A multipoint stress mixed finite element method for elasticity on simplicial grids

Ambartsumyan, Ilona; Khattatov, Eldar; Nordbotten, Jan Martin; Yotov, Ivan

doi:10.48550/arxiv.1805.09920

Cited by 4 publications

(21 citation statements)

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“…( 33) has the structure of a doubly non-linear evolution equation. The existence (and uniqueness) of a solution to (33) follows by employing Thm. 2; under above assumptions, it is sufficient to check that E red complies with the assumptions of Thm.…”

Section: Abstract Well-posedness Results For Degenerate Doubly Non-li...mentioning

confidence: 99%

“…We note, that the balance of linear momentum is incorporated intrinsically in Ṡ, which is characteristic for the dual formulation. Imposing only weak symmetry of stress tensors however is our choice, which is motivated by current advances in the robust discretization of the mixed formulation of elasticity and poro-elasticity, cf., e.g., [59,33,60,61,32]; imposing strong symmetry is also possible.…”

Section: Dual Formulation Of Linear Poro-elasticitymentioning

confidence: 99%

“…So far, no minimization formulation presented above lays a foundation for a fully structurepreserving, conforming Galerkin finite element discretization, which is conserving locally both mass and linear momentum. In order to achieve this, a fully mixed five-field formulation can be used, i.e., mixed formulations for both the mechanical and the fluid flow subproblems, cf., e.g., [59,33,32]. Consequently, both independent subproblems incorporate themselves a saddle point structure; however, different to the two-and three-field formulations, the coupling of the two subproblems is symmetric.…”

Section: Five-field Formulation Of Time-discrete Linear Poro-elasticitymentioning

confidence: 99%

“…We are not aware of any explicit result on the well-posedness of linear poro-viscoelasticity models, which consider strains composed of an elastic and a visco-elastic contribution as modelled by [22], or non-linear poro-elasticity under an infinitesimal strain assumption as studied in a fully discretized setting by [30]. In terms of numerical discretization and solution of the linear Biot model, stable, spatial discretizations for various choices of primary variables have been introduced [31,32,33,34]. Furthermore, physically motivated, robust operator splittings have been of great, recent interest, allowing for either using independent, tailored simulators for different physics or developing good block preconditioners for monolithic Krylov subspace methods.…”

Section: Introductionmentioning

confidence: 99%

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The gradient flow structures of thermo-poro-visco-elastic processes in porous media

Both,

Kumar,

Nordbotten

et al. 2019

Preprint

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In this paper, the inherent gradient flow structures of thermo-poro-visco-elastic processes in porous media are examined for the first time. In the first part, a modelling framework is introduced aiming for describing such processes as generalized gradient flows requiring choices of physical states, corresponding energies, dissipation potentials and external work rates. It is demonstrated that various existing models can be in fact written within this framework. Ultimately, the particular structure allows for a unified well-posedness analysis performed for different classes of linear and non-linear models. In the second part, the gradient flow structures are utilized for constructing efficient discrete approximation schemes for thermo-poro-visco-elasticity -in particular robust, physical splitting schemes. Applying alternating minimization to naturally arising minimization formulations of (semi-)discrete models is proposed. For such, the energy decrease per iteration is quantified by applying abstract convergence theory only utilizing convexity and Lipschitz continuity properties of the problem -a fairly simple but flexible machinery. By this approach, e.g., the widely used undrained and fixed-stress splits for the linear Biot equations are derived and analyzed. By application of the framework to more advanced models, novel splitting schemes with guaranteed theoretical convergence rates are naturally derived. Moreover, based on the minimization character of the (semi-)discrete equations, relaxation of splitting schemes by line search is proposed; numerical results show a potentially great impact on the acceleration of splitting schemes for both linear and nonlinear problems.

