A Post-Treatment of the Homogenization Method for Shape Optimization

Pantz, Olivier; Trabelsi, Karim

doi:10.1137/070688900

Cited by 120 publications

(106 citation statements)

References 11 publications

(15 reference statements)

Supporting

Mentioning

101

Contrasting

Order By: Relevance

“…The unit cell with a rectangular hole, used in the topology optimization problem, is simple enough to be represented by just two orthogonal cosine waves [13,14]. The first cosine wave describes the part of the unit cell aligned with y 1 , while the second cosine wave describes the part aligned with y 2 .…”

Section: Projecting a Uniform Micro-structurementioning

confidence: 99%

“…In a very appealing approach Pantz and Trabelsi introduced a method to project the microstructures from the relaxed design space to obtain a solid-void design with finite length-scale [13,14]. The local structure is oriented along the directions of lamination such that a well-connected design is achieved.…”

Section: Introductionmentioning

confidence: 99%

“…In a related study Rumpf and Pazos show that any type of (also spatially varying) unit-cell, represented by a Fourier series, can be projected on a fine scale mesh [15,16]. This paper shall be seen as a simplification and improvement of the approach introduced by Pantz and Trabelsi [13,14]. We simplify the projection approach and introduce procedures for controlling the size and shape of the projected design, such that high-resolution (e.g.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Homogenization‐based topology optimization for high‐resolution manufacturable microstructures

Groen

Sigmund

2017

Numerical Meth Engineering

266

215

View full text Add to dashboard Cite

This paper presents a projection method to obtain high-resolution, manufacturable structures from efficient and coarse-scale, homogenization-based topology optimization results. The presented approach bridges coarse and fine scale, such that the complex periodic micro-structures can be represented by a smooth and continuous lattice on the fine mesh. A heuristic methodology allows control of the projected topology, such that a minimum length-scale on both solid and void features is ensured in the final result. Numerical examples show excellent behavior of the method, where performances of the projected designs are almost equal to the homogenization-based solutions. A significant reduction in computational cost is observed compared to conventional topology optimization approaches.

show abstract

Section: Projecting a Uniform Micro-structurementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Homogenization‐based topology optimization for high‐resolution manufacturable microstructures

Groen

Sigmund

2017

Numerical Meth Engineering

266

215

View full text Add to dashboard Cite

show abstract

“…We observe that for T large, the optimal direction of lamination is close to the direction of lamination associated with the elliptic case. The knowledge of the optimal density and of the lamination allows to construct a minimizing sequence of classical designs (see [14]). with a support arranged as two vertical strips, positive on the left, negative on the right.…”

mentioning

confidence: 99%

Long Time Behavior of a Two-Phase Optimal Design for the Heat Equation

Allaire¹,

Münch²,

Periago³

2010

SIAM J. Control Optim.

View full text Add to dashboard Cite

Abstract. We consider a two-phase isotropic optimal design problem within the context of the transient heat equation. The objective is to minimize the average of the dissipated thermal energy during a fixed time interval [0, T ] . The time-independent material properties are taken as design variables. A full relaxation for this problem was established in [Relaxation of an optimal design problem for the heat equation, JMPA 89 (2008)] by using the homogenization method. In this paper, we study the asymptotic behavior as T goes to infinity of the solutions of the relaxed problem and prove that they converge to an optimal relaxed design of the corresponding two-phase optimization problem for the stationary heat equation. Next, we study necessary optimality conditions for the relaxed optimization problem under the transient heat equation and use those to characterize the micro-structure of the optimal designs, which appears in the form of a sequential laminate of rank at most N , the spatial dimension. An asymptotic analysis of the optimality conditions let us prove that, for T large enough, the order of lamination is in fact of at most N − 1. Several numerical experiments in 2D complete our study.

show abstract

“…A simple approach, using a local mean argument on s, is proposed in [20] to approximate such a sequence. Further, it would be interesting to use the information of the normal of the first-order laminate at each point given by λ β − λ α (we refer to [29] for such analysis in the context of Homogeneisation): iso-values of the vector λ β − λ α are given in Figure 10. We also observe that the energy release rate is reduced but not arbitrarily small in spite of the important degree of freedom contained by the shape of ω.…”

Section: Numerical Experimentsmentioning

confidence: 99%

Relaxation of an optimal design problem in fracture mechanic: the anti-plane case

Münch

Pedregal²

2009

ESAIM: COCV

View full text Add to dashboard Cite

Abstract. In the framework of the linear fracture theory, a commonly-used tool to describe the smooth evolution of a crack embedded in a bounded domain Ω is the so-called energy release rate defined as the variation of the mechanical energy with respect to the crack dimension. Precisely, the well-known Griffith's criterion postulates the evolution of the crack if this rate reaches a critical value. In this work, in the anti-plane scalar case, we consider the shape design problem which consists in optimizing the distribution of two materials with different conductivities in Ω in order to reduce this rate. Since this kind of problem is usually ill-posed, we first derive a relaxation by using the classical non-convex variational method. The computation of the quasi-convex envelope of the cost is performed by using div-curl Young measures, leads to an explicit relaxed formulation of the original problem, and exhibits fine microstructure in the form of first order laminates. Finally, numerical simulations suggest that the optimal distribution permits to reduce significantly the value of the energy release rate.Mathematics Subject Classification. 49J45, 65N30.

show abstract

A Post-Treatment of the Homogenization Method for Shape Optimization

Cited by 120 publications

References 11 publications

Homogenization‐based topology optimization for high‐resolution manufacturable microstructures

Homogenization‐based topology optimization for high‐resolution manufacturable microstructures

Long Time Behavior of a Two-Phase Optimal Design for the Heat Equation

Relaxation of an optimal design problem in fracture mechanic: the anti-plane case

Contact Info

Product

Resources

About