TY - JOUR
T1 - A modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free Galerkin method
AU - Zhu, T.
AU - Atluri, S. N.
PY - 1998
Y1 - 1998
N2 - 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 flexible than the conventional finite element method. Direct imposition of essential boundary conditions for the element free Galerkin (EFG) 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 û used as undetermined coefficients in the MLS approximation, uh(x) [uh(x) = Φ · û], was used to enforce the essential boundary conditions. A modified collocation method using the actual nodal values of the trial function uh (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. 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 definite system matrix.
AB - 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 flexible than the conventional finite element method. Direct imposition of essential boundary conditions for the element free Galerkin (EFG) 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 û used as undetermined coefficients in the MLS approximation, uh(x) [uh(x) = Φ · û], was used to enforce the essential boundary conditions. A modified collocation method using the actual nodal values of the trial function uh (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. 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 definite system matrix.
UR - http://www.scopus.com/inward/record.url?scp=0032048673&partnerID=8YFLogxK
U2 - 10.1007/s004660050296
DO - 10.1007/s004660050296
M3 - Article
AN - SCOPUS:0032048673
SN - 0178-7675
VL - 21
SP - 211
EP - 222
JO - Computational Mechanics
JF - Computational Mechanics
IS - 3
ER -