13th International Conference on Fracture June 16–21, 2013, Beijing, China -8- Therefore, to ensure the error does not grow during the iterations (with increasing i ), the spectral norm of Gshould be 1 1 1 ' ' 1 T f t G AB H SSSQKQ (25) This stability criterion is identical to that of (17). 5.3 Staggered stability This part is to prove that the error does not grow during the time marching (i.e. from 0 1 n t t t ). The error in (24) can be further written as 1 1 1 ,( 1) 1 ,(0) 1 1 1 1 1 ,( 1) ,(0) 1 1 1 1 1 1 ,( 1) ,(0) ,(0) 1 1 1 1 1 0 1 0 0 1 1 0 ' ' ' ' ' p i i p i p i n n n n p i p p n n n n n n n n p i p p n n n n n l n l n l l n p n n n l n l n l l e G e IG IGHe IG IG A r e L e M e N r e L e M M L e M M e M M N r 1 1 ' n n N r (26) If a full iteration (i.e. large value of i ) is adopted, 0 i L G ; or in the special case that 0 G , 0L , then the first two terms on the right side in the last equation of (26) vanish. Also considering that the truncation error 2 ' ( ) O t r [11], to ensure the stability of the staggered procedure, it requires that 1 1 1 1 1 ' 1. i T T f t M IG IG H H SQKQ SQKQ (27) Therefore, the staggered procedure is unconditionally stable as long as (25) is satisfied. 6. Numerical Simulations In this section, a 1D consolidation with elastic medium is carried out to verify the iterative coupling scheme described before. The effects of support sizes on the number of iterations are also demonstrated. 6.1 1D Consolidation with stabilization term As shown in Figure 1, a 10-m thick saturated elastic medium on an impervious base is subjected to a surface surcharge of 20kPa. Impervious boundaries are assigned to two sides and the base of the domain. A free flow boundary, i.e. 0 p kPa , is assigned to the top surface. Young's modulus of the elastic medium is E=10Mpa, Poisson's ratio is 0.2 , and the permeability is 8 5 10 / m s . The initial pore water pressure 0p is 20 kPaand effective stress is zero. Support size of 1.5 times of the particle interval is used in the simulations. As derived before, the stabilization term 1 T S Q K Q should be used to achieve an optimal convergence rate. However, for mathematical simplicity, 1 ( )p T pd K S N N is used in the simulation presented here to demonstrate the convergence behavior of the iterative scheme. Kis the bulk modulus. The error tolerance are set as 3 10 p p TOL TOL . The pore water pressure distribution along the mid-column nodes in the simulation is presented and compared with the analytical solution in Figure 2. The simulation and
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