Interaction and multiscale mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

A study on convergence and complexity of reproducing kernel collocation method

  • Hu, Hsin-Yun (Department of Mathematics, Tunghai University) ;
  • Lai, Chiu-Kai (Department of Mathematics, Tunghai University) ;
  • Chen, Jiun-Shyan (Department of Civil and Environmental Engineering, University of California)
  • Received : 2009.06.01
  • Accepted : 2009.08.14
  • Published : 2009.09.25
⟨ Previous Next ⟩

Abstract

In this work, we discuss a reproducing kernel collocation method (RKCM) for solving $2^{nd}$ order PDE based on strong formulation, where the reproducing kernel shape functions with compact support are used as approximation functions. The method based on strong form collocation avoids the domain integration, and leads to well-conditioned discrete system of equations. We investigate the convergence and the computational complexity for this proposed method. An important result obtained from the analysis is that the degree of basis in the reproducing kernel approximation has to be greater than one for the method to converge. Some numerical experiments are provided to validate the error analysis. The complexity of RKCM is also analyzed, and the complexity comparison with the weak formulation using reproducing kernel approximation is presented.

Keywords

References

  1. Ciarlet, P.G. (1978), The finite Element Method for Elliptic Problem, North-Holland, Inc., New York, NY.
  2. Belytschko, T., Krongauz, Y., Organ, D., Fleming, M. and Krysl, P. (1996), "Meshless methods: an overview and recent development", Comput. Method. Appl. M., 139, 3-49. https://doi.org/10.1016/S0045-7825(96)01078-X
  3. Gingold, R.A. and Monaghan, J.J. (1977), "Smoothed particle hydrodynamics: theory and application to non-spherical stars", Mon. Not. R. Astron. Soc., 181, 375-389. https://doi.org/10.1093/mnras/181.3.375
  4. Nayroles, B., Touzot, G. and Villon, P. (1992), "Generalizing the finite element method: diffuse approximation and diffuse elements", Comput. Mech., 10, 301-318.
  5. Belytschko, T., Lu, Y.Y. and Gu, L. (1994), "Element-free Galerkin methods", Int. J. Numer. Meth. Eng., 37, 229-256. https://doi.org/10.1002/nme.1620370205
  6. Melenk, J.M. and Babuska, I. (1996), "The partition of unity finite element method: basic theory and applications", Comput. Method. Appl. M., 139, 289-314. https://doi.org/10.1016/S0045-7825(96)01087-0
  7. Duarte, C.A. and Oden, J.T. (1996), "An h-p adaptive method using clouds", Comput. Method. Appl. M., 139, 237-262. https://doi.org/10.1016/S0045-7825(96)01085-7
  8. Liu, W.K., Jun, S. and Zhang, Y.F. (1995), "Reproducing kernel particle methods", Int. J. Numer. Meth. Fl., 20, 1081-1106. https://doi.org/10.1002/fld.1650200824
  9. Chen, J.S., Pan, C., Wu, C.T. and Liu, W.K. (1996), "Reproducing kernel particle methods for large deformation analysis of nonlinear structures", Comput. Method. Appl. M., 139, 195-227. https://doi.org/10.1016/S0045-7825(96)01083-3
  10. Chen, J.S., Yoon, S., Wang, H.P. and Liu, W.K. (2000), "An improved reproducing kernel particle method for nearly incompressible hyperelastic solids", Comput. Method. Appl. M., 181, 117-145. https://doi.org/10.1016/S0045-7825(99)00067-5
  11. Chen, J.S., Han, W., You, Y. and Meng, X. (2003), "A reproducing kernel method with nodal interpolation property", Int. J. Numer. Meth. Eng., 56, 935-960. https://doi.org/10.1002/nme.592
  12. Sukumar, N., Moran, B. and Belytschko, T. (1998), "The natural element method in solid mechanics", Int. J. Numer. Meth. Eng., 43, 839-887. https://doi.org/10.1002/(SICI)1097-0207(19981115)43:5<839::AID-NME423>3.0.CO;2-R
  13. Atluri, S.N. and Zhu, T.L. (2000) "The meshless local Petrov-Galerkin (MLPG) approach for solving problems in elasto-statics", Comput. Mech., 25, 169-179. https://doi.org/10.1007/s004660050467
  14. Kansa, E.J. (1992a), "Multiqudrics - a scattered data approximation scheme with applications to computational fluid dynamics - I. Surface approximations and partial derivatives", Comput. Math. Appl., 19, 127-145.
  15. Kansa, E.J. (1992b), "Multiqudrics - a scattered data approximation scheme with applications to computational fluid dynamics - II. Solutions to parabolic, hyperbolic and elliptic partial differential equations", Comput. Math. Appl., 19, 147-161.
  16. Bernardi, C. and Maday, Y. (1997), Handbook of Numerical Analysis, Techniques of Scientific Computing (Part 2), Vol. V., (Eds. Ciarlet, P.G. and Lions, J.L.), Elsevier Science Pub. Co., New York, NY.
  17. Han, W. and Meng, X. (2001), "Error analysis of the reproducing kernel particle method", Comput. Method. Appl. M., 190, 6157-6181. https://doi.org/10.1016/S0045-7825(01)00214-6
  18. Hu, H.Y. and Li, Z.C. (2006), "Collocation methods for Poisson's equation", Comput. Method. Appl. M., 195, 4139-4160. https://doi.org/10.1016/j.cma.2005.07.019
  19. Li, Z.C., Lu, T.T., Hu, H.Y. and Cheng, A.H.D. (2008), Tretftz and Collocation Methods, WIT press, Southampton, Boston.
  20. Hardy, R.L. (1971), "Multiquadric equations of topography and other irregular surfaces", J. Geophys. Res., 176, 1905-1915.
  21. Hardy, R.L. (1990), "Theory and applications of multiquadric-biharmonic method: 20 years of discovery", Comput. Math. Appl., 19, 163-208. https://doi.org/10.1016/0898-1221(90)90272-L
  22. Golub, G.H. and Van Loan, C.F. (1996), Matrix Computations (3rd edition), The Johns Hopkins University Press, Baltimore and London.
  23. Hu, H.Y. and Chen, J.S. (2009), "An error analysis of collocation method based on reproducing kernel approximation", Numer. Meth. Part. D. E.,(in press).
  24. Burden, R.L. and Faires, J.D. (2005), Numerical Analysis (8th edition), Thomson Brooks/Cole, Australia, Canada, Mexico, Singapore, Spain, UK, USA.
  25. Zhang, X., Chen, J.S. and Osher, S. (2008), "A multiple level set method for modeling grain boundary evolution of polycrystalline materials", Interact. Mult. Mech., 1, 178-191.
  26. Rojek, J. and Onate, E. (2008), "Multiscale analysis using a couple discrete/finite element model", Interact. Mult. Mech., 1, 1-31. https://doi.org/10.12989/imm.2008.1.1.001

