30,188 research outputs found
Multiscale Finite Element Methods for Nonlinear Problems and their Applications
In this paper we propose a generalization of multiscale finite element methods (Ms-FEM) to nonlinear problems. We study the convergence of the proposed method for nonlinear elliptic equations and propose an oversampling technique. Numerical examples demonstrate that the over-sampling technique greatly reduces the error. The application of MsFEM to porous media flows is considered. Finally, we describe further generalizations of MsFEM to nonlinear time-dependent equations and discuss the convergence of the method for various kinds of heterogeneities
Metric based up-scaling
We consider divergence form elliptic operators in dimension with
coefficients. Although solutions of these operators are only
H\"{o}lder continuous, we show that they are differentiable ()
with respect to harmonic coordinates. It follows that numerical homogenization
can be extended to situations where the medium has no ergodicity at small
scales and is characterized by a continuum of scales by transferring a new
metric in addition to traditional averaged (homogenized) quantities from
subgrid scales into computational scales and error bounds can be given. This
numerical homogenization method can also be used as a compression tool for
differential operators.Comment: Final version. Accepted for publication in Communications on Pure and
Applied Mathematics. Presented at CIMMS (March 2005), Socams 2005 (April),
Oberwolfach, MPI Leipzig (May 2005), CIRM (July 2005). Higher resolution
figures are available at http://www.acm.caltech.edu/~owhadi
A posteriori modeling error estimates in the optimization of two-scale elastic composite materials
The a posteriori analysis of the discretization error and the modeling error
is studied for a compliance cost functional in the context of the optimization
of composite elastic materials and a two-scale linearized elasticity model. A
mechanically simple, parametrized microscopic supporting structure is chosen
and the parameters describing the structure are determined minimizing the
compliance objective. An a posteriori error estimate is derived which includes
the modeling error caused by the replacement of a nested laminate
microstructure by this considerably simpler microstructure. Indeed, nested
laminates are known to realize the minimal compliance and provide a benchmark
for the quality of the microstructures. To estimate the local difference in the
compliance functional the dual weighted residual approach is used. Different
numerical experiments show that the resulting adaptive scheme leads to simple
parametrized microscopic supporting structures that can compete with the
optimal nested laminate construction. The derived a posteriori error indicators
allow to verify that the suggested simplified microstructures achieve the
optimal value of the compliance up to a few percent. Furthermore, it is shown
how discretization error and modeling error can be balanced by choosing an
optimal level of grid refinement. Our two scale results with a single scale
microstructure can provide guidance towards the design of a producible
macroscopic fine scale pattern
A Framework for Modeling Subgrid Effects for Two-Phase Flows in Porous Media
In this paper, we study upscaling for two-phase flows in strongly heterogeneous porous media. Upscaling a hyperbolic convection equation is known to be very difficult due to the presence of nonlocal memory effects. Even for a linear hyperbolic equation with a shear velocity field, the upscaled equation involves a nonlocal history dependent diffusion term, which is not amenable to computation. By performing a systematic multiscale analysis, we derive coupled equations for the average and the fluctuations for the two-phase flow. The homogenized equations for the coupled system are obtained by projecting the fluctuations onto a suitable subspace. This projection corresponds exactly to averaging along streamlines of the flow. Convergence of the multiscale analysis is verified numerically. Moreover, we show how to apply this multiscale analysis to upscale two-phase flows in practical applications
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