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

Abstract

Computing the macroscopic material response of a continuum body commonly involves the formulation of a phenomenological constitutive model. However, the response is mainly influenced by the heterogeneous microstructure. Computational homogenisation can be used to determine the constitutive behaviour on the macro-scale by solving a boundary value problem at the micro-scale for every so-called macroscopic material point within a nested solution scheme. Hence, this procedure requires the repeated solution of similar microscopic boundary value problems. To reduce the computational cost, model order reduction techniques can be applied. An important aspect thereby is the robustness of the obtained reduced model. Within this study reduced-order modelling (ROM) for the geometrically nonlinear case using hyperelastic materials is applied for the boundary value problem on the micro-scale. This involves the Proper Orthogonal Decomposition (POD) for the primary unknown and hyper-reduction methods for the arising nonlinearity. Therein three methods for hyper-reduction, differing in how the nonlinearity is approximated and the subsequent projection, are compared in terms of accuracy and robustness. Introducing interpolation or Gappy-POD based approximations may not preserve the symmetry of the system tangent, rendering the widely used Galerkin projection sub-optimal. Hence, a different projection related to a Gauss-Newton scheme (Gauss-Newton with Approximated Tensors-GNAT) is favoured to obtain an optimal projection and a robust reduced model.

Efficient assembly of linearized equations in nonlinear homogenization

“…This is a key contribution to the field of hyperelastic finite strain homogenization, considering that previous methods have suggested to use a numerical perturbation method [3,5] or a rough approximation by considering only one phase of the heterogeneous microstructure [3]. This is a key contribution to the field of hyperelastic finite strain homogenization, considering that previous methods have suggested to use a numerical perturbation method [3,5] or a rough approximation by considering only one phase of the heterogeneous microstructure [3].…”

“…The RB method based on an approximation of the displacement, u, is well-established, see for instance [3][4][5]. In the offline phase, displacement fluctuation snapshots are collected in a manner analogous to the procedure described in Section 2.1.1.…”

Challenges of order reduction techniques for problems involving polymorphic uncertainty

“…The KLE was used within the stochastic FEM from the first introduction of this method in to obtain some parametric representation of random material properties. Mathematically the KLE is identical to the POD, also known as the singular value decomposition . All mentioned spectral techniques provide in some sense an optimal basis for the model reduction and preserve basic model properties and there is a clear relation between nonlinear modes, eigenmodes and POD modes: the eigenmodes represent the tangent to the nonlinear modes, while the POD modes represent a linear fit of the nonlinear modes …”

“…Mathematically the KLE is identical to the POD, also known as the singular value decomposition. [16][17][18] All mentioned spectral techniques provide in some sense an optimal basis for the model reduction and preserve basic model properties and there is a clear relation between nonlinear modes, eigenmodes and POD modes: the eigenmodes represent the tangent to the nonlinear modes, while the POD modes represent a linear fit of the nonlinear modes. [19,20] The reduction of computational costs was addressed for stochastic PDEs by many authors.…”

Integration efficiency for model reduction in micro-mechanical analyses

“…The reconstruction of gappy data using modes is pioneered by Everson and Sirovich [34] from which several techniques originated such as the Empirical Interpolation Method [35], and later the Best Point Interpolation Method (BPIM) [36], Missing Point Estimation (MPE) [37], Discrete Empirical Interpolation Method (DEIM) [38]. A comparison between different interpolating methods is given by Soldner et al [39]. This review considers a geometrically non-linear hyper-elastic RVE which is reduced using three hyper-reduction methods, namely Discrete Empirical Interpolation Method (DEIM), Gappy-POD and Gauss-Newton with Approximated Tensors (GNAT).…”