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**A modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free Galerkin method.**
*(English)*
Zbl 0947.74080

Summary: The Element Free Galerkin (EFG) method, which is based on the Moving Least Squares (MLS) approximation, requires only nodal data and no element connectivity, and therefore is more flexible than the conventional finite element method. Direct imposition of essential boundary conditions for the element free Galerkin method is always difficult 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 fictitious nodal values \(\widehat{\mathbf u}\) used as undetermined coefficients in the MLS approximation, \(u^h({\mathbf x}) [u^h({\mathbf x})= \Phi\cdot \widehat{\mathbf u}]\), was used to enforce the essential boundary conditions.

A modified collocation method using the actual nodal values of the trial function \(u^h ({\mathbf x})\) is presented here, to enforce the essential boundary conditions. This modified 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 definite 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.

A modified collocation method using the actual nodal values of the trial function \(u^h ({\mathbf x})\) is presented here, to enforce the essential boundary conditions. This modified 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 definite 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.

### MSC:

74S30 | Other numerical methods in solid mechanics (MSC2010) |