On nonstandard Padé approximants suitable for effective properties of two-phase composite materials

This article investigates the existence of the nonstandard Padé approximants introduced by Cherkaev and Zhang [D.-L. Zhang and E. Cherkaev, Reconstruction of spectral function from effective permittivity of a composite material using rational function approximations, J. Comput. Phys. 228 (2009), pp. 5390–5409] for approximating the spectral function of composites from effective properties at different frequencies. The spectral functions contain information on microstructure of composites. Since this reconstruction problem is ill-posed Cherkaev [Inverse homogenization for evaluation of effective properties of a mixture, Inverse Probl. 17 (2001), pp. 1203–1218], the well-performed Padé approach is noteworthy and deserves further investigations. In this article, we validate the assumption that the effective dielectric component of interest of all two-phase composites can be approximated by Padé approximants whose denominator has nonzero power one term. We refer to this as the nonstandard Padé approximant, in contrast to the standard approximants whose denominators have nonzero constant terms. For composites whose spectral function assumes infinitely many different values such as the checkerboard microstructure, the proof is carried by using classical results for Markov–Stieltjes functions (also referred to as Stieltjes functions) Golden and Papanicolaou [Bounds on effective parameters of heterogeneous media by analytic continuation, Commun. Math. Phys. 90 (1983), pp. 473–491] and Cherkaev and Ou [De-homogenization: Reconstruction of moments of the spectral measure of the composite, Inverse Probl. 24 (2008), p. 065008]. However, it is well-known that spectral functions for microstructure such as rank-n laminates assume only finitely many different values, i.e. the measure in the Markov–Stieltjes function is supported at only finitely many points. For this case, we cannot find any existence results for nonstandard Padé approximants in the literature. The proof for this case is the focus of this article. It is done by utilizing a special product decomposition of the coefficient matrix of the Padé system. The results in this article can be considered as an extension of the Padé theory for Markov–Stieltjes functions whose spectral function take infinitely many different values to those taking only finitely many values. In the literature, the latter is usually excluded from the definition of Markov–Stieltjes functions because they correspond to rational functions, hence convergence of their Padé approximants is trivial. However, from an inverse problem point of view, we need to assure both the existence and convergence of the nonstandard Padé approximants, for all microstructures. The results in this article provide a mathematical foundation for applying the Padé approach for reconstructing the spectral functions of composites whose microstructure is not a priori known.     

Two-scale analysis method for predicting heat transfer performance of composite materials with random grain distribution

A two-scale analysis (TSA) method for predicting the heat transfer performance of composite materials with the random distribution of same-scale grains is presented. First the representation of the materials with the random distribution is briefly described. Then the two-scale analysis formulation of heat transfer behavior of the materials with random grain distribution of small periodicity is formally derived by means of     

A method of calculating the critical dimensions of composite operators in the massless nonlinear σ model

For an arbitrary system of composite operators that mix on renormalization, a general scheme is formulated for calculating the critical dimensions directly in the massless model with dimensional and minimal-subtraction type regularization. The critical dimensions of arbitrary powers of the auxiliary field and analogous operators with two additional derivatives are calculated as examples.St Petersburg State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 95, No. 1, pp. 160–175, April, 1993.     

Treatment of the composite fabric’s shaping using a Lagrangian formulation

In this paper, we are interested in the simulation of prepreg composite deformation by deep-drawing and laying-up. It uses new bi-component finite elements made of woven material in which the nodal interior loads are deduced from fibre tensile strain energy and not polymerized resin membrane energy. Specific treatment is used to analyze the frictional-contact problem between the deformable prepreg composite and the steel rigid tools. The frictional-contact method is based on the Lagrangian formulation and the preconditioned conjugate gradient method. Some numerical tests are given to investigate the performance of the numerical strategies.     

On the nature of “quasiphotons”

Reference solutions for the class of particle-like asymptotic solutions of the wave equation suggested by V. M. Babich and V. V. Ulin, are found.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 117–122.The authors are indebted to V. M. Babich for helpful discussions.This research was supported in part by INTAS-RFBR under grant 95-0012.     

Interaction of π and K mesons and nucleons in the model of composite superconformal strings

We construct a model of composite superconformal strings to describe hadron interaction amplitudes and the hadron spectrum. It belongs to a new class of dual (string) models and falls outside the scope of the usual approaches. In the framework of this model, the ordinary hadron scale of the order of 1GeV^?2 is regarded as a string scale α′. The supersymmetry requirements are satisfied only on the two-dimensional world surface. We calculate interaction amplitudes of π and K mesons in the framework of the theory. We find a method for removing the baryon parity degeneracy.     

On the discovery of the “good Broyden” method

This short note traces the events that led to the unsymmetric rank one formula known as the “good Broyden” update [5, 6], which is widely used within derivative-free mathematical software for solving a system of nonlinear equations. Received: February 5, 2000 / Accepted: February 19, 2000?Published online March 15, 2000     

On the convergence of the multivariate “homogeneous”qd-algorithm

The convergence of columns in the univariateqd-algorithm to reciprocals of polar singularities of meromorphic functions has often proved to be very useful. A multivariateqd-algorithm was discovered in 1982 for the construction of the so-called homogeneous Padé approximants.In the first section we repeat the univariate convergence results. In the second section we summarize the homogeneous multivariateqd-algorithm. In the third section a multivariate convergence result is proved by combining results from the previous sections. This convergence result is compared with another theorem for the general order multivariateqdg-algorithm. The main difference lies in the fact that the homogeneous form detects the polar singularities pointwise while the general form detects them curvewise.     

Commons dilemma and the “Tragedy of the Commons”

There are a lot of studies on commons dilemma. But few studies succeed to formulate it adequately. Moreover, although there are some situations where the dilemma does not occur, these studies focuses only the case that the dilemma does occur. In the present paper, we formulate the commons dilemma as an N‐person game and examine what situations our formulation represents. We show that there are six regions with respect to the value decrement induced by one additional animal and that commons dilemma does not occur in some of these regions.     

Numerical implementation of coherence for the example of the “Swiss Cross”

Summary We consider the two-dimensional Helmholtz equation u+u=0 inD with the boundary conditionsu=0 on D. D is the Swiss Cross — a region consisting of five unit squares. A method based on the concept of Coherence is utilized to determine an approximation for the first eigenvalue= ₁ more accurate than calculated by classical difference methods. The numerical result is used to illustrate isoperimetric upper and lower bounds for ₁, and to test some conjectures on its relations with torsional rigidity.Dedicated to the memory of Professor Lathar Collatz     

Correlation of experimental flap-lag-torsion damping—a case study

In 1982, carefully done experiments measured the flap-lag-torsion damping of hingeless rotor at various RPM's and for several combinations of flexure geometry and stiffness. The best elastic-blade stability codes of that time were unable to predict this damping and typically gave errors of the order of 100% (as percentage of the largest measured damping, 5 rad/s). Through the past ten years, various unsuccessful efforts have been made to improve those predictions. Now, improved correlation has been performed with errors of 0% to 20% of 5 rad/s. The results show that most of the former discrepancies can be attributed to aerodynamic effects, the largest of which is the dynamics of the three-dimensional wake.     

Graphlike Manifolds with the Type of the 1—skeleton of a 4—simplex

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