Properties of constitutive matrices for electrostatic and magnetostatic problems

Constitutive matrices allow one to write the constitutive relations, in an algebraic form, in some finite-volume-based numerical methods, such as the cell method and the finite integration technique. This paper shows some properties that link a general class of constitutive matrices to the finite-element stiffness matrices. One of these properties can be exploited to provide an alternative sufficient condition for the stability of a generalized finite-difference time-domain algorithm set up by these numerical methods.     

A precise time-step integration method by step-response and impulsive-response matrices for dynamic problems

In this paper, a precise time-step integration method for dynamic problems is presented. The second-order differential equations for dynamic problems are manipulated directly. A general damping matrix is considered. The transient responses are expressed in terms of the steady-state responses, the given initial conditions and the step-response and impulsive-response matrices. The steady-state responses for various types of excitations are readily obtainable. The computation of the step-response and impulsive-response matrices and their time derivatives are studied in this paper. A direct computation of these matrices using the Taylor series solutions is not efficient when the time-step size Δt is not small. In this paper, the recurrence formulae relating the response matrices at t=Δt to those at t=Δt/2 are constructed. A recursive procedure is proposed to evaluate these matrices at t=Δt from the matrices at t=Δt/2^m. The matrices at t=Δt/2^m are obtained from the Taylor series solutions. To improve the computational efficiency, the relations between the response matrices and their time derivatives are investigated. In addition, these matrices are expressed in terms of two symmetric matrices that can also be evaluated recursively. Besides, from the physical point of view, these matrices should be banded for small Δt. Both the stability and accuracy characteristics of the present algorithm are studied. Three numerical examples are used to illustrate the highly precise and stable algorithm. © 1997 John Wiley & Sons, Ltd.     

Axial interface optical phonon modes in a double-nanoshell system

Within the framework of the dielectric continuum (DC) model, we analyze the axial interface optical phonon modes in a double system of nanoshells. This system is constituted by two identical equidistant nanoshells which are embedded in an insulating medium. To illustrate our results, typical II-VI semiconductors are used as constitutive polar materials of the nanoshells. Resolution of Laplace's equation in bispherical coordinates for the potentials derived from the interface vibration modes is made. By imposing the usual electrostatic boundary conditions at the surfaces of the two-nanoshell system, recursion relations for the coefficients appearing in the potentials are obtained, which entails infinite matrices. The problem of deriving the interface frequencies is reduced to the eigenvalue problem on infinite matrices. A truncating method for these matrices is used to obtain the interface phonon branches. Dependences of the interface frequencies on the ratio of inter-nanoshell separation to core size are obtained for different systems with several values of nanoshell interdistance. Effects due to the change of shell and embedding materials are also investigated in interface phonon modes.     

Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part I. Method

We present a formulation for wave propagation and scattering through stacked gratings comprising metallic and dielectric cylinders. By modeling a photonic crystal as a grating stack of this type, we thus formulate an efficient and accurate method for photonic crystal calculations that allows us to calculate reflection and transmission matrices. The stack may contain an arbitrary number of gratings, provided that each has a common period. The formulation uses a Green's function approach based on lattice sums to obtain the scattering matrices of each layer, and it couples these layers through recurrence relations. In a companion paper [J. Opt Soc. Am. A 17, 2177 (2000)] we discuss the numerical implementation of the method and give a comprehensive treatment of its conservation properties.     

Accelerated bisection techniques for tri- and quindiagonal matrices

This paper considers accelerated bisection methods for calculating the eigenvalues of symmetric tridiagonal and quindiagonal matrices using cubic polynomial interpolation, as well as first and second order Newton iteration. Recursive relations based on the Sturm sequences are presented for the Newton type methods. The relations use a convenient scaling which retains the relative magnitudes in the iterative schemes but which avoids numerical overflow. Numerical examples show a significant increase in convergence rate.     

Phenomenological derivation of the Onsager reciprocal relations

The Onsager reciprocal relation for a linear dissipative system with two fluxes is derived within the framework of a phenomenological approach using the property of nonnegative definiteness of the dissipative function. It is established that the Onsager relations are valid when the generalized fluxes are zero for nonzero generalized thermodynamic forces. In the general case, the Onsager reciprocal relations can be phenomenologically derived based on the theory of matrices and determinants or on the Prigogine principle of the entropy production minimum.     

