Topological sensitivity analysis revisited for time-harmonic wave scattering problems. Part II: recursive computations by the boundary integral equation method

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Frédérique Le Louër ◽  
María-Luisa Rapún
Keyword(s):  
Acoustic Waves ◽  
Boundary Integral Equations ◽  
Order Term ◽  
Elliptic Boundary ◽  
Harmonic Wave ◽  
Boundary Integral ◽  
Neumann Condition ◽  
Content Type ◽  
Time Harmonic ◽  
Topological Gradient

PurposeThe purpose of this paper is to revisit the recursive computation of closed-form expressions for the topological derivative of shape functionals in the context of time-harmonic acoustic waves scattering by sound-soft (Dirichlet condition), sound-hard (Neumann condition) and isotropic inclusions (transmission conditions).Design/methodology/approachThe elliptic boundary value problems in the singularly perturbed domains are equivalently reduced to couples of boundary integral equations with unknown densities given by boundary traces. In the case of circular or spherical holes, the spectral Fourier and Mie series expansions of the potential operators are used to derive the first-order term in the asymptotic expansion of the boundary traces for the solution to the two- and three-dimensional perturbed problems.FindingsAs the shape gradients of shape functionals are expressed in terms of boundary integrals involving the boundary traces of the state and the associated adjoint field, then the topological gradient formulae follow readily.Originality/valueThe authors exhibit singular perturbation asymptotics that can be reused in the derivation of the topological gradient function in the iterated numerical solution of any shape optimization or imaging problem relying on time-harmonic acoustic waves propagation. When coupled with converging Gauss−Newton iterations for the search of optimal boundary parametrizations, it generates fully automatic algorithms.

Topological sensitivity analysis revisited for time-harmonic wave scattering problems. Part I: the free space case

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Frédérique Le Louër ◽  
María-Luisa Rapún
Keyword(s):  
Free Space ◽  
Order Term ◽  
Harmonic Wave ◽  
Dirichlet Condition ◽  
Point Sources ◽  
Neumann Condition ◽  
Scattering Problems ◽  
Content Type ◽  
Time Harmonic ◽  
Topological Gradient

PurposeIn this paper, the authors revisit the computation of closed-form expressions of the topological indicator function for a one step imaging algorithm of two- and three-dimensional sound-soft (Dirichlet condition), sound-hard (Neumann condition) and isotropic inclusions (transmission conditions) in the free space.Design/methodology/approachFrom the addition theorem for translated harmonics, explicit expressions of the scattered waves by infinitesimal circular (and spherical) holes subject to an incident plane wave or a compactly supported distribution of point sources are available. Then the authors derive the first-order term in the asymptotic expansion of the Dirichlet and Neumann traces and their surface derivatives on the boundary of the singular medium perturbation.FindingsAs the shape gradient of shape functionals are expressed in terms of boundary integrals involving the boundary traces of the state and the associated adjoint field, then the topological gradient formulae follow readily.Originality/valueThe authors exhibit singular perturbation asymptotics that can be reused in the derivation of the topological gradient function that generates initial guesses in the iterated numerical solution of any shape optimization problem or imaging problems relying on time-harmonic acoustic wave propagation.

scholarly journals Integral equation methods in inverse obstacle scattering

2000 ◽  
Vol 42 (1) ◽  
pp. 65-78 ◽  
Author(s):  
Rainer Kress
Keyword(s):  
Boundary Integral Equations ◽  
Plane Waves ◽  
Research Work ◽  
Two Dimensions ◽  
Boundary Integral ◽  
Field Pattern ◽  
Integral Equation Methods ◽  
Time Harmonic ◽  
Domain With A Corner ◽  
Far Field Pattern

AbstractIn this survey we consider a regularized Newton method for the approximate solution of the inverse problem to determine the shape of an obstacle from a knowledge of the far field pattern for the scattering of time-harmonic acoustic or electromagnetic plane waves. Our analysis is in two dimensions and the numerical scheme is based on the solution of boundary integral equations by a Nyström method. We include an example of the reconstruction of a planar domain with a corner both to illustrate the feasibility of the use of radial basis functions for the reconstruction of boundary curves with local features and to connect the presentation to some of the research work of Professor David Elliott.

