Multiscale imaging-through-analysis methods in computational mechanics

Abstract

The goal of the thesis is to develop a methodology which enables an autonomous computational framework for the high-fidelity and efficient analysis of imaging-based geometries with complex microstructures. The thesis presents three new techniques: the diffuse boundary condition approach, the multiscale predictor/corrector scheme, and the residual-driven iterative corrector scheme. These methods are developed to target major challenges in the analysis of imaging-based geometries. The diffuse boundary approach overcomes the obstacle in the imposition of boundary conditions in the context of the finite cell method, which requires the explicit parametrization of boundary surfaces. Its essential component is a diffuse geometry model generated from metastable phase-field solutions of the Allen-Cahn problem. Phase-field approximations of the boundary and its gradient are then employed to transfer all boundary terms in the variational formulation into volumetric terms. The multiscale predictor/corrector scheme is developed to tackle the high computational cost in nonlinear voxel finite elements. The core components of our method are top-down displacement and bottom-up stress projectors for the exchange of information between coarse and fine scales. These projectors enable the solution in terms of a series of small uncoupled systems at a fraction of the computing power and memory required by the fully coupled fine-scale system. Although the multiscale predictor/corrector approach yields the fine-mesh accuracy, it requires the balance of the approximation power of coarse-scale and fine-scale meshes. The residual-driven corrector scheme is developed to completely resolve this issue. The method is based on the concept of the multiscale finite element method. The definition of a local corrector problem for each multiscale basis function that mitigates the error introduced by local boundary conditions is the core idea of our approach. Each corrector problem results in a local corrector solution that improves the accuracy of the corresponding multiscale basis function at element interfaces. We employ several carefully chosen numerical examples in two and three dimensions, covering both linear and nonlinear problems, to demonstrate the accuracy, robustness, and versatility of these methods.