13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- ( ) (mass matrix); (stiffness matrix) ( ) ( ) (external forces ) (coupling matrix); ( ) (permeability matrix); ( ) (compressibility t u T u T u u T u T t T p p T p p T p f d d d d d d n d K M N N K B CB f N t N b Q B mN H N k N S N N matrix); ( )( ) ( ) w p T p p T f w d qd f N k b N and [1110 0 0]T m , Bis strain-displacement matrix (the spatial derivatives of the shape functions). The boundary conditions are u u on displacement boundary u and t t on traction boundary t , p p on pressure boundary p and ( ) T f q q n k p b on flux boundary w . The total boundary u t p w . Proper initial boundary conditions should also be given for the numerical simulation to start off. RKPM shape functions ([6,7]) are used in this study. The RKPM shape functions interpolate the solution over discrete nodes, which, unlike the conventional FEM, have no topological connectivity relationship. Here only the formulation of shape functions and essential boundary imposition are briefly introduced. Readers may refer to [6-8] for the detailed formulations and implementations. Function ( ) u x in the domain can be approximated by the following formulation ( ) ( ) ( ; ) u u K dV ñ x x x x (6) where ( ) u x are the values of field variables at particles. ( ; ) K ñ x x is the compactly-supported kernel function formulated in RKPM as multiplication of a correction function ( , ) C x x and a window function ( ) x x , i.e., ( ) ( , ) ( ) K C x x x x x x . The correction function ( , ) C x x is assumed to be linear with respect to( ) x x . The window function may take the form of a cubic spline or Gaussian function and has a rectangular or circular support in 2D case. Accordingly, the displacement can be interpolated as 1 ( ) ( ) NP h J J J u N u x x (7) where the reproducing kernel approximant (shape function) ( ) JN x is given by ( ) ( , ) ( ) J J J J N C V x x x x x (8) Jx and JV are the position and contributing volume of the Jth node, respectively. The RKPM shape function does not possess Kronecker delta property, i.e. ( ) h I I u u x . Therefore, special treatment is required to impose the essential boundary conditions. In this paper, the essential boundary condition is reinforced by transformation method ([7]). 4. Sequential Coupling Scheme There are numerous sequential coupling schemes with varied degrees of success in stability ([4]). This study introduces a simple stabilization term by considering the difference of pore fluid pressures between successive iterations. Rewrite the mechanics solver and fluid solver as follows: Mechanics Solver : (i+1,k+1) (i+1,k) (i+1,k) (i+1,k) (i+1,k) (i+1) u n+1 n+1 n+1 n+1 n+1 n+1 n+1 + + =0 M C u K u Qp f u (9)
RkJQdWJsaXNoZXIy MjM0NDE=