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A numerical study of different projection-based model reduction techniques applied to computational homogenisation.

Dominic Soldner1, Benjamin Brands1, Reza Zabihyan1

  • 1Chair of Applied Mechanics, Friedrich-Alexander-Universität Erlangen-Nürnberg, Egerlandstrasse 5, 91058 Erlangen, Germany.

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Summary

This study introduces robust reduced-order modeling (ROM) for hyperelastic materials in geometrically nonlinear analysis. A Gauss-Newton with Approximated Tensors (GNAT) projection method enhances accuracy and stability for computational homogenization.

Keywords:
Computational homogenisationHyper-reductionHyperelasticityReduced-order modelling

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Area of Science:

  • Computational mechanics
  • Materials science
  • Applied mathematics

Background:

  • Macroscopic material response is often dictated by heterogeneous microstructures.
  • Computational homogenization requires solving numerous micro-scale boundary value problems, leading to high computational costs.
  • Model order reduction techniques (MOR) can mitigate these costs, but model robustness is crucial.

Purpose of the Study:

  • To apply reduced-order modeling (ROM) to geometrically nonlinear, hyperelastic boundary value problems at the micro-scale.
  • To investigate and compare different hyper-reduction methods for handling nonlinearities within ROM.
  • To develop a robust reduced model for computational homogenization.

Main Methods:

  • Proper Orthogonal Decomposition (POD) for the primary unknown reduction.
  • Application of hyper-reduction techniques to address nonlinearities.
  • Comparison of three hyper-reduction strategies, focusing on nonlinearity approximation and projection methods.
  • Implementation and evaluation of a Gauss-Newton with Approximated Tensors (GNAT) projection scheme.

Main Results:

  • Standard hyper-reduction methods using interpolation or Gappy-POD may compromise the symmetry of the system tangent.
  • The widely used Galerkin projection can be sub-optimal due to loss of tangent symmetry.
  • The GNAT projection method demonstrated superior performance in preserving system properties and achieving a robust reduced model.

Conclusions:

  • The GNAT projection method offers an optimal approach for projecting nonlinear reduced-order models in hyperelasticity.
  • This method ensures the robustness of the reduced model, crucial for accurate computational homogenization.
  • The study provides a more efficient and reliable computational approach for predicting material behavior at the macro-scale.