Leonid Berlyand

Generalized Clausius–Mossotti Formula for Random Composite with Circular Fibers

An important area of materials science is the study of effective dielectric, thermal and electrical properties of two phase composite materials with very different properties of the constituents. The case of small concentration is well studied and analytical formulas such as Clausius–Mossotti (Maxwell–Garnett) are successfully used by physicists and engineers. We investigate analytically the case of an arbitrary number of unidirectional circular fibers in the periodicity cell when the concentration of the fibers is not small, i.e., we account for interactions of all orders (pair, triplet, etc.). We next consider transversely-random unidirectional composite of the parallel fibers and obtain a closed form representation for the effective conductivity (as a power series in the concentration v). We express the coefficients in this expansion in terms of integrals of the elliptic Eisenstein functions. These integrals are evaluated and the explicit dependence of the parameter d, which characterizes random position of the fibers centers, is obtained. Thus we have extended the Clausius–Mossotti formula for the non dilute mixtures by adding the higher order terms in concentration and qualitatively evaluated the effect of randomness in the fibers locations. In particular, we have proven that the periodic array provides extremum for the effective conductivity in our class of random arrays (“shaking” geometries). Our approach is based on complex analysis techniques and functional equations, which are solved by the successive approximations method.

Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization

We introduce a new variational method for the numerical homogenization of divergence form elliptic, parabolic and hyperbolic equations with arbitrary rough (L^∞) coefficients. Our method does not rely on concepts of ergodicity or scale-separation but on compactness properties of the solution space and a new variational approach to homogenization. The approximation space is generated by an interpolation basis (over scattered points forming a mesh of resolution H) minimizing the L^2 norm of the source terms; its (pre-)computation involves minimizing O(H^(-d)) quadratic (cell) problems on (super-)localized sub-domains of size O(H ln(1/H)). The resulting localized linear systems remain sparse and banded. The resulting interpolation basis functions are biharmonic for d ≤ 3, and polyharmonic for d ≥ 4, for the operator -div(a∇.) and can be seen as a generalization of polyharmonic splines to differential operators with arbitrary rough coefficients. The accuracy of the method (O(H)) in energy norm and independent from aspect ratios of the mesh formed by the scattered points) is established via the introduction of a new class of higher-order Poincare inequalities. The method bypasses (pre-)computations on the full domain and naturally generalizes to time dependent problems, it also provides a natural solution to the inverse problem of recovering the solution of a divergence form elliptic equation from a finite number of point measurements.

Flux Norm Approach to Finite Dimensional Homogenization Approximations with Non-Separated Scales and High Contrast

We consider linear divergence-form scalar elliptic equations and vectorial equations for elasticity with rough (L∞(Ω),

Strong and weak blow-up of the viscous dissipation rates for concentrated suspensions

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Symmetry Breaking in Annular Domains for a Ginzburg-Landau Superconductivity Model

) coefficients a(x) that, in particular, model media with non-separated scales and high contrast in material properties. While the homogenization of PDEs with periodic or ergodic coefficients and well separated scales is now well understood, we consider here the most general case of arbitrary bounded coefficients. For such problems, we introduce explicit and optimal finite dimensional approximations of solutions that can be viewed as a theoretical Galerkin method with controlled error estimates, analogous to classical homogenization approximations. In particular, this approach allows one to analyze a given medium directly without introducing the mathematical concept of an

Correlation properties of collective motion in bacterial suspensions.

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