Multi-dimensional wavelet reduction for the homogenisation of microstructures

One of the recent fields of interest in computational homogenisation is the development of model order reduction frameworks to address the significant computational costs enabling fast and accurate evaluation of the microstructural volume element. Model order reduction techniques are applied to computationally challenging analyses of detailed micro- and or macro-structural problems to reduce both computational time and memory usage. In order to alleviate the costly integration, a wavelet-reduced order model for one-dimensional microstructural problems was presented in van Tuijl et al. (2019). This novel approach addresses both the large number of degrees of freedom and integration costs and provides control on errors in the microstructural fields. In this work, this wavelet reduced order model is extended to a multi-dimensional framework and benchmarked for more realistic multi-scale problems. The Wavelet-Reduced Order Model consists of two reduction steps. First, a Reduced Order Model is constructed to reduce the dimensionality of the microstructural model. Second, a wavelet representation is applied to reduce the integration costs of the microstructural model, whilst maintaining control over the local integration error. The multi-dimensional Wavelet-Reduced Order Model is demonstrated for a set of two-dimensional path-dependent microstructural models, evaluating their accuracy and reduction with respect to the full order models on the microstructural and homogenised fields.