Support Size Adjustment Algorithm for Reproducing Kernel Particle Method with Semi-Lagrangian Formulation

Abstract:

As one of meshfree methods, reproducing kernel particle method (RKPM) is usually associated with semi-Lagrangian formulation for large deformation problem to avoid the failure of one-to-one mapping from current configuration to reference configuration. However, numerical crack may happen for large deformation problem working with semi-Lagrangian formulation, if we keep the support size of reproducing kernel shape function as constant. This paper proposed an algorithm to adjust the support size at every step and some numerical results are presented to demonstrate the improvement by the proposed algorithm. Meanwhile, this algorithm is very easy to implement for coding, which does not add much computational cost.

• Periodical: Advanced Materials Research (Volumes 368-373)
• Pages: 1660-1666
• DOI: https://doi.org/10.4028/www.scientific.net/AMR.368-373.1660
• Online since: October 2011
• Authors: Han Zhong Luo, Xue Wen Liu, Xing Chun Huang
• Keywords: Large Deformation Problem, Numerical Crack, Reproducing Kernel Particle Method (RKPM), Semi-Lagrangian Formulation, Support Size Adjustment

References

[1] R.A. Gingold, J.J. Monaghan, Smoothed particle hydrodynamics: theory and application to nonspherical stars. Monthly Notices Royal astron. Soc. 181 (1977) 375-389.

DOI: 10.1093/mnras/181.3.375

[2] L.M. Watson, M.R. Scott, Solving spline-collocation approximations to nonlinear two-point boundary-value problems by a homotopy method. Appl. Math. Comput. 24 (1987) 333-357.

DOI: 10.1016/0096-3003(87)90015-4

[3] B. Nayroles, G. Touzot, P. Villon, Generalizing the finite element method: diffuse approximation and diffuse elements. Comput. Mech. 10 (1992) 307-318.

DOI: 10.1007/bf00364252

[4] T. Belytschko, Y.Y. Lu, L. Gu, Element-free Galerkin methods. Int. J. Numer. Methods Engrg. 37 (1994) 229-256.

DOI: 10.1002/nme.1620370205

[5] J.M. Melenk, I. Babuska, The partition of unity finite element method: basic theory and applications. Comput. Methods Appl. Mech. Engrg. 139 (1996) 289-314.

DOI: 10.1016/s0045-7825(96)01087-0

[6] C.A. Duarte, J.T. Oden, An h-p adaptive method using clouds. Comput. Methods Appl. Mech. Eng. 139 (1996) 237-262.

[7] W.K. Liu, S. Jun, Y.F. Zhang, Reproducing kernel particle methods, Int. J. Numer. Methods Fluids 20 (1995) 1081-1106.

DOI: 10.1002/fld.1650200824

[8] J.S. Chen, C. Pan, C.T. Wu, W.K. Liu, Reproducing Kernel Particle Methods for large deformation analysis of non-linear structures. Comput. Methods Appl. Mech. Eng. 139 (1996) 195-227.

DOI: 10.1016/s0045-7825(96)01083-3

[9] N. Sukumar, B. Moran, T. Belytschko, The natural element method in solid mechanics. Int. J. Numer. Methods Engrg. 43 (1998) 839-887.

DOI: 10.1002/(sici)1097-0207(19981115)43:5<839::aid-nme423>3.0.co;2-r

[10] S.N. Atluri, T.L. Zhu, The meshless local Petrov-Galerkin (MLPG) approach for solving problems in elasto-statics. Comput. Mech. 25 (2000) 169-179.

DOI: 10.1007/s004660050467

[11] E.J. Kansa, Multiqudrics - A scattered data approximation scheme with applications to computational fluid-dynamics-I Surface approximations and partial derivatives. Comput. Math. Appl. 19 (1990) 127-145.

DOI: 10.1016/0898-1221(90)90270-t

[12] E.J. Kansa, 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 (1990) 147-161.

DOI: 10.1016/0898-1221(90)90271-k

[13] X. Zhang, J.S. Chen, S. Osher, A multiple level set method for modeling grain boundary evolution of polycrystalline materials. Int. Multi. Mech. 1 (2008) 178-191.

DOI: 10.12989/imm.2008.1.2.191

[14] R.P. Agarwal, D. O'Regan, Multiple nonnegative solutions for second order impulsive differential equations. Appl. Math. Comput. 114 (2000) 51-59.

DOI: 10.1016/s0096-3003(99)00074-0

[15] M.K. Kadalbajoo, Y.N. Reddy, Numerical treatment of singularly perturbed two point boundary value problems. Appl. Math. Comput. 21 (1987) 93-110.

DOI: 10.1016/0096-3003(87)90020-8

[16] H.M. Srivastava, R.K. Saxena, Operators of fractional integration and their applications. Appl. Math. Comput. 118 (2001) 1-52.

[17] J.S. Chen, C. Pan, C.T. Wu, Reproducing kernel particle methods for rubber hyperelasticity. Comput. Mech. 19 (1997) 211-227.

[18] J.S. Chen, S. Yoon, W.K. Liu, An improved reproducing kernel particle method for nearly incompressible hyperelastic solids, Comput. Methods. Appl. Mech. Eng., 181 (1997) 117-145.

DOI: 10.1016/s0045-7825(99)00067-5

[19] J.S. Chen, K.H. Chang, E. Hardee, S. Yoon, M. Kaneko, I. Grindeanu, A structural nonlinear analysis workspace (SNAW) based on meshless methods, Adv. Eng. Software, 30 (1997) 153-175.

DOI: 10.1016/s0965-9978(98)00072-6

[20] J.S. Chen, C. Pan, C.T. Wu, Applications of reproduction kernel particle method to large deformation and contact analysis of elastomers. Rubber Chem. Technol., 71 (1998) 191-213.

DOI: 10.5254/1.3538479

[21] J.S. Chen, C. Pan, M.O.L. Roque, et al., A Lagrangian reproducing kernel particle method for metal forming analysis, Comput. Mech. 22 (1998) 289-307.

DOI: 10.1007/s004660050361

[22] J.S. Chen, Y.C. Wu, Stability in Lagrangian and semi-Lagrangian reproducing kernel discretizations using nodal integration in nonlinear solid mechanics. Comput. Methods Appl. Sci., 5 (2007) 55-76.

DOI: 10.1007/978-1-4020-6095-3_4