8,211 research outputs found
The asymptotic homogenization elasticity tensor properties for composites with material discontinuities
The classical asymptotic homogenization approach for linear elastic composites with discontinuous material properties is considered as a starting point. The sharp length scale separation between the fine periodic structure and the whole material formally leads to anisotropic elastic-type balance equations on the coarse scale, where the arising fourth rank operator is to be computed solving single periodic cell problems on the fine scale. After revisiting the derivation of the problem, which here explicitly points out how the discontinuity in the individual constituents’ elastic coefficients translates into stress jump interface conditions for the cell problems, we prove that the gradient of the cell problem solution is minor symmetric and that its cell average is zero. This property holds for perfect interfaces only (i.e., when the elastic displacement is continuous across the composite’s interface) and can be used to assess the accuracy of the computed numerical solutions. These facts are further exploited, together with the individual constituents’ elastic coefficients and the specific form of the cell problems, to prove a theorem that characterizes the fourth rank operator appearing in the coarse-scale elastic-type balance equations as a composite material effective elasticity tensor. We both recover known facts, such as minor and major symmetries and positive definiteness, and establish new facts concerning the Voigt and Reuss bounds. The latter are shown for the first time without assuming any equivalence between coarse and fine-scale energies (Hill’s condition), which, in contrast to the case of representative volume elements, does not identically hold in the context of asymptotic homogenization. We conclude with instructive three-dimensional numerical simulations of a soft elastic matrix with an embedded cubic stiffer inclusion to show the profile of the physically relevant elastic moduli (Young’s and shear moduli) and Poisson’s ratio at increasing (up to 100 %) inclusion’s volume fraction, thus providing a proxy for the design of artificial elastic composites
Diffusion in quantum geometry
The change of the effective dimension of spacetime with the probed scale is a
universal phenomenon shared by independent models of quantum gravity. Using
tools of probability theory and multifractal geometry, we show how dimensional
flow is controlled by a multiscale fractional diffusion equation, and
physically interpreted as a composite stochastic process. The simplest example
is a fractional telegraph process, describing quantum spacetimes with a
spectral dimension equal to 2 in the ultraviolet and monotonically rising to 4
towards the infrared. The general profile of the spectral dimension of the
recently introduced multifractional spaces is constructed for the first time.Comment: 5 pages, 1 figure. v2: title slightly changed, discussion improve
Proceedings of the Symposium on Concrete Modelling, CONMOD2018
CONMOD2018 is a symposium on Concrete Modelling which is jointly organised by Delft University and Ghent University as part of the RILEM week 2018 in Delft, The Netherlands.
The symposium is the 5th in a series dealing with all aspects concerning modelling of concrete at various scales.
The symposium consist of 3 key-note papers and 62 regular papers presented over 3 days.
Parallel to the CONMOD2018 symposium a conference on Service Life Design (SLD4) and a workshop honouring Professor Klaas van Breugel were organised with topics that are related to concrete modelling.
In total more than 350 participants took part in the events organised during the RILEM week 2018
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
Tensor-based multiscale method for diffusion problems in quasi-periodic heterogeneous media
This paper proposes to address the issue of complexity reduction for the
numerical simulation of multiscale media in a quasi-periodic setting. We
consider a stationary elliptic diffusion equation defined on a domain such
that is the union of cells and we
introduce a two-scale representation by identifying any function defined
on with a bi-variate function , where relates to the
index of the cell containing the point and relates to a local
coordinate in a reference cell . We introduce a weak formulation of the
problem in a broken Sobolev space using a discontinuous Galerkin
framework. The problem is then interpreted as a tensor-structured equation by
identifying with a tensor product space of
functions defined over the product set . Tensor numerical methods
are then used in order to exploit approximability properties of quasi-periodic
solutions by low-rank tensors.Comment: Changed the choice of test spaces V(D) and X (with regard to
regularity) and the argumentation thereof. Corrected proof of proposition 3.
Corrected wrong multiplicative factor in proposition 4 and its proof (was 2
instead of 1). Added remark 6 at the end of section 2. Extended remark 7.
Added references. Some minor improvements (typos, typesetting
Multiscale asymptotic homogenization analysis of thermo-diffusive composite materials
In this paper an asymptotic homogenization method for the analysis of
composite materials with periodic microstructure in presence of thermodiffusion
is described. Appropriate down-scaling relations correlating the microscopic
fields to the macroscopic displacements, temperature and mass concentration are
introduced. The effects of the material inhomogeneities are described by
perturbation functions derived from the solution of recursive cell problems.
Exact expressions for the overall elastic and thermodiffusive constants of the
equivalent first order thermodiffusive continuum are derived. The proposed
approach is applied to the case of a two-dimensional bi-phase orthotropic
layered material, where the effective elastic and thermodiffusive properties
can be determined analytically. Considering this illustrative example and
assuming periodic body forces, heat and mass sources acting on the medium, the
solution performed by the first order homogenization approach is compared with
the numerical results obtained by the heterogeneous model.Comment: 40 pages, 13 figure
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