237 research outputs found
Numerical approximation of poroelasticity with random coefficients using Polynomial Chaos and Hybrid High-Order methods
In this work, we consider the Biot problem with uncertain poroelastic
coefficients. The uncertainty is modelled using a finite set of parameters with
prescribed probability distribution. We present the variational formulation of
the stochastic partial differential system and establish its well-posedness. We
then discuss the approximation of the parameter-dependent problem by
non-intrusive techniques based on Polynomial Chaos decompositions. We
specifically focus on sparse spectral projection methods, which essentially
amount to performing an ensemble of deterministic model simulations to estimate
the expansion coefficients. The deterministic solver is based on a Hybrid
High-Order discretization supporting general polyhedral meshes and arbitrary
approximation orders. We numerically investigate the convergence of the
probability error of the Polynomial Chaos approximation with respect to the
level of the sparse grid. Finally, we assess the propagation of the input
uncertainty onto the solution considering an injection-extraction problem.Comment: 30 pages, 15 Figure
Two-field finite element solver for linear poroelasticity, A
Includes bibliographical references.2020 Summer.Poroelasticity models the interaction between an elastic porous medium and the fluid flowing in it. It has wide applications in biomechanics, geophysics, and soil mechanics. Due to difficulties of deriving analytical solutions for the poroelasticity equation system, finite element methods are powerful tools for obtaining numerical solutions. In this dissertation, we develop a two-field finite element solver for poroelasticity. The Darcy flow is discretized by a lowest order weak Galerkin (WG) finite element method for fluid pressure. The linear elasticity is discretized by enriched Lagrangian () elements for solid displacement. First order backward Euler time discretization is implemented to solve the coupled time-dependent system on quadrilateral meshes. This poroelasticity solver has some attractive features. There is no stabilization added to the system and it is free of Poisson locking and pressure oscillations. Poroelasticity locking is avoided through an appropriate coupling of finite element spaces for the displacement and pressure. In the equation governing the flow in pores, the dilation is calculated by taking the average over the element so that the dilation and the pressure are both approximated by constants. A rigorous error estimate is presented to show that our method has optimal convergence rates for the displacement and the fluid flow. Numerical experiments are presented to illustrate theoretical results. The implementation of this poroelasticity solver in deal.II couples the Darcy solver and the linear elasticity solver. We present the implementation of the Darcy solver and review the linear elasticity solver. Possible directions for future work are discussed
Weakly imposed symmetry and robust preconditioners for Biot's consolidation model
We discuss the construction of robust preconditioners for finite element
approximations of Biot's consolidation model in poroelasticity. More precisely,
we study finite element methods based on generalizations of the
Hellinger-Reissner principle of linear elasticity, where the stress tensor is
one of the unknowns. The Biot model has a number of applications in science,
medicine, and engineering. A challenge in many of these applications is that
the model parameters range over several orders of magnitude. Therefore,
discretization procedures which are well behaved with respect to such
variations are needed. The focus of the present paper will be on the
construction of preconditioners, such that the preconditioned discrete systems
are well-conditioned with respect to variations of the model parameters as well
as refinements of the discretization. As a byproduct, we also obtain
preconditioners for linear elasticity that are robust in the incompressible
limit.Comment: 21 page
Coupling nonconforming and enriched Galerkin methods for robust discretization and fast solvers of poroelasticity problems
In this paper we propose a new finite element discretization for the
two-field formulation of poroelasticity which uses the elastic displacement and
the pore pressure as primary variables. The main goal is to develop a numerical
method with small problem sizes which still achieve key features such as
parameter-robustness, local mass conservation, and robust preconditionor
construction. For this we combine a nonconforming finite element and the
interior over-stabilized enriched Galerkin methods with a suitable
stabilization term. Robust a priori error estimates and parameter-robust
preconditioner construction are proved, and numerical results illustrate our
theoretical findings
FETI-DP algorithms for 2D Biot model with discontinuous Galerkin discretization
Dual-Primal Finite Element Tearing and Interconnecting (FETI-DP) algorithms
are developed for a 2D Biot model. The model is formulated with mixed-finite
elements as a saddle-point problem. The displacement and the Darcy
flux flow are represented with piecewise continuous elements
and pore-pressure with piecewise constant elements, {\it i.e.},
overall three fields with a stabilizing term. We have tested the functionality
of FETI-DP with Dirichlet preconditioners. Numerical experiments show a
signature of scalability of the resulting parallel algorithm in the
compressible elasticity with permeable Darcy flow as well as almost
incompressible elasticity.Comment: Accepted to the 27th International Conference on Domain Decomposition
Methods (DD27), 8 pages. arXiv admin note: text overlap with arXiv:2211.1502
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