Regular boundary element steady-state solutions of the traveling magnetic field problems

Regular boundary element method (R-BEM) is applied to analyze steady-state traveling magnetic field problems for which convective diffusion equation is considered as governing equation. We deal with a three-dimensional rectangular prism as a simple example in order to study stability and accuracy of regular boundary element (R-BE) solutions. It is found that R-BE solutions are unconditionally stable for a rectangular prism whose sides parallel to a traveling velocity are longer than those perpendicular to the velocity. Furthermore, we can show that R-BE solutions as well as conventional BE solutions have second-order accuracy. Finally, numerical precision is studied through the condition number of the system matrices used in the analysis for a few parameters. It is shown that the R-BEM is available for the analysis of three-dimensional steady-state convective diffusion equations.     

多变量矩阵方程异类约束解的修正共轭梯度法

基于求解线性代数方程组的共轭梯度法,通过对相关矩阵和系数的修改,建立了一种求多矩阵变量矩阵方程异类约束解的修正共轭梯度法.该算法不要求等价线性代数方程组的系数矩阵具备正定性、可逆性或者列满秩性,因此算法总是可行的.利用该算法不仅可以判断矩阵方程的异类约束解是否存在,而且在有异类约束解,不考虑舍入误差时,可在有限步计算后求得矩阵方程的一组异类约束解;选取特殊初始矩阵时,可求得矩阵方程的极小范数异类约束解.另外,还可求得指定矩阵在异类约束解集合中的最佳逼近.算例验证了该算法的有效性.     

一般振动系统的多输入状态反馈部分极点配置

本文讨论具有非对称阻尼、刚度矩阵的一般振动系统多输入部分极点配置问题。基于系统的矩阵二阶微分方程而非一阶状态方程形式,采用多输入状态反馈控制,使系统的部分极点得以配置,同时使其它的极点保持不变。对于该配置问题,本文给出了控制增益矩阵的一种显式解,最后用一个算例说明本文方法。     

Wave features related to a model of compressible immiscible mixtures of two perfect fluids

Summary The balance equations of a compressible immiscible mixture of two perfect fluids are derived by using a variational principle. It is shown that the equations of such a mixture obey Truesdell's third metaphysical principle. Then the governing equations, specialized by assuming that both phases have the same motion, are studied by means of a suitable asymptotic approach. It is derived an evolution equation representing a generalized Korteweg-de Vries equation containing also nonlinear terms in the higher order derivatives. Finally, the travelling wave solutions of this evolution equation are discussed.     

An efficient time-truncated boundary element formulation applied to the solution of the two-dimensional scalar wave equation

An efficient time-truncation algorithm applied to the boundary element solution of the two-dimensional scalar wave equation is proposed. Rational interpolation functions are here employed to compute time-domain influence matrices, in appropriate instants of time, improving previous time-truncation strategies presented in the literature. Two numerical examples are considered at the end of the paper, illustrating that the present scheme is fairly suitable for both bounded and unbounded models.     

A new efficient numerical method and the dynamic analysis of composite laminates with piezoelectric layers

Firstly, a numerical method for the inversion of Laplace transform is developed and its accuracy is shown through examples. Then, a state-vector equation for the dynamic problems of piezoelectric plates is deduced directly from a modified mixed variational principle for piezoelectric bodies and its exact solution for the dynamic problems of simply supported rectangle piezoelectric plate is simply given. For multilayered hybrid plates, we derive the solution in terms of the propagator matrices. The techniques accounts for the compatibility of generalized displacements and generalized stresses on the interface both the elastic layers and piezoelectric layers, and the transverse shear deformation and the rotary inertia of laminate are also considered in the global algebraic equation of structure. Meanwhile, there is no restriction on the thickness and the number of layers. As an application of the numerical inversion of Laplace transform presented in this paper, typical numerical examples of the harmonic vibration and transient response are proposed and discussed. Since the highly accurate numerical results, they can serve as benchmarks to test various thick plate theories and various numerical methods, such as the finite and boundary element methods for transient response problems.     