Dynamic Crack Analysis in Layered Piezoelectric Composites under Time-Harmonic Loading

2013 ◽  
Vol 577-578 ◽  
pp. 449-452
Author(s):  
Michael Wünsche ◽  
Felipe García-Sánchez ◽  
Chuan Zeng Zhang ◽  
Andrés Sáez
Keyword(s):  
Frequency Domain ◽  
Boundary Integral Equations ◽  
Piezoelectric Materials ◽  
Iterative Solution ◽  
Boundary Integral ◽  
Dynamic Crack ◽  
Crack Face ◽  
Piezoelectric Composites ◽  
Crack Analysis ◽  
Time Harmonic

In this Paper, Time-Harmonic Dynamic Crack Analysis in Two-Dimensional (2D), Layered and Linear Piezoelectric Composites is Presented. A Frequency-Domain Symmetric Galerkin Boundary Element Method (SGBEM) is Developed for this Purpose. the Piecewise Homogeneous Sub-Layers of the Piezoelectric Composites are Modeled by the Multi-Domain BEM Formulation. the Frequency-Domain Dynamic Fundamental Solutions for Linear Piezoelectric Materials are Applied in the Present BEM. the Boundary Integral Equations are Solved Numerically by a Galerkin-Method Using Quadratic Elements. an Iterative Solution Algorithm is Implemented to Consider the Non-Linear Semi-Permeable Electrical Crack-Face Boundary Conditions. Numerical Examples will be Presented and Discussed to Show the Influences of the Location and Size of the Crack, the Material Combination of the Sub-Layers, the Piezoelectric Effect and the Time-Harmonic Dynamic Loading on the Dynamic Intensity Factors.

scholarly journals Simulation of electrical machines: a FEM-BEM coupling scheme

2017 ◽  
Vol 36 (5) ◽  
pp. 1540-1551 ◽  
Author(s):  
Lars Kielhorn ◽  
Thomas Rüberg ◽  
Jürgen Zechner
Keyword(s):  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Boundary Element Methods ◽  
Electrical Machines ◽  
Special Treatment ◽  
Coupling Scheme ◽  
Computational Domain ◽  
Efficient Treatment ◽  
Content Type ◽  
Moving Parts

Purpose Electrical machines commonly consist of moving and stationary parts. The field simulation of such devices can be demanding if the underlying numerical scheme is solely based on a domain discretization, such as in the case of the finite element method (FEM). This paper aims to present a coupling scheme based on FEM together with boundary element methods (BEMs) that neither hinges on re-meshing techniques nor deals with a special treatment of sliding interfaces. While the numerics are certainly more involved, the reward is obvious: the modeling costs decrease and the application engineer is provided with an easy-to-use, versatile and accurate simulation tool. Design/methodology/approach The authors present the implementation of a FEM-BEM coupling scheme in which the unbounded air region is handled by the BEM, while only the solid parts are discretized by the FEM. The BEM is a convenient tool to tackle unbounded exterior domains, as it is based on the discretization of boundary integral equations (BIEs) that are defined only on the surface of the computational domain. Hence, no meshing is required for the air region. Further, the BIEs fulfill the decay and radiation conditions of the electromagnetic fields such that no additional modeling errors occur. Findings This work presents an implementation of a FEM-BEM coupling scheme for electromagnetic field simulations. The coupling eliminates problems that are inherent to a pure FEM approach. In detail, the benefits of the FEM-BEM scheme are: the decay conditions are fulfilled exactly, no meshing of parts of the exterior air region is necessary and, most importantly, the handling of moving parts is incorporated in an intriguingly simple manner. The FEM-BEM formulation in conjunction with a state-of-the-art preconditioner demonstrates its potency. The numerical tests not only reveal an accurate convergence behavior but also prove the algorithm to be suitable for industrial applications. Originality/value The presented FEM-BEM scheme is a mathematically sound and robust implementation of a theoretical work presented a decade ago. For the application within an industrial context, the original work has been extended by higher-order schemes, periodic boundary conditions and an efficient treatment of moving parts. While not intended to be used under all circumstances, it represents a powerful tool in case that high accuracies together with simple mesh-handling facilities are required.

Micromechanical Dynamic Influence of Rigid Disk-Shaped Inclusion on Neighboring Crack in 3D Elastic Matrix

2007 ◽  
Vol 567-568 ◽  
pp. 133-136 ◽  
Author(s):  
Victor V. Mykhas'kiv ◽  
O. Khay ◽  
Jan Sladek ◽  
Vladimir Sladek ◽  
Chuan Zeng Zhang
Keyword(s):  
Boundary Integral Equations ◽  
Crack Opening ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Elastic Matrix ◽  
Opening Displacement ◽  
Rigid Disk ◽  
Time Harmonic ◽  
Penny Shaped Crack ◽  
Shaped Inclusion

A 3D time-harmonic problem for an infinite elastic matrix with an arbitrarily located interacting rigid disk-shaped inclusion and a penny-shaped crack is analyzed by the boundary integral equation method. Perfect bonding between the matrix and the moving inclusion is assumed. The crack faces are subjected to time-harmonic loading. The boundary integral equations (BIEs) obtained are solved numerically by the implementation of regularization and discretization procedures. Numerical calculations are carried out for a crack under tensile loading of constant amplitude, where an interacting inclusion is perpendicular to the crack and has the same radius. Both the normal crack-opening-displacement and the mode-I stress intensity factor are investigated for different wave numbers and distances between the crack and the inclusion.

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