# Discretization Methods

The Finite Element Method (FEM) has become the de facto procedure for the computational modeling of problems in solid mechanics analysis. The method has been extensively used to obtain approximate solutions to boundary value problems that describe various physical phenomena. In simple words, the procedure subdivides a complex structure into an ensemble of smaller components called finite elements. Through the solution of a discrete system of equations, the joint behavior of the ensemble approximates that of whole system. The more refined the underlying mesh used, the smaller the error obtained by the method. The approximation is said to approach the actual solution as the mesh size h vanishes. Yet, to obtain an accurate solution via the standard FEM, the faces of finite elements must align to the geometry of the discretized objects or discontinuities. Failing to do so reduces the accuracy of the simulation and may render its results unreliable. In practice, however, the creation of a matching FE mesh is a time consuming process that requires the use of powerful and robust mesh generators. Still, complex geometries can render the creation of a matching mesh an obnoxious endeavor, and at worst, the simulation of physical problems involving moving boundaries or phases requires a completely different approach.

Motivated by these obstacles, enriched finite element methods aim at simulating problems using meshes that do not match any geometric entities. In simple terms, some a priori knowledge about the solution is incorporated into the analysis by means of enrichment functions which try to regain the behavior otherwise missing by the use of a non-matching mesh. The resulting methodologies are then able to recover optimal rates of convergence, i.e., the error in the approximate solution decreases at the same rate as that of the standard FEM as the mesh size h is reduced. Enriched FEM has been successfully applied, among others, to the modeling of fracture, composite materials, phase interfaces, and multi-scale problems.

## Fracture mechanics

SOM is currently working on creating new finite element techniques based on “Partition of Unity (PoU)” methods. A result of these efforts is the “Discontinuity-Enriched Finite Element Method (DE-FEM)” as a new enriched procedure for analyzing problems with both weak and strong discontinuities, i.e., material interfaces and cracks, respectively. This technique, which can fully decouple the discontinuities from the finite element mesh, can be extremely useful for analyzing problems in fracture mechanics. In fact, given an object with any geometry, and with an arbitrary number of material interfaces and cracks, you can then immerse that problem in an FE mesh that is completely independent of all those discontinuities, and get an accurate solution as if you had done the analysis with a matching discretization.

## Contact mechanics

By decoupling the finite element mesh from the geometric features of the problem, the enriched formulation developed at SOM can also be used to resolve contact. Contact constraints can be enforced by using simple multiple-point constraints or other schemes such as Lagrangian or augmented Lagrangian.