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A Moving Least Squares Approach to the Construction of Discontinuous Enrichment Functions

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Singular Phenomena and Scaling in Mathematical Models

Abstract

In this paper we are concerned with the construction of a piecewise smooth field from scattered data by a moving least squares approach. This approximation problem arises when so-called enrichment functions for a generalized finite element method are computed by a particle scheme on a finer scale. The presented approach is similar in spirit to the so-called visibility criterion but avoids the explicit reconstruction of the location of the discontinuity.

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References

  1. Aragón, A.M., Duarte, C.A.M., Geubelle, P.H.: Generalized finite element enrichment functions for discontinuous gradient fields. Int. J. Numer. Methods Eng. 82(2), 242–268 (2010)

    MATH  Google Scholar 

  2. Babuška, I., Banerjee, U.: Stable generalized finite element method. Technical report, ICES (2011)

    Google Scholar 

  3. Babuška, I., Lipton, R.: Optimal local approximation spaces for generalized finite element methods with application to multiscale problems. SIAM Multiscale Model. Simul. 9(1), 373–406 (2011)

    Article  MATH  Google Scholar 

  4. Babuška, I., Melenk, J.M.: The partition of unity finite element method: basic theory and applications. Comput. Methods Appl. Mech. Eng. 139, 289–314 (1996). Special issue on meshless methods

    Google Scholar 

  5. Babuška, I., Melenk, J.M.: The partition of unity method. Int. J. Numer. Methods Eng. 40, 727–758 (1997)

    Article  MATH  Google Scholar 

  6. Babuška, I., Caloz, G., Osborn, J.E.: Special finite element methods for a class of second order elliptic problems with rough coefficients. SIAM J. Numer. Anal. 31, 945–981 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  7. Babuška, I., Banerjee, U., Osborn, J.E.: 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, pp. 1–20. Springer, Berlin/Heidelberg (2003)

    Chapter  Google Scholar 

  8. Belytschko, T., Black, T.: Elastic crack growth in finite elements with minimal remeshing. Int. J. Numer. Methods Eng. 45, 601–620 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. Belytschko, T., Moës, N., Usui, S., Parimi, C.: Arbitrary discontinuities in finite elements. Int. J. Numer. Methods Eng. 50, 993–1013 (2001)

    Article  MATH  Google Scholar 

  10. Duarte, C.A., Kim, D.J.: Analysis and applications of a generalized finite element method with global-local enrichment functions. Comput. Mech. 50, 563–578 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Duarte, C.A.M., Babuška, I., Oden, J.T.: Generalized finite element methods for three dimensional structural mechanics problems. Comput. Struct. 77, 215–232 (2000)

    Article  Google Scholar 

  12. Duarte, C.A.M., Hamzeh, O.N., Liszka, T.J., Tworzydlo, W.W.: A generalized finite element method for the simulation of three-dimensional dynamic crack propagation. Int. J. Numer. Methods Eng. 190, 2227–2262 (2001)

    MATH  Google Scholar 

  13. Duarte, C.A., Reno, L.G., Simone, A.: A higher order generalized FEM for through-the-thickness branched cracks. Int. J. Numer. Methods Eng. 72(3), 325–351 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Farwig, R.: Multivariate interpolation of arbitrarily spaced data by moving least squares methods. J. Comput. Appl. Math. 16(1), 79–93 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fasshauer, G.F.: Meshfree Approximation Methods with MATLAB. World Scientific, River Edge (2007)

    MATH  Google Scholar 

  16. Fries, T.P., Belytschko, T.: The extended/generalized finite element method: an overview of the method and its applications. Int. J. Numer. Methods Eng. 84(3), 253–304 (2010)

    MathSciNet  MATH  Google Scholar 

  17. Kim, D.J., Duarte, C.A., Proenca, S.P.: Generalized finite element method with global-local enrichments for nonlinear fracture analysis. In: da Costa Mattos, H.S., Alves, M. (eds.) Mechanics of Solids in Brazil 2009, Rio de Janeiro (2009)

