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 L^{2} (instead of H^{-1}
with respect to Euclidean coordinates) and the
solution itself is in L^{∞}(0,T,H^{2}(Ω)) (instead of L^{∞}(0,T,H^{1}(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 L^{2}(Ω ×
(0,T))
into a space of
functions which at small (time and space) scales are close in H^{1} 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 L^{2} (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.