Transactions of the Korean Society of Mechanical Engineers A (대한기계학회논문집A)

The Korean Society of Mechanical Engineers (대한기계학회)
권호 표지
DOI QR코드

DOI QR Code

The Meshfree Method Based on the Least-Squares Formulation for Elasto-Plasticity

탄소성 최소 제곱 수식화와 이를 이용한 무요소법

  • 윤성기 (한국과학기술원 기계공학과) ;
  • 권기찬 (한국원자력연구소 사용후핵연료기술개발부)
  • Published : 2005.06.01
Download PDF
⟨ Previous Next ⟩

Abstract

A new meshfree method for the analysis of elasto-plastic deformations is presented. The method is based on the proposed first-order least-squares formulation, to which the moving least-squares approximation is applied. The least-squares formulation for the classical elasto-plasticity and its extension to an incrementally objective formulation for finite deformations are proposed. In the formulation, the equilibrium equation and flow rule are enforced in least-squares sense, while the hardening law and loading/unloading condition are enforced exactly at each integration point. The closest point projection method for the integration of rate-form constitutive equation is inherently involved in the formulation, and thus the radial-return mapping algorithm is not performed explicitly. Also the penalty schemes for the enforcement of the boundary and frictional contact conditions are devised. The main benefit of the proposed method is that any structure of cells is not used during the whole process of analysis. Through some numerical examples of metal forming processes, the validity and effectiveness of the method are presented.

