Stress and flux reconstruction in Biot’s poro-elasticity problem with application to a posteriori error analysis

Riedlbeck, Rita; Pietro, Daniele Antonio Di; Ern, Alexandre; Granet, S.; Kazymyrenko, Kyrylo

doi:10.1016/j.camwa.2017.02.005

Cited by 21 publications

(22 citation statements)

References 41 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…1) A computable upper bound for ||R P (p k hτ ,û k hτ )|| X T and ||R U (p k hτ ,û k hτ )|| X T . Proceeding as in [2,40], adding (ŵ k,n h , ∇q) to (6.13), choosing q ∈ X T with ||q|| X T = 1 and applying the Green theorem, and using (5.6), we obtain…”

Section: Guaranteed Reliabilitymentioning

confidence: 99%

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

View full text Add to dashboard Cite

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.

show abstract

Section: Guaranteed Reliabilitymentioning

confidence: 99%

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

View full text Add to dashboard Cite

show abstract

“…Let us prove (19). Since (15) holds for all a ∈ V int h , we infer that (14b) or (18b) is actually true for all…”

Section: Properties Of the Stress Reconstructionsmentioning

confidence: 86%

“…The condition (15) for all a ∈ V int h ensures that the constrained minimization problem (18) is well-posed.…”

Section: Arnold-falk-winther Stress Reconstructionmentioning

confidence: 99%

“…Both use mixed finite elements on cell patches around mesh vertices, as proposed for the Poisson problem in [8,5]. The first one was introduced in [15] and uses the Arnold-Winther (AW) mixed finite element spaces [3] providing a symmetric stress tensor. The second one follows the same approach, but imposing the symmetry only weakly and using the ArnoldFalk-Winther (AFW) mixed finite element spaces [2].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Equilibrated Stress Reconstructions for Linear Elasticity Problems with Application to a Posteriori Error Analysis

Riedlbeck

Pietro

Ern

2017

Springer Proceedings in Mathematics &Amp; Statistics

Self Cite

View full text Add to dashboard Cite

We present an a posteriori error estimate for the linear elasticity problem. The estimate is based on an equilibrated reconstruction of the Cauchy stress tensor, which is obtained from mixed finite element solutions of local Neumann problems. We propose two different reconstructions: one using Arnold-Winther mixed finite element spaces providing a symmetric stress tensor, and one using Arnold-FalkWinther mixed finite element spaces with a weak symmetry constraint. The performance of the estimate is illustrated on a numerical test with analytical solution.

show abstract

“…Let us for now suppose that u h solves (2.13) exactly, before considering iterative linearization methods such as (2.14) in Section 3.2. For the stress reconstruction we will use mixed finite element formulations on patches around mesh vertices in the spirit of [37,38]. The mixed finite elements based on the dual formulation of (1.1a) will provide a stress tensor lying in Hpdiv, Ωq.…”

Section: Patchwise Construction In the Arnold-falk-winther Mixed Finimentioning

confidence: 99%

Equilibrated Stress Tensor Reconstruction and A Posteriori Error Estimation for Nonlinear Elasticity

Botti

Riedlbeck²

2018

Computational Methods in Applied Mathematics

Self Cite

View full text Add to dashboard Cite

We consider hyperelastic problems and their numerical solution using a conforming finite element discretization and iterative linearization algorithms. For these problems, we present equilibrated, weakly symmetric, Hpdivq-conforming stress tensor reconstructions, obtained from local problems on patches around vertices using the Arnold-Falk-Winther finite element spaces. We distinguish two stress reconstructions, one for the discrete stress and one representing the linearization error. The reconstructions are independent of the mechanical behavior law. Based on these stress tensor reconstructions, we derive an a posteriori error estimate distinguishing the discretization, linearization, and quadrature error estimates, and propose an adaptive algorithm balancing these different error sources. We prove the efficiency of the estimate, and confirm it on a numerical test with analytical solution for the linear elasticity problem. We then apply the adaptive algorithm to a more application-oriented test, considering the Hencky-Mises and an isotropic damage models.

show abstract

Stress and flux reconstruction in Biot’s poro-elasticity problem with application to a posteriori error analysis

Cited by 21 publications

References 41 publications

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

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

Equilibrated Stress Reconstructions for Linear Elasticity Problems with Application to a Posteriori Error Analysis

Equilibrated Stress Tensor Reconstruction and A Posteriori Error Estimation for Nonlinear Elasticity

Contact Info

Product

Resources

About