Sequential Methods for Coupled Geomechanics and Multiphase Flow

128 CHAPTER 5. FIXED-STRAIN AND FIXED-STRESS SPLITS
We employ B-stability for unconditional stability of the nonlinear problem, which is
expressed as
dχ n+1 N ≤ dχ n N . (5.36)
First, we show B-stability when solving the flow problem. We solve the flow problem
first based on the maximum plastic work, for which we adopt the generalized midpoint
rule described in Simo (1991) and Simo and Govindjee (1991). The algorithmic maximum
plastic work is written as
≪ dΣ n − dΣ n+α , −dΣ n+α ≫
+ ≪ (αCdr∆dε n , 0), (−dσ ′n+α , −dκ n+α ) ≫ ≤ 0, (5.37)
where again d(·) = (·) − ˜
(·), (e.g. ∆dε n = ∆ε n − ∆˜ε n ). The flow problem A p ss also has the
constraint of δd ˙σ = 0, expressed as
dσ n+1 − dσ n = dσ n − dσ n−1 = · · · = dσ 1 − dσ 0
The discrete counterpart of the initial conditions in Equation 5.31 yields
From Equations 5.38 and 5.39, we obtain
which yields
(5.38)
Div dσ 1 − dσ 0 = 0, Div dσ 0 = 0. (5.39)
Div dσ n+1 = Div dσ n = · · · = 0, (5.40)
Div dσ n+α = 0. (5.41)