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Meshless and Generalized Finite Element Methods: A Survey of Some Major Results

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Meshfree Methods for Partial Differential Equations

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 26))

Abstract

In this lecture, we discuss Meshless and Generalized Finite Element Methods. We survey major results in this area with a unified approach.

The work of this author was partially supported by Office of Naval Research Grant N00014-99-1-0724.

The work of this author was partially supported by the Texas Institute for Computational and Applied Mathematics, University of Texas at Austin.

The work of this author was partially supported by the Texas Institute for Computational and Applied Mathematics, University of Texas at Austin.

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References

  1. Babuška, I. (1970): Approximation by Hill Functions, Comment Math. Univ. Carolinae, 11, pp. 787–811.

    MathSciNet  MATH  Google Scholar 

  2. Babuška, I. (1971): The Finite Element Method for Elliptic Equations, in Numerical Solution of Partial Differential Equations II, SYNSPADE 1970, B. Hubbard, ed., Academic Press, London, pp. 69–106.

    Google Scholar 

  3. Babuška, I. (1972): Approximation by Hill Functions II, Comment Math. Univ. Carolinae, 13, pp. 1–22.

    MathSciNet  MATH  Google Scholar 

  4. Babuška, I. (1973): The Finite Element Method with Penalty, Math. Comp, 27, pp. 221–228.

    MathSciNet  MATH  Google Scholar 

  5. Babuška, I. (1994): Courant Element: Before and After, in Finite Element Methods: Fifty Years of Courant Element, Lecture Notes in Pure and Applied Mathematics, Vol. 164, Marcel Dekker, pp. 37–51.

    Google Scholar 

  6. Babuška, I. Aziz, K. A. (1972): Survey Lectures on the Mathematical Foundations of the Finite Element Method, in The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential, A. K. Aziz, ed., Academic Press, pp. 3–345.

    Google Scholar 

  7. Babuška, I., Caloz, G. Osborn, J. (1994): Special Finite Element Methods for a class of second order elliptic problems with rough coefficients, SIAM J. Numer. Anal., 31, pp. 945–981.

    Article  MathSciNet  MATH  Google Scholar 

  8. Babuška, I., Kellogg, R. B. Pitkaranta, J. (1979): Direct and Inverse Error Estimates for Finite Elements with Mesh Refinements, Numer. Math., 33, pp. 447–471.

    Article  MathSciNet  MATH  Google Scholar 

  9. Babuška, I. Melenk, J. M. (1997): The Partition of Unity Finite Element Method, Int. J. Numer. Meth. Engng., 40, pp. 727–758.

    Article  MATH  Google Scholar 

  10. Babuška, I., Banerjee, U. and Osborn, J.: On Principles for the Selection of Shape Functions for the Generalized Finite Element Method, to appear.

    Google Scholar 

  11. Babuška, I., Banerjee, U. and Osborn, J., (2001): On Principles for the Selection of Shape Functions for the Generalized Finite Element Method, Technical Report #01-16, ticam, University of Texas at Austin.

    Google Scholar 

  12. Babuška, I., Banerjee, U. and Osborn, J.: Survey of Meshless and Generalized Finite Element Method: A Unified Approach, in preparation.

    Google Scholar 

  13. Babuška, I., Banerjee, U. and Osborn, J.: On the Approximability and the Selection of Particle Shape Functions, in preparation.

    Google Scholar 

  14. Buhmann, M. D. (2000): Radial Basis Functions, Acta Numerica, 9, pp. 1–38.

    Article  MathSciNet  Google Scholar 

  15. Chen, J. S., Wu, C. T. and You, Y. (2001): A Stabilized Conforming Nodal Integration for Galerkin Meshfree Methods, Int. J. Numer. Meth. Engng., 50, pp. 435–466.

    Article  MATH  Google Scholar 

  16. Ciarlet, P. G. (1991): Basic Error Estimates for Elliptic Problems, in Handbook of Numerical analysis, Vol. II, Part 1, P. G. Ciarlet and J. L. Lions eds., Elsivier Science Publ., pp. 19–351.

    Google Scholar 

  17. Daux, C., Moes, N., Dolbow, J., Sukumar, N. and Belytschko, T. (2001): Arbitrary Branched and Intersecting Cracks with the Extended Finite Element Method, Int. J. Numer. Meth. Engng., 48, pp. 1741–1760.

    Article  Google Scholar 

  18. Dolbow, J. and Belytschko, T. (1999): Numerical Integration of the Galerkin Weak Form in Meshfree Methods, Comp. Mech., 23, pp. 219–230.

    Article  MathSciNet  MATH  Google Scholar 

  19. Duarte, C. A. and Babuška, I. (2001): Mesh Independent p-Orthotropic Enrichment using Generalized Finite Element Method, Technical Report, TICAM, University of Texas at Austin.

