A Cahn–Hilliard–Biot system and its generalized gradient flow structure

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

doi:10.1016/j.aml.2021.107799

Cited by 9 publications

(5 citation statements)

References 14 publications

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“…The proposed mathematical framework represents an extension that seamlessly combines elements from the Cahn-Hilliard model and the quasi-static linear Biot equations. In [31], it was shown that the mathematical model possesses a generalized gradient flow structure, affirming the thermodynamic consistency of the model. The Cahn-Hilliard component governs solid phase changes within the system through a smooth phase-field variable, while the Biot equations oversee fluid flow and elasticity.…”

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confidence: 76%

“…Further, the interfacial parameter γ is selected as γ = 10 −4 . Following [31], the permeability κ(φ), compressibility M (φ), Biot-Willlis coefficient α(φ) and elasticity tensor C(φ) are depending on the phasefield variable φ through the interpolation function π(φ) that is defined by…”

Section: Thus We Have By the Young Inequalitymentioning

confidence: 99%

“…In this paper, we undertake a comprehensive well-posedness analysis of the Cahn-Hilliard-Biot model that is capable of capturing complex scenarios involving fluid flow through deformable porous media at the Darcy scale. This model has been derived recently in [31] and it exhibits the capacity to adapt to changing material characteristics, encompassing variations in stiffness, permeability, compressibility, and poroelastic coupling strength. These variations arise from Cahn-Hilliard-type phase changes occurring within the solid matrix, which is a phenomenon with diverse applications.…”

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confidence: 99%

“…The solid phases are represented using a diffuse interface approach of Cahn-Hilliard type, where surface tension, solid material deformation, and pore pressure act as driving forces. We follow the derivation and explanations in [31].…”

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confidence: 99%

“…where σ is the stress tensor. One may consider a non-zero right-hand side as originally derived in [31]. However, following the usual assumptions of the Cahn-Larché equations in [15,17], we consider no external body forces.…”

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confidence: 99%

See 4 more Smart Citations

Time-fractional Cahn–Hilliard equation: Well-posedness, degeneracy, and numerical solutions

Fritz¹,

Rajendran²,

Wohlmuth³

2022

Computers & Mathematics with Applications

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mentioning

confidence: 76%

Section: Thus We Have By the Young Inequalitymentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Time-fractional Cahn–Hilliard equation: Well-posedness, degeneracy, and numerical solutions

Fritz¹,

Rajendran²,

Wohlmuth³

2022

Computers & Mathematics with Applications

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Weak and strong solutions for a fluid‐poroelastic‐structure interaction via a semigroup approach

Avalos,

Gurvich,

Webster

2024

Math Methods in App Sciences

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A filtration system comprising a Biot poroelastic solid coupled to an incompressible Stokes free‐flow is considered in 3D. Across the flat 2D interface, the Beavers‐Joseph‐Saffman coupling conditions are taken. In the inertial, linear, and non‐degenerate case, the hyperbolic‐parabolic coupled problem is posed through a dynamics operator on a chosen energy space, adapted from Stokes‐Lamé coupled dynamics. A semigroup approach is utilized to circumvent issues associated to mismatched trace regularities at the interface. The generation of a strongly continuous semigroup for the dynamics operator is obtained via a non‐standard maximality argument. The latter employs a mixed‐variational formulation in order to invoke the Babuška‐Brezzi theorem. The Lumer‐Philips theorem then yields semigroup generation, and thereby, strong and generalized solutions are obtained. For the linear dynamics, density obtains the existence of weak solutions; we extend to the case where the Biot compressibility of constituents degenerates. Thus, for the inertial linear Biot‐Stokes filtration system, we provide a clear elucidation of strong solutions and a construction of weak solutions, as well as their regularity through associated estimates.

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Structure‐Preserving Approximation of the Cahn‐Hilliard‐Biot System

Brunk,

Fritz

2024

Numerical Methods Partial

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In this work, we propose a structure‐preserving discretization for the recently studied Cahn‐Hilliard‐Biot system using conforming finite elements in space and problem‐adapted explicit‐implicit Euler time integration. We prove that the scheme is thermodynamically consistent, that is, the balance of global phase and global volumetric fluid content and the energy dissipation balance. The existence of discrete solutions is established under suitable growth conditions. Furthermore, it is shown that the algorithm can be realized as a splitting method, that is, decoupling the Cahn‐Hilliard subsystem from the poro‐elasticity subsystem, while the first one is nonlinear and the second subsystem is linear. The schemes are illustrated by numerical examples and a convergence test.

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A Cahn–Hilliard–Biot system and its generalized gradient flow structure

Cited by 9 publications

References 14 publications

Time-fractional Cahn–Hilliard equation: Well-posedness, degeneracy, and numerical solutions

Time-fractional Cahn–Hilliard equation: Well-posedness, degeneracy, and numerical solutions

Weak and strong solutions for a fluid‐poroelastic‐structure interaction via a semigroup approach

Structure‐Preserving Approximation of the Cahn‐Hilliard‐Biot System

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