On the Convergence of a Non-incremental Homogenization Method for Nonlinear Elastic Composite Materials

In order to simulate the nonlinear behaviour of elastomer composite materials, we use a homogenization technique which induces nonlinear problems. This paper presents a non-incremental solving method which allows the reduction of computational costs. In this paper the equivalence between the proposed solving method and a Newton-type method is proved, which allows us to prove the convergence under realistic assumptions. Numerical results on a composite illustrate the performances of this method.     

Delamination of Unidirectional Viscoelastic Composite Materials

The delamination of unidirectional composite materials takes place under compression along the reinforcing elements and often occurs near the free surfaces or planes of these materials. The first investigation in this field was made in 1976 and since then numerous studies have been carried out. However, the approaches developed in these investigations are applicable only to time-independent composite materials. In the present paper, an approach for investigating unidirectional viscoelastic composite materials is proposed, with a particular problem as an example. The investigations are performed within the framework of exact three-dimensional geometrically nonlinear equations of viscoelasticity theory.     

Stretching of an Elastoplastic Plate with an Arbitrarily Oriented Crack

The problem of determination of the crack resistance of an elastoplastic plate, weakened by a rectilinear slit in the form of a cut, under the conditions of uniaxial stretching is considered. The material of the body is assumed to be incompressible, reinforcing, and obeying the Mises condition of plasticity. The straining theory of plasticity is used. The solutions are obtained in the elastic and plastic regions in the form of asymptotic expansions in the neighborhood of the end of the crack. Based on the conditions that the crack borders are unloaded and the elastic and plastic solutions are conjugate on the contour of the plastic region, unknown constants of integration are found. The two-leafed contours of the plastic region are obtained. The critical load is determined.     

Homogenization of Monotone Operators by the Method of Two-Scale Convergence

Homogenization theorems are obtained for nonlinear second-order elliptic operators occurring in some models of media with double porosity. Basic homogenization properties are proved by the method of two-scale convergence combined with the technique of p-connectedness.__________Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 24, pp. 44–66, 2004.     

Computational Homogenization of Micromechanically Resolved Textile Materials

This contribution deals with textile materials. On the macroscopic level textiles are characterized by a large area-to-thickness ratio, such that it is numerically efficient to treat the textile structure as a shell. To capture the contact behavior, fibers within a representative volume element are explicitly modeled by means of one dimensional beam elements on the microscopic level. A suitable, shell-specific homogenization method is developed, which connects the homogeneous shell specific macro level to a fiber structured micro level. This contribution investigates the determination of the nonlinear constitutive behavior of textile materials. Selected examples for the macroscopic behavior of microscopic heterogeneous fiber structured materials are presented. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Characterizations of Some Free Random Variables by Properties of Conditional Moments of Third Degree Polynomials

We investigate Laha–Lukacs properties of noncommutative random variables (processes). We prove that some families of free Meixner distributions can be characterized by the conditional moments of polynomial functions of degree 3. We also show that this fact has consequences in describing some free Lévy processes. The proof relies on a combinatorial identity. At the end of this paper we show that this result can be extended to a \(q\) -Gausian variable.     

Optimization by the Homogenization Method for Nonlinear Elliptic Dirichlet Systems

We study the existence of solutions of control problems relative to a nonlinear elliptic system with Dirichlet boundary conditions. In this problem, the control variables are the coefficients of the equations and the open set where they are posed. It is known that this class of problems has no solution in general, but using homogenization results about elliptic systems we show the existence of solutions when the controls are searched in a bigger set. These results are related to the selection of optimal materials and shapes.     

Computational Homogenization of Piezoelectric Materials using FE2

Due to the growing interest in determining the macroscopic material response of inhomogeneous materials, computational methods are becoming increasingly concerned with the application of homogenization techniques. In this work, two-scale classical (first-order) homogenization of electro-mechanically coupled problems using a FE²-approach is discussed. We explicitly formulated the homogenized coefficients of the elastic, piezoelectric and dielectric tensors for small strain as well as the homogenized remanent strain and remanent polarization. The homogenization of the coupled problem is done using different representative volume elements (RVEs), which capture the microstructure of the inhomogeneous material, to represent the macro material response. Later this technique is used to determine the macroscopic and microscopic configurational forces on certain defects. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Homogenization of a Highly Heterogeneous Elastic-Viscoelastic Composite Materials

We consider the homogenization of a highly heterogeneous periodic medium made of two connected components filled with two linear elastic materials having elasticity tensors of order O(1), separated by a third viscoelastic layer, having a thickness of the same order ${\varepsilon}$ as the basic periodicity cell, and a stiffness (resp. viscosity) tensor of order ${\varepsilon^r ({\rm resp.}, \varepsilon^s) (r, s) \in \mathbb{R}_{d}x+^{2}}$ . Using the method of two-scale convergence, a unified approach yielding rigorous proofs is given covering different scalings. In particular, it is shown that the medium behaves asymptotically as an elastic homogeneous one, except in the case m : = min(r, s) = 0 where it becomes viscoelastic with long memory.     

