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2014 | Book

Structural Topology Optimization (STO) – Exact Analytical Solutions: Part I Read chapter Read first chapter

Topology Optimization in Structural and Continuum Mechanics

Editors: George I. N. Rozvany, Tomasz Lewiński

Publisher: Springer Vienna

Book Series : CISM International Centre for Mechanical Sciences

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

The book covers new developments in structural topology optimization. Basic features and limitations of Michell’s truss theory, its extension to a broader class of support conditions, generalizations of truss topology optimization, and Michell continua are reviewed. For elastic bodies, the layout problems in linear elasticity are discussed and the method of relaxation by homogenization is outlined. The classical problem of free material design is shown to be reducible to a locking material problem, even in the multiload case. For structures subjected to dynamic loads, it is explained how they can be designed so that the structural eigenfrequencies of vibration are as far away as possible from a prescribed external excitation frequency (or a band of excitation frequencies) in order to avoid resonance phenomena with high vibration and noise levels. For diffusive and convective transport processes and multiphysics problems, applications of the density method are discussed. In order to take uncertainty in material parameters, geometry, and operating conditions into account, techniques of reliability-based design optimization are introduced and reviewed for their applicability to topology optimization.

Table of Contents

Frontmatter
Structural Topology Optimization (STO) – Exact Analytical Solutions: Part I
Abstract
As mentioned in the Preface, on the subject of STO the author has organized two previous CISM Advanced Courses in Udine (Rozvany 1992 and 1997), and with Niels Olhoff a NATO ARW in Budapest (Rozvany and Olhoff 2000).
George I. N. Rozvany
Structural Topology Optimization (STO) – Exact Analytical Solutions: Part II
Abstract
In this lecture we review the origin and basic features of the optimal layout theory (Prager and Rozvany 1977a), including optimal regions and difficulties with so-called O-regions. Then we discuss various extensions of this theory, including multiple load conditions and multiple design constraints, probabilistic design and pre-existing members.
George I. N. Rozvany, Erika Pinter
Some Fundamental Properties of Exact Optimal Structural Topologies
Abstract
In Lecture 1 of this author we discussed Michell's (1904) theory of optimal truss design, examined its range of validity, and looked at extended optimality criteria for a broader class of boundary conditions.
In Lecture 2 the optimal layout theory (Prager and Rozvany 1977) as well as optimal regions in exact truss topologies were reviewed and several extensions of the layout theory presented.
Whilst these two lectures touched on some basic features of optimal structural topologies, fundamental properties of these will be examined in detail in this lecture.
George I. N. Rozvany
Validation of Numerical Methods by Analytical Benchmarks, and Verification of Exact Solutions by Numerical Methods
Abstract
In this lecture, we discuss the validation of various numerical methods in structural topology optimization. This is done by computing numerically optimal topologies (e.g. for perforated plates in plane stress), using various numbers of ground elements and various volume fractions. Then the structural volume value is extrapolated for (theoretically) zero volume fraction and infinite number of ground elements, and this extrapolated value is compared with that of the analytical solution. As a related subject, volume-increasing effect of topology simplification is also discussed.
In the second part of the lecture new and highly efficient numerical methods are discussed, which are capable of optimizing truss topologies with up to over one billion potential truss elements, and thereby providing the verification of optimal volumes with an accuracy of up to four significant digits.
The first author will present the first part of this lecture, and the second author the second part.
George I. N. Rozvany, Tomasz Sokół
A Brief Review of Numerical Methods of Structural Topology Optimization
Abstract
In his first four lectures, the author discussed exact analytical solutions of structural topology optimization. Most real-world problems can only be solved by numerical, discretized methods, which are therefore of considerable practical importance.
George I. N. Rozvany
On Basic Properties of Michell’s Structures
Abstract
The paper is an introduction towards Michell’s optimum design problems of pin-jointed frameworks. Starting from the optimum design problem in its discrete setting we show the passage to the discrete-continuous setting in the kinematic form and then the primal stress-based setting. The classic properties of Hencky nets are consistently derived from the kinematic optimality conditions. The Riemann method of the net construction is briefly recalled. This is the basis for finding the analytical solutions to the optimum design problems. The paper refers mainly to the well known solutions (e.g. the cantilevers) yet discusses also the open problems concerning those classes of solutions in which the kinematic approach cannot precede the static analysis.
Tomasz Lewiński, Tomasz Sokół
Structural Shape and Topology Optimization
Abstract
This chapter is the printed matter of a series of lectures on shape optimization given in Udine (Italy) in June 2012. It is mainly focused on the optimal design problem in elasticity.
It is well known, since the pioneer work of Tartar and Murat in the late 70’s, that this problem – as well as the scalar shape optimization problem in conductivity – is ill posed and does not always admit solutions. If the goal is to construct numerical methods to address this optimal design problem, this illposedness will lead to algorithms showing instabilities and erratic behaviours.
