A symmetry reduction method for continuum structural topology optimization

doi:10.1016/S0045-7949(98)00158-8

Computers & Structures

Volume 70, Issue 1, January 1999, Pages 47-61

https://doi.org/10.1016/S0045-7949(98)00158-8 Get rights and content

Abstract

It is considered that asymmetrical material layout design solutions are caused by numerical roundoff and the convexity characteristics of alternative topology design formulations. Emphasis is placed here not on analyzing potential instabilities that lead to asymmetrical designs, but on a method to stabilize topology design formulations. A novel symmetry reduction method is proposed, implemented and studied. While enforcing symmetry and significantly reducing the size of the optimization problem, the symmetry reduction method is shown to have the added benefit of greatly simplified design sensitivity analysis of non-simple repeated vibrational eigenvalues which occur in many symmetrical structures.

Section snippets

Introduction and motivation

Continuum structural topology optimization is an increasingly powerful design tool which can be used to optimize the material arrangements in structural systems to achieve a wide variety of performance objectives, including as examples: minimal compliance1, 2, 3, 4; optimal strength5, 6; viscoelastic damping[7]; and compliant mechanisms8, 9, 10. For all of these structural or material topology design optimization applications, if the design domain, the boundary conditions and the loading are

Distribution of materials

The complete undeformed spatial domain of the structure being designed is denoted by Ω_S; its designable subset by Ω_D; and its non-designable subset, in which the spatial/topological arrangement of materials is taken to be fixed, by Ω_N. The arrangement of two pre-selected candidate materials $A$ and $B$ in Ω_D remains to be determined and so this region is called designable. A set of single or multiple loading/boundary conditions to which Ω_S will be subjected are specified and a starting design b⁽⁰⁾

Motivation

There are two primary reasons for imposing geometrical symmetry on material layout designs. First, even when the design loads are asymmetrical the designer may still seek a symmetric design so that reversed loading cases can be accommodated equally well. Second, even when both the design loads and boundary conditions applied to Ω_S are symmetrical, the resulting material layout field b as obtained through standard optimization techniques, will not necessarily feature the expected geometrical

Motivation

Many structures whose structural stiffnesses and mass distributions are the same and symmetrical in two or more orthogonal directions invariably encounter the phenomenon of repeated, non-simple vibrational eigenvalues. It is recognized that the design sensitivity analysis of repeated eigenvalues is difficult due to the fact that they are generally not continuously differentiable, but are only directionally differentiable[17]. Accordingly, design sensitivities for repeated vibrational

Summary and conclusions

It has been hypothesized that asymmetrical material layout solutions are caused by the non-convexity of highly penalized continuum topology design formulations coupled with limited precision in numerical computations. In this study, a novel symmetry reduction method to control and stabilize non-convex topology design formulations has been investigated and demonstrated on simple, representative examples.

Benefits of the proposed symmetry reduction method are that it produces symmetrical material

References (37)

H.P. Mlejnek et al.
An engineer’s approach to optimal material distribution and shape finding
Computer Methods in Applied Mechanics and Engineering
(1993)
C.S. Jog et al.
Stability of finite element models for distributed-parameter optimization and topology design
Computer Methods in Applied Mechanics and Engineering
(1996)
R.J. Yang et al.
Optimal topology design using linear programming
Computers & Structures
(1994)
K. Suzuki et al.
A homogenization method for shape and topology optimization
Computer Methods in Applied Mechanics and Engineering
(1991)
C.S. Jog et al.
Topology design with optimized, self-adaptive materials
International Journal for Numerical Methods in Engineering
(1994)
Sigmund O. Design of material structures using topology optimization, Report S69. Danish Center for Applied Mathematics...
C.C. Swan et al.
Voigt–Reuss topology optimization for structures with linear elastic material behaviours
International Journal for Numerical Methods in Engineering
(1997)
C.C. Swan et al.
Voigt–Reuss topology optimization for structures with nonlinear material behaviours
International Journal for Numerical Methods in Engineering
(1997)
C.C. Swan et al.
Topology design of material layout in structured composites of high stiffness and strength
Structural Optimization
(1997)
Kosaka I, Swan CC. Continuum topology optimization of viscoelastic systems of high stiffness and high damping. (to...

Ananthasuresh GK, Kota S, Gianchandani Y. A methodical approach to design of compliant micromechanisms. In: Proceedings...

Sigmund O. On the design of compliant mechanisms using topology optimization. Mechanics of Structures and Machinery,...

Kosaka I, Swan CC. Design of compliant structural mechanisms with Voigt–Reuss topology optimization. (to appear),...

A.R. Diaz et al.

