Stabilization procedures in coupled poromechanics problems: A critical assessment

Abstract

SUMMARYNumerical solutions for problems in coupled poromechanics suffer from spurious pressure oscillations when small time increments are used. This has prompted many researchers to develop methods to overcome these oscillations. In this paper, we present an overview of the methods that in our view are most promising. In particular we investigate several stabilized procedures, namely the fluid pressure Laplacian stabilization (FPL), a stabilization that uses bubble functions to resolve the fine-scale solution within elements, and a method derived by using finite increment calculus (FIC). On a simple one-dimensional test problem, we investigate stability of the three methods and show that the approach using bubble functions does not remove oscillations for all time step sizes. On the other hand, the analysis reveals that FIC stabilizes the pressure for all time step sizes, and it leads to a definition of the stabilization parameter in the case of the FPL-stabilization. Numerical tests in one and two dimensions on 4-noded bilinear and linear triangular elements confirm the effectiveness of both the FPL-and the FIC-stabilizations schemes for linear and nonlinear problems.

Coupled multi-phase thermo-poromechanical effects. Case study: CO2 injection at In Salah, Algeria

“…We use quasi-Newton updates to improve convergence while avoiding having to reform the Jacobian matrix at every iteration (Broyden, 1965;Matthies and Strang, 1979) and line search techniques for improving accuracy and stability while keeping the computational cost low. A more detailed description is available in Prévost (1981). This procedure permits the coupling of any two or more field equations without the need for implementing a special element that contains all equations.…”

Performance of mixed formulations for the particle finite element method in soil mechanics problems

“…Several techniques have been developed to address this problem introducing a characteristic stabilization parameter which depends on the element size and/or material properties, see [23]. The fluid pressure Laplacian stabilization (FPL) technique is used in this work because of its good solution accuracy and of the symmetry of the resulting system.…”

“…The expression chosen for the τ is the one proposed by Preisig et al in [23]. Then, the stabilized semi-discrete equations of the primal formulation, Equation (24), are given by…”

“…The reason behind this behaviour is the failure to satisfy the Babuska-Brezzy conditions [2,8] or the equivalent inf-sup condition [3] due to an improper finitedimensional space in the finite element discretization. To avoid this problem two strategies are common: either to use more complex, but stable, finite elements [38,26], or to apply stabilization procedures to originally unstable finite elements [23,39,21]. In the later approach, locking is mitigated at the cost of using a mixed formulation, thereby introducing extra degrees of freedom per node with respect to the primal formulation.…”

“…There are several possible mixed stabilized formulations which are relevant for these problems [22,23]. In this work alternative mixed stabilized formulations both for total stress and for coupled hydromechanical analyses at large strains are explored.…”