Date of Award
Doctor of Philosophy (PhD)
Aerospace Engineering Sciences
The focus of this research is uncertainty modeling for problems with random geometry. This dissertation develops a computational framework, based on the eXtended Finite Element Method (XFEM) and a Polynomial Chaos Expansion (PCE), for modeling heterogeneous materials with uncertain material interfaces. The uncertain geometry is characterized based on a finite set of random parameters, which requires a collection of measurement data or images. The XFEM is particularly useful for problems with changing interface geometries, as remeshing is avoided since a conforming mesh is not required. The XFEM is extended to the probability domain by a PCE based on the random parameters defining the uncertain geometry, and a random level set function implicitly defines the uncertain geometry. An intrusive PCE is implemented, which integrates the expansion within the deterministic model. Problems with continuous and discontinuous solutions at the material interface are solved, which utilize different enrichment functions. An accurate integration approach is introduced for the stochastic domain for both types of solutions. For problems with continuous solutions at the interface, a strategy for choosing a proper C0-continuous enrichment function is presented. A PCE is best suited to approximating a smooth behavior of the degrees of freedom, and this research shows that a proper C0-continuous enrichment function leads to a smooth behavior of the degrees of freedom when the spatial mesh is converged. To address solving problems with discontinuous solutions at the interface, an implementation of the Heaviside enriched XFEM is presented which provides a robust approach for handling complex interface configurations. A preconditioning scheme was developed to avoid ill-conditioning due to small intersected element volumes. The Heaviside enriched XFEM extended to the probability domain leads to a smooth behavior of the degrees of freedom regardless of the spatial mesh size. The C0-continuous enrichment requires simultaneous spatial and stochastic refinement to reduce the approximation error, while the Heaviside enrichment function leads to a solution that converges at low stochastic approximation orders for each spatial mesh size. Numerical examples include heat diffusion and linear elasticity for problems containing a single inclusion with random geometry.
Lang, Christapher, "An XFEM Approach to Modeling Material Interface Uncertainty" (2015). Aerospace Engineering Sciences Graduate Theses & Dissertations. 111.