On the optimization of the fixed‐stress splitting for Biot's equations

Storvik, Erlend; Both, Jakub Wiktor; Kumar, Kundan; Nordbotten, Jan Martin; Radu, F.

doi:10.1002/nme.6130

Cited by 53 publications

(55 citation statements)

References 55 publications

(198 reference statements)

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“…Therein, we stop the algorithm when the adaptive stopping criteria is satisfied. Clearly, the estimator behaves very similarly to what is usually observed for the fixed-stress error (see, e.g., [43]). Moreover, the theoretical parameter (marked by a star) coincides with the numerically optimal value.…”

Section: Stopping Criteria Balancing the Error Componentssupporting

confidence: 71%

“…Adaptive stopping criteria via a posteriori error estimates in the context of other model problems are treated in [1,4,19,25], see also the references therein. Furthermore, the resulting algorithms involve tuning parameters that can be optimized (see [43]); the results show how a posteriori error estimates can help optimize these parameters. To the best of our knowledge, this combination of features in the adaptive fixed-stress algorithms is unique.…”

Section: Domentioning

confidence: 99%

“…Furthermore, Algorithm 3.2 is exploiting the different time scales for the flow and mechanics subsystems. We note also that the efficiency of the two algorithms can be improved when the free parameter β is well-chosen (see [43]) and that step 2. (d) of Algorithm 3.1 in practice, is done in parallel as in [13].…”

Section: Domentioning

confidence: 99%

“…, with δ = 2. The choice of the parameter δ is theoretically calculated in [11,43,32] and possibly should lead to the best performance of the fixed-stress method in terms of number of iterations. We first test the performance of the space-time fixed stress algorithm (Algo.…”

Section: Stopping Criteria Balancing the Error Componentsmentioning

confidence: 99%

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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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Section: Stopping Criteria Balancing the Error Componentssupporting

confidence: 71%

Section: Domentioning

confidence: 99%

Section: Domentioning

confidence: 99%

Section: Stopping Criteria Balancing the Error Componentsmentioning

confidence: 99%

See 2 more Smart Citations

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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“…For a discussion on the stabilization/tuning parameter used in the undrained split approach, we refer to [12,15]. A theoretical investigation on the optimal choice for this parameter is performed in [53]. The linearization is based on either Newton's method, or the L-scheme [37,43,47] or a combination of them [14,37].…”

Section: Introductionmentioning

confidence: 99%

Iterative solvers for Biot model under small and large deformations

et al. 2020

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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.

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Sequential formulation of all‐way coupled finite strain thermoporomechanics for largely deformable gas hydrate deposits

Kim,

Lee

2024

Num Anal Meth Geomechanics

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We develop a numerically stable sequential formulation of thermoporomechanics for largely deformable gas hydrate deposits, extended from the fixed stress split of infinitesimal transformation. Constitutive equations are based on the total Lagrangian approach for both flow and geomechanics, including dynamic full tensor permeability and thermal conductivity updated from the deformation gradient. For space discretization, we take the cell‐centered finite volume and node‐based finite element method for flow and geomechanics, respectively. Then, we propose a sequential implicit method for all‐way coupled thermoporomechanics, where the nonisothermal multiphase flow problem of gas hydrates is solved implicitly first and then the geomechanics problem is solved implicitly at the next step. During solution of the flow problem, we fix the rate of first Pioal total stress for numerical stability as well as apply porosity correction and entropy correction to account for geomechanical effects. We test numerical examples where flow and geomechanics parameters are based on deep oceanic gas hydrate deposits. When applying depressurization, even though the results between the infinitesimal transformation and finite strain geomechanics are similar in the early stages due to small deformation, we find differences between them in the late times as deformation becomes large. Accordingly, permeability and thermal conductivity tensors become nonisotropic full tensors although they are initially isotropic. We identify numerical stability of the developed sequential method from the test cases that exhibit the highly complex coupled gas hydrate systems with large deformation. Thus, the proposed sequential formulation can be applied in largely deformable gas hydrate systems.

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On the optimization of the fixed‐stress splitting for Biot's equations

Cited by 53 publications

References 55 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

Iterative solvers for Biot model under small and large deformations

Sequential formulation of all‐way coupled finite strain thermoporomechanics for largely deformable gas hydrate deposits

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