Computational aspects in 2D SBEM analysis with domain inelastic actions

Abstract

The Symmetric Boundary Element Method, applied to structures subjected to temperature and inelastic actions, shows singular domain integrals.\ud In the present paper the strong singularity involved in the domain integrals of the stresses and tractions is removed and, by means of a limiting operation, this traction is evaluated on the boundary. First the weakly singular domain integral in the Somigliana Identity of the displacements is regularized and the singular integral is transformed into a boundary one using the Radial Integration Method; subsequently, using the differential operator applied to the displacement field, the Somigliana Identity of the tractions inside the body is obtained and through a limit operation its expression is evaluated on the boundary. The latter operation makes it possible to substitute the strongly singular domain integral in a strongly singular boundary one, defined as a Cauchy Principal Value, with which the related free term is associated. The expressions thus obtained for the displacements and the tractions, in which domain integrals are substituted by boundary integrals, were utilized in the Galerkin approach, for the evaluation in closed form of the load coefficients connected to domain inelastic actions.\ud This strategy makes it possible to evaluate the load coefficients avoiding considerable difficulties due to the geometry of the solid analyzed; the obtained coefficients were implemented in the Karnak.sGbem calculus code

Numerical approximation strategy for solutions and their derivatives for two-dimensional solids

“…The same method can be applied in PIES. The application of the proposed in the paper strategy, however, allows for obtaining solutions on the boundary by the series (9) or the Lagrange polynomial (14), and direct obtaining the derivatives of solutions by the differentiated series (15) or the differentiated Lagrange polynomial (19). With their help we can obtain the derivatives of solutions at any point of both the boundary and the domain still using the same expression.…”

“…In order to calculate the derivatives of the function interpolated by the Lagrange polynomial (14) we just have to differentiate it with respect to required variables. As a result we have obtained polynomials approximating derivatives of solutions, such as:…”

“…Assuming that the solutions in the domain and on the boundary are obtained by the interpolation series (9) or the Lagrange polynomial (14), obtaining the derivatives of solutions reduces only to calculate the derivatives of basis functions of (9) and (14). Finally, the calculation of derivatives at any node is reduced only to the direct substitution of coordinates of this node to obtain mathematical expressions approximating the partial derivatives of the desired order.…”

“…Using both variants of the placement of interpolation nodes (only the domain and both the domain and the boundary), such solutions are received directly from the series (9) or the Lagrange polynomial (14) by substituting into them values of Cartesian coordinates of points lying on the boundary, at which we want to get the solution. It should be noted however, that using the variant with interpolation nodes placed only in the domain, solutions are obtained by extrapolation.…”

“…Such situation, among others, is when we want to determine stresses on the boundary in elasticity problems [2,6]. There are several approaches to eliminate this difficulty, including the use of special methods for calculating singular integrals or the transformation of the troublesome equation to the nonsingular version [13,7,14]. We can also determine stresses on the basis of already obtained solutions on the boundary in conjunction with equations of elasticity [6].…”