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1
Content available remote

Topological derivatives for semilinear elliptic equations

100%
Iguernane M., Nazarov S., Roche J.-R., Sokolowski J., Szulc K.
International Journal of Applied Mathematics and Computer Science
|
2009
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tom 19
|
nr 2
191-205
EN
The form of topological derivatives for an integral shape functional is derived for a class of semilinear elliptic equations. The convergence of finite element approximation for the topological derivatives is shown and the error estimates in the L∞ norm are obtained. The results of numerical experiments which confirm the theoretical convergence rate are presented.
2
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A nonlinear model of a turbine blade by asymptotic analysis

100%
Rodríguez J. M.
International Journal of Applied Mathematics and Computer Science
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2002
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tom 12
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nr 1
101-113
EN
In this paper we obtain a limit model for a turbine blade fixed to a 3D solid. This model is a three-dimensional linear elasticity problem in the 3D part of the piece (the rotor) and a two-dimensional problem (the nonlinear shallow shell equations) in the 2D part (the turbine blade), with junction conditions in the part of the turbine blade fixed to the rotor. To obtain this model, we perform an asymptotic analysis, starting with the nonlinear three-dimensional elasticity equations on all the pieces and taking as a small parameter the thickness of the blade.
3
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The island model as a Markov dynamic system

100%
Schaefer R., Byrski A., Smołka M.
International Journal of Applied Mathematics and Computer Science
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2012
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tom 22
|
nr 4
971-984
EN
Parallel multi-deme genetic algorithms are especially advantageous because they allow reducing the time of computations and can perform a much broader search than single-population ones. However, their formal analysis does not seem to have been studied exhaustively enough. In this paper we propose a mathematical framework describing a wide class of island-like strategies as a stationary Markov chain. Our approach uses extensively the modeling principles introduced by Vose, Rudolph and their collaborators. An original and crucial feature of the framework we propose is the mechanism of inter-deme agent operation synchronization. It is important from both a practical and a theoretical point of view. We show that under a mild assumption the resulting Markov chain is ergodic and the sequence of the related sampling measures converges to some invariant measure. The asymptotic guarantee of success is also obtained as a simple issue of ergodicity. Moreover, if the cardinality of each island population grows to infinity, then the sequence of the limit invariant measures contains a weakly convergent subsequence. The formal description of the island model obtained for the case of solving a single-objective problem can also be extended to the multi-objective case.
4
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A level set method in shape and topology optimization for variational inequalities

100%
Fulmański P., Laurain A., Scheid J.-F., Sokołowski J.
International Journal of Applied Mathematics and Computer Science
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2007
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tom 17
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nr 3
413-430
EN
The level set method is used for shape optimization of the energy functional for the Signorini problem. The boundary variations technique is used in order to derive the shape gradients of the energy functional. The conical differentiability of solutions with respect to the boundary variations is exploited. The topology modifications during the optimization process are identified by means of an asymptotic analysis. The topological derivatives of the energy shape functional are employed for the topology variations in the form of small holes. The derivation of topological derivatives is performed within the framework proposed in (Sokołowski and Żochowski, 2003). Numerical results confirm that the method is efficient and gives better results compared with the classical shape optimization techniques.
5
Content available remote

Asymptotic analysis of the Askey-scheme I: from Krawtchouk to Charlier

86%
Dominici D.
Open Mathematics
|
2007
|
tom 5
|
nr 2
280-304
EN
We analyze the Charlier polynomials C n(χ) and their zeros asymptotically as n → ∞. We obtain asymptotic approximations, using the limit relation between the Krawtchouk and Charlier polynomials, involving some special functions. We give numerical examples showing the accuracy of our formulas.
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