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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.
2
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On the analysis of boundary value problems in nonsmooth domains

96%
Frémiot G., Horn W., Laurain A., Rao M., Sokołowski J.
Instytut Matematyczny Polskiej Akademii Nauk
EN
Problems involving cracks are of particular importance in structural mechanics, and gave rise to many interesting mathematical techniques to treat them. The difficulties stem from the singularities of domains, which yield lower regularity of solutions. Of particular interest are techniques which allow us to identify cracks and defects from the mechanical properties. Long before advent of mathematical modeling in structural mechanics, defects were identified by the fact that they changed the sound of a piece of material when struck. These techniques have been refined over the years. This volume gives a compilation of recent mathematical methods used in the solution of problems involving cracks, in particular problems of shape optimization. It is based on a collection of recent papers in this area and reflects the work of many authors, namely Gilles Frémiot (Nancy), Werner Horn (Northridge), Jiří Jarušek (Prague), Alexander Khludnev (Novosibirsk), Antoine Laurain (Graz), Murali Rao (Gainesville), Jan Sokołowski (Nancy) and Carol Ann Shubin (Northridge). We review the techniques which can be used for numerical analysis and shape optimization of problems with cracks and of the associated variational inequalities. The mathematical results include sensitivity analysis of variational inequalities, based on the concept of conical differential introduced by Mignot. We complete results on conical differentiability obtained for obstacle problems, by results derived for cracks with non-penetration condition and parabolic variational inequalities. Numerical methods for some problems are given as an illustration. From the point of view of applied mathematics numerical analysis is a necessary ingredient of applicability of the models proposed. We also extend the result on conical differentiability to the case of some evolution variational inequalities. The same mathematical model can be represented in different ways, like primal, dual or mixed formulations for an elliptic problem. We use such possibilities for models with cracks. For the shape sensitivity analysis, in Chapters 1 to 3 we give a thorough introduction to the use of first and second order shape derivatives and their application to problems involving cracks. In Chapter 1, for the convenience of the reader, we provide classical results on shape sensitivity analysis in smooth domains. In Chapter 2, the results on the first order Eulerian semi-derivative in domains with cracks are presented. Of particular interest is the so-called structure theorem for the shape derivative. In Chapter 3, the results on the Fréchet derivative in domains with cracks are presented as well, for first and second order derivatives, using a technique different from that in Chapter 2. In Chapter 4, we extend those ideas to Banach spaces, and give some applications of this extended theory. The polyhedricity of convex sets is considered in the spirit of [71], [87], in the most general setting. These abstract results can be applied to sensitivity analysis of crack problems with non-linear boundary conditions. The results obtained use non-linear potential theory and are interesting on their own. In Chapter 5, several techniques for the study of cracked domains with non-penetration conditions on the crack faces in elastic bodies are presented. The classical crack theory in elasticity is characterized by linear boundary conditions which do not correspond to the physical reality since the crack faces can penetrate each other in this model. In this chapter, non-penetration conditions on the crack faces are considered, which leads to a non-linear problem. The model is presented and the shape sensitivity analysis is performed. Chapter 6 is devoted to the newly developed smooth domain method for cracks. In that chapter the problem on a domain with a crack is transformed into a new problem on a smooth domain. This approach is useful for numerical methods. In \cite{belh} this formulation is used combined with mixed finite elements, and some error estimates are derived for the finite element approximation of variational inequalities with non-linear condition on the crack faces. We give applications of this method to some classical problems. Finally, in Chapter 7 we study integro-differential equations arising from bridged crack models. This is a classical technique, but we introduce a few modern approaches to it for completeness sake.
3
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Electrical Impedance Tomography: from topology to shape

74%
Hintermüller M., Laurain A.
Control and Cybernetics
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2008
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tom 37
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nr 4
913-933
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