Fully and semi-automated shape differentiation in NGSolve

Gangl, Peter; Sturm, Kevin; Neunteufel, Michael; Schöberl, Joachim

doi:10.1007/s00158-020-02742-w

Cited by 22 publications

(9 citation statements)

References 32 publications

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“…Dapogny et al (2018) developed a FreeFem++ code for Navier-Stokes fluid design problems using shape optimization, aiming to either minimize the dissipated energy or achieve a targeted velocity profile. Gangl et al (2021) conducted the shape optimization using the FE software package NGSolve (Schöberl 2014), which can solve a large number of boundary value problems efficiently, considering both unconstrained and PDE-constrained cases. Both semiautomatic and fully automatic approaches for calculating the first-and second-order shape derivatives are presented 2021) introduced an open-source shape optimization toolbox (Fireshape) built upon the FE software Firedrake (Rathgeber et al 2016), which is capable of calculating the shape derivatives automatically.…”

Section: Shape Optimizationmentioning

confidence: 99%

A comprehensive review of educational articles on structural and multidisciplinary optimization

Wang

Zhao

Zhou

et al. 2021

Struct Multidisc Optim

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Ever since the publication of the 99-line topology optimization MATLAB code (top99) by Sigmund in 2001, educational articles have emerged as a popular category of contributions within the structural and multidisciplinary optimization (SMO) community. The number of educational papers in the field of SMO has been growing rapidly in recent years. Some educational contributions have made a tremendous impact on both research and education. For example, top99 (Sigmund in Struct Multidisc Optim 21(2):120-127, 2001) has been downloaded over 13,000 times and cited over 2000 times in Google Scholar. In this paper, we attempt to provide a systematic and comprehensive review of educational articles and codes in SMO, including topology, sizing, and shape optimization and building blocks. We first assess the papers according to the adopted methods, which include density-based, level-set, ground structure, and more. We then provide comparisons and evaluations on the codes from several key aspects, including techniques, efficiency, usability, readability, environment, and compatibility. In addition, we conduct numerical experiments on the reviewed codes using the benchmark cantilever beam example to provide feedback on the overall user experience. With a systematic review and comparison, this paper aims to offer insights on the educational values and practicality for employing these codes. We try to provide not only guidance for beginners to approach various optimization methods, but also a dictionary to direct readers to effectively target the relevant codes and building blocks based on their demands. Finally, based on the findings in this review paper, we provide some perspectives and recommendations for future educational contributions.

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Section: Shape Optimizationmentioning

confidence: 99%

A comprehensive review of educational articles on structural and multidisciplinary optimization

Wang

Zhao

Zhou

et al. 2021

Struct Multidisc Optim

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“…The computation of shape derivatives can be challenging and also error prone due complicated expressions. In the recent publication [23] the fully automated and semi-automated computation of shape derivatives in NGSolve was presented. For instance, to compute the shape derivative (4.15), excluding the area and volume constraint, one can consider for fixed κ and σ the linear form (compare (3.28))…”

Section: Mesh S E T D E F O R M a T I O N ( D I S P L A C E M E N T ...mentioning

confidence: 99%

“…Note that NumPy 3 is required to execute the file. For further details concerning shape optimization in NGSolve we refer to [23]. Listing 1.…”

Section: Declaration Of Interestsmentioning

confidence: 99%

See 1 more Smart Citation

Numerical shape optimization of the Canham-Helfrich-Evans bending energy

Neunteufel¹,

Schöberl²,

Sturm³

2021

Preprint

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In this paper we propose a novel numerical scheme for the Canham-Helfrich-Evans bending energy based on a three-field lifting procedure of the distributional shape operator to an auxiliary mean curvature field. Together with its energetic conjugate scalar stress field as Lagrange multiplier the resulting fourth order problem is circumvented and reduced to a mixed saddle point problem involving only second order differential operators. Further, we derive its analytical first variation (also called first shape derivative), which is valid for arbitrary polynomial order, and discuss how the arising shape derivatives can be computed automatically in the finite element software NGSolve. We finish the paper with several numerical simulations showing the pertinence of the proposed scheme and method.

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“…The finite element software NGSolve allows to define PDEs in weak form in a symbolic way and also supports automated differentiation of expressions, see [16] for applications of these capabilities in the context of shape derivatives. Our implementation is available from [15].…”

mentioning

confidence: 99%

Automated computation of topological derivatives with application to nonlinear elasticity and reaction-diffusion problems

Gangl,

Sturm

2021

Preprint

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Purpose While topological derivatives have proven useful in applications of topology optimisation and inverse problems, their mathematically rigorous derivation remains an ongoing research topic, in particular in the context of nonlinear partial differential equation (PDE) constraints. Design/methodology/approach We present a systematic yet formal approach for the computation of topological derivatives of a large class of PDE-constrained topology optimization problems with respect to arbitrary inclusion shapes. Scalar and vector-valued as well as linear and nonlinear elliptic PDE constraints are considered in two and three space dimensions including a nonlinear elasticity model and nonlinear reaction-diffusion problems. The systematic procedure follows a Lagrangian approach for computing topological derivatives. Findings For problems where the exact formula is known, the numerically computed values show good coincidence. Moreover, by inserting the computed values into the topological asymptotic expansion, we verify that the obtained values satisfy the expected behaviour also for other, previously unknown problems, indicating the correctness of the procedure. Originality/value We present a systematic approach for the computation of topological derivatives that is applicable to a large class of problems. Most notably, our approach covers the topological derivative for a nonlinear elasticity problem, which has not been reported in the literature.

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Fully and semi-automated shape differentiation in NGSolve

Cited by 22 publications

References 32 publications

A comprehensive review of educational articles on structural and multidisciplinary optimization

A comprehensive review of educational articles on structural and multidisciplinary optimization

Numerical shape optimization of the Canham-Helfrich-Evans bending energy

Automated computation of topological derivatives with application to nonlinear elasticity and reaction-diffusion problems

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