Iterative solvers for Biot model under small and large deformations

Abstract

We consider L-scheme and Newton-based solvers for Biot model under large deformation. The mechanical deformation follows the Saint Venant-Kirchoff constitutive law. Furthermore, the fluid compressibility is assumed to be non-linear. A Lagrangian frame of reference is used to keep track of the deformation. We perform an implicit discretization in time (backward Euler) and propose two linearization schemes for solving the non-linear problems appearing within each time step: Newton's method and L-scheme. Each linearization scheme is also presented in a monolithic and a splitting version, extending the undrained split methods to non-linear problems. The convergence of the solvers, here presented, is shown analytically for cases under small deformation and numerically for examples under large deformation. Illustrative numerical examples are presented to confirm the applicability of the schemes, in particular, for large deformation.

A real‐time optimization algorithm for the fixed‐stress splitting scheme

“…Because the optimization strategy is only based on the convergence rates of the fixed-stress split, it does not matter, in principle, which mathematical models are considered for fluid flow and geomechanics. This means that non-linear poroelasticity models such as, for example, unsaturated flows, 40 non-linear Biot equations, 41 and large deformations, 42 to mention a few, could potentially benefit from the optimization strategy proposed here. The CGA (with AGS) can also be used for these other models provided that the convergence rate function is as well behaved as they are for the linear poroelasticity model.…”

A historical review of theory of deformable porous media, finite element modeling and solution algorithms

“…An important piece of work in the realm of theoretical convergence analysis of the fixed stress split technique is provided in [103]. The works of [104][105][106][107][108] provide a linear algebra point of view to the solution of the system of equations using the decoupling technique.…”

A moving finite element framework for fast infiltration in nonlinear poroelastic media

“…The well-known theory of large-deformation poroelasticity [28] combines Darcy's law with Terzaghi's effective stress and nonlinear elasticity in a rigorous kinematic framework, leading to a strongly nonlinear coupling between the pore structure and the fluid flow [18,37]. Another nonlinear poroelasticity model that takes large deformations into account is considered in [8]. In this model, the mechanical deformation follows the Saint Venant-Kirchhoff constitutive law for hyperelastic solid materials and the fluid compressibility in the fluid equation is assumed to be nonlinear.…”

“…The resulting system of nonlinearly coupled equations is solved by a standard Picard iterative procedure, which is linearly convergent. In the literature, this system is also solved by Newton's method [8], which is quadratically convergent. The drawbacks of the Newton-Raphson method are that the method is only locally convergent and that the computation of derivatives is needed.…”

“…The drawbacks of the Newton-Raphson method are that the method is only locally convergent and that the computation of derivatives is needed. Another valuable alternative to Picard's method is the L-scheme [8,9]. The L-method is robust and linearly convergent, and does not involve the computation of any derivatives.…”