A modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free Galerkin method

The Element free Galerkin method, which is based on the Moving Least Squares approximation, requires only nodal data and no element connectivity, and therefore is more ̄exible than the conventional ®nite element method. Direct imposition of essential boundary conditions for the element free Galerkin (EFG) method is always dif®cult because the shape functions from the Moving Least Squares approximation do not have the delta function property. In the prior literature, a direct collocation of the ®ctitious nodal values û used as undetermined coef®cients in the MLS approximation, uh…x† ‰uh…x† ˆ U ûŠ, was used to enforce the essential boundary conditions. A modi®ed collocation method using the actual nodal values of the trial function uh…x† is presented here, to enforce the essential boundary conditions. This modi®ed collocation method is more consistent with the variational basis of the EFG method. Alternatively, a penalty formulation for easily imposing the essential boundary conditions in the EFG method with the MLS approximation is also presented. The present penalty formulation yields a symmetric positive de®nite system stiffness matrix. Numerical examples show that the present penalty method does not exhibit any volumetric locking and retains high rates of convergence for both displacements and strain energy. The penalty method is easy to implement as compared to the Lagrange multiplier method, which increases the number of degrees of freedom and yields a non-positive de®nite system matrix. 1 Introduction Meshless approximations have become interesting and promising methods in solving partial differential equations, due to their ̄exibility in practical applications. The element free Galerkin method based on the Moving Least Squares approximation (Belytschko et al. 1994) is one of the meshless approaches. The original form of the element free Galerkin method was proposed by Nayroles et al. (1992), which they called the Diffuse Element Method. Belytschko and his research group have popularized the method in its current form. The Element free Galerkin method is almost identical to the conventional ®nite element method (FEM), as both of them are based on the Galerkin formulation, and employ local interpolation/approximation to approximate the trial functions. The key differences lie in the interpolation methods, integration schemes and in the enforcement of essential boundary conditions. The EFG method employs the Moving Least Squares (MLS) approximants to approximate the trial functions. One of the problems with the MLS approximants is that, in general, they do not pass through the data used to ®t the curve, i.e., they do not have the property of nodal interpolants as in the FEM, i.e., /i…xj† ˆ dij, where /i…xj† is the shape function corresponding to the node at xi, evaluated at a nodal point, xj, and dij is the Kronecker delta, unless the weight functions used in the MLS approximation are singular at nodal points. Also, the shape functions from the MLS approximation are not polynomials. Therefore, the essential boundary conditions in the EFG methods can not be easily and directly enforced. Several approaches have been studied for enforcing the essential boundary conditions in the EFG method ± such as the direct collocation method (Lu et al. 1994; Belytschko and Tabbara 1996; Mukherjee and Mukherjee 1997), Lagrange multipliers method (Belytschko et al. 1994; Krysl and Belytschko 1995), using tractions as Lagrange multipliers (Lu et al. 1994; Mukherjee and Mukherjee 1997), using the weak form of the essential boundary conditions (Lu, Belytschko and Tabbara 1995), and the combined FEM-EFG method (Belytschko et al. 1995; Krongauz and Belytschko 1996; Hegen 1996). Using independent Lagrange multipliers to enforce essential boundary conditions is common in structural analysis when boundary conditions can not be directly applied. However, this method leads to an awkward structure for the linear algebraic equations for the discrete system and increases the number of unknowns; moreover, the discrete linear system is no longer positive de®nite. Using tractions derived from the assumed displacement ®eld as Lagrange multipliers (in a solid mechanics problem) in the modi®ed variational principle can, of course, reduce the number of unknowns and therefore, reduce the computational costs, but it might bring unstable solutions (Xue 1984; Xue and Atluri 1985; Xue et al. 1985; Mang and Gallagher 1977, 1985), and is not as convenient as the direct imposition of the essential boundary conditions. Computational Mechanics 21 (1998) 211±222 Ó Springer-Verlag 1998