Interface dynamics in randomly heterogeneous porous media

Lazarev, Yu. N.; Petrov, P. V.; Tartakovsky, Daniel M.

doi:10.1016/j.advwatres.2004.11.003

Cited by 13 publications

(10 citation statements)

References 10 publications

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“…Similar mappings have been used extensively to solve numerically deterministic problems in structured body-fitted curvilinear coordinates [33], and more recently to analyze the interface dynamics in disordered media [19]. The properties and a more detailed Downloaded 11/27/14 to 130.159.70.209.…”

Section: Mapping Of the Random Domainmentioning

confidence: 99%

Numerical Methods for Differential Equations in Random Domains

Xiu¹,

Tartakovsky²

2006

SIAM J. Sci. Comput.

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157

148

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Physical phenomena in domains with rough boundaries play an important role in a variety of applications. Often the topology of such boundaries cannot be accurately described in all of its relevant detail due to either insufficient data or measurement errors or both. This topological uncertainty can be efficiently handled by treating rough boundaries as random fields, so that an underlying physical phenomenon is described by deterministic or stochastic differential equations in random domains. To deal with this class of problems, we propose a novel computational framework, which is based on using stochastic mappings to transform the original deterministic/stochastic problem in a random domain into a stochastic problem in a deterministic domain. The latter problem has been studied more extensively, and existing analytical/numerical techniques can be readily applied. In this paper, we employ both a stochastic Galerkin method and Monte Carlo simulations to solve the transformed stochastic problem. We demonstrate our approach by applying it to an elliptic problem in single-and double-connected random domains, and comment on the accuracy and convergence of the numerical methods.1. Introduction. Physical phenomena described by differential equations in domains with rough geometries are ubiquitous and include applications ranging from surface imaging [24] to manufacturing of nano-devices [5]. Indeed, given a proper spatial resolution, virtually any natural or manufactured surface becomes rough. Consequently, there is a growing interest in experimental, theoretical, and numerical studies of deterministic and probabilistic descriptions of such surfaces and of solutions of differential equations defined on the resulting domains.The early attempts to represent surface roughness and to study its effects on system behavior were based on simplified, easily parameterizable, deterministic surface inhomogeneities, such as symmetrical asperities to represent indentations [32,26] and semispheres to represent protrusions [17]. More general representations of surface roughness, which nevertheless admit simple parameterizations, include sinusoidal corrugations [13] and periodic linear segments [6]. A recent trend in describing surface roughness is to use fractal and fractal-like representation. These include deterministic Von Koch's [10, 6, 8] and Minkowski's [6] fractals, as well as their random generalizations [29].

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Section: Mapping Of the Random Domainmentioning

confidence: 99%

Numerical Methods for Differential Equations in Random Domains

Xiu¹,

Tartakovsky²

2006

SIAM J. Sci. Comput.

Self Cite

157

148

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“…It follows a procedure introduced in (10, 21, 24) and consists of two steps. First, the random flow domain is mapped onto a deterministic domain with smooth boundaries (Section A.1), the transformed Stokes equations become stochastic.…”

Section: Statistical Representation Of Random Surfacesmentioning

confidence: 99%

“…For the relatively simple flow domain under consideration, such a mapping can be defined analytically, for example, as ξ 1 = x 1 and ξ 2 = ( L y − x 2 )/[ L y − s ( x 1 , ω )]. For more complex geometries, a stochastic mapping ξ i = ξ i ( x 1 , x 2 ) ( i = 1, 2) and its inverse x i = x i ( ξ 1 , ξ 2 ) ( i = 1, 2) are constructed (e.g., 10, 24) by solving Laplace’s equations,

\frac{\partial^{2} x_{i}}{\partial ξ_{1}^{2}} + \frac{\partial^{2} x_{i}}{\partial ξ_{2}^{2}} = 0, (ξ_{1}, ξ_{2}) \in E, i = 1, 2

subject to the boundary conditions

x_{1} (0, ξ_{2}) = 0, x_{1} (L_{x}, ξ_{2}) = L_{x}, x_{1} (ξ_{1}, 0) = ξ_{1}, x_{1} (ξ_{1}, L_{y}) = ξ_{1};

x_{2} (0, ξ_{2}) = ξ_{2}, x_{2} (L_{x}, ξ_{2}) = ξ_{2}, x_{2} (ξ_{1}, 0) = s, x_{2} (ξ_{1}, L_{y}) = L_{y} .

Uncertainty (randomness) in domain geometry, s ( x 1 , ω ), manifests itself in the mapping problem through the boundary condition in Eq. 14.…”

Section: A Solving Stokes Equations On Random Domainmentioning

confidence: 99%

Impact of endothelium roughness on blood flow

Park

Intaglietta

Tartakovsky

2012

Journal of Theoretical Biology

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Cell free layer (CFL), a plasma layer bounded by the red blood cell (RBC) core and the endothelium, plays an important physiological role. Its width affects the effective blood viscosity as well as the scavenging and production of nitric oxide (NO). Measurements of the CFL and its spatio-temporal variability are highly uncertain, exhibiting random fluctuations. Yet traditional models of blood flow and NO scavenging treat the CFL’s bounding surfaces as deterministic and smooth. We investigate the effects of the endothelium roughness and uncertain (random) spatial variability on blood flow and estimates of effective blood viscosity.

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“…II D. Finally, simplification of complicated domain shape has motivated coordinate transforms based on Laplace equations solutions 15 and conformal mapping, the two-dimensional equivalent, in applications to problems of fluid flows, 16 diffusion, 17 electric potentials, 18 and random domains. 19,20 Although the method here entails the existence and computability of surface flattening transforms, establishing their practicability in acoustic propagation is the ultimate goal. To this end, Sec.…”

Section: Introductionmentioning

confidence: 99%

Global boundary flattening transforms for acoustic propagation under rough sea surfaces

Oba

2010

The Journal of the Acoustical Society of America

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This paper introduces a conformal transform of an acoustic domain under a one-dimensional, rough sea surface onto a domain with a flat top. This non-perturbative transform can include many hundreds of wavelengths of the surface variation. The resulting two-dimensional, flat-topped domain allows direct application of any existing, acoustic propagation model of the Helmholtz or wave equation using transformed sound speeds. Such a transform-model combination applies where the surface particle velocity is much slower than sound speed, such that the boundary motion can be neglected. Once the acoustic field is computed, the bijective (one-to-one and onto) mapping permits the field interpolation in terms of the original coordinates. The Bergstrom method for inverse Riemann maps determines the transform by iterated solution of an integral equation for a surface matching term. Rough sea surface forward scatter test cases provide verification of the method using a particular parabolic equation model of the Helmholtz equation.

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Interface dynamics in randomly heterogeneous porous media

Cited by 13 publications

References 10 publications

Numerical Methods for Differential Equations in Random Domains

Numerical Methods for Differential Equations in Random Domains

Impact of endothelium roughness on blood flow

Global boundary flattening transforms for acoustic propagation under rough sea surfaces

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