Two ways are possible to slightly change the initial problem so that it may be solved in an efficient and reliable way by numerical methods. The first one consists in enlarging the set of admissible shapes and allow, as solutions, “generalized shapes” that may contain fine mixtures of the initial material and void. This process is called “relaxation” and the main mathematical tool involved is the homogenization theory. It leads to the so called “homogenization method for shape optimization”. It is described in Section 2, both from theoretical and numerical point of views, including practical issues necessary to write a numerical code.
The second approach consists in restricting the set of admissible solutions. A few theorems prove the existence of solutions under further geometrical, topological or regularity constraints. The classical domain variation method is revisited using the level set representation, that may be seen as a constraint on the set of admissible solutions. The main tool is here the shape derivative. The level set algorithm is presented in Section 3.
Of course this chapter is too short to develop new results and methods and present extensive proofs. Its goal is rather to give an overview on different parts of our work on this topics. Interested readers should check the references cited to have further details.
François Jouve
Compliance Minimization of Two-Material Elastic Structures
Abstract
Minimum compliance problem set for a structure made of two elastic, isotropic materials is tackled in this paper. The relaxation by homogenization technique is used for obtaining its mathematically well-posed formulation. The problem is first discussed in general two-material context. Derivation of main results is recalled and supplemented with some explanations and remarks. Next, an important topic of one-material layout optimization (or shape optimization) is addressed. It is hampered by the non-smoothness of formula for relaxed stress energy hence its approximation is proposed which in turn makes the FEM easier to apply in solving the equilibrium problem. Shape optimization is then linked to a wellknown Michell problem of the lightest, fully stressed structures. Possible extension of the relaxation by homogenization method to other structures like thin or moderately thick plates in bending as well as thin plates or shells submerged to simultaneous in-plane and bending load are also commented.
Grzegorz Dzierżanowski, Tomasz Lewiński
The Free Material Design in Linear Elasticity
Abstract
The Free Material Design (FMD) is a branch of topology optimization. In the present article the FMD formulation is confined to the minimum compliance problem within the linear elasticity setting. The design variables are all elastic moduli, forming a Hooke tensor C at each point of the design domain. The isoperimetric condition concerns the integral of the p-norm of the vector of the eigenvalues of the tensor C. The most important version refers to p = 1, imposing the condition on the integral of the trace of C. The paper delivers explicit stress-based formulations and numerical solutions of the FMD problems in the case of a single load case as well as for a general case of a finite number of load conditions.
Sławomir Czarnecki, Tomasz Lewiński
Introductory Notes on Topological Design Optimization of Vibrating Continuum Structures
Abstract
This paper presents a brief introduction to topological design optimization, and together with five sequential papers gives an overview of the application of this rather novel method to problems of design of linearly elastic continuum-type structures against vibration and noise. The objective of such problems is often to drive the structural eigenfrequencies of vibration as far away as possible from a prescribed external excitation frequency - or band of excitation frequencies - in order to avoid resonance phenomena with high vibration and noise levels. This objective may, e.g., be achieved by (i) maximizing the fundamental eigenfrequency of the structure, (ii) maximizing the distance (gap) between two consecutive eigenfrequencies, (iii) maximizing the dynamic stiffness of the structure subject to forced vibration, or by (iv) minimizing the sound power flow radiated from the structural surface into an acoustic medium. The mathematical formulations of these optimization problems and several illustrative examples are presented in this series of papers.
Niels Olhoff, Jianbin Du
Structural Topology Optimization with Respect to Eigenfrequencies of Vibration
Abstract
A frequent goal of the design of vibrating structures is to avoid resonance of the structure in a given interval for external excitation frequencies. This can be achieved by, e.g., maximizing the fundamental eigenfrequency, an eigenfrequency of higher order, or the gap between two consecutive eigenfrequencies of given order, subject to a given amount of structural material and prescribed boundary conditions. Mathematical formulations and methods of numerical solution of these topology optimization problems are presented for linearly elastic structures without damping in this paper, and several illustrative results are shown.
Niels Olhoff, Jianbin Du
On Optimum Design and Periodicity of Band-gap Structures
Abstract
A band-gap structure usually consists of a periodic distribution of elastic materials or segments, where the propagation of waves is impeded or significantly suppressed for a range of external excitation frequencies. Maximization of the band-gap is therefore an obvious objective for optimum design. This problem is sometimes formulated by optimizing a parameterized design model which assumes multiple periodicity in the design. However, it is shown in the present paper that such an a priori assumption is not necessary since, in general, just the maximization of the gap between two consecutive eigenfrequencies leads to significant design periodicity.
Hence, it is the aim of this paper to apply the method presented in the preceding paper Olhoff and Du (2013B) to maximize gaps between two consecutive eigenfrequencies by shape optimization of transversely vibrating Bernoulli-Euler beams without damping, and to present and study the associated creation of periodicity in the optimized beam designs.
