Research Areas



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, and 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 Concentrated Media
II. Multiscale Modeling and Simulation of Heterogeneous Particulate Flows
III. Mathematical Biology/Nonlinear Elasticity, and Fracture Mechanics

The motivation for the studies described in 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 electrical 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 heterogeneous 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, that 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 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).


I. Asymptotic Analysis of Singular Phenomena in High Contrast Concentrated 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.

  Blow Up of Effective Properties  
  The first observation made about high contrast composites with inhomogeneities of concentration close to maximal was that their effective properties exhibit blow up as the typical interparticle distance gets small. Studies [I.4,I.6-I.8] 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 simulating 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 particles in a medium of finite conductivity and the singular behavior of the effective conductivity is captured for media with spherical inhomogeneities [I.6] and also for particles of optimal shape [I.7,I.8], whose geometry minimizes the energy among all composites made from the same components in the same volume 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 [I.4] 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 particles. 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 [I.2] 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 [I.2] 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 [I.3].  
  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 [I.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 [I.1].  
  [I.1] L. Borcea, Y. Gorb, Y. Wang: ''Asymptotic Approximation of the Dirichlet to Neumann map of High Contrast Conductive Media''. SIAM Multiscale Modeling and Simulation, Vol 12, No 4, 2014, pp. 1494-1532  
  [I.2] Y. Gorb: ''Singular Behavior of Electric Field of High Contrast Concentrated Composites''. Accepted in SIAM Multiscale Modeling and Simulations, 2015. 15 pages  
  [I.3] Y. Gorb, A. Novikov: ''Blow-up of Solutions to a p-Laplace Equation'', SIAM Multiscale Modeling and Simulations, Vol 10, No 3, 2012, pp. 727-743  
  [I.4] L. Berlyand, Y. Gorb, A. Novikov: ''Fictitious Fluid Approach and Anomalous Blow-up of the Dissipation Rate in a 2D Model of Concentrated Suspensions'', Arch. Rat. Mech. Anal., Vol 193, No 3, 2009, pp. 585-622  
  [I.5] L. Berlyand, G. Cardone, Y. Gorb, G. Panasenko: ''Asymptotic Analysis of an Array of Closely Spaced Absolutely Conductive Inclusions'', Networks and Heterogeneous Media, Vol 1, No 3, 2006, pp. 353-377  
  [I.6] L. Berlyand, Y. Gorb, A. Novikov: ''Discrete Network Approximation for Highly Packed Composites with Irregular Geometry in Three Dimensions'', Multiscale Methods in Science and Engineering, B. Engquist, P. Lotstedt, O. Runborg, eds., Lecture Notes in Computational Science and Engineering, Vol 44, Springer, 2005, pp. 21-58  
  [I.7] Y. Gorb, L. Berlyand: ''Asymptotics of the Effective Conductivity of Composites with Closely Spaced Inclusions of Optimal Shape'', The Quarterly Journal of Mechanics and Applied Mathematics, Vol 58, 2005, pp. 83-106  
  [I.8] Y. Gorb, L. Berlyand: ''The Effective Conductivity of Densely Packed High Contrast Composites with Inclusions of Optimal Shape'', Continuum Models and Discrete Systems, Kluwer Academic Publishers, D. Bergman et. al. eds., 2004, pp. 63-74  

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.

