Skip to main content
Log in

A fast object-oriented Matlab implementation of the Reproducing Kernel Particle Method

  • Original Paper
  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

Novel numerical methods, known as Meshless Methods or Meshfree Methods and, in a wider perspective, Partition of Unity Methods, promise to overcome most of disadvantages of the traditional finite element techniques. The absence of a mesh makes meshfree methods very attractive for those problems involving large deformations, moving boundaries and crack propagation. However, meshfree methods still have significant limitations that prevent their acceptance among researchers and engineers, namely the computational costs. This paper presents an in-depth analysis of computational techniques to speed-up the computation of the shape functions in the Reproducing Kernel Particle Method and Moving Least Squares, with particular focus on their bottlenecks, like the neighbour search, the inversion of the moment matrix and the assembly of the stiffness matrix. The paper presents numerous computational solutions aimed at a considerable reduction of the computational times: the use of kd-trees for the neighbour search, sparse indexing of the nodes-points connectivity and, most importantly, the explicit and vectorized inversion of the moment matrix without using loops and numerical routines.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Barbieri E, Meo M (2009) Evaluation of the integral terms in reproducing kernel methods. Comput Methods Appl Mech Eng 198(33–36): 2485–2507

    Article  MATH  Google Scholar 

  2. Belytschko T, Fleming M (1999) Smoothing, enrichment and contact in the element-free Galerkin method. Comput Struct 71(2): 173–195

    Article  MathSciNet  Google Scholar 

  3. Belytschko T, Tabbara M (1996) Dynamic fracture using element-free Galerkin methods. Int J Numer Methods Eng 39(6): 923–938

    Article  MATH  Google Scholar 

  4. Belytschko T, Gu L, Lu Y (1994) Fracture and crack growth by element-free Galerkin methods. Model Simul Mater Sci Eng 2: 519–534

    Article  Google Scholar 

  5. Belytschko T, Lu Y, Gu L (1994) Element-free Galerkin methods. Int J Numer Methods Eng 37(2): 229–256

    Article  MathSciNet  MATH  Google Scholar 

  6. Belytschko T, Lu Y, Gu L (1995) Crack propagation by element-free Galerkin methods. Eng Fract Mech 51(2): 295–315

    Article  Google Scholar 

  7. Belytschko T, Lu Y, Gu L, Tabbara M (1995) Element-free Galerkin methods for static and dynamic fracture. Int J Solids Struct 32(17): 2547–2570

    Article  MATH  Google Scholar 

  8. Belytschko T, Krongauz Y, Fleming M, Organ D, Liu WS (1996) Smoothing and accelerated computations in the element free Galerkin method. J Comput Appl Math 74(1): 111–126

    Article  MathSciNet  MATH  Google Scholar 

  9. Belytschko T, Krongauz Y, Organ D, Fleming M, Krysl P (1996) Meshless methods: an overview and recent developments. Comput Methods Appl Mech Eng 139(1–4): 3–47

    Article  MATH  Google Scholar 

  10. Breitkopf P, Rassineux A, Touzot G, Villon P (2000) Explicit form and efficient computation of MLS shape functions and their derivatives. Int J Numer Methods Eng 48(451): 466

    Google Scholar 

  11. Cartwright C, Oliveira S, Stewart D (2001) A parallel quadtree algorithm for efficient assembly of stiffness matrices in meshfree Galerkin methods. In: Proceedings of the 15th international parallel and distributed processing symposium, pp 1194–1198

  12. Chen J, Pan C, Wu C, Liu W (1996) Reproducing kernel particle methods for large deformation analysis of non-linear structures. Comput Methods Appl Mech Eng 139(1–4): 195–227

    Article  MathSciNet  MATH  Google Scholar 

  13. Chen J, Pan C, Wu C (1997) Large deformation analysis of rubber based on a reproducing kernel particle method. Comput Mech 19(3): 211–227

    Article  MathSciNet  MATH  Google Scholar 

  14. Collier N, Simkins D (2009) The quasi-uniformity condition for reproducing kernel element method meshes. Comput Mech 44(3): 333–342

    Article  MathSciNet  MATH  Google Scholar 

  15. Cormen T, Leiserson C, Rivest R, Stein C (2001) Introduction to algorithms. MIT Press, Cambridge

    MATH  Google Scholar 

  16. Dolbow J, Belytschko T (1998) An introduction to programming the meshless element free Galerkin method. Arch Comput Methods Eng 5(3): 207–241

    Article  MathSciNet  Google Scholar 

  17. Duarte C (1995) A review of some meshless methods to solve partial differential equations. TICAM Report 95-06

  18. Fasshauer G (2007) Meshfree approximation methods with Matlab. World Scientific Publishing Co. Inc., River Edge

    MATH  Google Scholar 

  19. Finkel R, Bentley J (1974) Quad trees a data structure for retrieval on composite keys. Acta Inform 4(1): 1–9

    Article  MATH  Google Scholar 

  20. Fries T, Matthies H (2004) Classification and overview of meshfree methods. Informatikbericht Nr. 2003-3, Scientific Computing University

  21. Griebel M, Schweitzer M (2002) A particle-partition of unity method–part II: efficient cover construction and reliable integration. SIAM J Sci Comput 23(5): 1655–1682

    Article  MathSciNet  MATH  Google Scholar 

  22. Idelsohn S, Oñate E (2006) To mesh or not to mesh. That is the question. Comput Methods Appl Mech Eng 195(37–40): 4681–4696

