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Journal ArticleDOI

Topology optimization of non-linear elastic structures and compliant mechanisms

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.
About: This article is published in Computer Methods in Applied Mechanics and Engineering.The article was published on 2001-03-16. It has received 1125 citations till now. The article focuses on the topics: Topology optimization & Optimization problem.
Citations
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Journal ArticleDOI
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

Journal ArticleDOI
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


Cites background or methods from "Topology optimization of non-linear..."

  • ...was suggested in Bruns and Tortorelli (2003); Wang and Wang (2005) for use with the density filtering (12)....

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  • ...Density filtering was introduced by Bruns and Tortorelli (2001) and mathematically proven as a viable approach by Bourdin (2001)....

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  • ...…1) mesh-independent filtering methods, constituting sensitivity filters (Sigmund, 1994, 1997; Sigmund and Petersson, 1998) and density filters (Bruns and Tortorelli, 2001; Bourdin, 2001); 2) constraint methods such as perimeter control (Ambrosio and Buttazzo, 1993; Haber et al, 1994), global…...

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  • ...was suggested in Bruns and Tortorelli (2003); Wang and Wang (2005) for use with the density filtering (12). In Bruns and Tortorelli (2003) the variance is defined by σd = R/3 and in Wang and Wang (2005) by σd = R/2....

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  • ...was suggested in Bruns and Tortorelli (2003); Wang and Wang (2005) for use with the density filtering (12). In Bruns and Tortorelli (2003) the variance is defined by σd = R/3 and in Wang and Wang (2005) by σd = R/2. In both cases, the filter is truncated at radius R. For the latter case, it means that the bell-shape curve is heavily truncated. In tests, the author did not experience any advantages of the smooth Gaussian function compared to the original linear function, however, it is possible that there is an advantage when the filtering scheme is used in connection with the element removal and re-introduction scheme suggested by Bruns and Tortorelli (2003). For completeness one may also consider a constant weighting function...

    [...]

Journal ArticleDOI
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 "Topology optimization of non-linear..."

  • ...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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Journal ArticleDOI
TL;DR: It is shown that simple projection methods do not ensure localMesh convergence and a modified robust topology optimization formulation based on erosion, intermediate and dilation projections is proposed that ensures both global and local mesh-convergence.
Abstract: Mesh convergence and manufacturability of topology optimized designs have previously mainly been assured using density or sensitivity based filtering techniques. The drawback of these techniques has been gray transition regions between solid and void parts, but this problem has recently been alleviated using various projection methods. In this paper we show that simple projection methods do not ensure local mesh-convergence and propose a modified robust topology optimization formulation based on erosion, intermediate and dilation projections that ensures both global and local mesh-convergence.

1,047 citations


Cites methods from "Topology optimization of non-linear..."

  • ...Filtering techniques originally comprised sensitivity filtering (Sigmund 1997) and density filtering approaches ( Bruns and Tortorelli 2001; Bourdin 2001) and these two approaches have produced globally meshconvergent designs that in many cases are fully satisfactory for practical purposes....

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  • ...Here the mesh independent density filtering ( Bruns and Tortorelli 2001; Bourdin 2001) is used as a basis to ensure existence of solutions....

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Journal ArticleDOI
TL;DR: The paper presents an efficient 88 line MATLAB code for topology optimization using the 99 line code presented by Sigmund as a starting point, and a considerable improvement in efficiency has been achieved, mainly by preallocating arrays and vectorizing loops.
Abstract: The paper presents an efficient 88 line MATLAB code for topology optimization. It has been developed using the 99 line code presented by Sigmund (Struct Multidisc Optim 21(2):120---127, 2001) as a starting point. The original code has been extended by a density filter, and a considerable improvement in efficiency has been achieved, mainly by preallocating arrays and vectorizing loops. A speed improvement with a factor of 100 is obtained for a benchmark example with 7,500 elements. Moreover, the length of the code has been reduced to a mere 88 lines. These improvements have been accomplished without sacrificing the readability of the code. The 88 line code can therefore be considered as a valuable successor to the 99 line code, providing a practical instrument that may help to ease the learning curve for those entering the field of topology optimization. The paper also discusses simple extensions of the basic code to include recent PDE-based and black-and-white projection filtering methods. The complete 88 line code is included as an appendix and can be downloaded from the web site www.topopt.dtu.dk .

998 citations


Cites methods from "Topology optimization of non-linear..."

  • ...In addition to the sensitivity filter (Sigmund, 1994, 1997), which is already implemented in the 99 line code, the new 88 line code also includes density filtering (Bruns and Tortorelli, 2001 ; Bourdin, 2001)....

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  • ...The results obtained with the PDE filter are similar to the ones obtained by using an exponentially decaying filter kernel (Bruns and Tortorelli, 2001)....

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References
More filters
Book
01 Jan 1989
TL;DR: In this article, the methodes are numeriques and the fonction de forme reference record created on 2005-11-18, modified on 2016-08-08.
Abstract: Keywords: methodes : numeriques ; fonction de forme Reference Record created on 2005-11-18, modified on 2016-08-08

17,327 citations

Journal ArticleDOI
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

01 Jan 1997

4,469 citations

Journal ArticleDOI
TL;DR: In this article, a new method for non-linear programming in general and structural optimization in particular is presented, in which a strictly convex approximating subproblem is generated and solved.
Abstract: A new method for non-linear programming in general and structural optimization in particular is presented. In each step of the iterative process, a strictly convex approximating subproblem is generated and solved. The generation of these subproblems is controlled by so called ‘moving asymptotes’, which may both stabilize and speed up the convergence of the general process.

4,218 citations

Journal ArticleDOI
TL;DR: In this article, the authors analyze and compare the various approaches to this concept in the light of variational bounds on effective properties of composite materials, and derive simple necessary conditions for the possible realization of grey-scale via composites, leading to a physical interpretation of all feasible designs as well as the optimal design.
Abstract: In topology optimization of structures, materials and mechanisms, parametrization of geometry is often performed by a grey-scale density-like interpolation function. In this paper we analyze and compare the various approaches to this concept in the light of variational bounds on effective properties of composite materials. This allows us to derive simple necessary conditions for the possible realization of grey-scale via composites, leading to a physical interpretation of all feasible designs as well as the optimal design. Thus it is shown that the so-called artificial interpolation model in many circumstances actually falls within the framework of microstructurally based models. Single material and multi-material structural design in elasticity as well as in multi-physics problems is discussed.

2,088 citations