| 1. |
- Berre, I., et al.
(författare)
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Verification benchmarks for single-phase flow in three-dimensional fractured porous media
- 2021
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Ingår i: Advances in Water Resources. - OXFORD ENGLAND : Elsevier BV. - 0309-1708 .- 1872-9657. ; 147
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Tidskriftsartikel (refereegranskat)abstract
- Flow in fractured porous media occurs in the earth's subsurface, in biological tissues, and in man-made materials. Fractures have a dominating influence on flow processes, and the last decade has seen an extensive development of models and numerical methods that explicitly account for their presence. To support these developments, four benchmark cases for single-phase flow in three-dimensional fractured porous media are presented. The cases are specifically designed to test the methods’ capabilities in handling various complexities common to the geometrical structures of fracture networks. Based on an open call for participation, results obtained with 17 numerical methods were collected. This paper presents the underlying mathematical model, an overview of the features of the participating numerical methods, and their performance in solving the benchmark cases.
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| 2. |
- Boon, Wietse M., et al.
(författare)
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An Adaptive Penalty Method for Inequality Constrained Minimization Problems
- 2021
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Ingår i: European Conference on Numerical Mathematics and Advanced Applications, ENUMATH 2019. - Cham : Springer Science and Business Media Deutschland GmbH. ; , s. 155-164
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Konferensbidrag (refereegranskat)abstract
- The primal-dual active set method is observed to be the limit of a sequence of penalty formulations. Using this perspective, we propose a penalty method that adaptively becomes the active set method as the residual of the iterate decreases. The adaptive penalty method (APM) therewith combines the main advantages of both methods, namely the ease of implementation of penalty methods and the exact imposition of inequality constraints inherent to the active set method. The scheme can be considered a quasi-Newton method in which the Jacobian is approximated using a penalty parameter. This spatially varying parameter is chosen at each iteration by solving an auxiliary problem.
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| 3. |
- Boon, Wietse M., et al.
(författare)
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Convergence of a tpfa finite volume scheme for mixed-dimensional flow problems
- 2020
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Ingår i: Springer Proceedings in Mathematics and Statistics. - Cham : Springer. ; , s. 435-444
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Konferensbidrag (refereegranskat)abstract
- A two-point flux approximation (TPFA) finite volume method is considered for mixed-dimensional fracture flow problems. Its construction is based on applying a face-based quadrature rule to a conforming, mixed finite element scheme of lowest order. A concise argument shows linear convergence in theory, which we confirm in practice by a numerical experiment.
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| 4. |
- Boon, Wietse M., et al.
(författare)
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Functional analysis and exterior calculus on mixed-dimensional geometries
- 2020
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Ingår i: Annali di Matematica Pura ed Applicata. - : Springer. - 0373-3114 .- 1618-1891.
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Tidskriftsartikel (refereegranskat)abstract
- We are interested in differential forms on mixed-dimensional geometries, in the sense of a domain containing sets of d-dimensional manifolds, structured hierarchically so that each d-dimensional manifold is contained in the boundary of one or more d+ 1 -dimensional manifolds. On any given d-dimensional manifold, we then consider differential operators tangent to the manifold as well as discrete differential operators (jumps) normal to the manifold. The combined action of these operators leads to the notion of a semi-discrete differential operator coupling manifolds of different dimensions. We refer to the resulting systems of equations as mixed-dimensional, which have become a popular modeling technique for physical applications including fractured and composite materials. We establish analytical tools in the mixed-dimensional setting, including suitable inner products, differential and codifferential operators, Poincaré lemma, and Poincaré–Friedrichs inequality. The manuscript is concluded by defining the mixed-dimensional minimization problem corresponding to the Hodge Laplacian, and we show that this minimization problem is well-posed.
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| 5. |
- Boon, Wietse M., et al.
(författare)
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Mixed-dimensional poromechanical models of fractured porous media
- 2023
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Ingår i: Acta Mechanica. - : Springer. - 0001-5970 .- 1619-6937. ; 234:3, s. 1121-1168
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Tidskriftsartikel (refereegranskat)abstract
- We combine classical continuum mechanics with the recently developed calculus for mixed-dimensional problems to obtain governing equations for flow in, and deformation of, fractured materials. We present models in both the context of finite and infinitesimal strain, and discuss nonlinear (and non-differentiable) constitutive laws such as friction models and contact mechanics in the fracture. Using the theory of well-posedness for evolutionary equations with maximal monotone operators, we show well-posedness of the model in the case of infinitesimal strain and under certain assumptions on the model parameters.
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| 6. |
- Boon, Wietse M., et al.
