26,034 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
Importance Sampling: Intrinsic Dimension and Computational Cost
The basic idea of importance sampling is to use independent samples from a
proposal measure in order to approximate expectations with respect to a target
measure. It is key to understand how many samples are required in order to
guarantee accurate approximations. Intuitively, some notion of distance between
the target and the proposal should determine the computational cost of the
method. A major challenge is to quantify this distance in terms of parameters
or statistics that are pertinent for the practitioner. The subject has
attracted substantial interest from within a variety of communities. The
objective of this paper is to overview and unify the resulting literature by
creating an overarching framework. A general theory is presented, with a focus
on the use of importance sampling in Bayesian inverse problems and filtering.Comment: Statistical Scienc
Recent developments in quantum mechanics with magnetic fields
We present a review on the recent developments concerning rigorous
mathematical results on Schr\"odinger operators with magnetic fields. This
paper is dedicated to the sixtieth birthday of Barry Simon.Comment: Update of the previous versions; some more references added and typos
and some minor errors correcte
On entropy production for controlled Markovian evolution
We consider thermodynamic systems with finitely many degrees of freedom and
subject to an external control action. We derive some basic results on the
dependence of the relative entropy production rate on the controlling force.
Applications to macromolecular cooling and to controlling the convergence to
equilibrium rate are sketched. Analogous results are derived for closed and
open n-level quantum systems.Comment: 19 page
A rigorous proof of the Landau-Peierls formula and much more
We present a rigorous mathematical treatment of the zero-field orbital
magnetic susceptibility of a non-interacting Bloch electron gas, at fixed
temperature and density, for both metals and semiconductors/insulators. In
particular, we obtain the Landau-Peierls formula in the low temperature and
density limit as conjectured by T. Kjeldaas and W. Kohn in 1957.Comment: 30 pages - Accepted for publication in A.H.
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