Concurrent multiscale modelling for heterogeneous materials with CutFEM

Abstract

Computational modelling of heterogeneous materials with complex microstructures is challenging due to their multiscale nature. While direct numerical simulations lead to accurate results, it is not tractable for large-scale models. Therefore, in this thesis, two novel concurrent multiscale frameworks have been developed for tractable simulation of 2D/3D highly heterogeneous materials, including composites and trabecular bone materials. The difficulty of discretising such materials with complex microstructure is circumvented by using the cut finite element method (CutFEM). Then, two efficient zooming techniques are proposed for coupling micro and acroscale models. In our multiscale frameworks, the CutFEM technique is utilised to discretise the corresponding micro/macro interface besides the microstructure. In the first framework, the smooth transition concurrent multiscale method, the two models are blended in a transition region and discretised over a single fixed computational mesh. While in the second framework, the two models have different meshes and are coupled over a sharp interface using Nitsche’s method. In both frameworks, the CutFEM technology has been used for discretisation purposes that permits representing the microstructure and micro/macro interfaces in a mesh-independent fashion. This feature of CutFEM allows to (re)locate the zooming region(s) (the region(s) we require microscopic analysis) over a fixed background mesh arbitrarily, thus improving the robustness of multiscale modelling and analysis. In chapter 3, the efficiency and robustness of the smoothed concurrent multiscale method is demonstrated for 2D and 3D linear elasticity problems. Then, in chapter 5, the performance of the second concurrent multiscale framework with a sharp interface is tested for 2D linear elasticity and plasticity materials. In chapter 4, the smoothed concurrent multiscale method developed in Chapter 3 is extended for brittle fracture problems, which are a prevalent example of multiscale phenomena. According to the literature, fracture initiation starts in microscopic length scales by an accumulation of micro cracks in a process zone that eventually leads to the creation of macro cracks. In this thesis, the phase field model has been adopted for the fracture problem, which considers the fracture in a diffusive way. Since phase field models suffer from demanding extremely refined meshes to represent cracks, an efficient numerical framework is essential to balance accuracy and computational costs. In chapter 4, we show that our smoothed concurrent multiscale framework is a suitable choice for such problems