Stokes型积分微分方程的质量集中各向异性非协调有限元分析

本文将Crouzeix-Raviart型非协调三角形元应用到发展型Stokes积分微分方程,给出了其质量集中非协调有限元逼近格式.在各向异性网格下,导出了速度的L2模和能量模及压力的L2模的误差估计.     

非定常Navier-Stokes方程的质量集中各向异性非协调有限元逼近

该文将一个低阶Crouzeix-Raviart型非协调三角形元应用到非定常Navier-Stokes方程,给出了其质量集中有限元逼近格式.在不需要传统Ritz-Volterra投影下,通过引入两个辅助有限元空间对边界进行估计的技巧,在各向异性网格下导出了速度的L~2模和能量模及压力的L~2模的误差估计.     

双曲型方程的非协调变网格有限元方法

采用变网格的思想讨论了双曲型方程在各向异性网格下的Crouzeix-Raviart型非协调有限元逼近.在不需要引入传统分析中Riesz投影的情况下,得到了相应最优误差估计.     

变度量各向异性网格生成算法和各向异性椭圆偏微分方程匹配网格

对于各向异性问题而言,在进行数值求解时,一个相匹配的各向异性网格至关重要.本文针对二阶椭圆偏微分方程,当其系数矩阵为各向异性时,给出了系数矩阵的逆作为相匹配的各向异性度量,在此度量下生成的各向异性网格即为匹配网格.本文还提出了新的变度量各向异性网格生成算法,该算法通过在各向异性背景网格上结合各向异性Delaunay准则和基于力平衡的结点优化函数生成高质量的各向异性三角化网格.通过数值算例可知,在匹配的各向异性三角化网格上,不仅有条件数小的代数离散系统,还有高精度的数值解,更甚者,在匹配网格结点上的l2范数误差有超收敛现象.     

Lagrange中心型守恒格式

提出了Lagrange中心型守恒气体动力学格式．引入了当前时刻子网格密度与当前时刻网格声速产生的网格分片常数压力．初始网格密度乘以初始子网格体积得到子网格质量,这些子网格质量除以当前时刻子网格体积得到当前时刻子网格密度．应用网格分片常数压力,构造了满足动量守恒、总能量守恒的Lagrange中心型守恒气体动力学格式,格点速度以与网格面的数值通量相容的方式计算．对Saltzman活塞问题等进行了数值模拟,数值结果显示Lagrange中心型守恒气体动力学格式的有效性和精确性．     

Stokes问题在各向异性网格下的Bernadi-Raugel有限元逼近

在各向异性网格下,得到了Stokes问题著名的Bernadi-Raugel混合有限元格式的超逼近性质,而且通过构造插值后处理算子得到了关于速度的超收敛结果.     

特征值问题的Lagrange型各向异性有限元方法

在各向异性网格下首先研究了二阶椭圆特征值问题算子谱逼近的若干抽象结果.然后将这些结果具体应用于线性和双线性Lagrange型协调有限元,得到了与传统有限元网格剖分下相同的最优误差估计,从而拓宽了已有的成果.     

扭曲网格上带有非光滑系数扩散方程的加权DG方法

具有各向异性和间断扩散系数的椭圆型方程在辐射流体力学和油藏模拟等许多物理应用中发挥着重要作用.辐射扩散问题的计算通常基于流体的网格.在流体计算中,网格会随着流体的流动发生扭曲变形.间断Galerkin (discontinuous Galerkin, DG)方法是计算数学中一类重要方法,适用于间断系数和非规则网格等复杂情形.本文在对称内惩罚方法的基础上发展加权DG方法求解扭曲网格上的椭圆方程.在理论分析中,首先给出DG方法中双线性形式的强制性和连续性的证明,然后基于强制性和连续性给出能量范数的误差分析,最后采用对偶论证技巧给出L²范数下的误差估计.数值实验在随机网格、正弦曲线型网格、Shestakov型网格和Z字型扭曲网格上进行,数值结果验证了加权DG方法对具有间断和各向异性扩散系数的椭圆问题的有效性.     

各向异性非内积型网格下旋转Q1元的收敛性

主要研究非协调旋转Q₁元在各向异性非内积型网格,即各向异性平行四边形网格下的收敛性．该元的插值误差在各向异性网格剖分时是发散的,但是我们证明了它对于二阶椭圆问题的各向异性收敛性．为了克服该元插值误差的这个缺陷,构造了一个适当的算子来证明它的逼近性质．利用一定的分析技巧,在各向异性平行四边形网格剖分下,得到了旋转Q₁元的最优逼近误差和相容误差．该结果可以引发人们继续分析一些不满足各向异性插值性质单元的收敛性．文章的最后用一些算例来验证了理论分析的正确性．     

带弱奇异核非线性积分微分方程的有限元分析

讨论了带弱奇异核的非线性抛物积分微分方程的Hermite型各向异性矩形元逼近.在各向异性网格下导出了关于Riesz投影的L~2和H~1模的误差估计.在半离散和向后欧拉全离散格式下,基于Riesz投影的性质并利用平均值技巧,分别得到了L~2模意义下的最优误差估计.     

