Abstract
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.
Highlights
The ongoing development of so-called smart materials over the last decades has given rise to the quest for numerical models which enable predictive, fast and accurate simulations of engineering structures
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 Proper Orthogonal Decomposition (POD), Gauss–Newton Approximated Tensors (GNAT), Empirical Cubature (EC) and Reduced Integration Domain (RID)
We applied the tools of reduced-order modelling to the problems arising in computational homogenisation in magneto-mechanics
Summary
The ongoing development of so-called smart materials over the last decades has given rise to the quest for numerical models which enable predictive, fast and accurate simulations of engineering structures. A detailed survey of DEIM, Gappy POD and GNAT for homogenisation of hyper-elastic materials at finite strains that focus on accuracy and robustness was performed in [26] Cubature methods are another possibility of reducing the cost of computing the nonlinearity. An investigation of the MPE method is beyond the scope of this article and we refer the interested reader to [37], where a detailed comparison of MPE, DEIM and Gappy POD was performed for a predator–prey model In this contribution, we will show the first application of reduced-order modelling for computational homogenisation in magneto-mechanics at finite strains. We will focus on reducing the problem at the micro-scale, using POD to compute the reduced basis and applying following hyper-reduction methods: DEIM, Gappy POD, GNAT, EC and RID. A thorough comparison between the techniques with emphasis on accuracy and robustness will be drawn
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