Partition of Unity Methods

2 A Step-by-Step Introduction to Enrichment

Stéphane P. A. Bordas¹, Alexander Menk², and Sundararajan Natarajan³

¹ University of Luxembourg, Luxembourg, UK² Robert Bosch GmbH, Germany³ Indian Institute of Technology Madras, India

In the previous chapter, we discussed the approximation properties of the finite element method (FEM) and learnt that it is not ideally suited to treat problems involving “rough” solutions. This happens when the solution or its derivatives are of low-order continuity. Examples of problems with rough solutions include:

• partial differential equations (PDEs) with discontinuous coefficients (e.g., diffusion in multi-material domains, material interfaces, cracks, shocks, boundary layers, etc.);
• problems with singular solutions or steep gradients (e.g., cracks in linear elasticity, flow around obstacles, etc.).

We will require the following definitions.

Definition 1 A solution is said to be strongly discontinuous when the primary field is discontinuous. For example, a crack leads to strong discontinuities in the displacement field through the crack.

Definition 2 A solution is said to be weakly discontinuous when its first derivative is discontinuous. For example, perfect material interfaces lead to discontinuities in the strain or the flux field through the interface, i.e., discontinuities in the first (normal) derivative of the displacement or the temperature field.

From simple examples where the standard FEM is unable to reproduce certain features ...