    Article  MATH  Google Scholar 

  23. Klaas O, Shephard M (2000) Automatic generation of octree-based three-dimensional discretizations for partition of unity methods. Comput Mech 25(2): 296–304

    Article  MATH  Google Scholar 

  24. Krysl P, Belytschko T (2001) ESFLIB: a library to compute the element free Galerkin shape functions. Comput Methods Appl Mech Eng 190(15–17): 2181–2206

    Article  MATH  Google Scholar 

  25. Lancaster P, Salkauskas K (1981) Surfaces generated by moving least squares methods. Math Comput 37(155): 141–158

    Article  MathSciNet  MATH  Google Scholar 

  26. Li S, Liu W (1996) Moving least-square reproducing kernel method part II: Fourier analysis. Comput Methods Appl Mech Eng 139(1–4): 159–193

    Article  MATH  Google Scholar 

  27. Li S, Liu W (2004) Meshfree particle methods. Springer, Berlin

    MATH  Google Scholar 

  28. Libersky L, Petschek A (1990) Smooth particle hydrodynamics with strength of materials. In: Proceedings of the next free-Lagrange conference, June 1990, Jackson Lake Lodge, Moran, Wyoming

  29. Libersky L, Petschek A, Carney T, Hipp J, Allahdadi F (1993) High strain Lagrangian hydrodynamics: a three-dimensional SPH code for dynamic material response. J Comput Phys 109(1): 67–75

    Article  MATH  Google Scholar 

  30. Liu G (2003) Mesh free methods: moving beyond the finite element method. CRC Press, Boca Raton

    MATH  Google Scholar 

  31. Liu G, Tu Z (2002) An adaptive procedure based on background cells for meshless methods. Comput Methods Appl Mech Eng 191(17–18): 1923–1943

    Article  MATH  Google Scholar 

  32. Liu W, Jun S, Zhang Y (1995) Reproducing kernel particle methods. Int J Numer Methods Fluids 20(8–9): 1081–1106

    Article  MathSciNet  MATH  Google Scholar 

  33. Liu W, Chen Y, Uras R, Chang C (1996) Generalized multiple scale reproducing kernel particle methods. Comput Methods Appl Mech Eng 139(1–4): 91–157

    Article  MathSciNet  MATH  Google Scholar 

  34. Liu W, Hao W, Chen Y, Jun S, Gosz J (1997) Multiresolution reproducing kernel particle methods. Comput Mech 20(4): 295–309

    Article  MathSciNet  MATH  Google Scholar 

  35. Liu W, Li S, Belytschko T (1997) Moving least-square reproducing kernel methods. (I) Methodology and convergence. Comput Methods Appl Mech Eng 143(1–2): 113–154

    Article  MathSciNet  MATH  Google Scholar 

  36. Lucy L (1977) A numerical approach to the testing of the fission hypothesis. Astron J 82(12): 1013–1024

    Article  Google Scholar 

  37. Macri M, De S, Shephard M (2003) Hierarchical tree-based discretization for the method of finite spheres. Comput Struct 81(8–11): 789–803

    Article  Google Scholar 

  38. Monaghan J (1982) Why particle methods work. SIAM J Sci Stat Comput 3: 422

    Article  MathSciNet  MATH  Google Scholar 

  39. Monaghan J (1988) An introduction to SPH. Comput Phys Commun 48(1): 89–96

    Article  MATH  Google Scholar 

  40. Monaghan J (1992) Smoothed particle hydrodynamics. Annu Rev Astron Astrophys 30(1): 543–574

    Article  Google Scholar 

  41. Moore A (1991) A tutorial on kd-trees. University of Cambridge Computer Laboratory. Technical Report No. 209. Extract from PhD thesis. http://www.autonlab.org/autonweb/14665.html

  42. Nguyen V, Rabczuk T, Bordas S, Duflot M (2008) Meshless methods: a review and computer implementation aspects. Math Comput Simul 79: 763–813

    Article  MathSciNet  MATH  Google Scholar 

  43. Rabczuk T, Belytschko T (2005) Adaptivity for structured meshfree particle methods in 2D and 3D. Int J Numer Methods Eng 63(11): 1559–1582

    Article  MathSciNet  MATH  Google Scholar 

  44. Shaofan L, Liu W (2002) Meshfree and particle methods and their applications. Appl Mech Rev 55: 1–34

    Article  Google Scholar 

  45. Tabarraei A, Sukumar N (2005) Adaptive computations on conforming quadtree meshes. Finite Elements Anal Des 41(7–8): 686–702

    Article  Google Scholar 

  46. You Y, Chen J, Lu H (2003) Filters, reproducing kernel, and adaptive meshfree method. Comput Mech 31(3): 316–326

    MATH  Google Scholar 

  47. Zhou J, Wang X, Zhang Z, Zhang L (2005) Explicit 3-D RKPM shape functions in terms of kernel function moments for accelerated computation. Comput Methods Appl Mech Eng 194(9–11): 1027–1035

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ettore Barbieri.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Barbieri, E., Meo, M. A fast object-oriented Matlab implementation of the Reproducing Kernel Particle Method. Comput Mech 49, 581–602 (2012). https://doi.org/10.1007/s00466-011-0662-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-011-0662-x

Keywords

Navigation