  Simulations of Particulate Flows Using Mixture Methods  
  In [II.2] 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 [II.3], is employed. The presented in [II.3] 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, it is often sufficient to predict the large scale solutions to certain accuracy, and a common approach is to ''scale up'' a heterogeneous medium. This means to derive coarse-scale equations with a prescribed analytical form that may differ from the underlying fine-scale equations, and to find an effective representation of the permeability on a coarse mesh so that the large scale flow can be correctly computed on this mesh, hence, greatly reducing the computational cost. In [II.4,II.6] the problem of upscaling for fluid flow in deformable porous media is studied, hence, introducing a fully multiscale framework for fluid-structure interaction problems with large interface displacements. A rigorous analysis of the proposed algorithms is also performed.  
  Homogenization for Rigid Suspensions with Random Velocity-Dependent Interfacial Forces  
  In [II.1] we investigate the influence of highly oscillating random velocity-dependent surface forces on the constitutive behavior of a suspension consisting of a viscous incompressible fluid that carries rigid neutrally buoyant particles. An equivalent description of a homogeneous (or effective) medium for suspensions in the moderate concentration regime of particles is derived and justified. The obtained effective medium is a viscous anisotropic flow depending on local mictrostructure and velocity-depending forcing. Obtained results can be readily applied in the development of a multiscale numerical techniques for numerical treatment of heterogeneous mixtures with small particles.  
  Simulations of Diffusion in Cellular Flows at High Peclet Numbers  
  In [II.5] an efficient numerical scheme is proposed for turbulent diffusion transport governed by cellular flows, which are generated by two-dimensional, steady, divergence-free velocity field. The mathematical formulation of the problem involves the steady advection-diffusion problem at high Peclet number, and finds it applications in oceanography, meteorology, etc.  
  [II.1] Y. Gorb, R. F. Maris, B. Vernescu: ''Homogenization for Rigid Suspensions with Random Velocity-Dependent Interfacial Forces'', Journal of Mathematical Analysis and Applications, Vol 420, No 1, 2014, pp. 632-668.  
  [II.2] Y. Gorb, O. Mierka, L. Rivkind, D. Kuzmin: ''Finite Element Simulation of Three-Dimensional Particulate Flows Using Mixture Models'', Journal of Computational and Applied Mathematics, Vol 270, 2014, pp. 443-450.  
  [II.3] D. Kuzmin, Y. Gorb: ''A Flux-Corrected Transport Algorithm for Handling the Close-Packing Limit in Dense Suspensions'', Journal of Computational and Applied Mathematics, Vol 236, No 18, 2012, pp. 4944-4951  
  [II.4] P. Popov, Y. Gorb, Y. Efendiev: ''Multiscale Finite Element Methods for Fluid-Structure Interaction Problems'', accepted at Computer Methods in Applied Mechanics and Engineering, 41 pages  
  [II.5] Y. Gorb, D. Nam, A. Novikov: ''Numerical Simulations of Diffusion in Cellular Flows at High Peclet Numbers'', Discrete and Continuous Dynamical Systems - B, Vol 15, No 1, 2011, pp. 75-92  
  [II.6] P. Popov, Y. Efendiev, Y. Gorb: ''Multiscale modeling and simulation of fluid flows in highly deformable porous media'', Large-Scale Scientific Computing: Lecture Notes in Computer Science, Vol 5910, 2010, pp. 148-156  

III. Nonlinear Solid Mechanics
  Mathematical Biology/Nonlinear Elasticity  
  In [III.1], the high frequency vibrational response of nonlinear elastic bodies subjected to large deformations and residual stresses is modeled and analyzed. An important application of the study is to the detection of a vulnerable atherosclerotic plaque in a large artery. While the effect of material parameters and body geometry on sound wave propagation is well studied, such is less the case for pre-stress and residual stress. It is also well known that the impact of residual stress on the stress distribution in a deformed nonlinear elastic body can be quite significant. To that end, the study of [III.1] investigates whether it is also significant on the small amplitude ultrasound vibrational frequency spectrum.  
  Fracture Mechanics  
  The main subject of Fracture Mechanics projects collected in [III.2-III.4] is the dynamic, transient propagation of a semi-infinite, mode I fracture considered for an infinite elastic body. While the bulk material behavior is assumed to be that of an isotropic, homogeneous, linear elastic body, the crack tip is modeled through inclusion of a nonlinear, elastic or viscoelastic, cohesive zone. The model is motivated by dynamic fracture in brittle polymers in which crack propagation is preceded by significant crazing in a thin region surrounding the crack tip.  
  [III.1] Y. Gorb, J. R. Walton: ''Dependence of the Vibrational Frequency Spectrum of a Nonlinear, Inhomogeneous Residually Stressed Elastic Body on Residual Stress'', International J. of Engineering Science, Vol. 48, 2010, pp. 1289-1312  
  [III.2] T. L. Leise, J. R. Walton, Y. Gorb: ''A Boundary Integral Method for a Dynamic, Transient Mode I Crack Problem with Viscoelastic Cohesive Zone'', International Journal of Fracture, Vol.162, No.1-2, 2010, pp. 69-76  
  [III.3] T. L. Leise, J. R. Walton, Y. Gorb: ''Reconsidering the Boundary Conditions for a Dynamic, Transient, Mode I Crack Problem'', J. of Mechanics of Materials and Structures, Vol.3, No.9, 2008, pp. 1797-1807  
  [III.4] Y. Gorb, T. L. Leise, J. R. Walton: ''Dynamic, Transient Mode I Crack Propagation with a Nonlinear Viscoelastic Cohesive'', Proceedings of the 6th International Conference on Mechanics of Time-Dependent Materials}, R. B. Hall, H. Lu, H. J. Qi, eds., Curran Associates, 57 Morehouse Lane, Red Hook, NY 12571, 2008, pp. 265-268