Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima
TL;DR: The current knowledge about numerical instabilities such as checkerboards, mesh-dependence and local minima occurring in applications of the topology optimization method are summarized and the methods with which they can be avoided are listed.
Abstract: In this paper we seek to summarize the current knowledge about numerical instabilities such as checkerboards, mesh-dependence and local minima occurring in applications of the topology optimization method. The checkerboard problem refers to the formation of regions of alternating solid and void elements ordered in a checkerboard-like fashion. The mesh-dependence problem refers to obtaining qualitatively different solutions for different mesh-sizes or discretizations. Local minima refers to the problem of obtaining different solutions to the same discretized problem when choosing different algorithmic parameters. We review the current knowledge on why and when these problems appear, and we list the methods with which they can be avoided and discuss their advantages and disadvantages.
Citations
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TL;DR: It is shown that only 49 Matlab input lines are required for solving a well-posed topology optimization problem and by adding three additional lines, the program can solve problems with multiple load cases.
Abstract: The paper presents a compact Matlab implementation of a topology optimization code for compliance minimization of statically loaded structures. The total number of Matlab input lines is 99 including optimizer and Finite Element subroutine. The 99 lines are divided into 36 lines for the main program, 12 lines for the Optimality Criteria based optimizer, 16 lines for a mesh-independency filter and 35 lines for the finite element code. In fact, excluding comment lines and lines associated with output and finite element analysis, it is shown that only 49 Matlab input lines are required for solving a well-posed topology optimization problem. By adding three additional lines, the program can solve problems with multiple load cases. The code is intended for educational purposes. The complete Matlab code is given in the Appendix and can be down-loaded from the web-site http://www.topopt.dtu.dk.
1,956 citations
TL;DR: An overview, comparison and critical review of the different approaches to topology optimization, their strengths, weaknesses, similarities and dissimilarities and suggests guidelines for future research.
Abstract: Topology optimization has undergone a tremendous development since its introduction in the seminal paper by Bendsoe and Kikuchi in 1988. By now, the concept is developing in many different directions, including “density”, “level set”, “topological derivative”, “phase field”, “evolutionary” and several others. The paper gives an overview, comparison and critical review of the different approaches, their strengths, weaknesses, similarities and dissimilarities and suggests guidelines for future research.
1,816 citations
TL;DR: In this article, the physical stiffness of an element is based on a function of the design variables of the neighboring elements, and a new class of morphology-based restriction schemes that work as density filters is introduced.
Abstract: To ensure manufacturability and mesh independence in density-based topology optimization schemes, it is imperative to use restriction methods. This paper introduces a new class of morphology-based restriction schemes that work as density filters; that is, the physical stiffness of an element is based on a function of the design variables of the neighboring elements. The new filters have the advantage that they eliminate grey scale transitions between solid and void regions. Using different test examples, it is shown that the schemes, in general, provide black and white designs with minimum length-scale constraints on either or both minimum hole sizes and minimum structural feature sizes. The new schemes are compared with methods and modified methods found in the literature.
1,305 citations
TL;DR: In this paper, the material density field is filtered to enforce a length scale on the field variation and is penalized to remove less effective intermediate densities to resolve the non-existent solution to the solid void topology problem.
1,125 citations
TL;DR: Topology optimization is the process of determining the optimal layout of material and connectivity inside a design domain this paper, which is the same as the problem of finding the optimal configuration of a set of components.
Abstract: Topology optimization is the process of determining the optimal layout of material and connectivity inside a design domain. This paper surveys topology optimization of continuum structures from the year 2000 to 2012. It focuses on new developments, improvements, and applications of finite element-based topology optimization, which include a maturation of classical methods, a broadening in the scope of the field, and the introduction of new methods for multiphysics problems. Four different types of topology optimization are reviewed: (1) density-based methods, which include the popular Solid Isotropic Material with Penalization (SIMP) technique, (2) hard-kill methods, including Evolutionary Structural Optimization (ESO), (3) boundary variation methods (level set and phase field), and (4) a new biologically inspired method based on cellular division rules. We hope that this survey will provide an update of the recent advances and novel applications of popular methods, provide exposure to lesser known, yet promising, techniques, and serve as a resource for those new to the field. The presentation of each method's focuses on new developments and novel applications.
1,052 citations
Cites background from "Numerical instabilities in topology..."
