AM  Vol.6 No.1 , January 2015
Element Free Gelerkin Method for 2-D Potential Problems
ABSTRACT
A meshfree method namely, element free Gelerkin (EFG) method, is presented in this paper for the solution of governing equations of 2-D potential problems. The EFG method is a numerical method which uses nodal points in order to discretize the computational domain, but where the use of connectivity is absent. The unknowns in the problems are approximated by means of connectivity-free technique known as moving least squares (MLS) approximation. The effect of irregular distribution of nodal points on the accuracy of the EFG method is the main goal of this paper as a complement to the precedent researches investigated by proposing an irregularity index (II) in order to analyze some 2-D benchmark examples and the results of sensitivity analysis on the parameters of the method are presented.

Cite this paper
Firoozjaee, A. , Hendi, E. and Farvizi, F. (2015) Element Free Gelerkin Method for 2-D Potential Problems. Applied Mathematics, 6, 149-162. doi: 10.4236/am.2015.61015.
References
[1]   Kreyszig, E. (2010) Advanced Engineering Mathematics. Wiley, Hoboken.

[2]   Liu, G.R. and Gu, Y. (2005) An Introduction to Meshfree Methods and Their Programming, Vol. 1. Springer, Berlin.

[3]   Zhuang, X. (2010) Meshless Methods: Theory and Application in 3D Fracture Modelling with Level Sets. Durham University, Durham.

[4]   Monaghan, J.J. (1988) An Introduction to SPH. Computer Physics Communications, 48, 89-96.
http://dx.doi.org/10.1016/0010-4655(88)90026-4

[5]   Onate, E., et al. (1996) A Finite Point Method in Computational Mechanics. Applications to Convective Transport and Fluid Flow. International Journal for Numerical Methods in Engineering, 39, 3839-3866.
http://dx.doi.org/10.1002/(SICI)1097-0207(19961130)39:22<3839::AID-NME27>3.0.CO;2-R

[6]   Nayroles, B., Touzot, G. and Villon, P. (1992) Generalizing the Finite Element Method: Diffuse Approximation and Diffuse Elements. Computational Mechanics, 10, 307-318.
http://dx.doi.org/10.1007/BF00364252

[7]   Belytschko, T., Lu, Y.Y. and Gu, L. (1994) Element-Free Galerkin Methods. International Journal for Numerical Methods in Engineering, 37, 229-256.
http://dx.doi.org/10.1002/nme.1620370205

[8]   Liu, G.R. and Gu, Y. (2001) A Point Interpolation Method for Two-Dimensional Solids. International Journal for Numerical Methods in Engineering, 50, 937-951.
http://dx.doi.org/10.1002/1097-0207(20010210)50:4<937::AID-NME62>3.0.CO;2-X

[9]   Liszka, T.J., Duarte, C.A.M. and Tworzydlo, W.W. (1996) hp-Meshless Cloud Method. Computer Methods in Applied Mechanics and Engineering, 139, 263-288.
http://dx.doi.org/10.1016/S0045-7825(96)01086-9

[10]   Babuska, I. and Melenk, J.M. (1997) The Partition of Unity Method. International Journal for Numerical Methods in Engineering, 40, 727-758.
http://dx.doi.org/10.1002/(SICI)1097-0207(19970228)40:4<727::AID-NME86>3.0.CO;2-N

[11]   Atluri, S. and Zhu, T. (1998) A New Meshless Local Petrov-Galerkin (MLPG) Approach in Computational Mechanics. Computational Mechanics, 22, 117-127.
http://dx.doi.org/10.1007/s004660050346

[12]   Gu, Y. and Liu, G.R. (2001) A Local Point Interpolation Method for Static and Dynamic Analysis of Thin Beams. Computer Methods in Applied Mechanics and Engineering, 190, 5515-5528.
http://dx.doi.org/10.1016/S0045-7825(01)00180-3

[13]   Rahmani Firoozjaee, A. and Afshar, M.H. (2011) Discrete Least Squares Meshless (DLSM) Method for Simulation of Steady State Shallow Water Flows. Scientia Iranica, 18, 835-845.
http://dx.doi.org/10.1016/j.scient.2011.07.016

[14]   Gu, Y. and Liu, G. (2002) A Boundary Point Interpolation Method for Stress Analysis of Solids. Computational Mechanics, 28, 47-54.
http://dx.doi.org/10.1007/s00466-001-0268-9

[15]   Liew, K., Cheng, Y. and Kitipornchai, S. (2007) Analyzing the 2D Fracture Problems via the Enriched Boundary Element-Free Method. International Journal of Solids and Structures, 44, 4220-4233.
http://dx.doi.org/10.1016/j.ijsolstr.2006.11.018

[16]   Liew, K., Cheng, Y. and Kitipornchai, S. (2005) Boundary Element-Free Method (BEFM) for Two-Dimensional Elastodynamic Analysis Using Laplace Transform. International Journal for Numerical Methods in Engineering, 64, 1610-1627.
http://dx.doi.org/10.1002/nme.1417

[17]   Liew, K., Cheng, Y. and Kitipornchai, S. (2006) Boundary Element-Free Method (BEFM) and Its Application to Two-Dimensional Elasticity Problems. International Journal for Numerical Methods in Engineering, 65, 1310-1332.
http://dx.doi.org/10.1002/nme.1489

[18]   Liew, K., Sun, Y. and Kitipornchai, S. (2007) Boundary Element-Free Method for Fracture Analysis of 2-D Anisotropic Piezoelectric Solids. International Journal for Numerical Methods in Engineering, 69, 729-749.
http://dx.doi.org/10.1002/nme.1786

[19]   Zhang, Z., Zhao, P. and Liew, K. (2009) Improved Element-Free Galerkin Method for Two-Dimensional Potential Problems. Engineering Analysis with Boundary Elements, 33, 547-554.
http://dx.doi.org/10.1016/j.enganabound.2008.08.004

[20]   Young, D., Chen, K. and Lee, C. (2005) Novel Meshless Method for Solving the Potential Problems with Arbitrary Domain. Journal of Computational Physics, 209, 290-321.
http://dx.doi.org/10.1016/j.jcp.2005.03.007

[21]   Firoozjaee, A.R. and Afshar, M.H. (2009) Discrete Least Squares Meshless Method with Sampling Points for the Solution of Elliptic Partial Differential Equations. Engineering Analysis with Boundary Elements, 33, 83-92.
http://dx.doi.org/10.1016/j.enganabound.2008.03.004

[22]   Singh, I. and Singh, A. (2009) A Meshfree Solution of Tow-Dimensional Potential Flow Problems. World Academy of Science, Engineering and Technology, International Science Index, 27, 587-597.

[23]   Lancaster, P. and Salkauskas, K. (1981) Surfaces Generated by Moving Least Squares Methods. Mathematics of Computation, 37, 141-158.
http://dx.doi.org/10.1090/S0025-5718-1981-0616367-1

 
 
Top