Sequential Methods for Coupled Geomechanics and Multiphase Flow

5.4. DISCRETE STABILITY OF THE NONLINEAR PROBLEM 131
leading to the same computational cost as the fixed-strain split. Then, the return mapping
is used only when solving the mechanical problem.
After we solve the flow problem, the discrete stability for mechanics A u ss is examined for
the following problem.
Div dσ n+α = 0, dp n+α = 0 ⇒ Div dσ ′n+α = 0. (5.50)
Equation 5.37 is applied again with maximum plastic dissipation. Under A u ss, the first
term of Equation 5.37 is the same as Equation 5.43, and the second term of Equation 5.37
is written as
≪ (αCdr∆dε n , 0), (−dσ ′n+α , −dκ n+α ) ≫
= −
α∆dε n : dσ ′n+α dΩ = 0 (from Equation 5.50). (5.51)
From Equations 5.37, 5.43, and 5.51, the algorithmic dissipation is given by
dΣ n+1 2
E − dΣn 2
E + (2α − 1) dΣ n+1 − dΣ n 2
≤ 0. (5.52)
E
From Equation 5.47, dp n+α = 0 in Equation 5.50 provides
1
2M ( n+1
dp 2 L2 − dp n 2
1
L2) = − (2α − 1)
2M
dp n+1 − dp n 2
L 2 . (5.53)
Using Equations 5.52 and 5.53, the evolution of the norm dχ N for A u ss is written as
dχ n+1 N − dχ n N
dp n+1 2
− 1
2M dpn 2
= dΣ n+1 2 1
+ E 2M L2 − dΣ n 2
E L2
dΣ
≤ −(2α − 1)
n+1 − dΣ n 2 1
+ dp E 2M
n+1 − dp n 2
L2
, (5.54)
from which the condition for unconditional stability of A u ss is 0.5 ≤ α ≤ 1. Therefore,