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.

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应用于计算均质化的不同基于投影的模型缩减技术的数值研究

计算连续体的宏观材料响应通常涉及到现象学本构模型的制定。然而，响应主要受异质微观结构的影响。计算均质化可用于通过在嵌套解方案中为每个所谓的宏观物质点在微观尺度上求解边界值问题来确定宏观尺度上的本构行为。因此，这个过程需要重复求解类似的微观边值问题。为了降低计算成本，可以应用模型降阶技术。一个重要的方面是所获得的简化模型的鲁棒性。在这项研究中，使用超弹性材料的几何非线性情况的降阶建模 (ROM) 应用于微观尺度上的边界值问题。这涉及用于主要未知和超归约方法的适当正交分解 (POD)，用于产生非线性。其中三种超归约方法的不同之处在于非线性的近似方式和随后的投影，在准确性和鲁棒性方面进行了比较。引入插值或基于 Gappy-POD 的近似可能无法保持系统切线的对称性，从而导致广泛使用的伽辽金投影不是最佳的。因此，与高斯-牛顿方案（Gauss-Newton with Approximated Tensors-GNAT）相关的不同投影有利于获得最佳投影和稳健的简化模型。