A combined finite element and Bayesian optimization framework for shape optimization in spectral geometry

Domínguez, Sebastián; Nigam, Nilima; Shahriari, Bobak

doi:10.1016/j.camwa.2017.08.044

Cited by 5 publications

(3 citation statements)

References 13 publications

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“…Meanwhile, many numerical experiments have been performed for small values of n (see for instance [2], [8, Ch. 1], [22]) which all suggest the validity of the conjecture.…”

Section: Introductionmentioning

confidence: 60%

On the polygonal Faber-Krahn inequality

Bogosel,

Bucur

2023

Journal De L’École Polytechnique — Mathématiques

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It has been conjectured by Pólya and Szegö seventy years ago that the planar set which minimizes the first eigenvalue of the Dirichlet-Laplace operator among polygons with n sides and fixed area is the regular polygon. Despite its apparent simplicity, this result has only been proved for triangles and quadrilaterals. In this paper we prove that for each n ⩾ 5 the proof of the conjecture can be reduced to a finite number of certified numerical computations. Moreover, the local minimality of the regular polygon can be reduced to a single numerical computation. For n = 5, 6, 7, 8 we perform this computation and certify the numerical approximation by finite elements, up to machine errors.Résumé (Sur l'inégalité Faber-Krahn polygonale). -Il y a soixante-dix ans, Pólya et Szegö ont conjecturé que l'ensemble du plan qui minimise la première valeur propre du laplacien avec conditions de Dirichlet au bord parmi les polygones de n côtés et aire fixée est le polygone régulier. Malgré sa simplicité apparente, cette conjecture a été démontrée seulement pour les triangles et les quadrilatères. Dans cet article, nous démontrons que pour chaque n ⩾ 5 la preuve de la conjecture peut être réduite à un nombre fini de calculs numériques certifiés. En particulier, la minimalité locale du polygone régulier est réduite à un seul calcul certifié. Pour n = 5, 6, 7, 8 nous faisons ce calcul et nous certifions l'approximation par éléments finis, aux erreurs d'arrondi près.

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“…Meanwhile, many numerical experiments have been performed for small values of n (see for instance [2], [8, Ch. 1], [22]) which all suggest the validity of the conjecture.…”

Section: Introductionmentioning

confidence: 60%

On the polygonal Faber-Krahn inequality

Bogosel,

Bucur

2023

Journal De L’École Polytechnique — Mathématiques

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“…Especially, when it comes to expensive function evaluations with incomputable derivatives and unknown convexity properties, Bayesian optimization proves its advantages against other methods, such as L-BFGS or best09 (Riche and Picheny 2021;Diouane et al 2021). Additionally, its ability in balancing exploration and exploitation against each other during the optimization process, makes it an appropriate approach for structural shape optimization problems (Zacchei and Molina 2018;Ghosh et al 2019;Dominguez et al 2017).…”

Section: Bayesian Optimizationmentioning

confidence: 99%

Spline-based shape optimization of large-scale composite leaf spring models using Bayesian strategies with multiple constraints

Winter

Fiebig²,

Franke³

et al. 2022

Struct Multidisc Optim

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The presented paper describes a shape optimization workflow using Bayesian strategies. It is applied to a novel automotive axle system consisting of leaf springs made from glass fiber reinforced plastics (GFRP). Besides the primary objectives of cost and mass reduction, the assembly has to meet multiple technical constraints with respect to various loading conditions. The related large-scale finite element model is fully parameterized by splines, hence the general shape of the guide curve as well as the spring’s height, width and material properties can be altered by the corresponding workflow. For this purpose, a novel method is developed to automatically generate high-quality meshes depending on the geometry of the respective springs. The size and complexity of the model demands the implementation of efficient optimization techniques with a preferably small number of required response function evaluations. Therefore, an existing optimization framework is extended by state-of-the-art Bayesian methods, including different kernel combinations and multiple acquisition function approaches, which are then tested, evaluated and compared. To properly address the use of GFRP as spring material in the objective function, an appropriate cost model is derived. Emerging challenges, such as conflicting targets regarding direct material costs and potential lightweight measures, are considered and investigated. The intermediate steps of the developed optimization procedure are tested on various sample functions and simplified models. The entire workflow is finally applied to the complete model and evaluated. Concluding, ideas and possibilities in improving the optimization process, such as the use of models with varying complexity, are discussed.

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“…Zhang et al [35,36] implemented a Kriging-assisted multiscale topology optimization methodology for inhomogeneous cellular structures. Dominguez et al [37] applied BO to solve shape optimization problems in a non-linear finite element framework. Liu et al [38] also used BO in a finite element framework to design for structural crashworthiness.…”

Section: Applications Of Bomentioning

confidence: 99%

Bayesian topology optimization for efficient design of origami folding structures

Shende

Gillman

Yoo

et al. 2021

Struct Multidisc Optim

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Bayesian optimization (BO) is a popular method for solving optimization problems involving expensive objective functions. Although BO has been applied across various fields, its use in structural optimization area is in its early stages. Origami folding structures provide a complex design space where the use of an efficient optimizer is critical. In this research work for the first time the ability of BO to solve origami-inspired design problems is demonstrated. A Gaussian process (GP) is used as the surrogate model that is trained to mimic the response of the expensive finite element (FE) objective function. The ability of this BO-FE framework to find optimal designs is verified by applying it to two well known origami design problems: chomper and twist chomper.The performance of the proposed approach is compared to traditional gradient-based optimization techniques and genetic algorithm methods in terms of ability to discover designs, computational efficiency and robustness. BO has many user-defined components and parameters, and intuitions for these for structural optimization are currently limited. In this work, the role of hyperparameter tuning and the sensitivity of Bayesian optimization to the quality and size of the initial training set is studied.

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A combined finite element and Bayesian optimization framework for shape optimization in spectral geometry

Cited by 5 publications

References 13 publications

On the polygonal Faber-Krahn inequality

On the polygonal Faber-Krahn inequality

Spline-based shape optimization of large-scale composite leaf spring models using Bayesian strategies with multiple constraints

Bayesian topology optimization for efficient design of origami folding structures

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