Analytical and numerical methods in shape optimization

Harbrecht, Helmut

doi:10.1002/mma.1008

Cited by 18 publications

(18 citation statements)

References 30 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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“…We use the space-time cylinder as reference domain, which means that we need to introduce a finite element mesh the unit circle. This mesh then gets mapped onto the spatial domain Ω t described by the parametrization for every point of time t, similarly as in [22]. Then, standard piecewise linear finite elements can be used to solve the partial differential equation for every time step, when mapping the weak formulation back to the reference domain.…”

Section: Using Lemma 36 Then Givesmentioning

confidence: 99%

On the Reformulation of the Classical Stefan Problem as a Shape Optimization Problem

Brügger¹,

Harbrecht²

2022

SIAM J. Control Optim.

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This article is concerned with the multi-dimensional one-phase Stefan problem, which belongs to the class of moving boundary problems. We suggest to reformulate the classical Stefan problem as a shape optimization problem, consisting of an objective functional for the moving boundary and a partial differential equation corresponding to a heat type equation. Minimizing the objective functional subject to the differential equation under consideration is equivalent to solving the Stefan problem. In order to apply gradient-based optimization algorithms, we analytically compute the shape gradient of the objective functional. A numerical example justifies our approach.

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“…In the past, shape Hessians (i.e., second-order shape derivatives) have been used mainly as a tool to analyze the well-posedness of shape optimization problems [5,6,11,16]. More recently, approximations of shape Hessians have also been used to accelerate convergence of iterative methods for the solution of shape optimization problems, for example, in imaging [17], aerodynamic design [13,25], and elliptic shape optimization problems [11,12].…”

Section: Introductionmentioning

confidence: 99%

A Shape Hessian-Based Boundary Roughness Analysis of Navier–Stokes Flow

Yang¹,

Stadler²,

Moser³

et al. 2011

SIAM J. Appl. Math.

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The influence of boundary roughness characteristics on the rate of dissipation in a viscous fluid is analyzed using shape calculus from the theory of optimal control of systems governed by partial differential equations. To study the mapping D from surface roughness topography to the dissipation rate of a Navier-Stokes flow, expressions for the shape gradient and Hessian are determined using the velocity method. In the case of Couette and Poiseuille flows, a flat boundary is a local minimum of the dissipation rate functional. Thus, for small roughness heights the behavior of D is governed by the flat-wall shape Hessian operator, whose eigenfunctions are shown to be the Fourier modes. For Stokes flow, the shape Hessian is determined analytically and its eigenvalues are shown to grow linearly with the wavenumber of the shape perturbation. For Navier-Stokes flow, the shape Hessian is computed numerically, and the ratio of its eigenvalues to those of a Stokes flow depend only on the Reynolds number based on the wavelength of the perturbation. The consequences of these results on the analysis of the effects of roughness on fluid flows are discussed.

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Analytical and numerical methods in shape optimization

Cited by 18 publications

References 30 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

On the Reformulation of the Classical Stefan Problem as a Shape Optimization Problem

A Shape Hessian-Based Boundary Roughness Analysis of Navier–Stokes Flow

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