Summary of the course

An Introduction to Partition of Unity-Based
Finite Element Methods

Carlos Armando Duarte

University of Illinois at Urbana-Champaign, USA

Course Objectives

Introduce and develop a thorough understanding of partition of unity approximations – generalized and extended finite element approximations: Strengths, advantages, a-priori error estimates, implementation issues and applications.

Course Outline

  1. Partition of unity approximations

    1. Open cover, partition of unity and reproducing condition

    2. Finite element Partition of unity

    3. Shepard partition of unity

    4. Partition of unity shape functions

      • Reproducing condition

      • Pasting of local approximations

      • A-priori error estimates

    5. Generalized/eXtended FEM shape functions in 1-, 2- and 3-dimensions

      • h and p extensions

      • Completeness

      • Conditioning, linear dependence and solution of equations

  1. Applications of GFEM

      • Weak and strong discontinuities

      • Singularities in 2- and 3-dimensions

      • Arbitrary cracks and crack propagation

      • Multiscale problems

        • Construction of scale-bridging enrichment functions