...These include the sensitivity filter (Sigmund and Petersson 1998) and the density filter (Bourdin 2001; Bruns and Tortorelli 2001), which modify either the sensitivity or the density value of an element based on the sensitivity or density of elements in a localized neighborhood....
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References
More filters
TL;DR: In this article, the authors present a methodology for optimal shape design based on homogenization, which is related to modern production techniques and consists of computing the optimal distribution in space of an anisotropic material that is constructed by introducing an infimum of periodically distributed small holes in a given homogeneous, i.i.
Abstract: Optimal shape design of structural elements based on boundary variations results in final designs that are topologically equivalent to the initial choice of design, and general, stable computational schemes for this approach often require some kind of remeshing of the finite element approximation of the analysis problem. This paper presents a methodology for optimal shape design where both these drawbacks can be avoided. The method is related to modern production techniques and consists of computing the optimal distribution in space of an anisotropic material that is constructed by introducing an infimum of periodically distributed small holes in a given homogeneous, i~otropic material, with the requirement that the resulting structure can carry the given loads as well as satisfy other design requirements. The computation of effective material properties for the anisotropic material is carried out using the method of homogenization. Computational results are presented and compared with results obtained by boundary variations.
5,858 citations
TL;DR: In this article, various ways of removing this discrete nature of the problem by the introduction of a density function that is a continuous design variable are described. But none of these methods can be used for shape optimization in a general setting.
Abstract: Shape optimization in a general setting requires the determination of the optimal spatial material distribution for given loads and boundary conditions. Every point in space is thus a material point or a void and the optimization problem is a discrete variable one. This paper describes various ways of removing this discrete nature of the problem by the introduction of a density function that is a continuous design variable. Domains of high density then define the shape of the mechanical element. For intermediate densities, material parameters given by an artificial material law can be used. Alternatively, the density can arise naturally through the introduction of periodically distributed, microscopic voids, so that effective material parameters for intermediate density values can be computed through homogenization. Several examples in two-dimensional elasticity illustrate that these methods allow a determination of the topology of a mechanical element, as required for a boundary variations shape optimization technique.
3,434 citations
TL;DR: In this paper, the COC algorithm is applied to the simultaneous optimization of the topology and geometry of trusses with many thousand potential members, and numerical results obtained are shown to be in close agreement with analytical results.
1,412 citations
"Numerical instabilities in topology..." refers methods in this paper
...Another probably more popular method is the so-called SIMP [Simple Isotropic Material with Penalization ( Zhou and Rozvany 1991 )] approach (eft BendsCe 1989)....
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TL;DR: In this paper, the authors present a method for optimal design of compliant mechanism topologies based on continuum-type topology optimization techniques and find the optimal mechanism topology within a given design domain and a given position and direction of input and output forces.
Abstract: This paper presents a method for optimal design of compliant mechanism topologies. The method is based on continuum-type topology optimization techniques and finds the optimal compliant mechanism topology within a given design domain and a given position and direction of input and output forces. By constraining the allowed displacement at the input port, it is possible to control the maximum stress level in the compliant mechanism. The ability of the design method to find a mechanism with complex output behavior is demonstrated by several examples. Some of the optimal mechanism topologies have been manufactured, both in macroscale (hand-size) made in Nylon, and in microscale (<.5mm)) made of micromachined glass.
1,282 citations
"Numerical instabilities in topology..." refers methods in this paper
...For the mesh-independence filter Sigmund (1997) and Siground and Torquato (1997) suggest starting with a large value of the filter size rmi n ensuring a convex solution and gradually to decrease it, to end up with a 0-1 design....
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Book•
01 Jan 1995
TL;DR: The method presented in this book has been developed by Martin Bendsoe in co-operation with other researchers and can be considered as one of the most effective approaches to the optimization of layout and material design as discussed by the authors.
Abstract: In the past, the possibilities of structural optimization were restricted to an optimal choice of profiles and shape. Further improvement can be obtained by selecting appropriate advanced materials and by optimising the topology, i.e. finding the best position and arrangement of structural elements within a construction. The optimization of structural topology permits the use of optimization algorithms at a very early stage of the design process. The method presented in this book has been developed by Martin Bendsoe in co-operation with other researchers and can be considered as one of the most effective approaches to the optimization of layout and material design.
1,185 citations