Local convergence of the method of multipliers for variational and optimization problems under the noncriticality assumption |
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Authors: | A?F?Izmailov A?S?Kurennoy Email author" target="_blank">M?V?SolodovEmail author |
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Institution: | 1.Institute of Applied Mathematics,Heidelberg University,Heidelberg,Germany;2.Fraunhofer Institute for Industrial Mathematics,Kaiserslautern,Germany;3.Department of Mathematics,University of Kaiserslautern,Kaiserslautern,Germany |
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Abstract: | We present an efficient algorithm to find an optimal fiber orientation in composite materials. Within a two-scale setting fiber orientation is regarded as a function in space on the macrolevel. The optimization problem is formulated within a function space setting which makes the imposition of smoothness requirements straightforward and allows for rather general convex objective functionals. We show the existence of a global optimum in the Sobolev space H 1(Ω). The algorithm we use is a one level optimization algorithm which optimizes with respect to the fiber orientation directly. The costly solve of a big number of microlevel problems is avoided using coordinate transformation formulas. We use an adjoint-based gradient type algorithm, but generalizations to higher-order schemes are straightforward. The algorithm is tested for a prototypical numerical example and its behaviour with respect to mesh independence and dependence on the regularization parameter is studied. |
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