13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- E. Check whether unbal unbal tot TOL F F ‖ ‖ ‖ ‖ . If NO , 1 k k and go to (2)(b) iiA. F. End mechanics solver. Commit ( 1) ( 1, 1) 1 1 i i k n n u u . iii. Check whether both the solutions ( 1) 1 i n p and ( 1) 1 i n u satisfy the convergence criteria, i.e. ( 1) ( ) ( 1) 1 1 1 i i i n n n p TOL p p p and ( 1) ( ) ( 1) 1 1 1 i i i n n n u TOL u u u . If NO, 1 i i , and go to (2)(b)i. iv. End iteration scheme. (c) Update p and u: ( 1) 1 1 i n n p p and ( 1) 1 1 i n n u u . (d) If n < total time step, 1 n n , and go to (2)(a). (3). End time integration. where v u is the velocity vector, unbal F is the unbalanced force, tot F is the total applied force. , , unbal p u TOL TOL TOL are tolerance for the unbalance force, pore water pressure variation and displacement variation, respectively. and are the two numerical parameters for the integration method. In the simulations presented in this paper, 0.25, 0.5 are used. C is taken as the conventional Rayleigh damping given by a b C M K. As mentioned before, the mechanics and fluid solvers may employ different solution schemes. In the scheme proposed, the fluid solver is formulated using implicit integration for pressure, while mechanics solver solves displacement explicitly. The reason for this is that fluid solver is generally ‘more’ linear and can be efficiently solved while highly nonlinear constitutive model may be used in the mechanics solver which makes the implicit method much more difficult. Besides, in the mechanics solver, the mass matrix in step (2)(b)iiC can be approximated as a diagonally-lumped mass matrix, inversion of which can be readily obtained. The computation cost would be significantly reduced, especially for large-scale boundary value problems, where a large number of degrees of freedom are inevitably involved. 5. Stability Analysis Based on the sequential scheme described in Section 4, the stability analysis on three levels needs to be inspected, i.e. stability of individual solvers, stability of iterations within one step (i.e. 1 k k , called iteration stability hereafter), stability during time marching (i.e. 1 n n , called staggered stability hereafter). It should be noted that stability of the former is only a necessary but not sufficient condition for the stability of the latter. Stable individual solvers does not guarantee the stability of iterations between these solvers, nor does a stable pair of solutions at one step guarantee stable solutions during the time marching. A stable iterative scheme requires numerical stability on all the three levels. In this work, the stability analysis of iterations during one step is approached in two ways, one by perturbation theory [9] and the other by error propagation [10]. Detailed analysis is presented as follows. 5.1 Stability of Individual Solvers As shown in Section 4, the equation of momentum equilibrium is solved explicitly. As proved in [16], with Rayleigh damping used and 1 2 4 , the mechanics solver is unconditionally stable. The fluid solver with implicit time difference scheme is called once during every iteration, therefore
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