Construction and sharp consistency estimates for atomistic/continuum coupling methods with general interfaces: a 2D model problem
C. Ortner, L. Zhang
Accepted for publication in SIAM Numerical Analysis. Preprint.
Abstract: We present a new variant of the geometry reconstruction approach for the formulation of atomistic/continuum coupling methods (a/c methods). For multi-body nearest-neighbour interactions on the 2D triangular lattice, we show that patch test consistent a/c methods can be constructed for arbitrary interface geometries. Moreover, we prove that all methods within this class are first-order consistent at the atomistic/continuum interface and second-order consistent in the interior of the continuum region.

Localized bases for finite dimensional homogenization approximations with non-separated scales and high-contrast
H. Owhadi, L. Zhang
SIAM Multiscale Model. Simul. Volume 9, Issue 4, pp. 1373-1398 (2011). Preprint.
Abstract: We construct finite-dimensional approximations of solution spaces of divergence form operators with $L^\infty$-coefficients. Our method does not rely on concepts of ergodicity or scale-separation, but on the property that the solution space of these operators is compactly embedded in $H^1$ if source terms are in the unit ball of $L^2$ instead of the unit ball of $H^{-1}$. Approximation spaces are generated by solving elliptic PDEs on localized sub-domains with source terms corresponding to approximation bases for $H^2$. The proposed method can naturally be generalized to vectorial equations (such as elasto-dynamics).

Global Energy Matching Method for Atomistic-to-Continuum Modeling of Self-Assembling Biopolymer Aggregates
L. Zhang, L. Berlyand, M. V. Fedorov, and H. Owhadi
SIAM Multiscale Model. Simul. Volume 8, Issue 5, pp. 1958-1980 (2010). Preprint.
Abstract: This paper studies mathematical models of biopolymer supramolecular aggregates that are formed by the self-assembly of single monomers. We develop a new multiscale numerical approach to model the structural properties of such aggregates. This theoretical approach establishes micro-macro relations between the geometrical and mechanical properties of the monomers and supramolecular aggregates. The energy matching method developed in this paper does not require crystalline order and, therefore, can be applied to general microstructures with strongly variable spatial correlations. We use this method to compute the shape and the bending stiffness of their supramolecular aggregates from known chiral and amphiphilic properties of the short chain peptide monomers. Numerical implementation of our approach demonstrates consistency with results obtained by molecular dynamics simulations.

Numerical Homogenization of the Accoustic Wave Equation with a Continuum Scales
H. Owhadi, L. Zhang.
Computer Methods in Applied Mechanics and Engineering, Volume 198, Issues 3-4, Pages 397-406, Dec 15 2008. Preprint, arXiv, math.AP/0604380.
Abstract: Consider the acoustic wave equation in dimension n in situations where the bulk modulus and the density of the medium are only bounded. Under a Cordes type condition the second order derivatives of the solution with respect to harmonic coordinates are in L² (instead of H^-1 with respect to Euclidean coordinates) and the solution itself is in L^∞(0,T,H²(Ω)) (instead of L^∞(0,T,H¹(Ω)) with respect to Euclidean coordinates). It is possible to homogenize the wave equation without assumptions of scale separation or ergodicity by pre-computing n solutions of the associated elliptic problems.
Movies (YouTube): simulation with fine mesh dof, simulation with reduced dof.

Homogenization of Parabolic Equations with a Continuum Space and Time Scales
H. Owhadi, L. Zhang.
SIAM J. Numer. Anal. Volume 46, Issue 1, pp. 1-36, 2007. Preprint arXiv, math.AP/0512504.
Abstract:This paper addresses the issue of homogenization of linear divergence form parabolic operators with no ergodicity and no scale separation in time or space. It appears that the inverse operator maps the unit ball of L²(Ω × (0,T)) into a space of functions which at small (time and space) scales are close in H¹ norm to a functional space of dimension n. It follows that once one has solved these equations at least n-times it is possible to homogenize them both in space and in time, reducing the number of operations counts necessary to obtain further solutions. We show that under a Cordes type condition that the first order time derivatives and second order space derivatives of the solution of these operators with respect to caloric coordinates are in L² (instead of H^-1 with Euclidean coordinates). If the medium is time independent then it is sufficient to solve n times the associated elliptic equation in order to homogenize the original parabolic problem.
Movies (YouTube): u in euclidean coordinates, u in caloric coordinates, gradient of u in Euclidean coordinates, gradient of u in caloric cooridinates, image of fine mesh in caloric coordinates.

Metric Based Upscaling
H. Owhadi, L. Zhang.
Communications on Pure and Applied Mathematics, Vol. 60, Issue 5, p 675-723, 2007. Preprint arXiv, math.NA/0505223. Correction.
Abstract: Consider divergence form elliptic operators in dimension n ≥ 2 with L^∞ coefficients. Although solutions of these operators are only Hölder continuous, they are differentiable C^1,α with respect to harmonic coordinates. It follows that numerical homogenization can be extended to situations where the medium has no separated scales. This new numerical homogenization method is based on the transfer a new metric from subgrid scales into computational scales in addition to traditional averaged (homogenized) quantities. Error bounds can be given and this method can also be used as a compression tool for differential operators.