Concurrent multiscale modelling for heterogeneous materials with CutFEM

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

Concurrent multiscale analysis without meshing: Microscale representation with CutFEM and micro/macro model blending

In this paper, we develop a novel unfitted multiscale framework that combines two separate scales represented by only one single computational mesh. Our framework relies on a mixed zooming technique where we zoom at regions of interest to capture microscale properties and then mix the micro and macroscale properties in a transition region. Furthermore, we use homogenization techniques to derive macro model material properties. The microscale features are discretized using CutFEM. The transition region between the micro and macroscale is represented by a smooth blending function. To address the issues with ill-conditioning of the multiscale system matrix due to the arbitrary intersections in cut elements and the transition region, we add stabilization terms acting on the jumps of the normal gradient (ghost-penalty stabilization). We show that our multiscale framework is stable and is capable to reproduce mechanical responses for heterogeneous structures in a mesh-independent manner. The efficiency of our methodology is exemplified by 2D and 3D numerical simulations of linear elasticity problems

Numerical modeling of shear band propagation in porous plastic dilatant materials by XFEM

This paper studies mixed-mode shear band propagation behaviors in porous plastic dilatant materials by the extended finite element method (XFEM). The Drucker-Prager elastoplastic model is combined with the strong discontinuity method to simulate the dilatant shear band. First, the dissipative nature of the localized area with displacement jump is integrated into the constitutive model by introducing a cohesive law. A new contribution lies that the yielding function is modified in the localized region to calculate the cohesive traction within the framework of the XFEM. The shear band propagation direction is determined by the singularity of the acoustic tensor and the corresponding localization vector is computed by the eigenvalue analysis. Then, the XFEM is used to calculate the numerical dilation with both the normal and shear modes for the localization band. Finally, two typical cases for the shear band propagation are used and numerical results are compared with existing works to confirm the efficiency and robustness of the developed method

XFEM, strong discontinuities and second-order work in shear band modeling of saturated porous media

We investigate shear band initiation and propagation in fully saturated porous media by means of a combination of strong discontinuities (discontinuities in the displacement field) and XFEM. As a constitutive behavior of the solid phase, a Drucker–Prager model is used within a framework of non-associated plasticity to account for dilation of the sample. Strong discontinuities circumvent the difficulties which appear when trying to model shear band formation in the context of classical nonlinear continuum mechanics and when trying to resolve them with classical numerical methods like the finite element method. XFEM, on the other hand, is well suited to deal with problems where a discontinuity propagates, without the need of remeshing. The numerical results are confirmed by the application of Hill’s second-order work criterion which allows to evaluate the material point instability not only locally but also for the whole domain

A novel hierarchical multiresolution framework using CutFEM

In this paper, we propose a robust concurrent multiscale method for continuum-continuum coupling based on the cut finite element method. The computational domain is defined in a fully non-conforming fashion by approximate signed distance functions over a fixed background grid and decomposed into microscale and macroscale regions by a novel zooming technique. The zoom interface is represented by a signed distance function which is allowed to intersect the computational mesh arbitrarily. We refine the mesh inside the zooming region hierarchically for high-resolution computations. In the examples considered here, the microstructure can possess void, and hard inclusions and the corresponding geometry is defined by a signed distance function interpolated over the refined mesh. In our zooming technique, the zooming interface is allowed to intersect the microstructure interface in a arbitrary way. Then, the coupling between the subdomains is applied using Nitsche's method across interfaces. This multiresolution framework proposes an efficient stabilized algorithm to ensure the stability of elements cut by the zooming and the microstructure interfaces. It is tested for several multiscale examples to demonstrate its robustness and efficiency for elasticity and plasticity problems