Stochastic Generalized Method of Moments

The generalized method of moments (GMM) is a very popular estimation and inference procedure based on moment conditions. When likelihood-based methods are difficult to implement, one can often derive various moment conditions and construct the GMM objective function. However, minimization of the objective function in the GMM may be challenging, especially over a large parameter space. Due to the special structure of the GMM, we propose a new sampling-based algorithm, the stochastic GMM sampler, which replaces the multivariate minimization problem by a series of conditional sampling procedures. We develop the theoretical properties of the proposed iterative Monte Carlo method, and demonstrate its superior performance over other GMM estimation procedures in simulation studies. As an illustration, we apply the stochastic GMM sampler to a Medfly life longevity study. Supplemental materials for the article are available online.     

The Numerical Solution of Linear One-dimensional Parabolic Problems by the Method of Moments

^* Present address: Department of Mathematics, Carnegie Institute of Technology, Pittsburg, Pa., U.S.A. An account is given of the method of moments and its applicationto the numerical solution of the simple one-dimensional heatconduction equation. The rapid convergence of the method isdemonstrated by numerical examples and an estimate of the erroris given. Features in common with the matrix method of Wadsworth& Wragg are also discussed.     

热力耦合问题数学均匀化方法的计算精度

针对复合材料周期结构热力耦合问题,推导了数学均匀化方法(MHM)各阶摄动位移的全解耦格式和各阶影响函数控制方程,并使用加权残量方法将其转化为易于编程计算的有限元列式.在解耦格式中,各阶摄动位移是相应阶次的影响函数和宏观场导数的乘积,即影响函数和宏观场导数的计算精度共同决定摄动项的精度,其中影响函数的计算精度取决于单胞边界条件选取的适用性.针对2D复合材料周期结构静力学问题,使用超单胞边界条件和微分求积有限单元法,分别提高了影响函数和宏观场导数的求解精度.在此基础上,研究了高阶展开项对MHM真实位移精度的影响,确定了二阶摄动项的必要性.最后应用最小势能原理评估了各阶摄动MHM的计算精度,数值比较结果验证了结论的正确性.     

分布形式未知的矩法<英>

我们考虑分布形式未知的矩法,假设分布函数F_θ(x)可用一多项式表示或逼近。多项式的阶数r未知,其系数为未知参数θ_1,…,θ_r。我们给出了θ_1,…,θ_r的一种新的矩法估计,r可自动地求得,估计是强一致的。     

Method of Moments Using Monte Carlo Simulation

Abstract

We present a computational approach to the method of moments using Monte Carlo simulation. Simple algebraic identities are used so that all computations can be performed directly using simulation draws and computation of the derivative of the log-likelihood. We present a simple implementation using the Newton-Raphson algorithm with the understanding that other optimization methods may be used in more complicated problems. The method can be applied to families of distributions with unknown normalizing constants and can be extended to least squares fitting in the case that the number of moments observed exceeds the number of parameters in the model. The method can be further generalized to allow “moments” that are any function of data and parameters, including as a special case maximum likelihood for models with unknown normalizing constants or missing data. In addition to being used for estimation, our method may be useful for setting the parameters of a Bayes prior distribution by specifying moments of a distribution using prior information. We present two examples—specification of a multivariate prior distribution in a constrained-parameter family and estimation of parameters in an image model. The former example, used for an application in pharmacokinetics, motivated this work. This work is similar to Ruppert's method in stochastic approximation, combines Monte Carlo simulation and the Newton-Raphson algorithm as in Penttinen, uses computational ideas and importance sampling identities of Gelfand and Carlin, Geyer, and Geyer and Thompson developed for Monte Carlo maximum likelihood, and has some similarities to the maximum likelihood methods of Wei and Tanner.     

Increase of Dimension by Homogenization

The paper deals with the homogenization of a Neumann's problem in a thin periodic weakly connected domain of R ³. The domain _n is composed of a large number n of disjoint periodic connected components linked by a periodic lattice _n of very thin bridges. According to the distribution and to the size of the linking bridges, the limit problem as n tends to infinity is either a 4d Neumann's problem or a 4d nonlocal problem. The additional term corresponding to the increase of dimension is due to the connection effect of the bridges.     

On the Solution of Problems of Nonlinear Conditional Optimization on Arrangements by the Cut-Off Method

We propose an exact method for the solution of a minimization problem on arrangements of a linear objective function with linear and concave additional constraints. We prove the finiteness of the proposed algorithm of the cut-off method.