Low‐order optimal regulation of parabolic PDEs with time‐dependent domain

Observer and optimal boundary control design for the objective of output tracking of a linear distributed parameter system given by a two‐dimensional (2‐D) parabolic partial differential equation with time‐varying domain is realized in this work. The transformation of boundary actuation to distributed control setting allows to represent the system's model in a standard evolutionary form. By exploring dynamical model evolution and generating data, a set of time‐varying empirical eigenfunctions that capture the dominant dynamics of the distributed system is found. This basis is used in Galerkin's method to accurately represent the distributed system as a finite‐dimensional plant in terms of a linear time‐varying system. This reduced‐order model enables synthesis of a linear optimal output tracking controller, as well as design of a state observer. Finally, numerical results are prepared for the optimal output tracking of a 2‐D model of the temperature distribution in Czochralski crystal growth process which has nontrivial geometry. © 2014 American Institute of Chemical Engineers AIChE J, 61: 494–502, 2015     

求解平面转子本征问题的因式分解方法

讨论了平面转子的可因式分解性,用因式分解的方法求解了平面转子的本征问题,并计算了均匀强电场下平面转子的能级和波函数。     

Singular stress fields in interfacial notches of hybrid metal matrix composites

This paper presents results from a study on the singular stress fields in metal matrix composites in which dual matrices exist. The adjoining metallic matrices flanking the interface can deform plastically with powerlaw strain hardening. These matrices may have both different hardening exponents and different yield strengths. An asymptotic analysis coupled with numerical eigen-analysis solved the spatial structure of the singular stress field at radial- and angular-dependent parts: . The dependence of the strength of the singular stresses on the matrix properties is discussed. The effects of local geometry on the nature of singular stresses are addressed. Highlights for interfacial notches are reported here. The drivers for this study are interfacial notches and free-edges in hybrid metal matrix composites (Fig. 1). However, the results can also be applied to other advanced structures which are composed of two or more distinct components or phases such as bone-implant interfaces and surface mounts in electronic packages     

The dispersion relation and eigenfunction expansions for water waves in a porous structure

The Sollitt-and-Cross model of water-wave motion in a porous structure involves a free-surface condition which contains a complex parameter. This leads to two particular difficulties when this model is used in conjunction with eigenfunction expansion techniques. First of all the roots of the dispersion relation are themselves complex and therefore difficult to locate by standard numerical methods. Secondly, the vertical eigenfunction problem is not self-adjoint and standard expansion theorems do not apply. In this paper it is shown how these two difficulties may be resolved with the aid of the theories of, respectively, complex variables and non-self-adjoint differential operators. In particular, a method is described that allows the explicit calculation of the roots of the dispersion relation, and the appropriate expansion theorem is given.     

Higher‐order finite elements based on generalized eigenfunctions of the Laplacian

We present a new class of higher‐order finite elements based on generalized eigenfunctions of the Laplace operator, which are suitable for both product and simplicial geometries in ?^d. Due to simultaneous orthogonality of the generalized eigenfunctions under both the H and L² products and their almost negligible dependence on reference maps, such finite elements are an excellent choice for the discretization of second‐order elliptic problems by the hp‐FEM. Analysis is illustrated by numerical results and comparisons with other popular higher‐order finite elements are presented. The new elements are used to compute efficiently the model of an electrostatic micromotor. Copyright © 2007 John Wiley & Sons, Ltd.     

Singular eigenfunction analysis of transport problems with spatially dependent cross sections — An overview

A substantial amount of recent work has been done in applying singular eigenfunctions to transport problems whose cross sections vary continuously with position. In this article we survey the methodology and results so far obtained, and we briefly discuss some of the many open questions.     

磁单极系统的O(2,1)■O(3)群对称性及其群表示

本文讨论了磁单极系统的O(2,1)O(3)群对称性;利用球坐标,把一个(3+1)维空间约化为(1+1)维空间;利用对称群生成元的本征函数,得到了(1+1)维空间中对称群的表示;并发现磁单极系统的磁单极强度半整数量子元化.     

一种提高管道临界流速计算效率的方法

使用复模态法和Galerkin模态截断法分别计算了Pasternak地基上两端铰支输流管道的临界流速,并通过与复模态法计算得到的精确解的比较,证明了Galerkin模态截断法对临界流速计算结果具有一定的影响。研究了Pasternak地基剪切刚度和弹簧刚度对截断误差的影响。计算结果表明:地基剪切刚度对管道系统临界流速的影响比弹簧刚度要大,且前者的增加可以减小Galerkin法的截断误差,但后者的增加则会增大这一误差。在此基础上,提出了分区域系数确定方法,并针对不同管道地基刚度,通过合理选取Galerkin法阶数,提高了输流管道临界流速计算效率。     

用超几何级数方法解任意(l)态双曲正切势场

对于s态的双曲正切势场下的Schr(o)dinger的径向方程,则能精确求出能量本征值和对应的波函数的表达式.对于非零的l态,则不能精确求解双曲势场下的在Schr(o)dinger的径向方程.本文针对不能精确求解非零的l态,从另一方面出发,能利用近似的方法得到能量本征值和对应的本征函数的表达式.应用类似的Pekeris近似方法,将含有角动量的一项在r0处展开,省略高阶项,原双曲正切势场Schr(o)dinger的径向方程变化成为超几何方程.应用超几何级数方法,得到了能量本征值和对应的本征函数的表达式.最后进行数值的验算,取两组不同的数值来计算能量本征值,得出结果基本一致,数值结果表明此法是可靠和有效的.