8,274 research outputs found
Effective transient behaviour of heterogeneous media in diffusion problems with a large contrast in the phase diffusivities
This paper presents a homogenisation-based constitutive model to describe the
effective tran- sient diffusion behaviour in heterogeneous media in which there
is a large contrast between the phase diffusivities. In this case mobile
species can diffuse over long distances through the fast phase in the time
scale of diffusion in the slow phase. At macroscopic scale, contrasted phase
diffusivities lead to a memory effect that cannot be properly described by
classical Fick's second law. Here we obtain effective governing equations
through a two-scale approach for composite materials consisting of a fast
matrix and slow inclusions. The micro-macro transition is similar to
first-order computational homogenisation, and involves the solution of a
transient diffusion boundary-value problem in a Representative Volume Element
of the microstructure. Different from computational homogenisation, we propose
a semi-analytical mean-field estimate of the composite response based on the
exact solution for a single inclusion developed in our previous work [Brassart,
L., Stainier, L., 2018. Effective transient behaviour of inclusions in
diffusion problems. Z. Angew Math. Mech. 98, 981-998]. A key outcome of the
model is that the macroscopic concentration is not one-to-one related to the
macroscopic chemical potential, but obeys a local kinetic equation associated
with diffusion in the slow phase. The history-dependent macroscopic response
admits a representation based on internal variables, enabling efficient time
integration. We show that the local chemical kinetics can result in non-Fickian
behaviour in macroscale boundary-value problems.Comment: 36 pages, 14 figure
A jigsaw puzzle framework for homogenization of high porosity foams
An approach to homogenization of high porosity metallic foams is explored.
The emphasis is on the \Alporas{} foam and its representation by means of
two-dimensional wire-frame models. The guaranteed upper and lower bounds on the
effective properties are derived by the first-order homogenization with the
uniform and minimal kinematic boundary conditions at heart. This is combined
with the method of Wang tilings to generate sufficiently large material samples
along with their finite element discretization. The obtained results are
compared to experimental and numerical data available in literature and the
suitability of the two-dimensional setting itself is discussed.Comment: 11 pages, 7 figures, 3 table
Adaptive multiscale model reduction with Generalized Multiscale Finite Element Methods
In this paper, we discuss a general multiscale model reduction framework
based on multiscale finite element methods. We give a brief overview of related
multiscale methods. Due to page limitations, the overview focuses on a few
related methods and is not intended to be comprehensive. We present a general
adaptive multiscale model reduction framework, the Generalized Multiscale
Finite Element Method. Besides the method's basic outline, we discuss some
important ingredients needed for the method's success. We also discuss several
applications. The proposed method allows performing local model reduction in
the presence of high contrast and no scale separation
Localized bases for finite dimensional homogenization approximations with non-separated scales and high-contrast
We construct finite-dimensional approximations of solution spaces of
divergence form operators with -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 if source
terms are in the unit ball of instead of the unit ball of .
Approximation spaces are generated by solving elliptic PDEs on localized
sub-domains with source terms corresponding to approximation bases for .
The -error estimates show that -dimensional spaces
with basis elements localized to sub-domains of diameter (with ) result in an
accuracy for elliptic, parabolic and hyperbolic
problems. For high-contrast media, the accuracy of the method is preserved
provided that localized sub-domains contain buffer zones of width
where the contrast of the medium
remains bounded. The proposed method can naturally be generalized to vectorial
equations (such as elasto-dynamics).Comment: Accepted for publication in SIAM MM
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