A shape optimization method for nonlinear axisymmetric magnetostatics using a coupling of finite and boundary elements

Lukáš, Dalibor; Postava, Kamil; Životský, Ondřej

doi:10.1016/j.matcom.2011.01.015

Cited by 6 publications

(4 citation statements)

References 19 publications

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“…In this context boundary element methods have proved themselves to be an extremely effective tool for the solution of the associated state equation; see, e.g., [8,13,23,25,27]. The main reason is that no domain triangulation has to be computed.…”

Section: Introductionmentioning

confidence: 99%

“…Recently shape optimization for elliptic boundary value problems has become a well-established mathematical and computational tool; see, e.g., [3,15,19,26,30] and the references therein. In this context boundary element methods have proved themselves to be an extremely effective tool for the solution of the associated state equation; see, e.g., [8,13,23,25,27]. The main reason is that no domain triangulation has to be computed.…”

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confidence: 99%

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On the Numerical Solution of a Shape Optimization Problem for the Heat Equation

Harbrecht¹,

Tausch²

2013

SIAM J. Sci. Comput.

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Abstract. The present paper is concerned with the numerical solution of a shape identification problem for the heat equation. The goal is to determine of the shape of a void or an inclusion of zero temperature from measurements of the temperature and the heat flux at the exterior boundary. This nonlinear and ill-posed shape identification problem is reformulated in terms of three different shape optimization problems: (a) minimization of a least-squares energy variational functional, (b) tracking of the Dirichlet data, and (c) tracking of the Neumann data. The states and their adjoint equations are expressed as parabolic boundary integral equations and solved using a Nyström discretization and a space-time fast multipole method for the rapid evaluation of thermal potentials. Special quadrature rules are derived to handle singularities of the kernel and the solution. Numerical experiments are carried out to demonstrate and compare the different formulations. 1. Introduction. Recently shape optimization for elliptic boundary value problems has become a well-established mathematical and computational tool; see, e.g., [3,15,19,26,30] and the references therein. In this context boundary element methods have proved themselves to be an extremely effective tool for the solution of the associated state equation; see, e.g., [8,13,23,25,27]. The main reason is that no domain triangulation has to be computed. Moreover, fast methods to handle the associated dense matrix problems have reached a mature state and can be readily applied within the iterative solution of the minimization problem.In contrast to elliptic shape optimization, the literature on parabolic shape optimization problems is rather limited. There are theoretical results (see, e.g., [5,16,17,29] and the references therein), but the development of efficient numerical methods for shape optimization problems with a parabolic state equation is still in its beginning stages, especially for three-dimensional geometries. With the goal to develop such efficient methods, we consider in the present paper a shape identification problem for the heat equation. Specifically, we detect an inclusion or void of zero temperature inside a solid or liquid body by measurements of the temperature and the transient heat flux at the accessible outer boundary.This problem is severely ill-posed and was discussed in a two-dimensional setting in [1,2], where the uniqueness of the inclusion and the differentiability with respect to its boundary was established. The approach in these papers is to use Newton's method to solve the nonlinear functional equation that maps the shape of the inclusion to the measured data. Thus each step of the iteration involves one forward problem

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Section: Introductionmentioning

confidence: 99%

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confidence: 99%

On the Numerical Solution of a Shape Optimization Problem for the Heat Equation

Harbrecht¹,

Tausch²

2013

SIAM J. Sci. Comput.

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“…For a detailed description and rigorous analysis of this coupling we refer the reader to [8,9]. Note that we only have to discretize the ferromagnetic yoke domain O i .…”

Section: Combined Fem-bem Methodsmentioning

confidence: 99%

Optimization of electromagnet for high-field polar magneto-optical microscopy

Lukáš

Postava

Životský

et al. 2010

Journal of Magnetism and Magnetic Materials

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“…When compared to more popular volume discretization techniques, BEM eliminates interior degrees of freedom and reduces the problem formulation to the boundary, which is particularly efficient in case of unbounded computational domains or in shape optimization [7,12]. Unfortunately, when implementing BEM, we have to deal with two difficulties: an integration of singular kernels and a dense matter of system matrices.…”

Section: Introductionmentioning

confidence: 99%

A parallel fast boundary element method using cyclic graph decompositions

et al. 2015

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We propose a method of a parallel distribution of densely populated matrices arising in boundary element discretizations of partial differential equations. In our method the underlying boundary element mesh consisting of n elements is decomposed into N submeshes. The related N ×N submatrices are assigned to N concurrent processes to be assembled. Additionally we require each process to hold exactly one diagonal submatrix, since its assembling is typically most time consuming when applying fast boundary elements. We obtain a class of such optimal parallel distributions of the submeshes and corresponding submatrices by cyclic decompositions of undirected complete graphs. It results in a method the theoretical complexity of which is O((n/ √ N) log(n/ √ N)) in terms of time for the setup, assembling, matrix action, as well as memory consumption per process. Nevertheless, numerical experiments up to n = 2744832 and N = 273 on a real-world geometry document that the method exhibits superior parallel scalability O((n/N) log n) of the overall time, while the memory consumption scales accordingly to the theoretical estimate.

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A shape optimization method for nonlinear axisymmetric magnetostatics using a coupling of finite and boundary elements

Cited by 6 publications

References 19 publications

On the Numerical Solution of a Shape Optimization Problem for the Heat Equation

On the Numerical Solution of a Shape Optimization Problem for the Heat Equation

Optimization of electromagnet for high-field polar magneto-optical microscopy

A parallel fast boundary element method using cyclic graph decompositions

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