Relevance to WCCM–ECCOMAS

The Generalized Finite Element Method (GFEM) provides significant flexibility in defining shape functions for problems that are challenging for classical FEMs. With the proper selection of basis functions, the GFEM can overcome many of the limitations of the classical FEM, especially for problems with moving interfaces, discontinuities, singularities, multiple scales of interest, and various other applications.

Course description

The Generalized Finite Element Method (GFEM) provides significant flexibility in defining shape functions for problems that are challenging for classical FEMs. With the proper selection of basis functions, the GFEM can overcome many of the limitations of the classical FEM, especially for problems with moving interfaces, discontinuities, singularities, multiple scales of interest, and various other applications. Moreover, the GFEM also retains many of the attractive features of the FEM, such as the capability to handle complex geometries and the use of basis functions with compact support, which lead to sparse matrices.

Furthermore, because the GFEM can be formulated as a hierarchically enriched FEM, both methods can be concurrently adopted to define approximations over an analysis domain. However, GFEM approximations also present challenges, such as controlling the conditioning of GFEM matrices and numerically integrating weak forms in the presence of non-smooth shape functions.

In the first part of this short course, we introduce the fundamental concepts of the GFEM—how and why it works—and explore its applications to representative classes of problems. In the second part of the course, we focus on applying the GFEM to three-dimensional fracture problems. Participants will have access to the ISET executable, a C++ library developed for GFEM simulations with a scripting interface, which we will use to explore and apply the method to representative example problems. Selected chapters of [1] will be provided to attendees. In the third part of the course, interface- and discontinuity-enriched finite element formulations are introduced. Concretely, we study the Interface-enriched Generalized Finite Element Method (IGFEM), a technique that was devised for problems with weak discontinuities (discontinuous field gradient). In addition, we study the Discontinuity-Enriched Finite Element Method (DE-FEM), which is a generalization of IGFEM for solving problems with both weak and strong discontinuities (e.g., fracture) with a unified formulation. Both IGFEM and DE-FEM are actually derivatives of GFEM. We delve into their formulations, study their application to academic problems, their stability, and we look at their performance in more complex scenarios.

Objectives and target groups

Objectives

• Understand how the GFEM works;
• Apply GFEM to 3D fracture problems;
• Learn IGFEM for interface problems;
• Learn DE-FEM for fracture problems;
• Gain hands-on experience with a GFEM 3D software.

Target groups Graduate students (MSc/PhD); Researchers in computational mechanics; Engineers working with FEM/fracture/interface problems.

Scientific and technical areas covered

• Generalized/Extended Finite Element Methods (GFEM/XFEM);
• Fracture mechanics (2D and 3D);
• Numerical methods for discontinuities;
• Interface- and discontinuity-enriched generalized formulations;
• Computational solid mechanics.