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Optimization of Robin-type problems of elasticity via homogenization and beam models with the application to medical

textiles design.

Vladimir Shiryaev, Julia Orlik, Grigory Panassenko

4th International Workshop on the Multiscale Modeling and

Methods in Biology and Medicine

Moscow, 2014

Fraunhofer ITWM, Technische Universität Kaiserslautern, Université Jean Monnet Saint-Etienne

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Contents

1. Motivation

2. Direct problem

3. Reduction to beam models in cell problems

4. Design space

5. Stress profile optimization

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Motivation

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DIRECT PROBLEM

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ui(x) = 0; x 2 @­

Statement of periodic problem

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State of the art in homogenization

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Homogenization

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Homogenization

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Plain strain motivation

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REDUCTION TO BEAM MODELS IN THE CELL PROBLEMS

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Cell problems: reduction to beam models

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Cell problems: derivatives of the effective tensor

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DESIGN SPACE

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Design space

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The homogenized coefficients as functions of the params

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STRESS PROFILE OPTIMIZATION

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Stress profile optimization

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Adjoint approach for the homogenized equation

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Optimization loop scheme

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POISSON’S RATIO OPTIMIZATION

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Poisson’s ratio optimization

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Poisson’s ratio optimization

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Poisson’s ratios optimization: numerical example