73 research outputs found
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 () 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 ) minimizing the norm of the source terms; its
(pre-)computation involves minimizing quadratic (cell)
problems on (super-)localized sub-domains of size .
The resulting localized linear systems remain sparse and banded. The resulting
interpolation basis functions are biharmonic for , and polyharmonic
for , for the operator -\diiv(a\nabla \cdot) and can be seen as a
generalization of polyharmonic splines to differential operators with arbitrary
rough coefficients. The accuracy of the method ( 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 Poincar\'{e}
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.Comment: ESAIM: Mathematical Modelling and Numerical Analysis. Special issue
(2013
Rise of correlations of transformation strains in random polycrystals
We investigate the statistics of the transformation strains that arise in random martensitic polycrystals as boundary conditions cause its component crystallites to undergo martensitic phase transitions. In our laminated polycrystal model the orientation of the n grains (crystallites)
is given by an uncorrelated random array of the orientation angles Ξ_i, i = 1, . . . ,n. Under imposed boundary conditions the polycrystal grains may undergo a martensitic transformation. The associated transformation strains Δ_i, i = 1, . . . ,n depend on the array of orientation angles, and they can be obtained as a solution to a nonlinear optimization problem. While the random variables Ξ_i,
i = 1, . . . ,n are uncorrelated, the random variables Δ_i, i = 1, . . . ,n may be correlated. This issue is
central in our considerations. We investigate it in following three different scaling limits: (i) Infinitely
long grains (laminated polycrystal of height L = â); (ii) Grains of finite but large height (L » 1); and (iii) Chain of short grains (L = l_0/(2n), l_0 « 1). With references to de Finettiâs theorem, Rieszâ
rearrangement inequality, and near neighbor approximations, our analyses establish that under the
scaling limits (i), (ii), and (iii) the arrays of transformation strains arising from given boundary
conditions exhibit no correlations, long-range correlations, and exponentially decaying short-range
correlations, respectivel
Effective Rheological Properties in Semidilute Bacterial Suspensions
Interactions between swimming bacteria have led to remarkable experimentally
observable macroscopic properties such as the reduction of the effective
viscosity, enhanced mixing, and diffusion. In this work, we study an individual
based model for a suspension of interacting point dipoles representing bacteria
in order to gain greater insight into the physical mechanisms responsible for
the drastic reduction in the effective viscosity. In particular, asymptotic
analysis is carried out on the corresponding kinetic equation governing the
distribution of bacteria orientations. This allows one to derive an explicit
asymptotic formula for the effective viscosity of the bacterial suspension in
the limit of bacterium non-sphericity. The results show good qualitative
agreement with numerical simulations and previous experimental observations.
Finally, we justify our approach by proving existence, uniqueness, and
regularity properties for this kinetic PDE model
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