(författare)
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Stable mixed finite elements for linear elasticity with thin inclusions
- 2021
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Ingår i: Computational Geosciences. - : Springer Nature. - 1420-0597 .- 1573-1499. ; 25:2, s. 603-620
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Tidskriftsartikel (refereegranskat)abstract
- We consider mechanics of composite materials in which thin inclusions are modeled by lower-dimensional manifolds. By successively applying the dimensional reduction to junctions and intersections within the material, a geometry of hierarchically connected manifolds is formed which we refer to as mixed-dimensional. The governing equations with respect to linear elasticity are then defined on this mixed-dimensional geometry. The resulting system of partial differential equations is also referred to as mixed-dimensional, since functions defined on domains of multiple dimensionalities are considered in a fully coupled manner. With the use of a semi-discrete differential operator, we obtain the variational formulation of this system in terms of both displacements and stresses. The system is then analyzed and shown to be well-posed with respect to appropriately weighted norms. Numerical discretization schemes are proposed using well-known mixed finite elements in all dimensions. The schemes conserve linear momentum locally while relaxing the symmetry condition on the stress tensor. Stability and convergence are shown using a priori error estimates and confirmed numerically.
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| 7. |
- Boon, Wietse M.
(författare)
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A parameter-robust iterative method for Stokes-Darcy problems retaining local mass conservation
- 2020
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Ingår i: Mathematical Modelling and Numerical Analysis. - : EDP Sciences. - 0764-583X .- 1290-3841. ; 54:6, s. 2045-2067
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Tidskriftsartikel (refereegranskat)abstract
- We consider a coupled model of free-flow and porous medium flow, governed by stationary Stokes and Darcy flow, respectively. The coupling between the two systems is enforced by introducing a single variable representing the normal flux across the interface. The problem is reduced to a system concerning only the interface flux variable, which is shown to be well-posed in appropriately weighted norms. An iterative solution scheme is then proposed to solve the reduced problem such that mass is conserved at each iteration. By introducing a preconditioner based on the weighted norms from the analysis, the performance of the iterative scheme is shown to be robust with respect to material and discretization parameters. By construction, the scheme is applicable to a wide range of locally conservative discretization schemes and we consider explicit examples in the framework of Mixed Finite Element methods. Finally, the theoretical results are confirmed with the use of numerical experiments.
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| 8. |
- Boon, Wietse M., et al.
(författare)
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Flux-mortar mixed finite element methods on nonmatching grids
- 2022
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Ingår i: SIAM Journal on Numerical Analysis. - : Society for Industrial & Applied Mathematics (SIAM). - 0036-1429 .- 1095-7170. ; 60:3, s. 1193-1225
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Tidskriftsartikel (refereegranskat)abstract
- We investigate a mortar technique for mixed finite element approximations of a class of domain decomposition saddle point problems on nonmatching grids in which the variable associated with the essential boundary condition, referred to as flux, is chosen as the coupling variable. It plays the role of a Lagrange multiplier to impose weakly continuity of the variable associated with the natural boundary condition. The flux-mortar variable is incorporated with the use of a discrete extension operator. We present well-posedness and error analysis in an abstract setting under a set of suitable assumptions, followed by a nonoverlapping domain decomposition algorithm that reduces the global problem to a positive definite interface problem. The abstract theory is illustrated for Darcy flow, where the normal flux is the mortar variable used to impose continuity of pressure, and for Stokes flow, where the velocity vector is the mortar variable used to impose continuity of normal stress. In both examples, suitable discrete extension operators are developed and the assumptions from the abstract theory are verified. Numerical studies illustrating the theoretical results are presented for Darcy flow.
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| 9. |
- Boon, Wietse M., et al.
(författare)
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Parameter-robust methods for the Biot-Stokes interfacial coupling without Lagrange multipliers
- 2022
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Ingår i: Journal of Computational Physics. - : Elsevier BV. - 0021-9991 .- 1090-2716. ; 467, s. 111464-
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we advance the analysis of discretizations for a fluid-structure interaction model of the monolithic coupling between the free flow of a viscous Newtonian fluid and a deformable porous medium separated by an interface. A five-field mixed-primal finite element scheme is proposed solving for Stokes velocity-pressure and Biot displacement -total pressure-fluid pressure. Adequate inf-sup conditions are derived, and one of the distinctive features of the formulation is that its stability is established robustly in all material parameters. We propose robust preconditioners for this perturbed saddle-point problem using appropriately weighted operators in fractional Sobolev and metric spaces at the interface. The performance is corroborated by several test cases, including the application to interfacial flow in the brain.
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| 10. |
- Boon, Wietse M., et al.
(författare)
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Robust Monolithic Solvers For The Stokes-Darcy Problem With The Darcy Equation In Primal Form
- 2022
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Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial & Applied Mathematics (SIAM). - 1064-8275 .- 1095-7197. ; 44:4, s. B1148-B1174
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Tidskriftsartikel (refereegranskat)abstract
- We construct mesh-independent and parameter-robust monolithic solvers for the coupled primal Stokes-Darcy problem. Three different formulations and their discretizations in terms of conforming and nonconforming finite element methods and finite volume methods are considered. In each case, robust preconditioners are derived using a unified theoretical framework. In particular, the suggested preconditioners utilize operators in fractional Sobolev spaces. Numerical experiments demonstrate the parameter-robustness of the proposed solvers.
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