An efficient boundary element method for computing accurate stresses in two-dimensional anisotropic problems

Abstract

The stresses calculated by the boundary element method are accurate everywhere except in a narrow region near the boundaries. This “boundary layer effect” is due to the presence of hypersingularities in the relevant kernals. An efficient method that eliminates the boundary layer effect and yields accurate stresses everywhere in two-dimensional anisotropic material problems is presented. This method, the modified displacement gradient method, utilizes two identities—Somigliana's identity and a second identity in terms of displacement gradients and tractions. In this method, the Somigliana's identity is used as in the traditional BEM to determine the displacements and tractions at all nodes on the boundary. All the boundary data is then used to determine the displacement gradients at each of the boundary nodes. These displacement gradients and tractions are then used in the second identity to calculate the displacement gradients (and hence strains) at interior points. The stresses are then calculated using the constitutive relationships. The modified displacement gradient method is applied to several two-dimensional elasticity problems with isotropic and orthotropic materials with circular or elliptic cutouts. Numerical studies indicate that the present method gives accurate stresses even in the boundary layer region and is computationally efficient and attractive. In conjunction with the modified displacement gradient method, three approaches that use different evaluation procedures and locations for determining displacement gradients are used. In the first approach, the averaging approach, displacement gradients are averaged at nodes common to adjacent elements. In the second approach, the non-averaging approach, displacement gradients are not averaged but are stored element-wise. In the last approach, the discontinuous element approach, the gradients are evaluated at the nodes of discontinuous elements. Numerical studies indicate that all three approaches yield nearly same stress results, and hence, any one approach can be used.