轴向运动变长度悬臂梁的振动控制

研究具有轴向运动速度、变长度悬臂梁的横向振动控制,提出了两种主动控制方案.采用Hamilton原理得到了端部带主动振子或跨内含有主动控制力的轴向运动悬臂梁的振动方程和边界条件.运用修正的伽辽金法得到了求解系统响应的近似方程.运用LQR法设计了主动振子和主动力的控制器,其中采用加权系数法选择Q,R矩阵.用数值计算仿真了两种控制方案的效果.结果表明采用主动振子或主动力都能够有效地控制轴向运动悬臂梁的横向振动.在相同的初始条件下,主动力控制的效果比主动振子要好,但主动振子的物理可实现性要优于主动力.     

An implicit–explicit solution method for electro‐osmotic flow through three‐dimensional micro‐channels

This paper presents a comprehensive finite‐element modelling approach to electro‐osmotic flows on unstructured meshes. The non‐linear equation governing the electric potential is solved using an iterative algorithm. The employed algorithm is based on a preconditioned GMRES scheme. The linear Laplace equation governing the external electric potential is solved using a standard pre‐conditioned conjugate gradient solver. The coupled fluid dynamics equations are solved using a fractional step‐based, fully explicit, artificial compressibility scheme. This combination of an implicit approach to the electric potential equations and an explicit discretization to the Navier–Stokes equations is one of the best ways of solving the coupled equations in a memory‐efficient manner. The local time‐stepping approach used in the solution of the fluid flow equations accelerates the solution to a steady state faster than by using a global time‐stepping approach. The fully explicit form and the fractional stages of the fluid dynamics equations make the system memory efficient and free of pressure instability. In addition to these advantages, the proposed method is suitable for use on both structured and unstructured meshes with a highly non‐uniform distribution of element sizes. The accuracy of the proposed procedure is demonstrated by solving a basic micro‐channel flow problem and comparing the results against an analytical solution. The comparisons show excellent agreement between the numerical and analytical data. In addition to the benchmark solution, we have also presented results for flow through a fully three‐dimensional rectangular channel to further demonstrate the application of the presented method. Copyright © 2007 John Wiley & Sons, Ltd.     

求解四元数矩阵方程AHXA=B的基于矩阵半张量积的新方法

四元数线性系统在控制理论和工程中有广泛的应用。利用矩阵半张量积对四元数矩阵方程进行研究，提出四元数矩阵的一种实向量表示并研究其性质。结合实向量表示与矩阵半张量积，给出四元数矩阵方程A^HXA=B的极小范数Hermitian解的存在条件及通解表达式，并且给出相应算法。数值实验证明了实向量表示方法的可行性。     

A numerical integration scheme for special finite elements for the Helmholtz equation

The theory for integrating the element matrices for rectangular, triangular and quadrilateral finite elements for the solution of the Helmholtz equation for very short waves is presented. A numerical integration scheme is developed. Samples of Maple and Fortran code for the evaluation of integration abscissæ and weights are made available. The results are compared with those obtained using large numbers of Gauss–Legendre integration points for a range of testing wave problems. The results demonstrate that the method gives correct results, which gives confidence in the procedures, and show that large savings in computation time can be achieved. Copyright © 2002 John Wiley & Sons, Ltd.     

An improved continued‐fraction‐based high‐order transmitting boundary for time‐domain analyses in unbounded domains

A high‐order local transmitting boundary to model the propagation of acoustic or elastic, scalar or vector‐valued waves in unbounded domains of arbitrary geometry is proposed. It is based on an improved continued‐fraction solution of the dynamic stiffness matrix of an unbounded medium. The coefficient matrices of the continued‐fraction expansion are determined recursively from the scaled boundary finite element equation in dynamic stiffness. They are normalised using a matrix‐valued scaling factor, which is chosen such that the robustness of the numerical procedure is improved. The resulting continued‐fraction solution is suitable for systems with many DOFs. It converges over the whole frequency range with increasing order of expansion and leads to numerically more robust formulations in the frequency domain and time domain for arbitrarily high orders of approximation and large‐scale systems. Introducing auxiliary variables, the continued‐fraction solution is expressed as a system of linear equations in iω in the frequency domain. In the time domain, this corresponds to an equation of motion with symmetric, banded and frequency‐independent coefficient matrices. It can be coupled seamlessly with finite elements. Standard procedures in structural dynamics are directly applicable in the frequency and time domains. Analytical and numerical examples demonstrate the superiority of the proposed method to an existing approach and its suitability for time‐domain simulations of large‐scale systems. Copyright © 2011 John Wiley & Sons, Ltd.     