Creep of polymer concrete in the nonlinear region

Two polyester-based polymer concretes with various volume content of diabase as an extender and aggregate are tested in creep under compression at different stress levels. The phenomenological and structural approaches are both used to analyze the experimental data. Common features of changes in the instantaneous and creep compliances are clarified, and a phenomenological creep model which accounts for the changes in the instantaneous compliance and in the retardation spectrum depending on the stress level is developed. It is shown that the model can be used to describe the experimental results of stress relaxation and creep under repeated loading. Modeling of the composite structure and subsequent solution of the optimization problem confirm the possibility of the existence of an interphase layer more compliant than the binder. A direct correlation between the interphase volume content and the instantaneous compliance of the composite is revealed. It is found that the distinction in nonlinearity of the viscoelastic behavior of the two polymer concretes under investigation can be due to the difference in their porosity. Submitted to the 11th International Conference on Mechanics of Composite Materials (Riga, June 11–15, 2000.) Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 2, pp. 147–164, 2000.     

Gaussian Versus Optimal Integration of Analytic Functions

We consider error estimates for optimal and Gaussian quadrature formulas if the integrand is analytic and bounded in a certain complex region. First, a simple technique for the derivation of lower bounds for the optimal error constants is presented. This method is applied to Szeg?-type weight functions and ellipses as regions of analyticity. In this situation, the error constants for the Gaussian formulas are close to the obtained lower bounds, which proves the quality of the Gaussian formulas and also of the lower bounds. In the sequel, different regions of analyticity are investigated. It turns out that almost exclusively for ellipses, the Gaussian formulas are near-optimal. For classes of simply connected regions of analyticity, which are additionally symmetric to the real axis, the asymptotic of the worst ratio between the error constants of the Gaussian formulas and the optimal error constants is calculated. As a by-product, we prove explicit lower bounds for the Christoffel-function for the constant weight function and arguments outside the interval of integration. September 7, 1995. Date revised: October 25, 1996.     

Asymptotics of the distribution of the integral of the positive part of the Brownian bridge for large arguments

The double Laplace transform of the distribution function of the integral of the positive part of the Brownian bridge was determined by M. Perman and J.A. Wellner, as well as the moments of this distribution. The purpose of the present paper is to determine the asymptotics of this distribution for large values of the argument, and the corresponding asymptotics of the moments.     

The nonlinear and scaled growth of the ottoman and Roman empires

In this work, mathematical models for the growth of the Ottoman and Roman Empires are found. The time interval considered for both cases covers the time from the birth of the empire to the end of the fast expansion period. These empires are assumed to be nonlinearly growing and self-multiplying systems. This approach utilizes the concepts of chaos theory, and scaling. The area governed by the empire is taken as the measure of its growth. It was found that the expansion of each empire on lands, seas, and on both (i.e., lands+seas) can be expressed by power laws. In the Ottoman Empire, the nonlinear growth power of total area is approximately equal to the golden ratio, and the nonlinear growth power of the expansion on lands is approximately equal to the square root of 2. In the case of the Romans, some numbers associated with the golden ratio, or the square root of 2, appear as the power of the nonlinear growth term. The appearance of both the golden ratio and the square root of 2 show that both empires had intention on achieving stability during their growth.     

Formulations and valid inequalities for the node capacitated graph partitioning problem

We investigate the problem of partitioning the nodes of a graph under capacity restriction on the sum of the node weights in each subset of the partition. The objective is to minimize the sum of the costs of the edges between the subsets of the partition. This problem has a variety of applications, for instance in the design of electronic circuits and devices. We present alternative integer programming formulations for this problem and discuss the links between these formulations. Having chosen to work in the space of edges of the multicut, we investigate the convex hull of incidence vectors of feasible multicuts. In particular, several classes of inequalities are introduced, and their strength and robustness are analyzed as various problem parameters change.     

基于偏序集的PROMETHEE方法优化研究

尽管PROMETHEE是当前最受欢迎的多准则决策方法之一,但在实践应用过程中,模型的应用范围与质量依然受制于指标权重问题。一些常用的赋权方法,不仅没有解决不确定权重问题,反而增加了决策风险。在偏序集相关定理的基础上,给出权重的定性信息即权重次序,由流出矩阵、流入矩阵和净流矩阵等定义,得到了PROMETHEE的偏序集表达形式。当流入和流出之和为常数时,证明了模型存在对偶性质。根据对偶性质,简化了PROMETHEE方法的分析步骤,删减模型冗余信息。应用偏序集表示的PROMETHEE,突破了模型没有具体权重便无法应用的思维定势,解决了模型赋权困难,增强了模型的鲁棒性,拓展了模型处理数据类型的范围。     

