A quadratically convergent unstructured remeshing strategy for shape optimization

Wilke, Daniel N.; Kok, Schalk; Groenwold, Albert A.

doi:10.1002/nme.1430

Cited by 22 publications

(40 citation statements)

References 13 publications

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“…Our mesh generator solves for the equilibrium position of a truss structure , which doubles as the FE mesh, using the ideal element length field

h^{{k}} MathClass-punc, k MathClass-rel= 1 MathClass-punc, 2 MathClass-punc, 3 MathClass-punc, MathClass-op\dots

as the unloaded truss lengths, as opposed to directly optimizing the mesh according to some optimality criterion . It has been demonstrated that this approach generates ‘good’ meshes.…”

Section: Adaptive Mesh Generatormentioning

confidence: 99%

“…We start with an initial uniform mesh at k = 0, using a uniform ideal element length field h {0} . After each analysis, we compute the ideal element length field

{\overset{h}{true}}^{{k}}

using the refinement strategy described in Section 4.1.…”

Section: Adaptive Mesh Generatormentioning

confidence: 99%

“…Then, the sensitivity of the displacement u F w.r.t. the control variables x is obtained by computing The computation of is obtained by direct differentiation of the FE equilibrium equations K u = f , given by

\frac{normald bold-italicK}{normald scriptX} bold-italicu MathClass-bin+ bold-italicK \frac{normald bold-italicu}{normald scriptX} MathClass-rel= \frac{normald bold-italicf}{normald scriptX} MathClass-punc.

For the constant external loads f we restrict ourselves to in this study,

\frac{normald bold-italicf}{normald scriptX} MathClass-rel= 0

and reduces to

bold-italicK \frac{normald bold-italicu}{normald scriptX} MathClass-rel= MathClass-bin- \frac{normald bold-italicK}{normald scriptX} bold-italicu MathClass-punc.

We compute

\frac{normald bold-italicK}{normald scriptX}

by direct differentiation of the analytical stiffness matrices for the linear strain triangle elements , which allows to solve for

\frac{normald bold-italicu}{normald scriptX}

, to then obtain .…”

Section: Sensitivity Analysismentioning

confidence: 99%

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Relaxed error control in shape optimization that utilizes remeshing

Wilke

Kok

Groenwold

2013

Numerical Meth Engineering

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SUMMARYShape optimization strategies based on error indicators usually require strict error control for every computed design during the optimization run. The strict error control serves two purposes. Firstly, it allows for the accurate computation of the structural response used to define the shape optimization problem itself. Secondly, it reduces the discretization error, which in turn reduces the size of the step discontinuities in the objective function that result from remeshing in the first place. These discontinuities may trap conventional optimization algorithms, which rely on both function and gradient evaluations, in local minima. This has the drawback that multiple analyses and error computations are often required per design to control the error.In this study we propose a methodology that relaxes the requirements for strict error control for each design. Instead, we rather control the error as the iterations progress. Our approach only requires a single analysis and error computation per design. Consequently, large discontinuities may initially be accommodated; their intensities reduce as the iterations progress. To circumvent the difficulties associated with local minima due to remeshing, we rely on gradient-only optimization algorithms, which have previously been shown to be able to robustly overcome these discontinuities.

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“…Our mesh generator solves for the equilibrium position of a truss structure , which doubles as the FE mesh, using the ideal element length field

h^{{k}} MathClass-punc, k MathClass-rel= 1 MathClass-punc, 2 MathClass-punc, 3 MathClass-punc, MathClass-op\dots

as the unloaded truss lengths, as opposed to directly optimizing the mesh according to some optimality criterion . It has been demonstrated that this approach generates ‘good’ meshes.…”

Section: Adaptive Mesh Generatormentioning

confidence: 99%

“…We start with an initial uniform mesh at k = 0, using a uniform ideal element length field h {0} . After each analysis, we compute the ideal element length field

{\overset{h}{true}}^{{k}}

using the refinement strategy described in Section 4.1.…”

Section: Adaptive Mesh Generatormentioning

confidence: 99%

\frac{normald bold-italicK}{normald scriptX} bold-italicu MathClass-bin+ bold-italicK \frac{normald bold-italicu}{normald scriptX} MathClass-rel= \frac{normald bold-italicf}{normald scriptX} MathClass-punc.

