Discretization Methods

The finite element method

Known about since the mid-twentieth century, the Finite Element Method (FEM) is a numerical procedure used to analyze problems in physics and engineering. FEM works by subdividing a large problem into smaller, simpler parts called finite elements. This discretization process results in a so-called finite element mesh that is subsequently used for the analysis. Although the behavior within each element is simple, once reassembled, they can reproduce very complex behavior.

Advanced finite element methods

In order to preserve accuracy, FEM relies on a "fitted mesh," where the sides of elements align with the problem geometry. For complex problems, creating the computational model—which includes the geometry and the mesh—can take up to 80% of the total analysis time. Therefore, my research focuses on creating new finite element technology that can entirely bypass this process. By decoupling the finite element discretization from the problem's geometry, "enriched finite element methods" provide the analyst with unprecedented flexibility. High accuracy is preserved by enhancing the finite element formulation with so-called enrichment functions. Enriched finite element methods can be applied to a large set of problems in computational mechanics, including:

• Material interfaces (weak discontinuities, where the gradient of the field is discontinuous);
• Cracks (strong discontinuities, where the field itself is discontinuous);
• Immersed boundary (fictitious domain) problems;
• Contact and coupling of non-matching meshes;
• Topology optimization.

In addition to enriched finite element methods, I have also investigated the virtual element method (VEM) and the spectral element method (SEM). Below you will find a brief account on my research efforts on method development.

Projects