16,736 research outputs found
Multiscale simulations of porous media flows in flow-based coordinate system
In this paper, we propose a multiscale technique for the simulation of porous media flows in a flow-based coordinate system. A flow-based coordinate system allows us to simplify the scale interaction and derive the upscaled equations for purely hyperbolic transport equations. We discuss the applications of the method to two-phase flows in heterogeneous porous media. For two-phase flow simulations, the use of a flow-based coordinate system requires limited global information, such as the solution of single-phase flow. Numerical results show that one can achieve accurate upscaling results using a flow-based coordinate system
A multiscale flux basis for mortar mixed discretizations of reduced Darcy-Forchheimer fracture models
In this paper, a multiscale flux basis algorithm is developed to efficiently
solve a flow problem in fractured porous media. Here, we take into account a
mixed-dimensional setting of the discrete fracture matrix model, where the
fracture network is represented as lower-dimensional object. We assume the
linear Darcy model in the rock matrix and the non-linear Forchheimer model in
the fractures. In our formulation, we are able to reformulate the
matrix-fracture problem to only the fracture network problem and, therefore,
significantly reduce the computational cost. The resulting problem is then a
non-linear interface problem that can be solved using a fixed-point or
Newton-Krylov methods, which in each iteration require several solves of Robin
problems in the surrounding rock matrices. To achieve this, the flux exchange
(a linear Robin-to-Neumann co-dimensional mapping) between the porous medium
and the fracture network is done offline by pre-computing a multiscale flux
basis that consists of the flux response from each degree of freedom on the
fracture network. This delivers a conserve for the basis that handles the
solutions in the rock matrices for each degree of freedom in the fractures
pressure space. Then, any Robin sub-domain problems are replaced by linear
combinations of the multiscale flux basis during the interface iteration. The
proposed approach is, thus, agnostic to the physical model in the fracture
network. Numerical experiments demonstrate the computational gains of
pre-computing the flux exchange between the porous medium and the fracture
network against standard non-linear domain decomposition approaches
A Bayesian numerical homogenization method for elliptic multiscale inverse problems
A new strategy based on numerical homogenization and Bayesian techniques for
solving multiscale inverse problems is introduced. We consider a class of
elliptic problems which vary at a microscopic scale, and we aim at recovering
the highly oscillatory tensor from measurements of the fine scale solution at
the boundary, using a coarse model based on numerical homogenization and model
order reduction. We provide a rigorous Bayesian formulation of the problem,
taking into account different possibilities for the choice of the prior
measure. We prove well-posedness of the effective posterior measure and, by
means of G-convergence, we establish a link between the effective posterior and
the fine scale model. Several numerical experiments illustrate the efficiency
of the proposed scheme and confirm the theoretical findings
Multiscale Finite Element Modeling of Nonlinear Magnetoquasistatic Problems Using Magnetic Induction Conforming Formulations
In this paper we develop magnetic induction conforming multiscale
formulations for magnetoquasistatic problems involving periodic materials. The
formulations are derived using the periodic homogenization theory and applied
within a heterogeneous multiscale approach. Therefore the fine-scale problem is
replaced by a macroscale problem defined on a coarse mesh that covers the
entire domain and many mesoscale problems defined on finely-meshed small areas
around some points of interest of the macroscale mesh (e.g. numerical
quadrature points). The exchange of information between these macro and meso
problems is thoroughly explained in this paper. For the sake of validation, we
consider a two-dimensional geometry of an idealized periodic soft magnetic
composite.Comment: Paper accepted for publication in the SIAM MMS journa
Numerical homogenization of elliptic PDEs with similar coefficients
We consider a sequence of elliptic partial differential equations (PDEs) with
different but similar rapidly varying coefficients. Such sequences appear, for
example, in splitting schemes for time-dependent problems (with one coefficient
per time step) and in sample based stochastic integration of outputs from an
elliptic PDE (with one coefficient per sample member). We propose a
parallelizable algorithm based on Petrov-Galerkin localized orthogonal
decomposition (PG-LOD) that adaptively (using computable and theoretically
derived error indicators) recomputes the local corrector problems only where it
improves accuracy. The method is illustrated in detail by an example of a
time-dependent two-pase Darcy flow problem in three dimensions
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