For the constant external loads f we restrict ourselves to in this study,

\frac{normald bold-italicf}{normald scriptX} MathClass-rel= 0

and reduces to

bold-italicK \frac{normald bold-italicu}{normald scriptX} MathClass-rel= MathClass-bin- \frac{normald bold-italicK}{normald scriptX} bold-italicu MathClass-punc.

We compute

\frac{normald bold-italicK}{normald scriptX}

by direct differentiation of the analytical stiffness matrices for the linear strain triangle elements , which allows to solve for

\frac{normald bold-italicu}{normald scriptX}

, to then obtain .…”

Section: Sensitivity Analysismentioning

confidence: 99%

See 1 more Smart Citation

Relaxed error control in shape optimization that utilizes remeshing

Wilke

Kok

Groenwold

2013

Numerical Meth Engineering

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“…For the material properties we use Young's modulus E = 200GPa and Poisson's ratio ν = 0.3. The meshes required for the finite element analyses are generated using a quadratic convergent remeshing strategy [24] with ideal element length h 0 = 1mm. To illustrate the discontinuous nature of the objective function, the two control variables x 8 and x 9 are perturbed around the reference configuration depicted in …”

Section: Multivariate Example Problem: Shape Optimizationmentioning

confidence: 99%

Gradient-only approaches to avoid spurious local minima in unconstrained optimization

et al. 2011

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We reflect on some theoretical aspects of gradient-only optimization for the unconstrained optimization of objective functions containing nonphysical step or jump discontinuities. This kind of discontinuity arises when the optimization problem is based on the solutions of systems of partial differential equations, in combination with variable discretization techniques (e.g. remeshing in spatial domains, and / or variable time stepping in temporal domains). These discontinuities, which may cause local minima, are artifacts of the numerical strategies used and should not influence the solution to the optimization problem. Although the discontinuities imply that the gradient field is not defined everywhere, the gradient field associated with the computational scheme can nevertheless be computed everywhere; this field is denoted the associated gradient field. We demonstrate that it is possible to overcome attraction to the local minima if only associated gradient information is used. Various gradient-only algorithmic options are discussed.

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“…For this, design element concept (Imam, 1982), 0020-7683/$ -see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijsolstr.2010.03.004 adaptive meshing (Bennet and Botkin, 1984) and remeshing (Wilke et al, 2006) techniques were used. A comprehensive review of the early development of shape optimization can be found in the survey literatures (Braibant and Fleury, 1984;Haftka and Grandhi, 1986;Ding, 1986;Hsu, 1994).…”

Section: Introductionmentioning

confidence: 99%

Shape optimization and its extension to topological design based on isogeometric analysis

Seo

Kim

Youn

2010

International Journal of Solids and Structures

121

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a b s t r a c tIn most of structural optimization approaches, finite element method (FEM) has been employed for structural response analysis and sensitivity calculation. However, the approaches generally suffer certain drawbacks. In shape optimization, cumbersome parameterization of design domain is required and time consuming remeshing task is also necessary. In topology optimization, design results are generally restricted on the initial design space and additional post-processing is required for communication with CAD systems. These drawbacks are due to the use of different mathematical languages in design or geometric modeling and numerical analysis: spline basis functions are used in design and geometric modeling whereas Lagrangian and Hermitian polynomials in analysis. Isogeometric analysis is a very attractive and promising alternative to overcome the limitations resulting from the use of the conventional FEM in structural optimization. In isogeometric analysis, the same spline information such as control points and spline basis functions which represent geometries in CAD systems are also used in numerical analysis. Such unification of the mathematical languages in CAD, analysis and design optimization can resolve the issues mentioned above. In this work, structural shape optimization using isogeometric analysis is studied on 2D and shell problems. The proposed framework is extended to topology optimization using trimming techniques. New inner fronts are introduced by trimming spline curves in topology optimization. Trimmed surface analysis which was recently proposed to analyze arbitrary complex topology problems is employed for topology optimization. Some benchmarking problems in shape and topology optimization are treated using the proposed approach.

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A quadratically convergent unstructured remeshing strategy for shape optimization

Cited by 22 publications

References 13 publications

Relaxed error control in shape optimization that utilizes remeshing

Relaxed error control in shape optimization that utilizes remeshing

Gradient-only approaches to avoid spurious local minima in unconstrained optimization

Shape optimization and its extension to topological design based on isogeometric analysis

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