Journal of the Chinese Institute of Engineers, Vol. 27, No. 4, pp. 473-479 (2004) 473 MOVING ELEMENT FREE PETROV-GALERKIN VISCOUS METHOD Mehrzad Ghorbany* and Ali Reza Soheili ABSTRACT Moving meshless methods are new generation of numerical methods for time dependent partial differential equations that have shock or high gradient region. These methods couple the moving finite element methods (MFE) with meshless methods. Here, grid coordinates are time dependent, unknown and are found together with an approximate solution to time dependent PDE. Weak form system is an stiff ODE system and here, it will be found with Galerkin and Petrov-Galerkin method. A pen- alty is appended to the energy functional for preventing high velocity, colliding and collapsing of nodes and prevention of concentration of all the nodes in the shock region. It controls their motion and also causes better conditioning of the mass matrix. Nu- merical solution of two examples demonstrates the accuracy of the approximation. Key Words: r-refinement, adaptive grids, moving finite element, moving least square method, diffuse element method, element free Galerkin and Petrov- Galerkin method. *Corresponding author. (Email: ghorbany@hamoon.usb.ac.ir) M. Ghorbany and A. R. Soheili are with the Department of Mathematics, Sistan and Baluchestan University, Zahedan, Iran. A. R. Soheili is currently at Simon Fraser University, B.C., Canada. I. INTRODUCTION Numerical solution of time dependent partial dif- ferential equations with shock, boundary layer, high gradient region and high oscillatory region such as gas dynamics problems, large deformation process, explosion and underwater bubble explosion, plasticity, elasticity, crack propagation phenomenon, wave propa- gation and penetration are subject of research for many years. These problems have some wild small region in which the solution has not good activity and this region moves with time. Approximation of this re- gion needs special techniques. Up to this time, there are two main r-refinement methods for numerical so- lution of time dependent PDE’s with shock: (a) mov- ing mesh methods and (b) moving finite element meth- ods with time dependent or moveable nodes. In these methods, the aim is moving of a fixed number of nodes and finding their adaptive coordinates to handle the activity of the problem and then finding approximate solution at them. In these methods, one gets an optimal process with minimal work. There is no ad- dition or deletion of nodes and there is no need to raise the computation order. Progressing the meshless methods, such as Shepard Interpolant (Shepard, 1968), Smoothed Particle Hydrodynamics (SPH) (Lucy, 1977; Gingold and Monaghan, 1982; Monaghan, 1988), Moving Least Square method (MLS) for interpola- tion and non-interpolation (approximation) (Lancaster and Salkauskas, 1981), Generalized Finite Difference (GFD) method (Liszka and Orkisz, 1984), Kansa’s method based on radial basis functions (Kansa, 1990), Diffuse Element Method (DEM) and Diffuse Approxi- mation Method (DAM) (Nayroles et al., 1992), Ele- ment Free Galerkin (EFG) method (Belytschko et al. , 1995), Element Free Petrov-Galerkin method (EFPG) (Krongauz and Belytschko, 1997), Reproducing Kernel Method (RKM) and Reproducing Kernel Particle Method (RKPM) (Sukky et al. , 1998; Aluru, 2000), Partition of Unity Method (PUM) (Babuska and Melenk, 1997), h-p clouds and h-p meshless method (Duarte and Oden, 1996), Wavelet Galerkin method (WGM) (Glowinsky et al. , 1990), and the methods for problems with shocks were explored. Two main advantages of the meshless methods are: (a) compu- tational efficiency by avoiding the mesh generation and remeshing process which explored high volume of computational work and ( b ) flexibility and