Early Publications on Partition of Unity Methods

Partition of unity methods:  There has been an increased interest on numerical techniques based on the so-called partition of unity. A brief summary of the origin of this class of methods can be found in  Duarte, C. A., Babuska, I. and Oden, J. T.,“Generalized Finite Element Methods for Three Dimensional Structural Mechanics Problems”. Computers and Structures, v77, 215-232, 2000

The generalized finite element method (GFEM) was proposed independently by Babuska and colleagues [2,3,21] (under the names “special finite element methods”, “generalized finite element method” and “finite element partition of unity method”) and by Duarte and Oden [9-12,25] (under the names “hp clouds” and “cloud-based hp finite element method”). Several of the so-called meshless methods proposed in recent years can also be viewed as special cases of the generalized finite element method. Recent surveys on meshless methods can be found in [6,8]. The key feature of these methods is the use of a partition of unity (PU), which is a set of functions whose values sum to the unity at each point x in a domain.

My early work on partition of unity methods appeared in the publications listed below.  Partition of unity approximations built with Westergaard near
crack tip expansions was first proposed in [4,5,7,8]. Today, this approach is widely used to solve fracture mechanics problems.  The first three-dimensional Generalized  Finite Element Method was presented in [7]. An hp-adaptive Generalized Finite Element Method was presented [6].

  1. Duarte, C. A., “The hp Cloud Method” . PhD Dissertation, The University of Texas at Austin, 1996.
  2. C. A. M. Duarte and J. T. Oden. Hp Clouds–an hp Meshless Method. Numerical Methods for Partial Differential Equations, v12, 673-705, 1996.
  3. C. A. M. Duarte and J. T. Oden. An hp Adaptive Method Using Clouds . Computer Methods in Applied Mechanics and Engineering, v139, 237-262, 1996.
  4. J. T. Oden and C. A. M. Duarte. Solution of Singular Problems Using h-p Clouds. In The Mathematics of Finite Elements and Applications, John Wiley & Sons, New York, NY, pp. 35-54, 1997. J. R. Whiteman, editor. Presented in MAFELAP 1996.
  5. Oden, J. T. and Duarte, C. A. “Cloud, Cracks and FEMs” . In Recent Developments in Computational and Applied Mechanics, CIMNE, Barcelona, Spain, pp. 302-321, 1997. B. Daya Reddy, editor. Dedicated to professor John Martin on the occasion of his sixtieth birthdate.
  6. Oden, J. T., Duarte, C. A. and Zienkiewicz, O. C., “A New Cloud-Based hp Finite Element Method”. Computer Methods in Applied Mechanics and Engineering, v153, 117-126, 1998. Color pre-print.
  7. C. A. Duarte, I. Babuska, and J. T. Oden. Generalized finite element methods for three dimensional structural mechanics problems. In S. N. Atluri and P. E. O’Donoghue, editors, Modeling and Simulation Based Engineering, volume I, pages 53-58. Tech Science Press, October 1998. Proceedings of the International Conference on Computational Engineering Science, Atlanta, GA, October 5-9, 1998.
  8. Duarte, C. A., Babuska, I. and Oden, J. T.,“Generalized Finite Element Methods for Three Dimensional Structural Mechanics Problems”. Computers and Structures, v77, 215-232, 2000.
  9. C. A. Duarte, O. N. Hamzeh, T. J. Liszka, and W. W. Tworzydlo. A generalized finite element method for the simulation of three-dimensional dynamic crack propagation. ComputerMethods in Applied Mechanics and Engineering, 190:2227–2262, 2001. Pre-print available.
  10. C. A. Duarte and I. Babuska. Mesh-independent directional p-enrichment using the generalized finite element method. International Journal for Numerical Methods in Engineering, 55(12):1477–1492, 2002. Pre-print available.

Related Reports

  1. C. A. Duarte and J. T. Oden. An hp Adaptive Method Using Clouds . Technical Report 96-07, TICAM, The University of Texas at Austin, Feb. 1996.
  2. C. A. Duarte and J. T. Oden. Hp Clouds — A Meshless Method to Solve Boundary-Value Problems. Technical Report 95-05, TICAM, The University of Texas at Austin, May 1995.
  3. C. A. Duarte and J. T. Oden. A Review of Some Meshless Methods to Solve Partial Differential Equations. Technical Report 95-06, TICAM, The University of Texas at Austin, May 1995.