--

3 (1) 2013

A linearly conforming point interpolation method (LC-PIM) for perfect Visco-Elastoplastic analysis of 2D solids


Author - Affiliation:
Bui Xuan Thang - Department of Mechanics, Faculty of Mathematics & Computer Science, University of Science, Vietnam National University – HCMC , Vietnam
Nguyen Thoi Trung - Division of Computational Mechanics, Ton Duc Thang University , Vietnam
Nguyen Xuan Hung - Division of Computational Mechanics, Ton Duc Thang University , Vietnam
Phung Van Phuc - Division of Computational Mechanics, Ton Duc Thang University , Vietnam
Corresponding author: Phung Van Phuc - kim.npt@ou.edu.vn

Abstract
A linearly conforming point interpolation method (LC-PIM) was recently proposed for the solid mechanics problems. In this paper, the LC-PIM is further extended to perfect visco-elastoplastic analyses of 2D solids. A dual formulation for the LC-PIM with displacements and stresses as the main variables is performed. The von-Mises yield function and the Prandtl-Reuss flow rule are used. In the numerical procedure, however, the stress variables are eliminated and the problem becomes only displacementdependent. The numerical results show that the LC-PIM is much more accurate than the FEM and possesses the upper bound property which is very meaningful for the viscoelastoplastic analyses which almost have not got the analytical solutions. This suggests that we can use two models, LC-PIM and FEM, to bound the solution, and can even estimate the global relative error of numerical solutions.

Keywords
Numerical methods; mesh-free methods; linearly conforming point interpolation method (LC-PIM); upper bound; visco-elastoplastic analyses.

Full Text:
PDF

References

Lucy LB. A numerical approach to testing the fission hypothesis. The Astronomical Journal 1977; 8(12):1013–1024.


Liu GR, Liu MB. Smoothed Particle Hydrodynamics—A Meshfree Practical Method. World Scientific: Singapore, 2003.


Liszka T, Orkisz J. The finite difference methods at arbitrary irregular grids and its applications in applied mechanics. Computers and Structures 1980; 11:83–95.


Nayroles B, Touzot G, Villon P. Generalizing the finite element method: diffuse approximation and diffuse elements. Computational Mechanics 1992; 10:307–318.


Belytschko Y, Lu YY, Gu L. Element-free Galerkin methods. International Journal for Numerical Methods in Engineering 1994; 37:229–256.


Liu WK, Jun S, Zhang YF. Reproducing kernel particle methods. International Journal for Numerical Methods in Engineering 1995; 20:1081–1106.


Atluri SN, Zhu T. A new meshless local Petrov–Galerkin (MLPG) approach in computational mechanics. Computational Mechanics 1998; 22:117–127.


Liu GR, Gu YT. A point interpolation method for two-dimensional solids. International Journal for Numerical Methods in Engineering 2001; 50:937–951.


Liu GR. Meshfree Methods: Moving Beyond the Finite Element Method. CRC Press: Boca Raton, FL, 2002.


Liu GR, Gu YT. An Introduction to Meshfree Methods and their Programming. Springer: Dordrecht, The Netherlands, 2005.


Wang JG, Liu GR. A point interpolation meshless method based on radial basis functions. International Journal for Numerical Methods in Engineering 2002; 54:1623–1648.


Liu GR, Zhang GY, Gu YT, Wang YY. A meshfree radial point interpolation method (RPIM) for three-dimensional solids. Computational Mechanics 2005; 36(6):421–430.


Chen JS, Wu CT, Yoon S, You Y. A stabilized conforming nodal integration for Galerkin mesh-free methods. International Journal for Numerical Methods in Engineering 2001; 50:435–466.


Liu GR, Zhang GY, Dai KY, Wang YY, Zhong ZH, Li GY, Han X. A linearly conforming point interpolation method (LC-PIM) for 2D solid mechanics problems. International Journal of Computational Methods 2005; 2(4):645–665.


Liu GR, Li Y, Dai KY, Luan MT, Xue W. A linearly conforming RPIM for 2D solid mechanics. International Journal of Computational Methods 2006.


Zhang GY, Liu GR, Wang YY, Huang HT, Zhong ZH, Li GY, Han X. A linearly conforming point interpolation method (LC-PIM) for three-dimensional elasticity problems. International Journal for Numerical Methods in Engineering 2007; 72:1524–1543.


J.S. Chen, C.T. Wu, S. Yoon, Y. You, A stabilized conforming nodal integration for Galerkin meshfree method, International Journal for Numerical Methods in Engineering. 50 (2001) 435-466.


T. Nguyen-Thoi, G.R. Liu, H.C. Vu-Do, H. Nguyen-Xuan, An edge-based smoothed finite element method (ES-FEM) for visco-elastoplastic analyses in 2D solids using triangular mesh, Computational Mechanics. 45 (2009) 23-44.


T. Nguyen-Thoi, G.R. Liu, H.C. Vu-Do, H. Nguyen-Xuan, A face-based smoothed finite element method (FS-FEM) for visco-elastoplastic analyses of 3D solids using tetrahedral mesh, Computer Methods in Applied Mechanics and Engineering. 198 (2009) 3479-3498.


G.R. Liu, T. Nguyen-Thoi, K.Y. Lam, A novel Alpha Finite Element Method (aFEM) for exact solution to mechanics problems using triangular and tetrahedral elements, Computer Methods in Applied Mechanics and Engineering. 197 (2008) 3883-3897.


C. Carstensen, R. Klose, Elastoviscoplastic Finite Element Analysis in 100 lines of Matlab, Journal of Numerical Mathematics. 10 (2002) 157-192.


W. Han, B.D. Reddy, Computational plasticity: The variational basis and numerical analysis, Computational Mechanics Advances. 2 (1995) 283:400.


C. Carstensen, S.A. Funken, Averaging technique for FE-a posteriori error control in elasticity. Part 1: Conforming FEM, Computer Methods in Applied Mechanics and Engineering. 190 (2001) 2483-2498.


G.L. Liu, Nguyen Thoi Trung, Smoothed Finite Element Methods. CRC Press: Boca Raton, Florida, 2010.



Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.