Shape optimization of structures for multiple loading conditions using a homogenization method

Structural Optimization

(1992)

M.P. Bendsoe

Optimal shape design as a material distribution problem

Structural Optimization

(1989)

Ramm E, Bletzinger KU, Reitinger R, Maute K. The challenge of structural optimization. In: Topping BHV, Papadrakakis...

Kosaka I. A conceptual design method for structures and structural materials. Ph.D thesis, College of Engineering, The...

M.P. Bendsoe et al.

Optimal design of material properties and material distribution for multiple loading conditions

International Journal for Numerical Methods in Engineering

(1995)

Cited by (85)

Structural topology optimisation based on a multi-agent model
2023, Engineering Structures
In this study, we propose to equip the basic vector-based multi-agent system with global structural awareness by introducing strain energy density gradient vectors to achieve structural topology optimisation. This multi-agent based topology optimisation method can conveniently generate diverse and competitive structural designs with different parameters of the agent swarm behavior rules (separation, alignment and cohesion). Several numerical examples are introduced to explore the relationship between parameters and resulting design diversities. It is demonstrated that the proposed method can produce a variety of designs that possess not only high structural performance but also distinctly different topologies. Taking advantages of the computational multi-agent system, this method has potential to be easily integrated with agent system algorithms in other research fields to generate diverse designs with high structural performance, elegant geometries and interesting features for architects, engineers and other designers.
Topology optimization for infinite fatigue life of cyclic symmetric structures subjected to non-proportional loading
2023, Computers and Structures
This paper presents a density based topology optimization method for infinite fatigue life constraints of non-proportional load cases, with a specific focus on parts with cyclic symmetry. Considering non-proportional loads in topology optimization significantly broadens the types of design problems that can be handled. The method estimates the local variation in Signed von Mises stress using a smooth min/max function and constrains the resulting stress amplitude using established stress based topology optimization methods. Accounting for non-proportionality of loading significantly increases the computation cost with respect to existing proportional methods, as the time-varying stress field needs to be computed. Inertia effects are neglected in the structural analysis. Therefore, a quasi-static analysis is used to obtain the stress history. To reduce the computational cost, advantage is taken of cyclic symmetric properties to reduce the number of necessary time steps to evaluate. This reduces the computational cost roughly proportional to the number of unique load time steps present in the repeated segments as opposed to a standard implementation. The method is tested on numerical examples in 2D and 3D for both proportional and non-proportional loads and was found to be locally accurate up to the accuracy of the constraint aggregation.
Improved adaptive growth method of stiffeners for three-dimensional box structures with respect to natural frequencies
2020, Computers and Structures
Citation Excerpt :
Firstly studied by Diaz and Kikuchi [3] in the form of optimizing a single frequency of plate disks, natural frequency optimization has been developed based on the homogenization design (HD) method, the solid isotropic material with penalization (SIMP) method, the evolutionary structural optimization (ESO)/bi-directional ESO (BESO) method, and others over decades of research. Ma et al. [22,23], Diaz et al. [4], and Kosaka and Swan [16] proposed different formulations for simultaneously maximizing several frequencies of disk and plate structures by building a scalar weighted function of eigenfrequencies as the objective. The paper of Krog and Olhoff [17] takes fundamental and higher order eigenfrequencies of disk and plate structures into the optimization and firstly solved it as a max–min problem of maximizing a lower bound design variable.
Adaptive growth method is a structural topology optimization approach based on the growth mechanism of natural biological branching systems, and it has been successfully applied for the static reinforcement of three-dimensional box structures. However, due to the effect of structural characteristics towards natural frequencies, it is difficult for adaptive growth method to effectively grow stiffeners inside three-dimensional box structures considering dynamic performance. And this paper proposes an improved version of adaptive growth method to solve this problem. Firstly, apart from the growth mechanism of natural branching systems in the initial adaptive growth method, the morphology of root tips is introduced into the growth process of internal stiffeners. The thickness of stiffeners during the optimization can be divided into four different zones: initial thickness zone, immediate thickness zone, allowable thickness zone and maturation zone. Then, a material interpolation scheme with different thickness zones is introduced to simultaneously penalize the elastic modulus and density of material of internal stiffeners according to their thicknesses. By applying this bio-inspired method, obstacles of the ground structure in dynamic design of stiffeners in three-dimensional box structures can be effectively eliminated. And notably, the internal stiffeners can successfully grow from “seed lines” and gradually bifurcate or degenerate along the optimal direction to improve the natural frequencies of structures with a clear layout pattern. Finally, the effectiveness and advantages of the proposed method are illustrated by several numerical examples.
