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The Least-Squares Meshfree Method for the Analysis of Rigid-Plastic Deformation

강소성 변형 해석을 위한 최소 제곱 무요소법

  • 윤성기 (한국과학기술원 기계공학과) ;
  • 권기찬 (한국과학기술원 기계공학과)
  • Published : 2004.12.01

Abstract

The least-squares formulation for rigid-plasticity based on J$_2$-flow rule and infinitesimal theory and its meshfree implementation using moving least-squares approximation are proposed. In the least-squares formulation the squared residuals of the constitutive and equilibrium equations are minimized. Those residuals are represented in a form of first-order differential system using the velocity and stress components as independent variables. For the enforcement of the boundary and frictional contact conditions, penalty scheme is employed. Also the reshaping of nodal supports is introduced to avoid the difficulties due to the severe local deformation near the contact interface. The proposed least-squares meshfree method does not require any structure of extrinsic cells during the whole process of analysis. Through some numerical examples of metal forming processes, the validity and effectiveness of the method are investigated.

Keywords

LSMFM;Least-Squares;Meshfree Method;Rigid-Plasticity;Metal Forming

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, 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, 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, Berlin, Spring-Verlag
  24. Kobayashi, S, Oh, S.l. and Altan, T., 1989, Metal Forming and the Finite-Element Method, Oxford University Press, New York
  25. Hill, R., 1950, The Mathematical Theory of Plasticity, Oxford University Press, New York
  26. Oh, S.l. and Kobayashi, S., 1980, 'Finite Element Analysis of Plane-strain Sheet Bending,' Int. J. Mech. Sci., Vol. 22, pp. 583-594 https://doi.org/10.1016/0020-7403(80)90020-X