Cited by

  1. Error analysis of collocation method based on reproducing kernel approximation vol.27, pp.3, 2011, https://doi.org/10.1002/num.20539
  2. Meshfree Methods: Progress Made after 20 Years vol.143, pp.4, 2017, https://doi.org/10.1061/(ASCE)EM.1943-7889.0001176
  3. Model order reduction for meshfree solution of Poisson singularity problems vol.102, pp.5, 2015, https://doi.org/10.1002/nme.4743
  4. Weighted Reproducing Kernel Collocation Method and Error Analysis for Inverse Cauchy Problems vol.08, pp.03, 2016, https://doi.org/10.1142/S1758825116500307
  5. An accelerated, convergent, and stable nodal integration in Galerkin meshfree methods for linear and nonlinear mechanics vol.107, pp.7, 2016, https://doi.org/10.1002/nme.5183
  6. Perturbation and stability analysis of strong form collocation with reproducing kernel approximation vol.88, pp.2, 2011, https://doi.org/10.1002/nme.3168
  7. A gradient reproducing kernel collocation method for boundary value problems vol.93, pp.13, 2013, https://doi.org/10.1002/nme.4432
  8. Stability and dispersion analysis of reproducing kernel collocation method for transient dynamics vol.32, pp.6, 2011, https://doi.org/10.1007/s10483-011-1457-6
  9. Improved Element-Free Galerkin method (IEFG) for solving three-dimensional elasticity problems vol.3, pp.2, 2009, https://doi.org/10.12989/imm.2010.3.2.123
  10. The coupling of complex variable-reproducing kernel particle method and finite element method for two-dimensional potential problems vol.3, pp.3, 2009, https://doi.org/10.12989/imm.2010.3.3.277
  11. Radial basis function collocation method for a rotating Bose-Einstein condensation with vortex lattices vol.5, pp.2, 2009, https://doi.org/10.12989/imm.2012.5.2.131
  12. An immersed transitional interface finite element method for fluid interacting with rigid/deformable solid vol.13, pp.1, 2009, https://doi.org/10.1080/19942060.2019.1586774
  13. Investigation of Multiply Connected Inverse Cauchy Problems by Efficient Weighted Collocation Method vol.12, pp.1, 2009, https://doi.org/10.1142/s175882512050012x
  14. Generalized reproducing kernel peridynamics: unification of local and non-local meshfree methods, non-local derivative operations, and an arbitrary-order state-based peridynamic formulation vol.7, pp.2, 2020, https://doi.org/10.1007/s40571-019-00266-9
  15. Detecting Inverse Boundaries by Weighted High-Order Gradient Collocation Method vol.8, pp.8, 2009, https://doi.org/10.3390/math8081297
  16. Gradient Enhanced Localized Radial Basis Collocation Method for Inverse Analysis of Cauchy Problems vol.12, pp.9, 2020, https://doi.org/10.1142/s1758825120501070
  17. Towards a general interpolation scheme vol.381, pp.None, 2021, https://doi.org/10.1016/j.cma.2021.113830