Reduced-Order Modelling and Homogenisation in Magneto-Mechanics: A Numerical Comparison of Established Hyper-Reduction Methods

The simulation of complex engineering structures built from magneto-rheological elastomers is a computationally challenging task. Using the FE 2 method, which is based on computational homogenisation, leads to the repetitive solution of micro-scale FE problems, causing excessive computational effort. In this paper, the micro-scale FE problems are replaced by POD reduced models of comparable accuracy. As these models do not deliver the required reductions in computational effort, they are combined with hyper-reduction methods like the Discrete Empirical Interpolation Method (DEIM), Gappy POD, Gauss–Newton Approximated Tensors (GNAT), Empirical Cubature (EC) and Reduced Integration Domain (RID). The goal of this work is the comparison of the aforementioned hyper-reduction techniques focusing on accuracy and robustness. For the application in the FE 2 framework, EC and RID are favourable due to their robustness, whereas Gappy POD rendered both the most accurate and efficient reduced models. The well-known DEIM is discarded for this application as it suffers from serious robustness deficiencies.

Multidimensional Hyper-Reduction of Large Mechanical Models Involving Internal Variables

We propose to incorporate a Response Surface (RS) approximation of variables over a parametric domain into a weak form of parametric Partial Differential Equations (PDEs). Hence a multidimensional model-reduction can be achieved. We propose a multidimensional a priori model reduction method to generate or to enrich RSs. It is coined multidimensional because the fields to forecast are defined over an augmented domain in term of dimension. They are functions of both space variables and parameters that simultaneously evolve in time. This changes the functional space related to the weak form of the PDEs and the definition of the reduced bases. It has a significant impact on the proposed model reduction method. In particular, a new point of view on interpolation of variables has to be addressed. A Multidimensional Reduced Integration Domain (MRID) is proposed to reduce the complexity of the reduced formulation. A multidimensional Hyper-Reduction method extract from the MRID truncated equilibrium equations, truncated residuals and a truncated error indicator.