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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.
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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
Content available remote Electrical impedance tomography: from topology to shape
100%
Hintermuller M. , Laurain A.
Control and Cybernetics
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2008
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tom 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.
3
Content available Optimal design of bar structures with their supports in problems of stability and free vibrations
100%
Bojczuk D. , Rębosz-Kurdek A.
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tom Vol. 52 nr 2
533--546
EN
The problem of maximization of the buckling load and the problem of maximization of the natural vibration frequency under a condition imposed on the global cost is discussed. Cross-sectional areas of bar structures and number of elastic supports, their positions and stiffnesses (or the number and positions of rigid supports) are selected as design parameters. The proposed here algorithm of optimization of bar structures with their supports is applied for analysis of some optimization problems. Illustrative examples confirm applicability of the proposed approach.
4
Content available Applying the topological derivative and the boundary element method to design of shape of the heat exchanger
100%
Freus K. , Freus S.
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tom Vol. 14, nr 2
5--12
EN
In this work, the topological derivative for the Laplace equation is used to solve a design problem. This derivative describes the sensitivity of the problem when a small hole is formed at an arbitrary point of the domain. The goal of this work is to design topology of the domain when the Robin condition is imposed on the holes. Physically, the holes can be construed as cooling channels. For finding the solution of the governing equation the boundary element method is applied. The final part of the paper presents the design of the heat exchanger and results of computations.
5
Content available remote A level set method in shape and topology optimization for variational inequalities
100%
Fulmański P. , Laurain A. , Scheid J.-F. , Sokołowski J.
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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.
6
Content available remote Topological derivatives for semilinear elliptic equations
72%
Iguernane M. , Nazarov S. A. , Roche J. R. , Sokolowski J. , Szulc K.
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tom Vol. 19, no 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.
7
Content available remote Optimal internal dissipation of a damped wave equation using a topological approach
72%
Münch A.
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tom Vol. 19, no 1
15-37
EN
We consider a linear damped wave equation defined on a two-dimensional domain [...], with a dissipative term localized in a subset [...]. We address the shape design problem which consists in optimizing the shape of [...] in order to minimize the energy of the system at a given time T. By introducing an adjoint problem, we first obtain explicitly the (shape) derivative of the energy at time T with respect to the variation in [...]. Expressed as a boundary integral on [...], this derivative is then used as an advection velocity in a Hamilton-Jacobi equation for shape changes. We use the level-set methodology on a fixed working Eulerian mesh as well as the notion of the topological derivative. We also consider optimization with respect to the value of the damping parameter. The numerical approximation is presented in detail and several numerical experiments are performed which relate the over-damping phenomenon to the well-posedness of the problem.
8
Content available Topological sensitivity analysis for a coupled nonlinear problem with an obstacle
58%
Abdelbari M. , Nachi K. , Sokolowski J.
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tom Vol. 46, no. 1
5--25
EN
The Topological Derivative has been recognized as a powerful tool in obtaining the optimal topology for several kinds of engineering problems. This derivative provides the sensitivity of the cost functional for a boundary value problem for nucleation of a small hole or a small inclusion at a given point of the domain of integration. In this paper, we present a topological asymptotic analysis with respect to the size of singular domain perturbation for a coupled nonlinear PDEs system with an obstacle on the boundary. The domain decomposition method, referring to the SteklovPoincar´epseudo-differential operator, is employed for the asymptotic study of boundary value problem with respect to the size of singular domain perturbation. The method is based on the observation that the known expansion of the energy functional in the ring coincides with the expansion of the Steklov-Poincar´e operator on the boundary of the truncated domain with respekt to the small parameter, which measures the size of perturbation. In this way, the singular perturbation of the domain is reduced to the regular perturbation of the Steklov-Poincar´e map ping for the ring. The topological derivative for a tracking type shape functional is evaluated so as to obtain the useful formula for application in the numerical methods of shape and topology optimization.
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