A Hybrid High-Order Method for Multiple-Network Poroelasticity

Botti, Lorenzo Alessio; Botti, Michele; Pietro, Daniele Di

doi:10.1007/978-3-030-69363-3_6

Cited by 6 publications

(12 citation statements)

References 33 publications

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“…The additional scalar equation is meant to ease the stability analysis of the problem without compromising the efficiency of its discretization in terms of computational cost. Following the idea of [12,35], we introduce the auxiliary variable ϕ = λ∇ • u − αp − βT , that represents a volumetric contribution to the total stress. We will refer to this variable as the pseudo-total pressure.…”

Section: Four-field Formulationmentioning

confidence: 99%

“…The aim of this Section and Section 5 is to perform a complete numerical analysis of the linearized PolyDG semi-discretization (12). Before establishing an a priori estimate we define the energy norms and present some preliminary results.…”

Section: Stability Analysismentioning

confidence: 99%

“…In this section we derive the a priori estimate for the semi-discrete problem (12). To ease the notation we define the norm…”

Section: Stability Estimatesmentioning

confidence: 99%

“…Moreover, the introduction of the pseudo-total pressure variable ensures inf-sup stability and robustness with respect to locking phenomena in the quasi-incompressible limit. In [12,34,35] an analogous approach is considered for the poroelastic problem. Note that, a further possibility for dealing with the quasi-incompressible case in the analysis of the Biot's system is the introduction of the solid pressure (cf.…”

Section: Introductionmentioning

confidence: 99%

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Discontinuous Galerkin approximation of the fully-coupled thermo-poroelastic problem

Antonietti¹,

Bonetti²,

Botti³

2022

Preprint

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We present and analyze a discontinuous Galerkin method for the numerical modelling of the non-linear fully-coupled thermo-poroelastic problem. For the spatial discretization, we design a high-order discontinuous Galerkin method on polygonal and polyhedral grids based on a novel four-field formulation of the problem. To handle the non-linear convective transport term in the energy conservation equation we adopt a fixed-point linearization strategy. We perform a robust stability analysis for the linearized semi-discrete problem under mild requirements on the problem data. A priori hp-version error estimates in suitable energy norms are also derived. A complete set of numerical simulations is presented in order to validate the theoretical analysis, to inspect numerically the robustness properties, and to test the capability of the proposed method in a practical scenario inspired by a geothermal problem.

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Section: Four-field Formulationmentioning

confidence: 99%

Section: Stability Analysismentioning

confidence: 99%

“…In this section we derive the a priori estimate for the semi-discrete problem (12). To ease the notation we define the norm…”

Section: Stability Estimatesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Discontinuous Galerkin approximation of the fully-coupled thermo-poroelastic problem

Antonietti¹,

Bonetti²,

Botti³

2022

Preprint

Self Cite

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show abstract

“…It is assumed in what follows that both 𝜕Ω D and 𝜕Ω N have non-zero (𝑑 − 1)-dimensional Hausdorff measure (otherwise, additional closure conditions are needed). HHO methods are gaining momentum in the field of continuum mechanics having been successfully applied to nonlinear elasticity problems [1,19], diffusion dominated incompressible flow problems [2,38], porous-media flows [18] and poro-elasticity problems [15,16]. Similarly to Hybridizable Discontinuous Galerkin (HDG) methods, HHO formulations rely on Degrees Of Freedom (DOFs) associated to polynomial functions defined over mesh elements and mesh faces, the so called elemental and skeletal DOFs.…”

Section: Introductionmentioning

confidence: 99%

HHO methods for the incompressible Navier-Stokes and the incompressible Euler equations

Botti¹,

Massa²

2021

Preprint

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We propose two Hybrid High-Order (HHO) methods for the incompressible Navier-Stokes equations and investigate their robustness with respect to the Reynolds number. While both methods rely on a HHO formulation of the viscous term, the pressure-velocity coupling is fundamentally different, up to the point that the two approaches can be considered antithetical. The first method is kinetic energy preserving, meaning that the skew-symmetric discretization of the convective term is guaranteed not to alter the kinetic energy balance. The approximated velocity fields exactly satisfy the divergence free constraint and continuity of the normal component of the velocity is weakly enforced on the mesh skeleton, leading to H-div conformity. The second scheme relies on Godunov fluxes for pressure-velocity coupling: a Harten, Lax and van Leer (HLL) approximated Riemann Solver designed for cell centered formulations is adapted to hybrid face centered formulations. The resulting numerical scheme is robust up to the inviscid limit, meaning that it can be applied for seeking approximate solutions of the incompressible Euler equations. The schemes are numerically validated performing steady and unsteady two dimensional test cases and evaluating the convergence rates on ℎ-refined mesh sequences. In addition to standard benchmark flow problems, specifically conceived test cases are conducted for studying the error behaviour when approaching the inviscid limit.

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HHO Methods for the Incompressible Navier-Stokes and the Incompressible Euler Equations

Botti

Massa

2022

J Sci Comput

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We propose two Hybrid High-Order (HHO) methods for the incompressible Navier-Stokes equations and investigate their robustness with respect to the Reynolds number. While both methods rely on a HHO formulation of the viscous term, the pressure-velocity coupling is fundamentally different, up to the point that the two approaches can be considered antithetical. The first method is kinetic energy preserving, meaning that the skew-symmetric discretization of the convective term is guaranteed not to alter the kinetic energy balance. The approximated velocity fields exactly satisfy the divergence free constraint and continuity of the normal component of the velocity is weakly enforced on the mesh skeleton, leading to H-div conformity. The second scheme relies on Godunov fluxes for pressure-velocity coupling: a Harten, Lax and van Leer approximated Riemann Solver designed for cell centered formulations is adapted to hybrid face centered formulations. The resulting numerical scheme is robust up to the inviscid limit, meaning that it can be applied for seeking approximate solutions of the incompressible Euler equations. The schemes are numerically validated performing steady and unsteady two dimensional test cases and evaluating the convergence rates on h-refined mesh sequences. In addition to standard benchmark flow problems, specifically conceived test cases are conducted for studying the error behaviour when approaching the inviscid limit.

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A Hybrid High-Order Method for Multiple-Network Poroelasticity

Cited by 6 publications

References 33 publications

Discontinuous Galerkin approximation of the fully-coupled thermo-poroelastic problem

Discontinuous Galerkin approximation of the fully-coupled thermo-poroelastic problem

HHO methods for the incompressible Navier-Stokes and the incompressible Euler equations

HHO Methods for the Incompressible Navier-Stokes and the Incompressible Euler Equations

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