    Google Scholar 

  20. Duarte, C. A., Babuška, I., and Oden J. T. (2000): Generalized Finite Element Methods for three dimensional structural mechanics problems, Computer and Stuctures, 77, pp. 215–232.

    Article  Google Scholar 

  21. Duarte, C. A., Hamzeh O. H., Liszka, T. J. and Tworzydlo, W. W. (2001): A Generalized Finite Element Method for the Simulation of Three-Dimensional Crack Propagation, Comput. Methods Appl. Mech. Engrg., 190, pp. 2227–2262.

    Article  MATH  Google Scholar 

  22. Gingold R. A. and Monaghan J. J. (1977): Smoothed Particle Hydrodynamics: Theory and Application to Non Spherical Stars, Mon. Not. R. astr. Soc., 181, pp. 375–389.

    MATH  Google Scholar 

  23. Griebel, M. and Schweitzer, M. A. (2001): A Particle-Partition of Unity Method, Part II, Efficient Cover Construction and Reliable Integration, Preprint, University of Bonn.

    Google Scholar 

  24. Griebel, M. and Schweitzer, M. A. (2001): A Particle-Partition of Unity Method, Part III, A Multilevel Solver, Preprint, University of Bonn.

    Google Scholar 

  25. Lancaster, P. and Salkauskas, K. (1981): Surfaces Generated by Moving Least Squares Method, Math. Comp, 37, pp. 141–158.

    Article  MathSciNet  MATH  Google Scholar 

  26. Li, S. and Liu, W. K. (1996): Moving Least Squares Reproducing Kernal Particle Method, Part II, Fourier Analysis, Comput. Methods Appl. Mech. Engrg., 139, pp. 159–194.

    Article  MathSciNet  MATH  Google Scholar 

  27. Li, S. and Liu, W. K., (2001): Meshfree and Particle Methods and Their Application, to appear in Applied Mechanics Review.

    Google Scholar 

  28. W. K. Liu, S. Jun, and Y. F. Zhang (1995): Reproducing Kernel Particle Methods, Int. J. Numer. Meth. Fluids, 20, pp. 1081–1106.

    Article  MathSciNet  MATH  Google Scholar 

  29. Liu, W. K., Li, S. and Belytschko, T. (1997): Moving Least Square Reproducing Kernel Particle Method, Methodology and Convergence, Comput. Methods Appl. Mech. Engrg., 143, pp. 422–453.

    Article  MathSciNet  Google Scholar 

  30. Melenk, J. M. and Babuška, I., (1996): The Partition of Unity Finite Element Method: Theory and Application, Comput. Methods Appl. Mech. Engrg., 139, pp. 289–314.

    Article  MathSciNet  MATH  Google Scholar 

  31. Mikhlin, S. G., (1971): The Numerical Performance of Variational Methods, Walkers-Noordhoff.

    Google Scholar 

  32. Nitsche, J., (1970/1971): Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind, Abh. Math. Univ. Hamburg, 36, pp. 9–15.

    Article  MathSciNet  Google Scholar 

  33. Stein, E. M., (1970): Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press.

    Google Scholar 

  34. Stenberg, R., (1995): On Some Techniques for approximating Boundary Conditions in the Finite Element Method, Journal of Computational and Applied Mathematics, 63, pp. 139–148.

    Article  MathSciNet  MATH  Google Scholar 

  35. Strang, G., (1971): The Finite Element Method and Approximation Theory, in Numerical Solution of Partial Differential Equations II, SYNSPADE 1970, B. Hubbard eds., Academic Press, London, pp. 547–584.

    Google Scholar 

  36. Strang, G. and Fix, G., (1973): A Fourier Analysis of Finite Element Variational Method, in Constructive Aspects of Functional analysis, Edizioni Cremonese, pp. 795–840.

    Google Scholar 

  37. Stroubolis, T., Babuška, I. and Copps, K., (2000): The Design and Analysis of the Generalized Finite Element Method, Comput. Methods Appl. Mech. Engrg., 181, pp. 43–69.

    Article  MathSciNet  Google Scholar 

  38. Stroubolis, T., Copps, K. and Babuška, I., (2001): The Generalized Finite Element Method, Comput. Methods Appl. Mech. Engrg., 190, pp. 4081–4193.

    Article  MathSciNet  Google Scholar 

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Babuška, I., Banerjee, U., Osborn, J.E. (2003). Meshless and Generalized Finite Element Methods: A Survey of Some Major Results. In: Griebel, M., Schweitzer, M.A. (eds) Meshfree Methods for Partial Differential Equations. Lecture Notes in Computational Science and Engineering, vol 26. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56103-0_1

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  • DOI: https://doi.org/10.1007/978-3-642-56103-0_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-43891-5

  • Online ISBN: 978-3-642-56103-0

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