Partition of Unity Methods

3 Partition of Unity Revisited

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

The previous chapter provided a first introduction to the concept of enrichment and introduced some key issues related to its use within a finite element framework. The goal of the present chapter is to take the concept one step further and analyze more in depth the notions of completeness, partition of unity, mesh-geometry interaction, and imposition of boundary conditions.

3.1 Completeness, Consistency, and Reproducing Conditions

The notion of partition of unity is easily introduced by defining the completeness of approximations, which is expressed in terms of the highest order of the polynomials that can be represented exactly.

Definition 1 [Reproducing condition] Let be a function defined over and a set of approximating functions defined over a set of nodes such that . The ’s reproduce on , if and only if:

(3.1)

For example, if the approximation is able to ...