Asymptotic analysis and topological derivative for 3D quasi-linear magnetostatics

In this paper we study the asymptotic behaviour of the quasilinear $\curl$-$\curl$ equation of 3D magnetostatics with respect to a singular perturbation of the differential operator and prove the existence of the topological derivative using a Lagrangian approach. We follow the strategy proposed in our recent previous work ( https://doi.org/10.1051/cocv/2020035 ) where a systematic and concise way for the derivation of topological derivatives for quasi-linear elliptic problems in $H^1$ is introduced. In order to prove the asymptotics for the state equation we make use of an appropriate Helmholtz decomposition. The evaluation of the topological derivative at any spatial point requires the solution of a nonlinear transmission problem. We discuss an efficient way for the numerical evaluation of the topological derivative in the whole design domain using precomputation in an offline stage. This allows us to use the topological derivative for the design optimization of an electrical machine.

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Sensitivity of the Second Order Homogenized Elasticity Tensor to Topological Microstructural Changes

Journal of Elasticity ◽

10.1007/s10659-021-09836-6 ◽

2021 ◽

Author(s):

V. Calisti ◽

A. Lebée ◽

A. A. Novotny ◽

J. Sokolowski

Keyword(s):

Circular Inclusion ◽

Elasticity Tensor ◽

Second Order ◽

Topological Derivative ◽

Microstructural Changes ◽

Topological Derivatives ◽

Spatial Dimensions ◽

Elasticity Tensors ◽

Derivatives Of ◽

Optimization Of Microstructures

AbstractThe multiscale elasticity model of solids with singular geometrical perturbations of microstructure is considered for the purposes, e.g., of optimum design. The homogenized linear elasticity tensors of first and second orders are considered in the framework of periodic Sobolev spaces. In particular, the sensitivity analysis of second order homogenized elasticity tensor to topological microstructural changes is performed. The derivation of the proposed sensitivities relies on the concept of topological derivative applied within a multiscale constitutive model. The microstructure is topologically perturbed by the nucleation of a small circular inclusion that allows for deriving the sensitivity in its closed form with the help of appropriate adjoint states. The resulting topological derivative is given by a sixth order tensor field over the microstructural domain, which measures how the second order homogenized elasticity tensor changes when a small circular inclusion is introduced at the microscopic level. As a result, the topological derivatives of functionals for multiscale models can be obtained and used in numerical methods of shape and topology optimization of microstructures, including synthesis and optimal design of metamaterials by taking into account the second order mechanical effects. The analysis is performed in two spatial dimensions however the results are valid in three spatial dimensions as well.

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Two perturbation results for nondegenerate solutions of some semilinear Dirichlet problems

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500024963 ◽

1989 ◽

Vol 111 (1-2) ◽

pp. 1-11 ◽

Cited By ~ 1

Author(s):

Mario Michele Coclite

Keyword(s):

Differential Operator ◽

Singular Perturbation ◽

Nonlinear Term ◽

Dirichlet Problems ◽

The Stability

SynopsisThe stability of nondegenerate solutions of some semilinear Dirichlet problems is studied. Two specific situations are considered: firstly, a singular perturbation of the differential operator; secondly, a perturbation of the nonlinear term using a term which also depends on the gradient of the solution.

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Topological Derivative Based Imaging of Small Inclusions in the Time-Harmonic Regime

Mathematical Methods in Elasticity Imaging ◽

10.23943/princeton/9780691165318.003.0007 ◽

2015 ◽

Author(s):

Habib Ammari ◽

Elie Bretin ◽

Josselin Garnier ◽

Hyeonbae Kang ◽

Hyundae Lee ◽

...

