Group Seminar: Computational Methods for PDEs - Michael Winkler & Maria Heigl
Maria Heigl/JKU. Title: Isogeometric Analysis for non-linear elasticity problems
Michael Winker (RICAM)
Date: Tuesday, June 27, 2023
Time: 1:45 pm, S2 416-1
Title: On finding multiple stationary points of nonconvex, box-constrained optimization problems
Abstract: Design optimization of an electric motor with respect to a certain physical quantity (e.g. the torque) typically gives rise to nonconvex optimization problems. Evolutionary algorithms are able to detect multiple candidates for local solutions of said problems but require a lot of computational effort and are bound to a parametrization of the design domain. A mathematically more rigorous approach in design optimization is density-based topology optimization where the design domain is represented by a density function which is additionally required to fulfil box-constraints. Varying initial guesses when using Newton-like methods to solve these problems may not necessarily lead to more solutions. A technique called 'Deflation' preventing iterative methods to converge to an already found solution
and hence opening up space for finding more distinct solutions is introduced. We give an outline on how 'Deflation' may be used in conjunction with Newton's method to solve unconstrained optimization problems and further with a semismooth Newton method to solve simple box-constrained optimization problems.
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Maria Heigl (NuMa)
Date: Tuesday, June 27, 2023
Time: 2:30 pm, S2 416-1
Title: Isogeometric Analysis for a non-linear elasticity problem
Abstract: We consider a non-linear elasticity problem for modelling a human artery. Therefore, we introduce the St. Venant-Kirchhoff material law and the corresponding stored energy function, which is used to define the energy potential of hyperelastic, isotropic materials. Based on the total energy potential, we derive a non-linear weak formulation and briefly mention existence and uniqueness results, mainly based on the theory of polyconvexity. For discretizing and solving our model problem numerically, we use an Isogeometric Analysis (IgA) approach. Motivated by the well-known locking effect, we shortly discuss the advantages and disadvantages of using high order IgA in contrast to high order Finite Element Method (FEM) and present some examples in two and three dimensions. Further, numerical results of those experiments will be provided.