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Section: Abstract Well-posedness Results For Degenerate Doubly Non-li...mentioning

confidence: 99%

Section: Dual Formulation Of Linear Poro-elasticitymentioning

confidence: 99%

Section: Five-field Formulation Of Time-discrete Linear Poro-elasticitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The gradient flow structures of thermo-poro-visco-elastic processes in porous media

Both,

Kumar,

Nordbotten

et al. 2019

Preprint

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“…Our approach is based on the five-field formulation proposed in [21]. We couple the recently developed multipoint stress mixed finite element (MSMFE) method for elasticity [2,3] with weak stress symmetry and the multipoint flux mixed finite element (MFMFE) method for Darcy flow [18,43,45]. The MFMFE method is related to the the finite volume multipoint flux approximation (MPFA) method [1,14].…”

Section: Introductionmentioning

confidence: 99%

A coupled multipoint stress -- multipoint flux mixed finite element method for the Biot system of poroelasticity

Ambartsumyan,

Khattatov,

Yotov

2020

Preprint

Self Cite

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We present a mixed finite element method for a five-field formulation of the Biot system of poroelasticity that reduces to a cell-centered pressure-displacement system on simplicial and quadrilateral grids. A mixed stress-displacement-rotation formulation for elasticity with weak stress symmetry is coupled with a mixed velocity-pressure Darcy formulation. The spatial discretization is based on combining the multipoint stress mixed finite element (MSMFE) method for elasticity and the multipoint flux mixed finite element (MFMFE) method for Darcy flow. It uses the lowest order Brezzi-Douglas-Marini mixed finite element spaces for the poroelastic stress and Darcy velocity, piecewise constant displacement and pressure, and continuous piecewise linear or bilinear rotation. A vertex quadrature rule is applied to the velocity, stress, and stress-rotation bilinear forms, which block-diagonalizes the corresponding matrices and allows for local velocity, stress, and rotation elimination. This leads to a cell-centered positive-definite system for pressure and displacement at each time step. We perform error analysis for the semidiscrete and fully discrete formulations, establishing first order convergence for all variables in their natural norms. The numerical tests confirm the theoretical convergence rates and illustrate the locking-free property of the method.

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Adaptive asynchronous time-stepping, stopping criteria, and a posteriori error estimates for fixed-stress iterative schemes for coupled poromechanics problems

Ahmed

Nordbotten

Radu

2020

Journal of Computational and Applied Mathematics

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In this paper we develop adaptive iterative coupling schemes for the Biot system modeling coupled poromechanics problems. We particularly consider the space-time formulation of the fixed-stress iterative scheme, in which we first solve the problem of flow over the whole space-time interval, then exploiting the space-time information for solving the mechanics. Two common discretizations of this algorithm are then introduced based on two coupled mixed finite element methods in-space and the backward Euler scheme in-time. Therefrom, adaptive fixed-stress algorithms are build on conforming reconstructions of the pressure and displacement together with equilibrated flux and stresses reconstructions. These ingredients are used to derive a posteriori error estimates for the fixed-stress algorithms, distinguishing the different error components, namely the spatial discretization, the temporal discretization, and the fixed-stress iteration components. Precisely, at the iteration k ≥ 1 of the adaptive algorithm, we prove that our estimate gives a guaranteed and fully computable upper bound on the energy-type error measuring the difference between the exact and approximate pressure and displacement. These error components are efficiently used to design adaptive asynchronous time-stepping and adaptive stopping criteria for the fixed-stress algorithms. Numerical experiments illustrate the efficiency of our estimates and the performance of the adaptive iterative coupling algorithms.

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A multipoint stress mixed finite element method for elasticity on simplicial grids

Cited by 4 publications

References 0 publications

The gradient flow structures of thermo-poro-visco-elastic processes in porous media

The gradient flow structures of thermo-poro-visco-elastic processes in porous media

A coupled multipoint stress -- multipoint flux mixed finite element method for the Biot system of poroelasticity

Adaptive asynchronous time-stepping, stopping criteria, and a posteriori error estimates for fixed-stress iterative schemes for coupled poromechanics problems

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