A consistent relaxation of optimal design problems for coupling shape and topological derivatives

Amstutz, Samuel; Dapogny, Charles; Ferrer, A.

doi:10.1007/s00211-018-0964-4

Cited by 14 publications

(24 citation statements)

References 40 publications

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“…Indeed, the measures of sensitivity of the optimized function with respect to the domain, and the update of the design between iterations are totally unrelated. • This algorithm has paved the way to recent variants, making the connection between density-based methods and shape and topological sensitivity analyses; see [31,99] for further details.…”

Section: Phase Field Methodsmentioning

confidence: 99%

Shape and topology optimization

Allaire

Dapogny

Jouve

2021

Handbook of Numerical Analysis

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This chapter is an introduction to shape and topology optimization, with a particular emphasis on the method of Hadamard for appraising the sensitivity of quantities of interest with respect to the domain, and on the level set method for the numerical representation of shapes and their evolutions. At the theoretical level, the method of Hadamard considers variations of a shape as "small" deformations of its boundary; this results in a mathematically convenient and versatile notion of differentiation with respect to the domain, which has historically often been associated with "body-fitted" geometric optimization methods. At the numerical level, the level set method features an implicit description of the shape, which arises as the negative subdomain of an auxiliary "level set function". This type of representation is well-known to be very efficient when it comes to describing dramatic evolutions of domains (including topological changes). The combination of these two ingredients is an ideal approach for optimizing both the geometry and the topology of shapes, and two related implementation frameworks are presented. The first and oldest one is a Eulerian shape capturing method, using a fixed mesh of a working domain in which the optimal shape is sought. The second and newest one is a Lagrangian shape tracking method, where the shape is exactly meshed at each iteration of the optimization process. In both cases, the level set algorithm is instrumental in updating the shapes, allowing for dramatic deformations between the iterations of the process, and even for topological changes. Most of our applicative examples stem from structural mechanics although some other physical contexts are briefly exemplified. Other topology optimization methods, like density-based algorithms or phase-field methods are also presented, at a lesser level of details, for comparison purposes.

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Section: Phase Field Methodsmentioning

confidence: 99%

Shape and topology optimization

Allaire

Dapogny

Jouve

2021

Handbook of Numerical Analysis

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“…In a first step, for a given applied macroscopic stress σ, a given volume fraction ρ and a given semi-axis ratio ξ, the smoothing exponent q is optimized for minimizing the maximum local stress by solving the optimization problem (33). Note that, since the macroscopic stress is aligned with the microstructure and its amplitude is irrelevant in linearized elasticity, this stress is parametrized by a single scalar parameter φ in (35). The optimal smoothing exponent q * = q * (ξ, ρ, φ) thus depends on three quantities (ξ, ρ, φ).…”

Section: Optimal Micro-structure For Stress Minimizationmentioning

confidence: 99%

“…The optimal exponent q * (ξ, ρ, σ) clearly depends not only on the geometrical parameters (ξ, ρ) but also on the applied macroscopic stress σ. Recall that σ is parametrized by a stress angle φ in (35). In this work, one of the goals is precisely to obtain a parametrized optimal geometry independent of the applied stress.…”

Section: Averaging With Respect To Stress Valuesmentioning

confidence: 99%

Stress minimization for lattice structures. Part I: Micro-structure design

Ferrer

Geoffroy-Donders

Allaire

2021

Phil. Trans. R. Soc. A.

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Lattice structures are periodic porous bodies which are becoming popular since they are a good compromise between rigidity and weight and can be built by additive manufacturing techniques. Their optimization has recently attracted some attention, based on the homogenization method, mostly for compliance minimization. The goal of our two-part work is to extend lattice optimization to stress minimization problems two-dimensionally. The present first part is devoted to the choice of a parametrized periodicity cell that will be used for structural optimization in the second part of our work. In order to avoid stress concentration, we propose a square cell microstructure with a super-ellipsoidal hole instead of the standard rectangular hole often used for compliance minimization. This type of cell is parametrized two-dimensionally by one orientation angle, two semi-axis and a corner smoothing parameter. We first analyse their influence on the stress amplification factor by performing some numerical experiments. Second, we compute the optimal corner smoothing parameter for each possible microstructure and macroscopic stress. Then, we average (with specific weights) the optimal smoothing exponent with respect to the macroscopic stress. Finally, to validate the results, we compare our optimal super-ellipsoidal hole with the Vigdergauz microstructure which is known to be optimal for stress minimization in some special cases. This article is part of the theme issue ‘Topics in mathematical design of complex materials’.

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“…The domain is equipped with a structured triangular mesh of 20 000 elements from a Cartesian grid by splitting each cell into four triangles. The density function is approximated with

{double-struckP}_{1}

Lagrangian finite element functions and the following filter is applied to the density function:

ρ_{k} = \frac{\underset{i \in S_{k}}{false\sum} \int_{normalΩ} N_{i} N_{j} ρ_{j}}{\underset{i \in S_{k}}{false\sum} \int_{normalΩ} N_{i}},

where ρ k is the density value in element k and S k is the set of nodes of element k , see the work of Amstutz et al for further details. This filter is defined in a discrete sense, in terms of the shape functions.…”

Section: Comparison Between Simp and Simp‐allmentioning

confidence: 99%

“…where k is the density value in element k and S k is the set of nodes of element k, see the work of Amstutz et al 22 for further details. This filter is defined in a discrete sense, in terms of the shape functions.…”

Section: Numerical Examples Comparisonmentioning

confidence: 99%

SIMP‐ALL: A generalized SIMP method based on the topological derivative concept

Ferrer

2019

Numerical Meth Engineering

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Summary Topology optimization has emerged in the last years as a promising research field with a wide range of applications. One of the most successful approaches, the SIMP method, is based on regularizing the problem and proposing a penalization interpolation function. In this work, we propose an alternative interpolation function, the SIMP‐ALL method that is based on the topological derivative concept. First, we show the strong relation in plane linear elasticity between the Hashin‐Shtrikman (H‐S) bounds and the topological derivative, providing a new interpretation of the last one. Then, we show that the SIMP‐ALL interpolation remains always in between the H‐S bounds regardless the materials to be interpolated. This result allows us to interpret intermediate values as real microstructures. Finally, we verify numerically this result and we show the convenience of the proposed SIMP‐ALL interpolation for obtaining auto‐penalized optimal design in a wider range of cases. A MATLAB code of the SIMP‐ALL interpolation function is also provided.

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A consistent relaxation of optimal design problems for coupling shape and topological derivatives

Cited by 14 publications

References 40 publications

Shape and topology optimization

Shape and topology optimization

Stress minimization for lattice structures. Part I: Micro-structure design

SIMP‐ALL: A generalized SIMP method based on the topological derivative concept

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