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I. Asymptotic Analysis of Singular Phenomena in High-Contrast Yuliya Gorb, Mathematics Department, University of Houston RESEARCH STATEMENT 1 My research interests lie in the fields of Asymptotic Analysis for Partial Differential Equations (PDEs), Homogenization Theory for Composite Materials, Multiscale Modeling, Analysis and Computation, Nonlinear Solid Mechanics. The ongoing research activities mainly focus on comprehensive study of high contrast heterogeneous/composite media, and, in particular, include: I. Asymptotic Analysis of Singular Phenomena Occurred in High Contrast Dense Media [1,3,5,12,15–18], II. Multiscale Modeling and Simulation of Heterogeneous Particulate Flows [2,4,6–8,10], III. Mathematical Biology/Nonlinear Elasticity and Fracture Mechanics [9, 11, 13, 14]. The motivation for the studies described in Sections I and II below is driven by the applications in which processes occur in media whose constituents have vastly different mechanical properties. Many natural and man-made materials exhibit vast spatial variability in most of their properties, such as hydraulic and electri- cal conductivities, dielectric permeability, etc. Mathematically this means that one has to solve PDEs with coefficients that take extremely large and/or very small values in the domain. Such a feature of a heteroge- neous medium is referred to as a high contrast in material properties. In addition to the high contrast, which is a physical condition, another common feature of all problems under consideration is of geometrical nature. Namely, the heterogeneous domain either has highly concentrated inhomogeneities or contains multiple scales. This feature corresponds to rapidly varying coefficients of the underlying PDEs. “Rapidly varying” means that this function fluctuates on a length scale that is much smaller than the size of the domain occupied by the composite. The corresponding PDEs pose difficult analytical and computational challenges. Indeed, the fields inside strongly heterogeneous high contrast concentrated media exhibit singular behavior that is laborious to capture analytically, or the convergence rates of iterative methods, such as conjugate gradients, when solving a problem numerically, deteriorate as the variations in problem parameters increase. Typical examples of applications of described models include but are not limited to: (a) particulate flows (particle sedimentation, fluidization and conveying), (b) subsurface flows in natural porous formations, (c) electrical conduction in composite materials, and (d) medical and geophysical imaging (impedance tomography). To that end, my current research interest pertains to the development of efficient analytical and numerical tools for describing strongly heterogeneous flows in media with large variations in material properties and complex geometry. I. Asymptotic Analysis of Singular Phenomena in High-Contrast Dense Heterogeneous Media Mentioned above physical and geometric conditions of the problem (that is, the large ratio between the greatest and smallest values and high oscillations of the function describing the medium) are the reason due to which the corresponding numerical approximation is prohibitively expensive. Indeed, in numerical treatment of these problems one needs very fine meshes to capture the correct behavior of medium flows which leads to large and ill-conditioned matrices that have to be inverted. As a result, iterative methods used in matrix inversion have poor convergence rates. On the other hand, exactly those two features of the underlying problem make it amenable to asymptotic analysis which is performed in [1,3,5,12,15–18]. ⋆ Blow Up of Effective Properties The first observation made about high contrast composites with inhomogeneities of concentration close to maximal is that their effective properties exhibit blow up as the typical interparticle distance gets small. Studies [12, 16–18] are devoted to capturing the singular behavior of effective properties by developing and justifying so-called network models leading to accurate and cost-efficient numerical methods for simulat- ing the properties of the corresponding heterogeneous media. We explore two mathematical formulations: the scalar and vectorial ones. The scalar problem represents densely spaced infinitely conducting parti- cles in a medium of finite conductivity and the singular behavior of the effective conductivity is captured for media with spherical inhomogeneities [16] and also for particles of optimal shape [17, 18], whose ge- ometry minimizes the energy among all composites made from the same components in the same volume Yuliya Gorb, Mathematics Department, University of Houston RESEARCH STATEMENT 2 fractions. The vectorial problem deals with understanding of the effective rheological behavior of complex fluids (mixtures/suspensions). One of the most important rheological properties is viscosity. In [12] the motion of an irregular array of highly packed solid particles in an incompressible Newtonian fluid is considered, and the effective viscosity of the suspension is captured revealing some interesting features seen in the two-dimensional case, namely, an anomalous blow up. ⋆ Blow Up of Electric Field Besides the effective properties of aforementioned composites the electric field, described by the solution gradient, also exhibits blow up at the points of the closest distance between neighboring inhomogeneities. We note that the uniform estimates for solution gradient was a subject of numerous studies in the last decade with only upper and lower bounds for it being constructed. However, the exact asymptotics has not been captured yet. In [3] such an asymptotics is derived precisely and parameters used in the obtained asymptotics are explicitly characterized. It is also important to mention that previous contributions on the subject that provided only bounds for the gradient’s blow up, had their limitations, e.g. some of them use methods that work only in two dimensions, some deal with inhomogeneities of spherical shape only, and all of them were designed to treat linear problems only, with no direct extension to a nonlinear case. In contrast, techniques developed and adapted in [3] work for any number of particles of arbitrary shape in any dimension, and allow for a straightforward generalization to a nonlinear case, such as p-Laplacian, see [5]. Despite the extensive work on estimates for the electric field in gaps between perfectly conducting particles, there has not been much progress made for the other “extreme” case of insulating particles. While in 2D the blow up of solution gradient can be obtained by making use of convex duality, the 3D case is a long standing open problem, which is one of the undertaken projects now. ⋆ Asymptotic Approximation of the Dirichlet-to-Neumann map Since design and analysis of preconditioners which converge independently of the variations in physical parameters is important for many applications, the study of [1] focuses on the development of a Dirichlet- to-Neumann (DtN) preconditioner. This preconditioner is used in non-overlapping domain decomposition methods for solving flows in high contrast heterogeneous media by the corresponding iterative method. Recall, the DtN operator maps the boundary trace of the solution to its normal derivative at the boundary, and is determined by the quadratic form that represent the system’s energy. However, the analysis of the DtN map is much more involved than that of the energy, because of the arbitrary boundary conditions. An explicit characterization of the DtNmap in the asymptotic limit of the typical distance between the particles tending to zero is obtained in [1] and the application of derived asymptotics to the development of domain decomposition methods is currently under investigation. II. Multiscale Modeling and Simulations for Heterogeneous Particulate Flows While described above studies gear towards asymptotic analysis, projects of this sections concern with the development of efficient solvers for highly heterogeneous formations. The past and ongoing research efforts of this Section were supported by NSF grants DMS-0811180 and DMS-1016531, and main results are collected in [2, 4, 6–8, 10]. ⋆ Simulations of Particulate Flows Using Mixture Methods In [4] we discuss the numerical treatment of three-dimensional models for (semi-)dilute and concentrated suspensions of particles in incompressible fluids. An Eulerian mixture model in which the effective density and viscosity depend on the local volume fraction of the disperse phase is used. When it comes to simulating dense suspensions, it is essential to enforcing physically-motivated upper bounds for scalar conservation laws describing the evolution of this volume fraction. A flux-corrected transport algorithm for handling the closepacking limit in concentrated suspensions, originally developed in [6], is employed. The presented in Yuliya Gorb, Mathematics Department, University of Houston RESEARCH STATEMENT 3 [6] scheme is nonlinear even for a linear transport equation, and its application is of particular importance in the numerical treatment of continuity equations in Eulerian two-phase flow models (granular materials, fluidized beds). ⋆ Multiscale FEM for Fluid-Structure Interaction Problems with Large Interface Displacements Flows through porous formations is a subject that has received increased attention in recent years due to its relevance in a wide range of applications in petroleum engineering and biotechnology. As in other undertaken projects, the medium here is described by the permeability field that might be highly oscillatory. Fortunately,
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