Advanced Finite Element Methods for Geometrically Complex Engineering Problems

Abstract

Geometric complexity remains a central obstacle in finite element analysis be-cause accuracy often depends on boundaries and microstructures that are difficult to mesh, move over time, or vary parametrically across a design. This dissertation de- velops two computational frameworks that preserve geometric fidelity while reducing the cost of standard body-fitted workflows. The first is an embedded ghost-node fi- nite element method for incompressible flow with moving boundaries that decouples the computational mesh from the interface while strongly enforcing velocity con- ditions on cut elements. A volume-fraction-based pressure treatment stabilizes the mixed formulation near immersed boundaries and suppresses spurious oscillations on small cut cells. The method is verified through manufactured-solution convergence studies and validated on steady and unsteady cylinder benchmarks, an oscillating- cylinder problem, and a flexible leaflet fluid-structure interaction case. These ex- amples demonstrate accurate treatment of curved immersed boundaries, transient wakes, prescribedinterfacemotion, andtwo-waycouplingonfixedstructuredmeshes. The second framework is a parametric Element Reduced Order Model for graded lattice structures with continuously varying unit-cell geometry. It combines polyno- mial regression for projection matrices with empirical interpolation for stiffness and load operators, allowing a single trained model to represent continuous geometric variation without libraries of precomputed reduced models. The formulation pre- serves the modular assembly structure of the full-order model while making many- query analysis of graded lattices practical. Across the tested configurations, the reduced model maintained below 1% displacement error while achieving up to 108 times online speedup. Together, these frameworks show that geometric complex- ity can be addressed by redesigning the numerical method rather than simplifying the geometry, spanning moving interfaces at the continuum scale and parametrically varying microstructures at the material scale.