Topological derivatives for elliptic problems

Sokołowski, Jan; Żochowski, Antoni

doi:10.1088/0266-5611/15/1/016

Cited by 74 publications

(48 citation statements)

References 4 publications

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“…Soko lowski andŻochowski [2][3][4] give the mathematical background underlying the technique and derive an explicit expression for the topological derivative in linear elasticity. Céa et al [5] discuss the relationship between topological and shape optimization in the context of optimal structural design, and demonstrate the methods' utility through numerical examples.…”

Section: Introductionmentioning

confidence: 99%

On the topological sensitivity of transient acoustic fields

Malcolm

Guzina

2008

Wave Motion

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The concept of topological sensitivity has been successfully employed as an imaging tool to obtain the correct initial topology and preliminary geometry of hidden obstacles for a variety of inverse scattering problems. In this paper, we extend these ideas to acoustic scattering involving transient waveforms and penetrable obstacles. Through a boundary integral equation framework, we present a derivation of the topological sensitivity for the featured class of problems and illustrate numerically the utility of the proposed method for preliminary geometric reconstruction of penetrable obstacles. For generality, we also cast the topological sensitivity in the so-called adjoint field setting that is amenable to a generic computational treatment using, for example, finite element or finite difference methods.

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Section: Introductionmentioning

confidence: 99%

On the topological sensitivity of transient acoustic fields

Malcolm

Guzina

2008

Wave Motion

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“…This issue is particularly pronounced in low regularity hyperbolic models such as wave equations. The reason is simple, the asymptotic analysis performed for the stationary problems [42][43][44][45][46][47] gives useful information for low frequencies only, one can see this phenomenon when dealing with the spectral problems.…”

Section: Modelling Of Imperfectionsmentioning

confidence: 99%

Sensitivity analysis of hyperbolic optimal control problems

2010

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The aim of this paper is to perform sensitivity analysis of optimal control problems defined for the wave equation. The small parameter describes the size of an imperfection in the form of a small hole or cavity in the geometrical domain of integration. The initial state equation in the singularly perturbed domain is replaced by the equation in a smooth domain. The imperfection is replaced by its approximation defined by a suitable Steklov's type differential operator. For approximate optimal control problems the well-posedness is shown. One term asymptotics of optimal control are derived and justified for the approximate model. The key role in the arguments is played by the so called "hidden regularity" of boundary traces generated by hyperbolic solutions.

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“…Fourth, one of the main advantages of the asymptotic model for the topological sensitivity is that it provides a more accurate tool for identification of cracks and cavities in the inverse problems (see, e.g., [16,17]). At the same time, the asymptotic model based topological sensitivity ) ( 0 x S defined by (32) is computed using only information of the non-damaged domain .…”

Section: Asymptotic Models For the Topological Sensitivity The Open Amentioning

confidence: 99%

“…Numerical results obtained by help of the topological derivative can be found in [3,12]. We refer to [16,17] for applications in inverse problems. The topological derivative was incorporated [18] into the level set method [19].…”

Section: Introductionmentioning

confidence: 99%