Keywords

References

  1. Chen, J.S., Pan, C., Wu, C.T. and Liu W.K., 1996, 'Reproducing Kernel Particle Methods for Large Deformation Analysis of Non-linear Structures,' Comput. Methods Appl. Meeh. Engrg., Vol. 139, pp. 195-227 https://doi.org/10.1016/S0045-7825(96)01083-3
  2. Chen, J.S., Pan, C., Rogue, C.M.O.L. and Wang H.P., 198, 'A Lagrangian Reproducing Kernel Particle Method for Metal Forming Analysis,' Comput. Mech., Vol. 22, pp. 289-307 https://doi.org/10.1007/s004660050361
  3. Li, S., Hao, W. and Liu, W.K., 2000, 'Numerical Simulations of Large Deformation of Thin Shell Structures using Meshfree Method,' Comput. Mech., Vol. 25, pp. 102-116 https://doi.org/10.1007/s004660050463
  4. Yoon, S. and Chen, J.S., 2002, 'Accelerated Meshfree Method for Metal Forming Simulation,' Finite Elem. Anal. Des., Vol. 38, pp. 937-948 https://doi.org/10.1016/S0168-874X(02)00086-0
  5. Liew, K.M., Ng, T.Y. and Wu, Y.C., 2002, 'Meshfree Method for Large Deformation Analysis-A Reproducing Kernel Particle Approach,' Eng. Struet., Vol. 24, pp. 543-551 https://doi.org/10.1016/S0141-0296(01)00120-1
  6. Beissel, S. and Belytschko, T., 1996, 'Nodal Integration of the Element-Free Galerkin Method,' Comput. Methods Appl. Mech. Engrg., Vol. 139, pp. 49-74 https://doi.org/10.1016/S0045-7825(96)01079-1
  7. Atluri, S.N. and Zhu, T., 1998, 'A New Meshless Local Petrov-Galerkin (MLPG) Approach in Computational Mechanics,' Comput. Mech., Vol. 22, pp.117-127 https://doi.org/10.1007/s004660050346
  8. Dolbow, J. and Belytschko, T., 1999, 'Numerical Integration of the Galerkin Weak Form in Meshfree Methods,' Comput. Mech., Vol. 23, pp. 219-230 https://doi.org/10.1007/s004660050403
  9. Chen, J.S., Wu, C.T., Yoon, S. and You, Y., 2001, 'A Stabilized Confonning Nodal Integration for Galerkin Mesh-free Methods,' Int. J. Numer. Meth. Engng., Vol. 50, pp. 435-466 https://doi.org/10.1002/1097-0207(20010120)50:2<435::AID-NME32>3.0.CO;2-A
  10. Carpinteri, A., Ferro, G. and Ventura, G., 2002, 'The Partition of Unity Quadrature in Meshless Methods,' Int. J. Numer. Meth. Engng., Vol. 54, pp. 987-1006 https://doi.org/10.1002/nme.455
  11. Dolbow, J. and Belytschko, T., 1999, 'Volumetric Locking in the Element Free Galerkin Method,' Int. J. Numer. Meth. Engng., Vol. 46, pp. 925-942 https://doi.org/10.1002/(SICI)1097-0207(19991030)46:6<925::AID-NME729>3.0.CO;2-Y
  12. Chen, J.S., Yoon, S., Wang, H.P. and Liu, W.K., 2000, 'An Improved Reproducing Kernel Particle Method for Nearly Incompressible Finite Elasticity,' Comput. Methods Appl. Mech. Engrg., Vol. 181, pp. 117-145 https://doi.org/10.1016/S0045-7825(99)00067-5
  13. Chen, J.S., Wang, H.P., Yoon, S. and You, Y., 2000, 'Some Recent Improvement in Meshfree Methods for Incompressible Finite Elasticity Boundary Value Problems with Contact,' Comput. Mech., Vol. 25, pp. 137-156 https://doi.org/10.1007/s004660050465
  14. Park, S.H. and Youn, S.K., 2001, 'The Least-Squares Meshfree Method,' Int. J. Numer. Meth. Engng., Vol. 52, pp. 997-1012 https://doi.org/10.1002/nme.248
  15. Park, S.H. and Youn, S.K, 2001, 'Least-Squares Meshfree Methods and Integration Error,' Transactions of the KSME(A), Vol. 25, No. 10, pp. 1605-1612
  16. Park, S.H., Kwon, K.C. and Youn, S.K., 2003, 'A Study on the Convergence of Least-Squares Meshfree Method under Inaccurate Integration,' Int. J. Numer. Meth. Engng., Vol. 56, pp. 1397-1419 https://doi.org/10.1002/nme.613
  17. Zhang, X., Liu, X.H., Song, K.Z. and Lu, M.W, 2001, 'Least-Squares Collocation Meshless Method,' Int. J. Numer. Meth. Engng., Vol. 51, pp. 1089-1100 https://doi.org/10.1002/nme.200
  18. Cai, Z., Manteuffel, T.A., McCormick, S.P. and Parter, S.Y., 1998, 'First-order System Least Squares (FOSLS) for Planar Linear Elasticity: Pure Traction Problem,' SIAM J. Numer. Anal., Vol. 35, pp. 320-335 https://doi.org/10.1137/S0036142995294930
  19. Kwon, K.C., Park, S.H., Jiang, B.N. and Youn, S.K., 2003, 'The Least-Squares Meshfree Method for Solving Linear Elastic Problems,' Comput. Mech., Vol. 30,pp.196-211 https://doi.org/10.1007/s00466-002-0379-y
  20. Kwon, K.C., Park, S.H. and Youn, S.K., 2002, 'The Least-Squares Meshfree Method for Linear Elasticity,' Transactions of the KSME(A), Vol. 26, No. 11, pp. 2312-2321 https://doi.org/10.3795/KSME-A.2002.26.11.2312
  21. Lancaster, P. and Salkauskas, K., 1981, 'Surfaces Generated by Moving Least-Squares Methods,' Math. Comput., Vol. 37, pp. 141-158 https://doi.org/10.2307/2007507
  22. Duarte, C.A. and Oden, J.T., 1995, 'Hp Clouds - A Meshless Method to Solve Boundary-Value Problems,' Technical Report 95-05, TICAM, University of Texas at Austin, 1995
  23. Jiang, B.N., 1998, The Least-Squares Finite Element Method - Theory and Applications in Computational Fluid Dynamics and Electromagnetics, Spring-Verlag, Berlin
  24. Simo, J.C. and Hughes, T.J.R., 1989, Computational Inelasticity, Springer-Verlag, New York
  25. Simo, J.C, Kennedy, J.G and Taylor, R.L., 1989, 'Complementary mixed finite element formulations for elastoplasticity,' Comput. Methods Appl. Mech. Engrg., Vol. 74, pp. 177-206 https://doi.org/10.1016/0045-7825(89)90102-3
  26. Kwon, K.C. and Youn, S.K, 2004, 'The Least-Squares Meshfree Method for the Analysis of Rigid-Plastic Deformation,' Transactions of the KSME(A), Vol. 28, No. 12, pp. 2019-2031 https://doi.org/10.3795/KSME-A.2004.28.12.2019