Introduction to Shape Optimization

Sokołowski, Jan; Zolésio, Jean-Paul

doi:10.1007/978-3-642-58106-9

Cited by 1,090 publications

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“…From the practical point of view, shape calculus plays an important role in structural design [12,26], iterative methods for image processing and metrology problems [2], and asymptotic regularization [15] among others. It often appears in connection with the level set method [24].…”

Section: Related Workmentioning

confidence: 99%

Second-order domain derivative of normal-dependent boundary integrals

Balzer

2010

J. Evol. Equ.

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Numerous reconstruction tasks in (optical) surface metrology allow for a variational formulation. The occurring boundary integrals may be interpreted as shape functions. The paper is concerned with the second-order analysis of such functions. Shape Hessians of boundary integrals are considered difficult to find analytically because they correspond to third-order derivatives of an, in a sense equivalent, domain integral. We complement previous results by considering cost functions depending explicitly on the surface normal. The correctness and practicability of our calculations are verified in the context of a Newton-type shape reconstruction method.

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Section: Related Workmentioning

confidence: 99%

Second-order domain derivative of normal-dependent boundary integrals

Balzer

2010

J. Evol. Equ.

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“…If the boundary value problem (2.2) is uniquely solvable, then the existence of the derivative u can be shown by applying the Implicit Function Theorem or regular perturbation techniques (see [4, Section 2.2], [5, Section 5.5], [6]). In general, the elliptic boundary value problem (2.2) has a finite dimensional kernel and the derivative u is not uniquely defined by the boundary value problem (2.3).…”

Section: Formulation Of the Problemmentioning

confidence: 99%

“…The numerical solution of this kind of nonlinear optimization problems requires the knowledge of the derivatives of the objective and constraint functionals with respect to small variations of the shape (see the monographs [3,6,2] for a detailed description of the problem and a review of known results). Therefore, there is a strong practical interest in methods for calculating the shape derivative dJ(Ω 0 ) = lim …”

Section: Introductionmentioning

confidence: 99%

“…For this kind of functionals it can be shown under some growth conditions for the functions f and g that the shape derivative dJ exists provided the material derivative u of the solution u with respect to shape variations exists [3,6]. If the elliptic boundary value problem is uniquely solvable, the existence of the material derivative u can be proved by several techniques like the Implicit Function Theorem [6,2] or with the help of a priori estimates (see e.g. [5, Section 5.5.1]).…”

Section: Introductionmentioning

confidence: 99%

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Linear elliptic boundary value problems in varying domains

Bochniak

2003

Mathematische Nachrichten

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The paper is devoted to the study of solutions to linear elliptic boundary value problems in domains depending smoothly on a small perturbation parameter. To this end we transform the boundary value problem onto a fixed reference domain and obtain a problem in a fixed domain but with differential operators depending on the perturbation parameter. Using the Fredholm property of the underlying operator we show the differentiability of the transformed solution under the assumption that the dimension of the kernel does not depend on the perturbation parameter. Furthermore, we obtain an explicit representation for the corresponding derivative.

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“…Even if numerous works exist on the calculus of shape derivatives of various shape functionals [11][12][13]15,34], in the framework of boundary integral equations the scientific literature is not extensive. However, one can cite the papers [29,31,30], where Potthast has considered the question, starting with his PhD thesis [32], for the Helmholtz equation with Dirichlet or Neumann boundary conditions and the perfect conductor problem, in spaces of continuous and Hölder continuous functions.…”

Section: Introductionmentioning

confidence: 99%

Shape Derivatives of Boundary Integral Operators in Electromagnetic Scattering. Part II: Application to Scattering by a Homogeneous Dielectric Obstacle

Costabel

Louër

2012

Integr. Equ. Oper. Theory

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We develop the shape derivative analysis of solutions to the problem of scattering of time-harmonic electromagnetic waves by a penetrable bounded obstacle. Since boundary integral equations are a classical tool to solve electromagnetic scattering problems, we study the shape differentiability properties of the standard electromagnetic boundary integral operators. The latter are typically bounded on the space of tangential vector fields of mixed regularity TH − 1 2 (divΓ, Γ). Using Helmholtz decomposition, we can base their analysis on the study of pseudodifferential integral operators in standard Sobolev spaces, but we then have to study the Gâteaux differentiability of surface differential operators. We prove that the electromagnetic boundary integral operators are infinitely differentiable without loss of regularity. We also give a characterization of the first shape derivative of the solution of the dielectric scattering problem as a solution of a new electromagnetic scattering problem.

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Introduction to Shape Optimization

Cited by 1,090 publications

References 0 publications

Second-order domain derivative of normal-dependent boundary integrals

Second-order domain derivative of normal-dependent boundary integrals

Linear elliptic boundary value problems in varying domains

Shape Derivatives of Boundary Integral Operators in Electromagnetic Scattering. Part II: Application to Scattering by a Homogeneous Dielectric Obstacle

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