A level set method for structural topology optimization
03 Jan 2003-Computer Methods in Applied Mechanics and Engineering (North-Holland)-Vol. 192, Iss: 1, pp 227-246
TL;DR: A new approach to structural topology optimization that represents the structural boundary by a level set model that is embedded in a scalar function of a higher dimension that demonstrates outstanding flexibility of handling topological changes, fidelity of boundary representation and degree of automation.
About: This article is published in Computer Methods in Applied Mechanics and Engineering.The article was published on 2003-01-03. It has received 2404 citations till now. The article focuses on the topics: Topology optimization & Topological derivative.
Citations
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TL;DR: A new numerical method based on a combination of the classical shape derivative and of the level-set method for front propagation, which can easily handle topology changes and is strongly dependent on the initial guess.
2,176 citations
Cites methods from "A level set method for structural t..."
...Keywords: Shape optimization; Topology optimization; Sensitivity analysis; Shape derivative; Level-set...
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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: Future directions such as the "print-it-all" paradigm, that have the potential to re-imagine current research and spawn completely new avenues for exploration are pointed out.
Abstract: Additive manufacturing (AM) is poised to bring about a revolution in the way products are designed, manufactured, and distributed to end users. This technology has gained significant academic as well as industry interest due to its ability to create complex geometries with customizable material properties. AM has also inspired the development of the maker movement by democratizing design and manufacturing. Due to the rapid proliferation of a wide variety of technologies associated with AM, there is a lack of a comprehensive set of design principles, manufacturing guidelines, and standardization of best practices. These challenges are compounded by the fact that advancements in multiple technologies (for example materials processing, topology optimization) generate a "positive feedback loop" effect in advancing AM. In order to advance research interest and investment in AM technologies, some fundamental questions and trends about the dependencies existing in these avenues need highlighting. The goal of our review paper is to organize this body of knowledge surrounding AM, and present current barriers, findings, and future trends significantly to the researchers. We also discuss fundamental attributes of AM processes, evolution of the AM industry, and the affordances enabled by the emergence of AM in a variety of areas such as geometry processing, material design, and education. We conclude our paper by pointing out future directions such as the "print-it-all" paradigm, that have the potential to re-imagine current research and spawn completely new avenues for exploration. The fundamental attributes and challenges/barriers of Additive Manufacturing (AM).The evolution of research on AM with a focus on engineering capabilities.The affordances enabled by AM such as geometry, material and tools design.The developments in industry, intellectual property, and education-related aspects.The important future trends of AM technologies.
1,792 citations
TL;DR: The state-of-the-art of topological design and manufacturing processes of various types of porous metals, in particular for titanium alloys, biodegradable metals and shape memory alloys are reviewed.
1,393 citations
Cites methods from "A level set method for structural t..."
...The level-set method provides an effective technique to represent smooth boundaries nd to control topology changes [111]....
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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 methods from "A level set method for structural t..."
...…and integral filtering (Poulsen, 2003); and 3) other methods like wavelet parameterizations (Kim and Yoon, 2000; Poulsen, 2002), phase-field approaches (Bourdin and Chambolle, 2003; Wang and Zhou, 2004) and level-set methods (Sethian and Wiegmann, 2000; Wang et al, 2003; Allaire et al, 2004)1....
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References
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TL;DR: The PSC algorithm as mentioned in this paper approximates the Hamilton-Jacobi equations with parabolic right-hand-sides by using techniques from the hyperbolic conservation laws, which can be used also for more general surface motion problems.
13,020 citations
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
3,822 citations
"A level set method for structural t..." refers background or methods or result in this paper
...The Courant–Friedrichs–Lewy condition requires Dt to satisfy Dtmax jVnijk j6Ds; ð42Þ where Ds stands for the minimum grid space among the three dimensions [18]....
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...This optimization procedure is derived from the fundamentals of curve and surface evolution of the level set methods [18] in terms of evolution of the level set surfaces described by Eq....
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...Further, the models can incorporate a large number of degrees of freedom and a number of numerical techniques have been developed [18] to make the initial value problem of Eq....
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...There is no need to re-parameterize the model as it undergoes significant changes in shape, in contrast to any conventional boundary shape design [18]....
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...It is often described as a rectangular sampling on a rectilinear grid of x over D [18]....
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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