A Hybrid High-Order Discretization Method for Nonlinear Poroelasticity

AbstractIn this work, we construct and analyze a nonconforming high-order discretization method for the quasi-static single-phase nonlinear poroelasticity problem describing Darcean flow in a deformable porous medium saturated by a slightly compressible fluid. The nonlinear elasticity operator is discretized using a Hybrid High-Order method, while the Darcy operator relies on a Symmetric Weighted Interior Penalty discontinuous Galerkin scheme. The method is valid in two and three space dimensions, delivers an inf-sup stable discretization on general meshes including polyhedral elements and nonmatching interfaces, supports arbitrary approximation orders, and has a reduced cost thanks to the possibility of statically condensing a large subset of the unknowns for linearized versions of the problem. Moreover, the proposed construction can handle both nonzero and vanishing specific storage coefficients.

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A Hybrid High-Order method for creeping flows of non-Newtonian fluids

ESAIM Mathematical Modelling and Numerical Analysis ◽

10.1051/m2an/2021051 ◽

2021 ◽

Author(s):

Michele Botti ◽

Daniel Castanon Quiroz ◽

Daniele Di Pietro ◽

André Harnist

Keyword(s):

Steady Motion ◽

Incompressible Fluids ◽

High Order ◽

Well Posedness ◽

Discretization Method ◽

Order Method ◽

Numerical Examples ◽

High Order Method ◽

Creeping Flows ◽

General Meshes

In this paper, we design and analyze a Hybrid High-Order discretization method for the steady motion of non-Newtonian, incompressible fluids in the Stokes approximation of small velocities. The proposed method has several appealing features including the support of general meshes and high-order, unconditional inf-sup stability, and orders of convergence that match those obtained for Leray-Lions scalar problems. A complete well-posedness and convergence analysis of the method is carried out under new, general assumptions on the strain rate-shear stress law, which encompass several common examples such as the power-law and Carreau-Yasuda models. Numerical examples complete the exposition.

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A High-Order Local Discontinuous Galerkin Scheme for Viscoelastic Fluid Flow

Lecture Notes in Computational Science and Engineering - Recent Advances in Computational Engineering ◽

10.1007/978-3-319-93891-2_4 ◽

2018 ◽

pp. 51-61

Author(s):

Anne Kikker ◽

Florian Kummer

Keyword(s):

Fluid Flow ◽

Discontinuous Galerkin ◽

Viscoelastic Fluid ◽

High Order ◽

Galerkin Scheme ◽

Local Discontinuous Galerkin ◽

Discontinuous Galerkin Scheme

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A positivity-preserving high order discontinuous Galerkin scheme for convection–diffusion equations

Journal of Computational Physics ◽

10.1016/j.jcp.2018.04.002 ◽

2018 ◽

Vol 366 ◽

pp. 120-143 ◽

Cited By ~ 6

Author(s):

Sashank Srinivasan ◽

Jonathan Poggie ◽

Xiangxiong Zhang

Keyword(s):

Discontinuous Galerkin ◽

High Order ◽

Diffusion Equations ◽

Positivity Preserving ◽

Galerkin Scheme ◽

Convection Diffusion ◽

Discontinuous Galerkin Scheme

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Utilization of the Brinkman Penalization to Represent Geometries in a High-Order Discontinuous Galerkin Scheme on Octree Meshes

Symmetry ◽

10.3390/sym11091126 ◽

2019 ◽

Vol 11 (9) ◽

pp. 1126 ◽

Cited By ~ 1

Author(s):

Nikhil Anand ◽

Neda Ebrahimi Pour ◽

Harald Klimach ◽

Sabine Roller

Keyword(s):

Discontinuous Galerkin ◽

High Order ◽

Essential Property ◽

Penalization Method ◽

Galerkin Scheme ◽

Discontinuous Galerkin Scheme ◽

Symmetry Properties ◽

Wall Boundary Conditions ◽

Curved Elements ◽

Brinkman Penalization Method

We investigate the suitability of the Brinkman penalization method in the context of a high-order discontinuous Galerkin scheme to represent wall boundaries in compressible flow simulations. To evaluate the accuracy of the wall model in the numerical scheme, we use setups with symmetric reflections at the wall. High-order approximations are attractive as they require few degrees of freedom to represent smooth solutions. Low memory requirements are an essential property on modern computing systems with limited memory bandwidth and capability. The high-order discretization is especially useful to represent long traveling waves, due to their small dissipation and dispersion errors. An application where this is important is the direct simulation of aeroacoustic phenomena arising from the fluid motion around obstacles. A significant problem for high-order methods is the proper definition of wall boundary conditions. The description of surfaces needs to match the discretization scheme. One option to achieve a high-order boundary description is to deform elements at the boundary into curved elements. However, creating such curved elements is delicate and prone to numerical instabilities. Immersed boundaries offer an alternative that does not require a modification of the mesh. The Brinkman penalization is such a scheme that allows us to maintain cubical elements and thereby the utilization of efficient numerical algorithms exploiting symmetry properties of the multi-dimensional basis functions. We explain the Brinkman penalization method and its application in our open-source implementation of the discontinuous Galerkin scheme, Ateles. The core of this presentation is the investigation of various penalization parameters. While we investigate the fundamental properties with one-dimensional setups, a two-dimensional reflection of an acoustic pulse at a cylinder shows how the presented method can accurately represent curved walls and maintains the symmetry of the resulting wave patterns.

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