Coupling Biot and Navier-Stokes problems for fluid-poroelastic structure interaction (original) (raw)
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We develop a computational model to study the interaction of a fluid with a poroelastic material. The coupling of Stokes and Biot equations represents a prototype problem for these phenomena, which feature multiple facets. On one hand, it shares common traits with fluid-structure interaction. On the other hand it resembles the Stokes-Darcy coupling. For these reasons, the numerical simulation of the Stokes-Biot coupled system is a challenging task. The need of large memory storage and the difficulty to characterize appropriate solvers and related preconditioners for the equations at hand are typical shortcomings of classical discretization methods applied to this problem, such as the finite element method for spatial discretization and finite differences for time stepping. The application of loosely coupled time advancing schemes mitigates these issues, because it allows to solve each equation of the system independently with respect to the others, at each time step. In this work, we develop and thoroughly analyze a loosely coupled scheme for Stokes-Biot equations. The scheme is based on Nitsche's method for enforcing interface conditions. Once the interface operators corresponding to the interface conditions have been defined, time lagging allows us to build up a loosely coupled scheme with good stability properties. The stability of the scheme is guaranteed provided that appropriate stabilization operators are introduced into the variational formulation of each subproblem. The error of the resulting method is also analyzed, showing that splitting the equations pollutes the optimal approximation properties of the underlying discretization schemes. In order to restore good approximation properties, while maintaining the computational efficiency of the loosely coupled approach, we consider the application of the loosely coupled scheme as a preconditioner for the monolithic approach. Both theoretical insight and numerical results confirm that this is a promising way to develop efficient solvers for the problem at hand.
A variational multiscale stabilized formulation for the incompressible Navier–Stokes equations
Computational Mechanics, 2009
This paper presents a variational multiscale residual-based stabilized finite element method for the incompressible Navier-Stokes equations. Structure of the stabilization terms is derived based on the two level scale separation furnished by the variational multiscale framework. A significant feature of the new method is that the fine scales are solved in a direct nonlinear fashion, and a definition of the stabilization tensor τ is derived via the solution of the fine-scale problem. A computationally economic procedure is proposed to evaluate the advection part of the stabilization tensor. The new method circumvents the Babuska-Brezzi (inf-sup) condition and yields a stable formulation for high Reynolds number flows. A family of equal-order pressurevelocity elements comprising 4-and 10-node tetrahedral elements and 8-and 27-node hexahedral elements is developed. Convergence rates are reported and accuracy properties of the method are presented via the lid-driven cavity flow problem. Keywords Multiscale finite element methods • Navier-Stokes equations • Convergence rates • Equal order interpolation functions • Tetrahedral elements • Hexahedral elements
Parameter-Robust Methods for the Biot-Stokes Interfacial Coupling Without Lagrange Multipliers
SSRN Electronic Journal, 2021
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 velocitypressure 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.
Computers & Structures, 2013
This paper presents a monolithic formulation frame work combined with an anisotropic mesh adaptation for fluid-structure inter action (FSI) applications with complex geometry. The fluid-solid interfaces are captured using a level-set method. A new a posteriori error estimate, based on the length distribution tensor approach and the associated edge based error analysis, is then used to ensure an accurate capturing of the discontin uities at the fluid-solid inter face. It enables to calculate a stretch ing factor providing a new edge length distribution, its associated tensor and the corresp onding metric. The optimal stretching factor field is obtained by solving an optimization problem under the constraint of a fixed number of edges in the mesh. The presence of the structure will be taken into account by means of an extra stress tensor in the Navier-Stokes equations. The system is solved using a stabilized three-field, stress, velocity and pressure finite element (FE) formulation. It consists in the decompo sition for both the velocity and the pressure fields into coarse/resolve d scales and fine/unresolved scales and also in the efficient enrichment of the extra constraint. We assess the accuracy of the proposed formulation by simulating 2D and 3D time-dependent numer ical examples such as: falling disk in a channel, turbulent flows behind an airfoil profile and flow behind an immers ed vehicle.
Variational Multiscale Finite Element Method for Flows in Highly Porous Media
Multiscale Modeling & Simulation, 2011
We present a two-scale finite element method for solving Brinkman's and Darcy's equations. These systems of equations model fluid flows in highly porous and porous media, respectively. The method uses a recently proposed discontinuous Galerkin FEM for Stokes' equations by Wang and Ye and the concept of subgrid approximation developed by Arbogast for Darcy's equations. In order to reduce the "resonance error" and to ensure convergence to the global fine solution the algorithm is put in the framework of alternating Schwarz iterations using subdomains around the coarse-grid boundaries. The discussed algorithms are implemented using the Deal.II finite element library and are tested on a number of model problems. 1 2 O. ILIEV, R. LAZAROV, AND J. WILLEMS introduced a new phenomenological relation between the velocity and the pressure gradient (see, also [23, page 94]):
Multiscale methods in computational fluid and solid mechanics
First, an attempt is made towards gaining a more systematic understanding of recent progress in multiscale modelling in computational solid and fluid mechanics. Sub-sequently, the discussion is focused on variational multiscale methods for the compressible and incompressible Navier-Stokes equations. Examples are given of the application of this class of methods to a turbulent channel flow. Finally, multigrid methods, which can also be conceived as a subclass of multiscale methods, are applied to fluid-structure interaction problems.
The Multiscale Finite Element Method (MsFEM) is developed in the vein of Crouzeix-Raviart element for solving viscous incompressible flows in genuine heterogeneous media. Such flows are relevant in many branches of engineering, often at multiple scales and at regions where analytical representations of the microscopic features of the flows are often unavailable. Full accounts to these problems heavily depend on the geometry of the system under consideration and are computationally expensive. Therefore, a method capable of solving multiscale features of the flow without confining itself to fine scale calculations is sought after. The approximation of boundary condition on coarse element edges when computing the multiscale basis functions critically influences the eventual accuracy of any MsFEM approaches. The weakly enforced continuity of Crouzeix-Raviart function space across element edges leads to a natural boundary condition for the multiscale basis functions which relaxes the sensitivity of our method to complex patterns of obstacles exempt from the needs of implementing any oversampling techniques. Additionally, the application of penalization method makes it possible to avoid complex unstructured domain and allows extensive use of simpler Cartesian meshes.