Eigen-frequencies and harmonic responses in topology optimisation: A CAD-compatible algorithm
2020, Engineering Structures
The formulation of Topology Optimisation (TO) problems related to dynamics is particularly challenging, due to some intrinsic difficulties of mathematical and numerical nature. This paper deals with the integration of specific physical quantities, such as eigen-frequencies and dynamic compliance, in a special TO algorithm, which combines a classical pseudo-density field with Non-Uniform Rational Basis Spline (NURBS) entities. In this framework, wherein some of the NURBS continuous parameters (i.e. control points and weights) are the new design variables, important advantages can be exploited. In particular, beyond the reduction of the number of design variables and the definition of an implicit filter zones, the post-processing phase involving the CAD reconstruction of the optimised geometry is immediate for 2D problems and it needs few operations in 3D. Classical TO problems dealing with structural dynamics, as the maximisation of the first eigen-frequency and the minimisation of the dynamic compliance, are formulated in the NURBS framework. Accordingly, the analytical expressions of the gradients of the considered physical quantities are derived in closed form. In order to show the effectiveness of the proposed approach, an exhaustive numerical campaign is proposed and the algorithm is applied to both 2D and 3D benchmarks. Moreover, a sensitivity analysis of the final optimised solutions to the NURBS discrete parameters is provided as well.
Topology optimization for eigenfrequencies of a rotating thin plate via moving morphable components
2019, Journal of Sound and Vibration
An efficient and explicit topology optimization approach is initially proposed for eigenfrequencies of a rotating thin plate in this study. First of all, an accurate dynamic model of the rotating thin plate is established via the thin plate elements of the absolute nodal coordinate formulation (ANCF). When performing the modal characteristic analysis of the rotating thin plate at a prescribed angular velocity, the linear perturbation analysis is employed, during which the coupling between the membrane and bending deformations is considered. The coupling term makes the dynamic model established in the study more accurate than conventional models, especially in the case of large deformations. Then, the moving morphable components (MMC) are used to describe the topology of the plate. In the frame of MMC-based topology optimization, explicit geometrical parameters, positions, and orientations of the components are taken as the design variables so that the total number of the design variables can be greatly reduced. For the topology optimization, the sensitivities of a simple eigenfrequency and multiply repeated eigenfrequencies with respect to a design variable are analytically derived. During the optimization, in order to remove the localized modes in the low-density areas, the mass and stiffness matrices of the thin plate elements of ANCF are carefully penalized. Finally, four numerical examples are presented to validate the proposed topology optimization approach and to demonstrate its effectiveness for two objectives, i.e., maximizing either the first eigenfrequency or the gap between two consecutive eigenfrequencies of a rotating thin plate.
Smoothing evolutionary structural optimization for structures with displacement or natural frequency constraints
2018, Engineering Structures
Citation Excerpt :
The works of Tenek and Hagiwara [51] and Ma et al. [32] developed frequency optimization problems using the homogenization method. Kosaka and Swan [26] have used the SIMP method. However, the SIMP model has been proven to be inadequate for frequency optimization due to local artificial modes in lower mass areas.
The Smoothing Evolutionary Structural Optimization (SESO) technique was extended to solve 2D elastic problems with constraint of displacements or natural frequencies. At the end of each finite element analysis, a scalar representing the sensitivity due to the removal of an element is calculated. Thus, the elements that have the lowest values are removed from the structure, while the displacements in prescribed locations are kept inside of limits stated or the first frequencies are maximized. The proposed technique proved to be adequate and efficient in the execution of shape and topological optimization.

View all citing articles on Scopus

¹: Now with VMA Engineering, Colorado Springs, Colorado, USA

View full text

A symmetry reduction method for continuum structural topology optimization

Abstract

Section snippets

Introduction and motivation

Distribution of materials

Motivation

Motivation

Summary and conclusions

Computer Methods in Applied Mechanics and Engineering

Computer Methods in Applied Mechanics and Engineering

Computers & Structures

A homogenization method for shape and topology optimization

Computer Methods in Applied Mechanics and Engineering

Topology design with optimized, self-adaptive materials

International Journal for Numerical Methods in Engineering

Voigt–Reuss topology optimization for structures with linear elastic material behaviours

International Journal for Numerical Methods in Engineering

Voigt–Reuss topology optimization for structures with nonlinear material behaviours

International Journal for Numerical Methods in Engineering

Topology design of material layout in structured composites of high stiffness and strength

Structural Optimization

Shape optimization of structures for multiple loading conditions using a homogenization method

Structural Optimization

Optimal shape design as a material distribution problem

Structural Optimization

Optimal design of material properties and material distribution for multiple loading conditions

International Journal for Numerical Methods in Engineering