Introduction and Applications of Generalized Finite Element Methods

Dates: July 7, 2025 (full day) and July 8, 2025 (morning only)

The Generalized Finite Element Method (GFEM) provides substantial flexibility in defining shape functions and their corresponding approximation spaces. With the proper selection of basis functions, the GFEM can overcome many of the limitations associated with classical FEM, particularly for problems involving moving interfaces, discontinuities, singularities, multiple scales of interest, and various other applications. Moreover, the GFEM also retain 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 results in 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 the control of the conditioning of GFEM matrices and numerically integrating weak forms in the presence of non-smooth shape functions.

In this short course, we present the basic ideas of GFEM approximations and discuss their applications to representative classes of problems. The approximation properties are presented both from an intuitive perspective—based on the reproducing properties of GFEM shape functions—and a more abstract viewpoint—focusing on patch approximation spaces and their combination using a partition of unity. This latter approach leads to a priori error estimates for partition of unity methods, which form the theoretical framework of GFEM. In the second part of the course, we will focus on applying the Generalized Finite Element Method to problems involving three-dimensional fracture propagation. Participants will have access to the ISET executable, a C++ library developed for GFEM simulations with a Tcl interface, which they will use to explore and apply the method to a series of representative example problems. Selected chapters of [1] will be provided to attendees.

Please note: The workshop will be conducted in English.

Reference

[1] C.A. Duarte and A.M. Aragón. Enriched Finite Element Methods, 2024.

https://www.sciencedirect.com/book/9780323855150/fundamentals-of-enriched-finite-element-methods

O workshop ocorrerá nos dias 07 e 08/07/2025, no LI312 da FECFAU. Faça sua inscrição através do Even3: https://www.even3.com.br/introduction-and-applications-of-generalized-finite-element-methods-596399/