545 research outputs found
Model Reduction for Multiscale Lithium-Ion Battery Simulation
In this contribution we are concerned with efficient model reduction for
multiscale problems arising in lithium-ion battery modeling with spatially
resolved porous electrodes. We present new results on the application of the
reduced basis method to the resulting instationary 3D battery model that
involves strong non-linearities due to Buttler-Volmer kinetics. Empirical
operator interpolation is used to efficiently deal with this issue.
Furthermore, we present the localized reduced basis multiscale method for
parabolic problems applied to a thermal model of batteries with resolved porous
electrodes. Numerical experiments are given that demonstrate the reduction
capabilities of the presented approaches for these real world applications
On Multiscale Methods in Petrov-Galerkin formulation
In this work we investigate the advantages of multiscale methods in
Petrov-Galerkin (PG) formulation in a general framework. The framework is based
on a localized orthogonal decomposition of a high dimensional solution space
into a low dimensional multiscale space with good approximation properties and
a high dimensional remainder space{, which only contains negligible fine scale
information}. The multiscale space can then be used to obtain accurate Galerkin
approximations. As a model problem we consider the Poisson equation. We prove
that a Petrov-Galerkin formulation does not suffer from a significant loss of
accuracy, and still preserve the convergence order of the original multiscale
method. We also prove inf-sup stability of a PG Continuous and a Discontinuous
Galerkin Finite Element multiscale method. Furthermore, we demonstrate that the
Petrov-Galerkin method can decrease the computational complexity significantly,
allowing for more efficient solution algorithms. As another application of the
framework, we show how the Petrov-Galerkin framework can be used to construct a
locally mass conservative solver for two-phase flow simulation that employs the
Buckley-Leverett equation. To achieve this, we couple a PG Discontinuous
Galerkin Finite Element method with an upwind scheme for a hyperbolic
conservation law
Robust error estimates in weak norms for advection dominated transport problems with rough data
We consider mixing problems in the form of transient convection--diffusion
equations with a velocity vector field with multiscale character and rough
data. We assume that the velocity field has two scales, a coarse scale with
slow spatial variation, which is responsible for advective transport and a fine
scale with small amplitude that contributes to the mixing. For this problem we
consider the estimation of filtered error quantities for solutions computed
using a finite element method with symmetric stabilization. A posteriori error
estimates and a priori error estimates are derived using the multiscale
decomposition of the advective velocity to improve stability. All estimates are
independent both of the P\'eclet number and of the regularity of the exact
solution
Numerical Homogenization of the Acoustic Wave Equations with a Continuum of Scales
In this paper, we consider numerical homogenization of acoustic wave
equations with heterogeneous coefficients, namely, when the bulk modulus and
the density of the medium are only bounded. We show that under a Cordes type
condition the second order derivatives of the solution with respect to harmonic
coordinates are (instead with respect to Euclidean coordinates)
and the solution itself is in (instead of
with respect to Euclidean coordinates). Then, we
propose an implicit time stepping method to solve the resulted linear system on
coarse spatial scales, and present error estimates of the method. It follows
that by pre-computing the associated harmonic coordinates, it is possible to
numerically homogenize the wave equation without assumptions of scale
separation or ergodicity.Comment: 27 pages, 4 figures, Submitte
Nonlinear nonlocal multicontinua upscaling framework and its applications
In this paper, we discuss multiscale methods for nonlinear problems. The main
idea of these approaches is to use local constraints and solve problems in
oversampled regions for constructing macroscopic equations. These techniques
are intended for problems without scale separation and high contrast, which
often occur in applications. For linear problems, the local solutions with
constraints are used as basis functions. This technique is called Constraint
Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM).
GMsFEM identifies macroscopic quantities based on rigorous analysis. In
corresponding upscaling methods, the multiscale basis functions are selected
such that the degrees of freedom have physical meanings, such as averages of
the solution on each continuum.
This paper extends the linear concepts to nonlinear problems, where the local
problems are nonlinear. The main concept consists of: (1) identifying
macroscopic quantities; (2) constructing appropriate oversampled local problems
with coarse-grid constraints; (3) formulating macroscopic equations. We
consider two types of approaches. In the first approach, the solutions of local
problems are used as basis functions (in a linear fashion) to solve nonlinear
problems. This approach is simple to implement; however, it lacks the nonlinear
interpolation, which we present in our second approach. In this approach, the
local solutions are used as a nonlinear forward map from local averages
(constraints) of the solution in oversampling region. This local fine-grid
solution is further used to formulate the coarse-grid problem. Both approaches
are discussed on several examples and applied to single-phase and two-phase
flow problems, which are challenging because of convection-dominated nature of
the concentration equation
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