Keyword(s):

Sensitivity Analysis ◽

Shear Waves ◽

Diffraction Limit ◽

Topological Derivative ◽

Helmholtz Decomposition ◽

Wave Speeds ◽

Time Harmonic ◽

Optimal Resolution ◽

Resolution Imaging ◽

Measurement Noises

This chapter introduces a topological derivative (TD) based imaging framework for detecting small inclusions in the time-harmonic regime. Based on a weighted Helmholtz decomposition of the TD based imaging functional, optimal resolution imaging is achieved. Its stability properties with respect to both medium and measurement noises are investigated. The chapter first considers the TD imaging functional resulting from the expansion of the filtered quadratic misfit with respect to the size of the inclusion. It shows that the imaging functional may not attain its maximum at the location of the inclusion. Moreover, the resolution of the image is below the diffraction limit. Both phenomena are due to the coupling of pressure and shear waves propagating with different wave speeds and polarization directions. The chapter concludes by presenting the sensitivity analysis of a modified imaging functional based on the weighted Helmholtz decomposition of the TD.

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Topological derivatives via one-sided derivative of parametrized minima and minimax

Engineering Computations ◽

10.1108/ec-06-2021-0318 ◽

2021 ◽

Vol ahead-of-print (ahead-of-print) ◽

Author(s):

Michel C. Delfour

Keyword(s):

Directional Derivative ◽

Topological Derivative ◽

Dimensional Euclidean Space ◽

Objective Functions ◽

State Equations ◽

Content Type ◽

Adjoint State ◽

Topological Derivatives ◽

Wide Range ◽

The One

PurposeThe object of the paper is to illustrate how to obtain the topological derivative as a semidifferential in a general and practical mathematical setting for d-dimensional perturbations of a bounded open domain in the n-dimensional Euclidean space.Design/methodology/approachThe underlying methodology uses mathematical notions and powerful tools with ready to check assumptions and ready to use formulas via theorems on the one-sided derivative of parametrized minima and minimax.FindingsThe theory and the examples indicate that the methodology applies to a wide range of problems: (1) compliance and (2) state constrained objective functions where the coupled state/adjoint state equations appear without a posteriori substitution of the adjoint state.Research limitations/implicationsDirect approach that considerably simplifies the analysis and computations.Originality/valueIt was known that the shape derivative was a differential. But the topological derivative is only a semidifferential, that is, a one-sided directional derivative, which is not linear with respect to the direction, and the directions are d-dimensional bounded measures.

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An introduction to the topological derivative

Engineering Computations ◽

10.1108/ec-07-2021-0433 ◽

2021 ◽

Vol ahead-of-print (ahead-of-print) ◽

Author(s):

Samuel Amstutz

Keyword(s):

Simple Model ◽

Design Methodology ◽

Topological Derivative ◽

Modern Approach ◽

Content Type ◽

Adjoint State ◽

Topological Derivatives ◽

State Technique ◽

Didactic Presentation

PurposeThis paper provides a self-contained introduction to the mathematical aspects of the topological derivative.Design/methodology/approachFull justifications are given on simple model problems following a modern approach based on the averaged adjoint state technique. Extensions are discussed in relation with the literature on the field.FindingsClosed expressions of topological derivatives are obtained and commented.Originality/valueSeveral cases are covered in a unified and didactic presentation. Some elements of proof are novel.

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A simplified derivation technique of topological derivatives for quasi-linear transmission problems

ESAIM Control Optimisation and Calculus of Variations ◽

10.1051/cocv/2020035 ◽

2020 ◽

Vol 26 ◽

pp. 106

Author(s):

Peter Gangl ◽

Kevin Sturm

Keyword(s):

Differential Operators ◽

Optimization Problems ◽

Fundamental Solutions ◽

Linear Constraints ◽

Topological Derivative ◽

Rigorous Derivation ◽

Transmission Problems ◽

Lagrangian Framework ◽

Topological Derivatives ◽

Systematic Derivation

In this paper we perform the rigorous derivation of the topological derivative for optimization problems constrained by a class of quasi-linear elliptic transmission problems. In the case of quasi-linear constraints, techniques using fundamental solutions of the differential operators cannot be applied to show convergence of the variation of the states. Some authors succeeded showing this convergence with the help of technical computations under additional requirements on the problem. Our main objective is to simplify and extend these previous results by using a Lagrangian framework and a projection trick. Besides these generalisations the purpose of this manuscript is to present a systematic derivation approach for topological derivatives.

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