Wyniki wyszukiwania - BazTech

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On shape sensitivity analysis of the cost functional without shape sensitivity of the state variable

Kasumba H., Kunisch K.

Control and Cybernetics

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2011

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Vol. 40, no 4

989-1017

EN

A general framework for calculating shape derivatives for domain optimization problems with partial differential equations as constraints is presented. The first order approximation of the cost with respect to the geometry perturbation is arranged in an efficient manner that allows the computation of the shape derivative of the cost without the necessity to involve the shape derivative of the state variable. In doing so, the state variable is only required to be Lipschitz continuous with respect to geometry perturbations. Application to shape optimization with the Navier-Stokes equations as PDE constraint is given.

2

Shape derivatives for general objective functions and the incompressible Navier-Stokes equations

Schmidt S., Schulz V.

Control and Cybernetics

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2010

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Vol. 39, no 3

677-713

EN

The aim of this paper is to present the shape derivative for a wide array of objective functions using the incompressible Navier-Stokes equations as a state constraint. Most real world applications of computational fluid dynamics are shape optimization problems in nature, yet special shape optimization techniques are seldom used outside the field of elliptic partial differential equations and linear elasticity. This article tries to be self contained, also presenting many useful results from the literature. We conclude with a comparison of different objective functions for the shape optimization of an obstacle in a channel, which can be done quite conveniently when one knows the general form of the shape gradient.

3

Electrical impedance tomography: from topology to shape

Hintermuller M., Laurain A.

Control and Cybernetics

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2008

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Vol. 37, no 4

913-933

EN

A level set based shape and topology optimization approach to electrical impedance tomography (EIT) problems with piecewise constant conductivities is introduced. The proposed solution algorithm is initialized by using topological sensitivity analysis. Then it relies on the notion of shape derivatives to update the shape of the domains where conductivity takes different values.

4

Shape sensitivity analysis of the Dirichlet Laplacian in a half-space

Amrouche Ch., Nečasová S., Sokołowski J.

Bulletin of the Polish Academy of Sciences. Mathematics

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2004

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Vol. 52, nr 4

365-380

EN

Material and shape derivatives for solutions to the Dirichlet Laplacian in a half-space are derived by an application of the speed method. The proposed method is general and can be used for shape sensitivity analysis in unbounded domains for the Neumann Laplacian as well as for the elasticity boundary value problems.

5

Topological derivative for optimal control problems

Sokołowski J., Żochowski A.

Control and Cybernetics

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1999

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Vol. 28, no 3

611-625

EN

The topological derivative is introduced for the extremal values of cost functionals for control problems. The optimal control problem considered in the paper is defined for the elliptic equation which models the deflection of an elastic membrane. The derivative measures the sensitivity of the optimal value of the cost with respect to changes in topology. A change in topology means removing a small ball from the interior of the domain of integration. The topological derivative can be used for obtaining the numerical solutions of the shape optimization problems.