A Generalized Enrichment Procedure for Finite Element Methods with Applications to Compressible Flows
Open Access
Author:
Schmidt, Matthew
Graduate Program:
Mechanical Engineering
Degree:
Master of Science
Document Type:
Master Thesis
Date of Defense:
March 21, 2025
Committee Members:
David M Williams, Thesis Advisor/Co-Advisor Xiang Yang, Committee Member Mary Frecker, Program Head/Chair
Keywords:
enrichment compressible Navier-Stokes finite element methods velocity temperature XFEM
Abstract:
In this work, we develop a systematic approach for selecting enrichment functions to enhance the solutions of finite element methods for compressible flows. This is accomplished by extending the definition of the Lebesgue constant for nodal polynomial interpolation to modal, transcendental enrichment functions. The enrichment functions are defined using the partition of unity method. Desirable properties for enrichment functions are proposed, and new classes of enrichment functions are introduced. To validate the approach, the enrichment functions are implemented in a discontinuous Galerkin Implicit Large Eddy Simulation solver, and enriched solutions are compared against standard (unenriched) polynomial solutions. In particular, we use the Method of Manufactured Solutions and the compressible Taylor-Green vortex test case to demonstrate that the proposed enrichment improves accuracy over the unenriched solution. These results provide a foundation for systematic enrichment in finite element methods for compressible flows.