Generalized derivatives and nonsmooth optimization, a finite dimensional tour

During the early 1960’s there was a growing realization that a large number of optimization problems which appeared in applications involved minimization of non-differentiable functions. One of the important areas where such problems appeared was optimal control. The subject of nonsmooth analysis arose out of the need to develop a theory to deal with the minimization of nonsmooth functions. The first impetus in this direction came with the publication of Rockafellar’s seminal work titledConvex Analysis which was published by the Princeton University Press in 1970. It would be impossible to overstate the impact of this book on the development of the theory and methods of optimization. It is also important to note that a large part of convex analysis was already developed by Werner Fenchel nearly twenty years earlier and was circulated through his mimeographed lecture notes titledConvex Cones, Sets and Functions, Princeton University, 1951. In this article we trace the dramatic development of nonsmooth analysis and its applications to optimization in finite dimensions. Beginning with the fundamentals of convex optimization we quickly move over to the path breaking work of Clarke which extends the domain of nonsmooth analysis from convex to locally Lipschitz functions. Clarke was the second doctoral student of R.T. Rockafellar. We discuss the notions of Clarke directional derivative and the Clarke generalized gradient and also the relevant calculus rules and applications to optimization. While discussing locally Lipschitz optimization we also try to blend in the computational aspects of the theory wherever possible. This is followed by a discussion of the geometry of sets with nonsmooth boundaries. The approach to develop the notion of the normal cone to an arbitrary set is sequential in nature. This approach does not rely on the standard techniques of convex analysis. The move away from convexity was pioneered by Mordukhovich and later culminated in the monographVariational Analysis by Rockafellar and Wets. The approach of Mordukhovich relied on a nonconvex separation principle called theextremal principle while that of Rockafellar and Wets relied on various convergence notions developed to suit the needs of optimization. We then move on to a parallel development in nonsmooth optimization due to Demyanov and Rubinov called Quasidifferentiable optimization. They study the class of directionally differentiable functions whose directional derivatives can be represented as a difference of two sublinear functions. On other hand the directional derivative of a convex function and also the Clarke directional derivatives are sublinear functions of the directions. Thus it was thought that the most useful generalizations of directional derivatives must be a sublinear function of the directions. Thus Demyanov and Rubinov made a major conceptual change in nonsmooth optimization. In this section we define the notion of a quasidifferential which is a pair of convex compact sets. We study some calculus rules and their applications to optimality conditions. We also study the interesting notion of Demyanov difference between two sets and their applications to optimization. In the last section of this paper we study some second-order tools used in nonsmooth analysis and try to see their relevance in optimization. In fact it is important to note that unlike the classical case, the second-order theory of nonsmoothness is quite complicated in the sense that there are many approaches to it. However we have chosen to describe those approaches which can be developed from the first order nonsmooth tools discussed here. We shall present three different approaches, highlight the second order calculus rules and their applications to optimization.     

Analysis of the Stress State of the Surface Layer of a Polymeric Mass upon Filling of Bulky Compression Molds

The stress state of the surface layer of a polymeric mass during filling of bulky compression molds is analyzed. It is shown that, at particular rheological characteristics of the mass, temperature, and filling rates, cracking of the surface layer occurs, which leads to defects in the finished products. A physical analysis of this process makes it possible to conclude that the cracks arise due to the normal stresses operating in the front region of the moving polymeric mass. It is found that, under certain flow conditions, areas with a pressure lower than the atmospheric one appear on the surface of the polymer. If the tensile stresses arising in these local regions are higher than the tensile strength of the mass, the continuity of the composition is broken in the direction determined by the greatest rate of the normal deformation. To confirm the reliability of the crack-formation mechanism proposed, the distribution of the pressure and normal stresses over the free surface is calculated based on a numerical method. These calculations show that, by comparing the stress level achieved in the front region with the tensile-strength characteristics of the polymeric composition, it is possible to predict, with a sufficient accuracy, the possibility of crack formation in the surface layer of such a mass under given flow conditions and thus to solve the question on flawless manufacturing of products.     

Mario Pieri’s address at the University of Catania

The contributions made by the Italian mathematician Mario Pieri (1860-1913) are well known in the field of geometry. Pieri was a member of the School of Peano at the University of Turin. There he became engaged both by the problems of logic and by the philosophical aspects of Peano’s epistemology. This article was motivated by Pieri’s address given at the University of Catania, at the inauguration of the 1906-1907 academic year. My aim is to identify Pieri’s philosophical premises as found in his works and to present them in the general framework of the historical development of the Peano School.     

Generating facets for the cut polytope of a graph by triangular elimination

The cut polytope of a graph arises in many fields. Although much is known about facets of the cut polytope of the complete graph, very little is known for general graphs. The study of Bell inequalities in quantum information science requires knowledge of the facets of the cut polytope of the complete bipartite graph or, more generally, the complete k-partite graph. Lifting is a central tool to prove certain inequalities are facet inducing for the cut polytope. In this paper we introduce a lifting operation, named triangular elimination, applicable to the cut polytope of a wide range of graphs. Triangular elimination is a specific combination of zero-lifting and Fourier–Motzkin elimination using the triangle inequality. We prove sufficient conditions for the triangular elimination of facet inducing inequalities to be facet inducing. The proof is based on a variation of the lifting lemma adapted to general graphs. The result can be used to derive facet inducing inequalities of the cut polytope of various graphs from those of the complete graph. We also investigate the symmetry of facet inducing inequalities of the cut polytope of the complete bipartite graph derived by triangular elimination.