树脂基体及其复合材料对拉伸冲击载荷的响应

本文在我校固体为学教研室自行设计研制的摆锤式单杆冲击拉伸试验装置上对用作复合材料基体的环氧树脂和不鲍和聚脂的动态力学性能进行实验研究,对试件的断口进行了电镜观察和分析,得到了在不同应变率下拉伸的应为应变关系和有关结果;进而讨论了这两种基体材料对复合材料动态力学性能的影响,同时提出了设计抗冲击的结构复合材料应遵循纤维和基体的动态最大应变匹配原则。      

Variational methods for selective mass scaling

A new variational method for selective mass scaling is proposed. It is based on a new penalized Hamilton’s principle where relations between variables for displacement, velocity and momentum are imposed via a penalty method. Independent spatial discretization of the variables along with a local static condensation for velocity and momentum yields a parametric family of consistent mass matrices. In this framework new mass matrices with desired properties can be constructed. It is demonstrated how usage of these non-diagonal mass matrices decreases the maximum frequency of the discretized system and allows for larger steps in explicit time integration. At the same time the lowest eigenfrequencies in the range of interest and global structural response are not significantly changed. Results of numerical experiments for two-dimensional and three-dimensional problems are discussed.     

Love waves on a half-space with a gradient piezoelectric layer by the geometric integration method

The propagation of Love waves on an elastic homogeneous half-space with a piezoelectric gradient covering layer is studied by the geometric integration method in this article. First, the state transfer equation of a Love wave is derived from the governing equations and constitutive relations. Then, the transfer matrix of the state vector is obtained by solving the state transfer equation of a Love wave and then the stiffness matrix is obtained. By combining transfer matrices and the stiffness matrices of the gradient covering layer and the homogeneous half-space, the total surface stiffness matrix of a Love wave is obtained. Lastly, the application of the electrically open circuit and short circuit conditions and mechanically traction-free conditions gives the frequency dispersive relation of a Love wave. For the gradient covering layer, the material constants at the bottom of the covering layer may be greater or smaller than that at the top of the covering layer. The two situations and three kinds of gradient profiles for each of these two situations are investigated. The numerical results show that the Love wave speed is sensitive to not only the material constants at the bottom and the top of the covering layer, but also the gradient profiles of the covering layer.     

基于柔度的网格截面非线性梁单元

梁单元材料非线性有限元求解时,材料进入非线性阶段后,难以通过梁理论准确描述截面的应力状态,该文据此提出了基于柔度法和分布式塑性理论的梁单元材料非线性方法-网格截面法,这种方法采用平面等参单元将梁单元网格化,由单元轴向积分点位置截面网格积分点的应力描述单元截面应力分布,并通过对截面网格材料的积分得到积分点位置的截面刚度,并运用基于柔度的有限元方法,通过力插值函数和能量原理得到梁单元的柔度矩阵,进而对柔度矩阵求逆以计算单元刚度矩阵。同时讨论了该方法在进行结构材料非线性有限元分析时的优越性。最后通过算例验证了上述结论。     

General parametric stiffness excitation – anti-resonance frequency and symmetry

Summary Stability investigations on vibration cancelling employing the concept of actuators with general stiffness elements are presented. Systems with an arbitrary number of degrees of freedom with linear spring- and damping elements are considered, that are subject to self-excitation as well as parametric excitation by stiffness variations with arbitrary phase relations. General conditions for full vibration suppression are derived analytically by applying a singular perturbation technique. These conditions naturally lead to the terms of parametric resonance and anti-resonance and enable a stability classification with respect to the parametric excitation matrices and their symmetry properties. The results are compared to former investigations of systems with a single or synchronous stiffness variation in time and geometrical interpretations are given. These basic results obtained can be used for design of a control strategy for actuators with periodically actuated stiffness elements and arbitrary phase relations.     

Non-iterative data-processing algorithms for source location on the basis of measured differences in distance

Different groups of direct non-interative algorithms for analyzing measurement data from a multiposition sensing system designed to locate sources of radiation are systematically derived and classified. The features and range of application of the algorithms are discussed. A power analyis is performed and analytic relations are obtained for matrices describing the covariance of source-coordinate estimates obtained by the algorithms within the context of first-order perturbation theory. Translated from Izmeritel'naya Tekhnika, No. 7, pp. 32– 36, July, 1996.     

A note on universal matrices for triangular finite elements for the quasi-harmonic equation

An algorithm to generate universal matrices for plane triangular finite elements for the general ‘quasi-harmonic’ equation is presented. For every member of the triangle family three numerical universal matrices are obtained which are independent of the size, shape and ‘material’ properties of the element. Of these, two are basic and the third can be generated from one of these two. The element ‘stiffness’ matrix is conveniently generated by manipulating these two basic matrices taking into account the size, shape and material properties of the element in a simple manner.     

