Ph.D. Thesis


An Extended Finite Element Method
with Discontinuous Enrichment
for Applied Mechanics

John E. Dolbow
December 1999


Abstract

The modeling of a discontinuous field with a standard finite element approximation presents unique challenges. The construction of an approximating space which is discontinuous across a given line or surface places strict restrictions on the finite element mesh. The simulation of an evolution of the discontinuity is in turn burdened by the requirement to remesh at each stage of the calculation. This work approaches the problem by locally enriching the standard element-based approximation with discontinuous functions. The enriched basis is formed from a union of the set of nodal shape functions with a set of products of nodal shape functions and enrichment functions. The construction of the approximating space in this fashion places the formulation in the class of partition of unity methods. In order to prevent linear dependencies in the approximation and spurious singular modes in the global system of equations, guidelines for the selection of enriched nodes and the construction of the bilinear form are developed. By aligning the discontinuities in the enrichment functions with a specified geometry, a discontinuous field is represented independently of the finite element mesh. This capability is shown to significantly extend the standard method for a number of applications in the field of applied mechanics.

The additional incorporation of near-tip functions allows for a natural application of the enriched approximation to fracture mechanics. The capability to accurately calculate stress intensity factors for a mesh which does not conform to the crack geometry is a distinct advantage of the method. In addition to two-dimensional linear elastic fracture, the extension of the method to the modeling of mixed-mode fracture in Mindlin-Reissner plates is examined. To this end, a domain form of the interaction integral is developed for the extraction of moment and shear force intensity factors. The application of discontinuous enrichment to model interfaces with nonlinear constitutive laws is also developed. In conjunction with a well developed iterative technique, several different constitutive laws are considered on the discontinuous interface, including frictional contact on the crack faces. The simulation of crack growth in this context is straightforward, as the enrichment functions alone model the crack geometry such that no remeshing is necessary. The simulated crack paths are shown to correlate well with experimental data, and consistent results are obtained with the method throughout.

The document is available in both pdf and postscript formats, as specified below.
PDF Format

  1. Preliminary Pages [Title, Abstract, Acknowledgements, TOC, etc.] [PDF, 13 pages]
  2. Chapter 1 [Introduction] [PDF, 9 pages]
  3. Chapter 2 [Overview of Partition of Unity Methods] [PDF, 20 pages]
  4. Chapter 3 [Modeling Arbitrary Discontinuities] [PDF, 42 pages]
  5. Chapter 4 [An Enriched Formulation for Plate Fracture] [PDF, 36 pages]
  6. Chapter 5 [Nonlinear Constitutive Laws on Interfaces] [PDF, 31 pages]
  7. Chapter 6 [Crack Growth Simulations] [PDF, 20 pages]
  8. Chapter 7 [Conclusions and Future Work] [PDF, 3 pages]
  9. References [References] [PDF, 5 pages]
  10. Appendices [Appendix A and B] [PDF, 10 pages]
Postscript Form
Ph.D. Thesis (Ph.D. Thesis, Gzipped PS, 176 pages, 2.6MB)

  1. Preliminary Pages [Title, Abstract, Acknowledgements, TOC, etc.] [PS, 13 pages, 858K]
  2. Chapter 1 [Introduction] [PS, 9 pages, 849K]
  3. Chapter 2 [Overview of Partition of Unity Methods] [PS, 20 pages, 993K]
  4. Chapter 3 [Modeling Arbitrary Discontinuities] [PS, 42 pages, 2.0MB]
  5. Chapter 4 [An Enriched Formulation for Plate Fracture] [PS, 36 pages, 1.1MB]
  6. Chapter 5 [Nonlinear Constitutive Laws on Interfaces] [PS, 31 pages, 4.4MB]
  7. Chapter 6 [Crack Growth Simulations] [PS, 20 pages, 6MB]
  8. Chapter 7 [Conclusions and Future Work] [PS, 3 pages, 831K]
  9. References [References] [PS, 5 pages, 840K]
  10. Appendices [Appendix A and B] [PS, 10 pages, 898K]