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Topology optimization with anisotropic materials, including a filter to smooth fiber pathways

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Abstract

In a recent publication, an approach to optimize the orientation of anisotropic materials was presented. This strategy was embedded into the thermodynamic topology optimization based on growth. In this paper, we show that the thermodynamic orientation optimization can also be used in more classical approaches to topology optimization. We furthermore enhance the approach by a novel filtering technique to provide control over the smoothness of the pathway of principal material directions, i.e., the curvature of fibers. The filter is based on a convolution operator and is applied to the material stiffness tensor, so that the filtering technique is not directly bounded to the actual parameterization for the design variables. To this end, the topology is defined by a continuous density approach with penalization of intermediate densities (SIMP) solved via the optimality criteria method (OCM). A set of three continuous Euler angles is used as additional design variables to describe the local material rotation of the anisotropic base material. The thermodynamic optimization of the material orientation is performed by evolution of the Euler angles to minimize the elastic energy. The related evolution equations are derived by means of the Hamilton principle, well-known from material modeling.

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Notes

  1. This dependence could be removed by filtering the rotated base material tensor \(\mathbb {E}^{\mathrm {R}}\) instead of the effective material tensor \(\mathbb {E}\).

  2. We tested our material orientation optimization for a homogenous density distribution without filtering: the results coincide with the principal stress direction with a relative compliance difference less then 0.5%

  3. A monolithic solution for the displacements and the design is not considered here which would result in a solution via an iterative non-linear FEM with increased number of degrees of freedom, and thus a more complex implementation. However, a monolithic solution could improve the convergence behavior and may require less iterations resulting in an overall reduced calculation time.

  4. The element-wise design variables are extrapolated to the nodes of the FE mesh and the isosurface for \(\hat {\chi } = 0.5\) of the linear interpolated density field is shown.

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Acknowledgments

The authors gratefully acknowledge financial support through the ZIM project with grant number AiF-ZIM (ZF4620401US8).

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Authors and Affiliations

  1. Institute of Mechanics of Materials, Ruhr-Universität Bochum, Bochum, Germany

    Dustin R. Jantos & Klaus Hackl

  2. Institute of Continuum Mechanics, Ruhr-Universität Bochum, Bochum, Germany

    Philipp Junker

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  1. Dustin R. Jantos
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  2. Klaus Hackl
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Correspondence to Dustin R. Jantos.

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Replication of results

All results can be reproduced by following the program structure in the flowchart in Fig. 1 and the parameters provided in Section 3.1. No additional information is needed.

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Jantos, D.R., Hackl, K. & Junker, P. Topology optimization with anisotropic materials, including a filter to smooth fiber pathways. Struct Multidisc Optim 61, 2135–2154 (2020). https://doi.org/10.1007/s00158-019-02461-x

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