    Google Scholar 

  18. Levin, D.: The approximation power of moving least-squares. Math. Comput. 67(224), 1517–1531 (1998)

    Article  MATH  Google Scholar 

  19. Li, S., Liu, W.K.: Meshfree Particle Methods. Springer, Berlin/New York (2004). Incorporated

    Google Scholar 

  20. López de Silanes, M.C., Parra, M.C., Pasadas, M., Torrens, J.J.: Spline approximation of discontinuous multivariate functions from scattered data. J. Comput. Appl. Math. 131(1–2), 281–298 (2001)

    Google Scholar 

  21. Moës, N., Dolbow, J., Belytschko, T.: A finite element method for crack growth without remeshing. Int. J. Numer. Methods Eng. 46, 131–150 (1999)

    Article  MATH  Google Scholar 

  22. Mohammadi, S.: Extended Finite Element Method. Blackwell Publishing, Oxford/Malden (2008)

    Book  MATH  Google Scholar 

  23. Pereira, J.P., Duarte, C.A.M., Guoy, D., Jiao, X.: Hp-generalized fem and crack surface representation for non-planar 3-d cracks. Int. J. Numer. Methods Eng. 77(5), 601–633 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes: The Art of Scientific Computing, 3rd edn. Cambridge University Press, Cambridge/New York (2007)

    Google Scholar 

  25. Schweitzer, M.A.: Generalized finite element and meshfree methods. Habilitation, Univeristät Bonn (2008)

    Google Scholar 

  26. Schweitzer, M.A.: Generalizations of the finite element method. Cent. Eur. J. Math. 10, 3–24 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. Shepard, D.: A two-dimensional interpolation function for irregularly-spaced data. In: Proceedings of the 1968 23rd ACM national conference, ACM’68, pp. 517–524. ACM, New York (1968)

    Google Scholar 

  28. Strouboulis, T., Babuška, I., Copps, K.: The design and analysis of the generalized finite element method. Comput. Methods Appl. Mech. Eng. 181(I 3), 43–69 (2000)

    Google Scholar 

  29. Strouboulis, T., Copps, K., Babuška, I.: The generalized finite element method. Comput. Methods Appl. Mech. Eng. 190, 4081–4193 (2001)

    Article  MATH  Google Scholar 

  30. Strouboulis, T., Zhang, L., Babuška, I.: Generalized finite element method using mesh-based handbooks: application to problems in domains with many voids. Comput. Methods Appl. Mech. Eng. 192, 3109–3161 (2003)

    Article  MATH  Google Scholar 

  31. Strouboulis, T., Zhang, L., Babuška, I.: p-Version of the generalized FEM using mesh-based handbooks with applications to multiscale problems. Int. J. Numer. Methods Eng. 60, 1639–1672 (2004)

    Google Scholar 

  32. Wendland, H.: Local polynomial reproduction and moving least squares approximation. IMA J. Numer. Anal. 21, 285–300 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  33. Xu, J., Belytschko, T.: Discontinuous radial basis function approximations for meshfree methods. In: Griebel, M., Schweitzer, M.A. (eds.) Meshfree Methods for Partial Differential Equations II. Lecture Notes in Computational Science and Engineering, vol. 43, pp. 231–253. Springer, Berlin (2005)

    Chapter  Google Scholar 

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Authors and Affiliations

  1. Institut für Parallele und Verteilte Systeme, Universität Stuttgart, Universitätsstraße 38, D-70569, Stuttgart, Germany

    Marc Alexander Schweitzer & Sa Wu

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  1. Marc Alexander Schweitzer
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  2. Sa Wu
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Correspondence to Marc Alexander Schweitzer .

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  1. Universität Bonn Institut für Numerische Simulation, Bonn, Germany

    Michael Griebel

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Schweitzer, M.A., Wu, S. (2014). A Moving Least Squares Approach to the Construction of Discontinuous Enrichment Functions. In: Griebel, M. (eds) Singular Phenomena and Scaling in Mathematical Models. Springer, Cham. https://doi.org/10.1007/978-3-319-00786-1_15

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