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The reproducing singularity particle shape functions for problems containing singularities

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Abstract

In this paper, we construct particle shape functions that reproduce singular functions as well as polynomial functions. We also construct piecewise polynomial convolution partition of unity functions by taking the convolution of the scaled conical window function with the characteristic functions of quadrangular patches (we provide the computer code for this construction). We demonstrate that the reproducing singular particle shape functions yield highly accurate numerical solutions for the singularity problems with crack singularity or a jump boundary data singularity.

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References

  1. Atluri S, Shen S (2002) The meshless method. Tech Science Press

  2. Babuska I, Banerjee U, Osborn JE (2003) Survey of meshless and generalized finite element methods: a unified appraoch. Acta Numer, pp. 1-125. Cambridge Press, Cambridge

  3. Babuska I, Banerjee U, Osborn JE (2004) On the approximability and the selection of particle shape functions. Numer math 96:601–640

    Article  MATH  MathSciNet  Google Scholar 

  4. Babuška I, Oh H-S (1990) The p-version of the finite element method for domains with corners and for infinite domains. Numer Methods PDEs 6:371–392

    Google Scholar 

  5. Babuška I, Rosenzzweig MR (1972) A finie element scheme for domains with corners. Numer Math 20:371–392

    Google Scholar 

  6. Duarte CA, Oden JT (1996) An hp adaptive method using clouds. Comput Methods Appl Mech Eng 139:237–262

    Article  MATH  MathSciNet  Google Scholar 

  7. Han W, Meng X (2001) Error alnalysis of reproducing kernel particle method. Comput Methods Appl Mech Eng 190:6157–6181

    Article  MATH  MathSciNet  Google Scholar 

  8. Han W, Meng X (2002) On a Meshfree method for singular problems. CMES(Tech Science Press) 3:65–76

    MATH  MathSciNet  Google Scholar 

  9. Kim H, Lee SJ, Oh H-S (2003) Numerical methods and error analysis for one diemensional elliptic problems containing singularities. Numer Methods PDEs 19:399–420

    MATH  MathSciNet  Google Scholar 

  10. Li S, Liu WK (2004) Meshfree particle methods. Springer, Heidelberg

    MATH  Google Scholar 

  11. Li S, Lu H, Han W, Liu WK, Simkins DC Jr (2004) Reproducing kernel element method: part II. Globally conforming I m/C n hierarchies. Comput Methods Appl Mech Eng 193:953–987

    Article  MATH  MathSciNet  Google Scholar 

  12. Liu WK, Han W, Lu H, Li S, Cao J (2004) Reproducing kernel element method: part I. Theoretical formulation. Comput Methods Appl Mech Eng 193:933–951

    Article  MATH  MathSciNet  Google Scholar 

  13. Liu WK, Jun S, Zhang YF (1995) Reproducing kernel particle methods. Int J Numer Methods Fluids 20:1081–1106

    Article  MATH  MathSciNet  Google Scholar 

  14. Liu WK, Liu S, Jun S, Li Adee J, Belytschko T (1995) Reproducing kernel particle methods for structural dynamics. Int J Numer Methods Eng 38:1655–1679

    Article  MATH  Google Scholar 

  15. Liu WK, Li S, Belytschko T (1997) Moving least square reproducing kernel method. Part I: Methodology and convergence. Comput Methods Appl Mech Eng 143:422–453

    Article  MathSciNet  Google Scholar 

  16. Lucas TR, Oh H-S (1993) The method of auxiliary mapping for the finite element solutions of elliptic problems containing singularities. J Comput Phys 108:327–342

    Article  MATH  MathSciNet  Google Scholar 

  17. Melenk JM, Babuška I (1996) The partition of unity finite element method: theory and application. Comput Methods Appl Mech Eng 139:239–314

    Article  Google Scholar 

  18. Oh H-S, Babuška I (1995) The method of auxiliary mapping for the finite element solutions of plane elasticity problems containing singularities. J Comput Phys 121:193-212

    Article  MATH  MathSciNet  Google Scholar 

  19. Oh H-S, Kim H, Lee S-J (2001) The numerical methods for oscillating singularities in elliptic boundary value problems. J Comput Phys 170:742–763

    Article  MATH  MathSciNet  Google Scholar 

  20. Oh H-S, Kim JG (2006) The piecewise polynomial partition of unity shape functions for the generalized finite element methods (revision process)

  21. Oh H-S, Kim JG, Jeong JW (2005) The Closed Form Reproducing polynomial particle shape functions for meshfree particle methods. Comput Methods Appl Mech Eng (to appear)

  22. Oh H-S, Kim JG, Jeong JW (2006) The smooth piecewise particle shape functions corresponding to patch-wise non-uniformly spaced particles for meshfree particles methods. Comput Mech (to appear)

  23. Parton VZ, Perli PI (1984) Mathematical methods of the theroy of elasticity, I. MIR Publisherd, Moscow

    Google Scholar 

  24. Stroubolis T, Copps K, Babuska I (2001) Generalized finite element method. Comput Methods Appl Mech Eng 190:4081–4193

    Article  Google Scholar 

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

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC, 28223, USA

    Hae-Soo Oh & Jae Woo Jeong

  2. Department of Mathematics, Kangwon National University, Chunchon, 200-701, South Korea

    June G. Kim

Authors
  1. Hae-Soo Oh
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  2. Jae Woo Jeong
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  3. June G. Kim
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Corresponding author

Correspondence to Hae-Soo Oh.

Additional information

H.-S. Oh was supported in part by funds provided by the University of North Carolina at Charlotte. J. G. Kim was supported in part by the Research Grant of the Kangwon National University. J. G. Kim is a Visiting Professor of the University of North Carolina at Charlotte.

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Oh, HS., Jeong, J.W. & Kim, J.G. The reproducing singularity particle shape functions for problems containing singularities. Comput Mech 41, 135–157 (2007). https://doi.org/10.1007/s00466-007-0174-x

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  • DOI: https://doi.org/10.1007/s00466-007-0174-x

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