surface of the neck against an embossing die placed outside of the neck. An in-house software is developed to build efficiently finite element analysis models, based on embossing pattern images, to simulate neck embossing process proposed. The numerical simulations are performed to investigate influences of the die fillet, the pressure and embossing pattern on the neck embossing results, such as the thickness reduction and the outlook of the embossed region. An optimum design flowchart is also proposed to design neck embossing decoration, and a structural optimization method is applied to obtain an optimum solution. The optimum design result shows that, by optimizing the die fillet and the pressure, subject to the maximum thickness reduction ratio and the range of outlook assess factor of the embossed region, the optimum embossed neck can be obtained under consideration of vision, touch and neck surface coating protection.

Jing Han, Koetsu Yamazaki, Akiyoshi Matsuzaki

Preliminary Study on Optimization of a Bulge Tool for Nuclear Fuel Manufacturing

A nuclear fuel assembly, manufactured through several processes, is composed of a range of components. Guide thimbles and spacer grid sleeves, among others, are particularly important to maintain the integrity of fuel assembly, and they are connected through a bulge forming process. The purpose of the bulge process for fuel assembly is to connect guide thimbles to spacer grids. The connecting load between tubes is affected by bulge joint strength, and this strength depends on bulge design. While reaching to a specified load, bulge equipment endures high working loads during the working process, and a cracked tool can be seen infrequently. The equipment consists of a bulge tool and a taper pin, and the neck of the bulge tool is most susceptible to damage. A crack may appear in fuel assembly when the bulge tool is broken during the bulge process. The optimization of bulge equipment is difficult because the bulge process has geometric nonlinearity, boundary nonlinearity, and material nonlinearity. The work velocity of the bulge process for nuclear fuel manufacturing is relatively slower than that of general forming processes, but a stroke is very important. Therefore, nonlinear analysis should be required in the optimization process. In this research, the design of experiments using an orthogonal array and the finite element analyses are employed to determine the optimal shape and material. Design variables are the material and three types of local shapes of a bulge tool, and the level of the design variables is three. The objective of the optimal design is to reduce the maximum stress imposed on a bulge tool. The commercial software, ABAQUS, is utilized for nonlinear static analysis of the bulge process, and L9 orthogonal array is used for the optimization of the bulge tool.

Jae-Jun Lee, Young-Duk Sim, Nam-Gyu Park, Se-Ick Son, Jong-Sung Yoo

Design of Bone Plates for Mandibular Reconstruction Using Topology and Shape Optimization

Tumors in the maxillofacial area and cancer therapy in general can lead to necrosis of the lower jaw (mandible). To prevent tumor spread and infection the necrotic bone and surrounding soft tissue needs to be surgically removed. To restore normal function and appearance of the jaw, the resected mandible is reconstructed using autologous (originating within the body) bone graft from the fibula and titanium bone plates for the osteosynthesis (fixation) of the bone fragments. Currently, standard bone plates are used that have a high volume or, in the case of mini plates, show a high risk of implant failure. The introduction of optimized, patient-specific bone plates helps to decrease intraoperative stress for the patient and facilitates an improved patient recovery by providing sufficient stabilization of the bone fragments as well as high stiffness and durability of the implants while minimizing the volume of the implanted material. Within this publication, the design process of the topology optimized bone plates is shown. First, a finite element model of the reconstructed mandible was created using standardized, synthetic bone models, boundary conditions based on a custom made, biomechanical testing rig for the mandible and different biomechanical load cases. Then, the volumes of the initial design areas of the bone plates were reduced using topology optimization and the design of the final bone plates was developed including shape optimization.Finally, a design validation of the optimized bone plate designs was conducted by a non-linear finite element analysis. A reduction of total volume of 44.9% was achieved by a maximum stress in the bone plates for the different load cases of 230.00 MPa to 797.10 MPa concluding high fatigue strength of the implants. The uneven load distribution over the screws of each bone plate could be reduced by up to 73.72% preventing overloading of one screw and thus reducing the risk of screw failure.

Michael Seebach, Felix Theurer, Peter Foehr, Constantin von Deimling, Rainer Burgkart, Michael Friedrich Zaeh

Comparative Study Between Different Strut’s Cross Section Shape on Minimizing Low Wall Shear Stress Along Stent Vicinity via Surrogate-Based Optimization

Endovascular stent has been employed to treat patients with intravascular diseases. Research on stent optimization is currently performed in order to find the best design in increasing the treatment efficacy. In this research, stent optimization is performed based on a finite element analysis method via Kriging surrogate model to observe the wall shear stress (WSS) conditions on the strut vicinity. Two configurations, rectangle and triangle are adopted as the cross section of a stent strut and compared to see the effects of the cross section on WSS condition. Strut gap in the range from 1 mm to 3 mm and the strut size length from 0.05 mm to 0.45 mm are considered as the design variables for each cross section. Structure contact simulation between stent and vessel wall is carried out to obtain the 5% vessel expansion. Afterward, computational fluid dynamics simulation is performed to analyze the hemodynamic effect of stent design along with wall deformation. Minimizing the percentage of low WSS area (WSS < 1 Pa) relative to the length of stent deployment area is set as the objective function of this optimization since low WSS is believed to promote some problems such as atherosclerosis. In total, 45 and 42 simulation iterations are conducted respectively for both cross sections to develop the Kriging surrogate models for efficient global optimization. Besides the prediction of the optimized configuration, broader observation on its behavior within the design range is also well predicted. The optimized configuration has 2.99 mm gap and 0.1 mm width for the rectangular strut, and 2.00 mm gap and 0.99 mm width for the triangular strut. The triangular strut has better performance in reducing the low WSS area with 14.6% of low WSS area on its optimized design, compared to 18.3% of the rectangular strut. Moreover, the triangular shape strut produces more stable performance; most design configuration with the strut width of less than 0.35 mm can keep low WSS area at the minimum value.

Narendra Kurnia Putra, Pramudita Satria Palar, Hitomi Anzai, Koji Shimoyama, Makoto Ohta


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