Topological Derivatives for Semilinear Elliptic Equations

Iguernane, Mohamed; Назаров, С. А.; Roche, Jean Rodolphe; Sokołowski, Jan; Szulc, Katarzyna

doi:10.2478/v10006-009-0016-4

Cited by 40 publications

(46 citation statements)

References 21 publications

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“…The standard approach would use the values of topological derivative for initial location of these holes, and then shape derivative would be applied for fine tuning their sizes and positions (Fulmanski et al 2007;Iguernane et al 2009). Such a method is fast, but there is a danger of landing in a local optimum, especially when the number of holes is bigger than one.…”

Section: Numerical Investigationsmentioning

confidence: 99%

Application of topological derivative to accelerate genetic algorithm in shape optimization of coupled models

Szulc

Żochowski

2014

Struct Multidisc Optim

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In the paper we consider a new method based on the genetic algorithm for finding the location and size of small holes in the domain, in which the coupled linear and nonlinear boundary value problems are defined. The linear and nonlinear components are connected by the transmission conditions on the interface boundary. The expansion with respect to small parameter of the shape functional for nonlinear component and the expansion of Steklov-Poincaré operator for linear component are derived in order to determine the form of topological derivative for shape functional defined for coupled model. Then the topological derivative is employed for evaluation of the probability density applied to generation of location of holes by the genetic algorithm.

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Section: Numerical Investigationsmentioning

confidence: 99%

Application of topological derivative to accelerate genetic algorithm in shape optimization of coupled models

Szulc

Żochowski

2014

Struct Multidisc Optim

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“…It should be emphasized that this paper does not intend to present any formal proof of the topological derivative to non-linear elastic problems. The topological derivative in the cases of semilinear problem and linear elasticity is demonstrated in [19,23], based on the asymptotic expansions which results in the topological derivatives of general shape functionals. Asymptotic analysis is also presented for nonlinear Helmholtz and Navies-Stokes equations in [1][2][3].…”

Section: Tsa For the Total Lagrangian Formulationmentioning

confidence: 99%

“…To obtain the sensitivity of the cost function (19), the lagrangian function for the hyperelastic problem, in the non-perturbed configuration Ω 0 τ , is written as…”

Section: Tsa In Nonlinear Hyperelastic Problemsmentioning

confidence: 99%

“…The directional derivative of the total potential energy functional in u τ in the directionu τ , according to (19), may be written as…”

Section: Tsa In Nonlinear Hyperelastic Problemsmentioning

confidence: 99%

“…Replacing the definition of the total potential energy for the non-linear nearly incompressible hyperelastic problem, given in (19), in (34) and using the Reynolds Transport Theorem, the partial derivative of the lagrangian function in the non-perturbed configuration Ω…”

Section: Tsa In Nonlinear Hyperelastic Problemsmentioning

confidence: 99%

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Topological sensitivity analysis for a two-parameter Mooney-Rivlin hyperelastic constitutive model

Pereira

Bittencourt

2010

Lat. Am. j. solids struct.

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The Topological Sensitivity Analysis (TSA) is represented by a scalar function, called Topological Derivative (TD), that gives for each point of the domain the sensitivity of a given cost function when an infinitesimal hole is created. Applications to the Laplace, Poisson, Helmoltz, Navier, Stokes and Navier-Stokes equations can be found in the literature. In the present work, an approximated TD expression applied to nonlinear hyperelasticity using the two parameter Mooney-Rivlin constitutive model is obtained by a numerical asymptotic analysis. The cost function is the total potential energy functional. The weak form of state equation is the constraint and the total Lagrangian formulation used. Numerical results of the presented approach are considered for hyperelastic plane problems.

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