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Optimizing fiber orientation in fiber-reinforced materials using efficient upscaling

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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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Acknowledgements

The present work was funded by the Stiftung Rheinland-Pfalz für Innovation within the project “Multiskalensimulation für die Entwicklung von Hochleistungsverbundwerkstoffen (MUSSEH)” at Fraunhofer Institute for Industrial Mathematics. The third author was supported by the DFG via SPP 1253.

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Authors and Affiliations

  1. Institute of Applied Mathematics, Heidelberg University, Im Neuenheimer Feld 293, 69120, Heidelberg, Germany

    S. Frei

  2. Fraunhofer Institute for Industrial Mathematics, Fraunhofer-Platz 1, 67663, Kaiserslautern, Germany

    H. Andrä

  3. Department of Mathematics, University of Kaiserslautern, 67663, Kaiserslautern, Germany

    R. Pinnau & O. Tse

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  1. S. Frei
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  2. H. Andrä
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  3. R. Pinnau
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  4. O. Tse
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Correspondence to S. Frei.

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Frei, S., Andrä, H., Pinnau, R. et al. Optimizing fiber orientation in fiber-reinforced materials using efficient upscaling. Comput Optim Appl 62, 111–129 (2015). https://doi.org/10.1007/s10589-013-9630-z

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