In the end of the present paper, in order to study the band-gap for travelling waves, a repeated inner segment of the optimized beams is analyzed using Floquet theory and the waveguide finite element (WFE) method. Finally, the frequency response is computed for the optimized beams when these are subjected to an external time-harmonic loading with different excitation frequencies, in order to investigate the attenuation levels in prescribed frequency band-gaps. The results demonstrate that there is almost perfect correlation between the band-gap size/location of the emerging band structure and the size/location of the corresponding eigenfrequency gap in the finite structure.
Niels Olhoff, Bin Niu
Topological Design for Minimum Dynamic Compliance of Structures under Forced Vibration
Abstract
This paper deals with topology optimization of elastic, continuum structures without damping that are subjected to time-harmonic, dynamic loading with prescribed excitation frequency and amplitude. An important objective of such a design problem is often to drive the eigenfrequencies of the structure as far away as possible from the excitation frequency in order to avoid resonance and reduce the vibration level of the structure. The total structural volume, the boundary conditions, and the material are given.
Niels Olhoff, Jianbin Du
Topological Design for Minimum Sound Emission from Structures under Forced Vibration
Abstract
This paper is devoted to topology optimization problems formulated with the design objective of minimizing the sound power radiated from the structural surface(s) into a surrounding acoustic medium. Bi-material elastic continuum structures without material damping are considered. The structural vibrations are excited by time-harmonic external mechanical loading with prescribed excitation frequency, amplitude, and spatial distribution. Several numerical results are presented and discussed for bi-material plate-like structures with different sets of boundary and loading conditions.
Niels Olhoff, Jianbin Du
Discrete Material Optimization of Vibrating Laminated Composite Plates for Minimum Sound Emission
Abstract
This paper deals with vibro-acoustic optimization of laminated composite plates without damping. The vibration is excited by time-harmonic external mechanical loading with prescribed frequency and amplitude, and the design objective is to minimize the total sound power radiated from the surface of the laminated plate to the surrounding acoustic medium. Instead of solving the Helmholtz equation for evaluation of the sound power, advantage is taken of the fact that the surface of the laminated plate is flat, which implies that Rayleigh’s integral approximation can be used to evaluate the sound power radiated from the surface of the plate. The novel Discrete Material Optimization (DMO) formulation has been applied to achieve the design optimization of fiber angles, stacking sequence and selection of material for laminated composite plates. Several numerical examples are presented in order to illustrate this approach.
Niels Olhoff, Bin Niu
Topology Optimization of Diffusive Transport Problems
Abstract
This lecture covers topology optimization methods for solving diffusive transport problems, in particular heat conduction in solids. The importance of understanding the underlying physical phenomena to properly define the optimization problem is emphasized. Numerical issues arising from large gradients in the diffusion coefficients along with remedies for mitigating theses problems will be discussed.
Kurt Maute
Topology Optimization of Flows: Stokes and Navier-Stokes Models
Abstract
This lecture will discuss density-based approaches for solving flow topology optimization problems. The focus is on low-Reynolds number fluid models, namely Stokes and laminar Navier-Stokes models, at steady-state conditions.
Kurt Maute
Topology Optimization of Coupled Multi-Physics Problems
Abstract
Topology optimization provides a promising approach to systematically design multi-physics problems, such as thermo-mechanical, electro-static, and fluid-structure interaction problems. This class of design problems is often dominated by nonlinear phenomena and is not well suited for intuitive design strategies. In this lecture we will discuss applications of topology optimization methods to coupled multi-physics problems, emphasizing the differences between volumetric and interface coupling in the context of topology optimization. Focusing on density methods, topology optimization of piezo-electric devices and fluid-structure interaction problems will be studied.
Kurt Maute
The Extended Finite Element Method
Abstract
This chapter discusses level-set topology optimization methods where the governing state equations are discretized by the Extended Finite Element Method (XFEM). In contrast to conventional methods that map the level-set function into a density distribution of a fictitious material, the XFEM allow preserving the crisp geometry information of level-sets in the mechanical model. A brief introduction into the XFEM will be provided and its application to topology optimization will be illustrated with problems in fluid mechanics.
Kurt Maute
Topology Optimization under Uncertainty
Abstract
Considering stochastic variations in material parameters, geometry, and boundary conditions is for the majority of engineering design problems of pivotal importance, in order to obtain robust and reliable designs. While design optimization under uncertainty has matured for sizing and shape optimization over the past two decades, accounting for stochastic variations in topology optimization is still in its infancy. This lecture will introduce basic approaches to include uncertainty models and predictions into the topology optimization process.
Kurt Maute
Metadata
Title
Topology Optimization in Structural and Continuum Mechanics
Editors
George I. N. Rozvany
Tomasz Lewiński
Copyright Year
2014
Publisher
Springer Vienna
Electronic ISBN
978-3-7091-1643-2
Print ISBN
978-3-7091-1642-5
DOI
https://doi.org/10.1007/978-3-7091-1643-2

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