Enriched finite element methods

While the finite element method (FEM) is the standard procedure for solving problems in solid mechanics, modeling problems with complex/evolving geometries rapidly exposes the main pitfalls of the method: creating matching meshes is a tedious and error-prone process that can take most of the time in a real-world simulation. In this advanced course on finite element analysis, we delve into enriched finite element formulations. Students will be exposed to state-of-the-art methodologies for solving challenging linear boundary value problems. Particular emphasis will be placed in methodologies for solving problems with discontinuities, e.g., material interfaces, cracks, voids, etc. For these problem, we focus in decoupling the analysis from its underlying finite element discretization. By enriching the finite element space, it is possible to analyze problems with the same accuracy and rate of convergence as those of standard FEM on meshes that align to the problem’s geometry. This is a hands-on course where students develop on the finite element code used in ME46050 (Advanced Finite Element Methods).

Course objectives

  • • Discuss the different state-of-the-art enriched finite element formulations and identify problems where such formulations can be used;
  • • Extend a standard FEM package to include enrichments;
  • • Evaluate the different enriched methodologies and judge their performance on real problems;
  • • Use the software package to simulate complex problems that require an enriched formulation.

Video lectures

  1.   Introduction
  2.   The Generalized Finite Element Method
  3.   Polynomial enrichments and Babuska's algorithm
  4.   GFEM for material interfaces
  5.   Computational aspects of GFEM
  6.   The Interface-enriched Generalized Finite Element Method
  7.   Review of IGFEM and advantages with respect to GFEM
  8.   GFEM for fracture mechanics
  9.   The Discontinuity-Enriched Finite Element Method
  10.   Introduction to Partition of Unity methods