13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- singularity subtraction technique and the variable transformation technique the strong and weak singularities in Eqs. (11) can be removed [2, 8, 9]. 3. Numerical solution procedure In order to avoid the domain discretization into internal cells for evaluating the domain integrals in Eqs. (12) and (13) the radial integration method (RIM) developed by Gao is applied [3, 4]. The functions (12) and (13) can be rewritten in matrix form [8] as ( , ) ( , , ) ( , ) ( , , ) ( , ) , p p p d p p d F x F x y u y G x y u y (14) where F is the vector of functions (1)( )u iF and (1)( ) F , u is the vector containing the displacements iu and the temperature , and the 3 3 matrices F and G are given in [8]. The unknown fields iu or are approximated by a series of prescribed basis functions and the linear polynomials 0 1 1 ( , ) ( ) ( ) ( ) ( ), ( ) ( ) 0, A A j A A A i i i j i i i j A A A u p p R a p x a p p p x x (15) where || || A R x x is the distance from the application point A to the field point x, Ai and j ia are the unknown expansion coefficients to be determined and A j x denotes the coordinates at the application point A, which consist of all boundary nodes and some selected internal nodes. The fourth order spline-type radial basis function [3, 4] is used 2 3 4 ( ) 1 6 8 3 . A R R R R (16) The unknown coefficients Ai and j ia can be determined by applying the application point A in Eq. (16) to every node. Then, a system of linear algebraic equations can be obtained in matrix form as , u (17) where is the vector consisting of the coefficients Ai for all points and j ia . If two application points do not coincide, the matrix is invertible and thereby 1 . u (18) Substitution of Eqs. (15) into the domain integrals of Eq. (14) yields 0 . a a k ij j j ij j ij k j ij G u d G d a G y d a G d (19) Applying the RIM [3, 4, 8] to the domain integrals in Eq. (19) results in , 1 0 0 1 1 ( ) k a a k k ij j j ij j ij j k j ij r r r r G u d F d a F d a x a F d r n r n r n (20) with the radial integrals 1 2 0 0 0 0 , . , r r r a a ij ij ij ij ij ij F rG dr F r G dr F rG dr (21)
RkJQdWJsaXNoZXIy MjM0NDE=