Complete matrix formulation of fault-position method for voltage-dip characterisation

The method of fault position is useful for characterising power-system performance in the presence of voltage dips due to faults. It is based on short-circuit simulations repeated for all the system nodes and for many points along the system lines: fault voltages that are below a preset threshold are the required voltage dips. These dips are stored in so-called dip matrices which contain only the dips in all the system nodes when faults occur at points along the lines. The paper proposes a new compact analytical formulation of dip matrices for balanced and unbalanced faults in terms of bidimensional vector relations and for site- and system-voltage dip indexes. Compact formulations are very useful tools when several sensitivity analyses have to be conducted to estimate variation of site- and system-voltage dip indexes in relation to possible reinforcement and/or compensation devices. Graphical presentation of dip matrices is also proposed as a valuable tool to ascertain the critical area for system performance. Numerical applications to an actual transmission system are presented to demonstrate the easy applicability of the model.     

Symmetry and reversibility in materials with hidden variables

Consequences of microscopic reversibility on the constitutive equations of materials with hidden variables are deduced by taking the view that the expectation value of the fluctuations is determined through the macroscopic behaviour of the body. As a result it is shown that reciprocity relations hold for after effect matrices connecting variables which are conjugate at equilibrium. In particular, the usual relaxation function of the linearised isothermal theory of viscoelasticity is found to be symmetric.     

混凝土杆系结构滞回全过程分析

本文基于非线性有限元原理,采取由材料本构关系直接形成单元M—N—φ关系的方法,推导了混凝土杆系结构的单元刚度矩阵,该矩阵考虑了材料非线性、几何非线性、轴力二次矩、混凝土的裂面效应、预应力的特点、钢筋的粘结滑移以及材料的双切线模量等的影响;编制了相应的分析程序,并对两榀混凝土门架的滞回性能进行了模拟计算,计算值与试验结果吻合较好。本文程序实现了混凝土杆系结构包括下降段在内的滞回全过程分析,从而为该类结构的抗震研究提供了一个准确、实用的工具。     

Q(M) and the depolarization index scalar metrics

A depolarization scalar metric for Mueller matrices, named Q(M), is derived from the degree of polarization. Q(M) has been recently reported, and it has been deduced from the nine bilinear constraints between the sixteen elements of the Mueller-Jones matrix. We discuss the relations between Q(M) and the depolarization index.     

Fault-tolerant homopolar magnetic bearings

This paper summarizes the development of a novel magnetic suspension that improves reliability via fault-tolerant operation. The suspension is suitable for flywheels used in satellites and space stations for attitude control and energy storage. Specifically, we show that flux coupling between poles of a homopolar magnetic bearing can deliver desired forces even after termination of coil currents to a subset of "failed poles". Linear, coordinate-decoupled force-voltage relations are also maintained before and after failure by bias linearization. We determined current distribution matrices that adjust the currents and fluxes following a pole set failure for many faulted pole combinations. We used one-dimensional magnetic circuit models with fringe and leakage factors derived from detailed, three-dimensional finite-element field models to obtain the current distribution matrices and the system response. Reliability is based on the success criterion that catcher bearing-shaft contact does not occur after pole failures. The magnetic bearing reliability is improved by increasing the number of the radial poles. An advantage of our method over other redundant approaches is a significantly reduced requirement for backup hardware such as additional actuators or power amplifiers.     

Coupled deflection analysis of thin-walled Timoshenko laminated composite beams

For the deflection analyses of thin-walled Timoshenko laminated composite beams with the mono- symmetric I-, channel-, and L-shaped sections, the stiffness matrices are derived based on the solutions of the simultaneous ordinary differential equations. A general thin-walled composite beam theory considering shear deformation effect is developed by introducing Vlasov’s assumptions. The shear stiffnesses of thin-walled composite beams are explicitly derived from the energy equivalence. The equilibrium equations and force-deformation relations are derived from energy principles. By introducing 14 displacement parameters, a generalized eigenvalue problem that has complex eigenvalues and multiple zero eigenvalues is formulated. Polynomial expressions are assumed as trial solutions for displacement parameters and eigenmodes containing undetermined parameters equal to the number of zero eigenvalues are determined by invoking the identity condition to the equilibrium equations. Then the displacement functions are constructed by combining eigenvectors and polynomial solutions corresponding to nonzero and zero eigenvalues, respectively. Finally, the stiffness matrices are evaluated by applying the member force-displacement relations to the displacement functions. In addition, the finite beam element formulation based on the classical Lagrangian interpolation polynomial is presented. In order to verify the validity and the accuracy of this study, the numerical solutions are presented and compared with the finite element results using the isoparametric beam elements and the detailed three-dimensional analysis results using the shell elements of ABAQUS. Particularly the effects of shear deformations on the deflection of thin-walled composite beams with the mono-symmetric I-, channel-, and L-shaped sections with various lamination schemes are investigated.