On a numerical subgrid upscaling algorithm for Stokes–Brinkman equations

Iliev, Oleg; Lakdawala, Zahra; Starikovičius, Vadimas

doi:10.1016/j.camwa.2012.05.011

Cited by 13 publications

(6 citation statements)

References 17 publications

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“…Due to the variable sizes and geometries of these perforations, solutions to these problems have multiscale features. One solution approach involves posing the problem in a domain without perforations but with a very high contrast penalty term representing the domain heterogeneities ( [31,43,28,32]). However, the void space can be a small portion of the whole domain and, thus, it is computationally expensive to enlarge the domain substantially.…”

Section: Introductionmentioning

confidence: 99%

Online adaptive local multiscale model reduction for heterogeneous problems in perforated domains

Chung

Efendiev²,

Leung³

et al. 2016

Applicable Analysis

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In this paper, we develop and analyze an adaptive multiscale approach for heterogeneous problems in perforated domains. We consider commonly used model problems including the Laplace equation, the elasticity equation, and the Stokes system in perforated regions. In many applications, these problems have a multiscale nature arising because of the perforations, their geometries, the sizes of the perforations, and configurations. Typical modeling approaches extract average properties in each coarse region, that encapsulate many perforations, and formulate a coarse-grid problem. In some applications, the coarse-grid problem can have a different form from the fine-scale problem, e.g., the coarse-grid system corresponding to a Stokes system in perforated domains leads to Darcy equations on a coarse grid. In this paper, we present a general offline/online procedure, which can adequately and adaptively represent the local degrees of freedom and derive appropriate coarse-grid equations. Our approaches start with the offline procedure (following [18]), which constructs multiscale basis functions in each coarse region and formulates coarse-grid equations. In [18], we presented the offline simulations without the analysis and adaptive procedures, which are needed for accurate and efficient simulations. The main contributions of this paper are (1) the rigorous analysis of the offline approach (2) the development of the online procedures and their analysis (3) the development of adaptive strategies. We present an online procedure, which allows adaptively incorporating global information and is important for a fast convergence when combined with the adaptivity. We present online adaptive enrichment algorithms for the three model problems mentioned above. Our methodology allows adding and guides constructing new online multiscale basis functions adaptively in appropriate regions. We present the convergence analysis of the online adaptive enrichment algorithm for the Stokes system. In particular, we show that the online procedure has a rapid convergence with a rate related to the number of offline basis functions, and one can obtain fast convergence by a sufficient number of offline basis functions, which are computed in the offline stage. The convergence theory can also be applied to the Laplace equation and the elasticity equation. To illustrate the performance of our method, we present numerical results with both small and large perforations. We see that only a few (1 or 2) online iterations can significantly improve the offline solution. *

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Section: Introductionmentioning

confidence: 99%

Online adaptive local multiscale model reduction for heterogeneous problems in perforated domains

Chung

Efendiev²,

Leung³

et al. 2016

Applicable Analysis

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“…We solved equations and numerically using Chorin's projection method (Chorin, ; Griebel et al, ) on a nonsteady state formulation (cf. Iliev et al, ) until convergence to steady state. Inside the pore‐scale domain, the hydraulic conductivity was assumed infinite resulting in the classical Stokes equations.…”

Section: Methodsmentioning

confidence: 99%

“…In this way, a uniform set of equations for all domains is provided and model coupling is avoided. Still, the full parameterization of the continuum‐scale domain requires information on hydraulic properties that is frequently obtained by upscaling methods (e.g., Iliev et al, ). However, the Stokes‐Brinkman equations are not generally valid for porous media (e.g., Auriault, ) and largely depend on the choice of an effective viscosity parameter (Brinkman, ).…”

Section: Introductionmentioning

confidence: 99%

Efficient Prediction of Multidomain Flow and Transport in Hierarchically Structured Porous Media

Ritschel

Schlüter

Köhne

et al. 2018

Water Resources Research

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Structural hierarchy is a fundamental characteristic of natural porous media. Yet it provokes one of the grand challenges for the modeling of fluid flow and transport since pore‐scale structures and continuum‐scale domains often coincide independent of the observation scale. Common approaches to represent structural hierarchy build, for example, on a multidomain continuum for transport or on the coupling of the Stokes equations with Darcy's law for fluid flow. These approaches, however, are computationally expensive or introduce empirical parameters that are difficult to derive with independent observations. We present an efficient model for fluid flow based on Darcy's law and the law of Hagen‐Poiseuille that is parameterized based on the explicit pore space morphology obtained, for example, by X‐ray μ‐CT and inherently permits the coupling of pore‐scale and continuum‐scale domain. We used the resulting flow field to predict the transport of solutes via particle tracking across the different domains. Compared to experimental breakthrough data from laboratory‐scale columns with hierarchically structured porosity built from solid glass beads and microporous glass pellets, an excellent agreement was achieved without any calibration. Furthermore, we present different test scenarios to compare the flow fields resulting from the Stokes‐Brinkman equations and our approach to comprehensively illustrate its advantages and limitations. In this way, we could show a striking efficiency and accuracy of our approach that qualifies as general alternative for the modeling of fluid flow and transport in hierarchical porous media, for example, fractured rock or karstic aquifers.

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“…L. Guta and S. Sundar [13] presented the wave-porous structure interaction by using the Navier-Stokes-Brinkman system and applied the Finite Volume Method (FVM) to calculate the numerical results. O. Iliev et al [15] presented a numerical subgrid resolution approach for solving the Stokes-Brinkman system of equations for various scientific and industrial problems. B.K.…”

Section: Introductionmentioning

confidence: 99%

The Flow in Periciliary Layer in Human Lungs with Navier-Stokes-Brinkman Equations

Wuttanachamsri

Oangwatcharaparkan

2021

Tamkang J. Math.

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In the human respiratory tract, air breathed in is often contaminated with strange particles such as dust and chemical spray, which may cause people respiratory diseases. However, the human body has an innate immune system that helps to trap the debris by secreting mucus to catch the foreign particles, which are removed from the body by the movement of tiny hairs lining on the surface of the epithelial cells in the immune system. The layer containing the tiny hairs or cilia is called Periciliary Layer (PCL). In this research, we find the velocity of the fluid in the PCL moved by a ciliary beating by using the Navier-Stokes-Brinkman equations. We apply the Galerkin finite element method to determine numerical solutions. For the steady linear case of the equation, the numerical result is in good agreement with an exact solution. Including the time derivative and nonlinear terms, we show that the velocity of the liquid is affected by the velocity of the solid, which follows the physical meaning of the fluid flow. The result can be applied as a bottom boundary condition of the mucous layer to be able to find the velocity of mucus in the human lungs.

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On a numerical subgrid upscaling algorithm for Stokes–Brinkman equations

Cited by 13 publications

References 17 publications

Online adaptive local multiscale model reduction for heterogeneous problems in perforated domains

Online adaptive local multiscale model reduction for heterogeneous problems in perforated domains

Efficient Prediction of Multidomain Flow and Transport in Hierarchically Structured Porous Media

The Flow in Periciliary Layer in Human Lungs with Navier-Stokes-Brinkman Equations

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