矩形贮箱类液固耦合系统的平动响应研

采用与以往不同的方式,综合运用两种不同形式的Ｌａｇｒａｎｇｅ函数,以简洁的方式建立了常重力下受到水平激励的矩形贮箱类液固耦合系统的耦合动力学方程。该方法大大减少了公式推导,且所得的动力学方程是降阶的。通过数值仿真发现,在一定的激励幅值和频率下,该耦合系统会出现零点漂移等非线性动力学现象。     

Equivalence of a Cauchy system and a class of boundary-value problems in thin plate theory

Summary In [1] we reduced the solution of a classical boundary-value problem, namely the biharmonic equation in a rectangular domain, to a Cauchy formulation. The theory was developed in the context of elementary thin plate theory. It was shown that a rectangular plate with three edges clamped and the fourth edge free can be completely described by a system of integro-differential equations subject to initial values. In this paper we prove the converse, i.e., that any solution of the Cauchy system is a solution of the biharmonic equation, completing the equivalence.     

The generalized boundary element approach to Burgers' equation

The generalized boundary element method is presented for the numerical solution of Burgers' equation. The new method is based on the set of boundary integral equations derived for each subdomain by using the fundamental solution for the linearized differential operator of the equation. The resulting system of quasi-non-linear equations is solved implicitly with use of a simple iterative procedure. The adaptability and the accuracy of the proposed method are demonstrated by three examples and a comparison of the numerical results with the exact solution or other existing solutions is shown for the first example.     

The numerical solution of the Navier-Stokes equations for a three-dimensional laminar flow in curved pipes using finite-difference methods

Summary The numerical method presented treats the primitive-variable form of the Navier-Stokes equations. It is shown how to treat the generalised orthogonal coordinate form of the equations in order to retain the numerical stability of the linearised equations when these are approximated by finite differences. A property analogous to diagonal dominance in more simple systems is shown to exist for the complete set of difference approximations to the flow equations so that the matrix of the finite-difference equations has all of its eigen values in the left-hand half-plane. It follows that the linearized equations are unconditionally stable. An entirely new difference scheme for the continuity equation is derived and shown to be superior to the more commonly used central-difference approximations for the high-Reynolds-number flow considered. The total package is tested against experiment on a shear flow through a 90° rectangular bend. The experimental measurements are of total-pressure distributions, and these indicate the presence of a strong secondary flow. The computed results give a close agreement to the experimental results.     

Boundary value techniques for finite element solutions of the diffusion-convection equation

In this paper, the formulation of six-point and nine-point finite element equations for the solution of the diffusion-convection equation is presented. The six-point equation requires the solution of a tridiagonal system of equations and the nine-point centred equation is treated as a solution of a boundary value problem which leads to a large linear system of equations. Some numerical experiments are presented and the comparison with existing methods is included.     

Accurate solution for free vibration analysis of functionally graded thick rectangular plates resting on elastic foundation

Vibration analysis of a functionally graded rectangular plate resting on two parameter elastic foundation is presented here. The displacement filed based on the third order shear deformation plate theory is used. By considering the in-plane displacement components of an arbitrary material point on the mid-plane of the plate and using Hamilton’s principle, the governing equations of motion are obtained which are five highly coupled partial differential equations. An analytical approach is employed to decouple these partial differential equations. The decoupled equations of functionally graded rectangular plate resting on elastic foundation are solved analytically for levy type of boundary conditions. The numerical results are presented and discussed for a wide range of plate and foundation parameters. The results show that the Pasternak (shear) elastic foundation drastically changes the natural frequency. It is also observed that in some boundary conditions, the in-plane displacements have significant effects on natural frequency of thick functionally graded plates and they cannot be ignored.     

A new discretization method and its application to solve incompressible Navier–Stokes equations

 A new numerical method is presented in this paper. This method directly solves partial differential equations in the Cartesian coordinate system. It can be easily applied to solve irregular domain problems without introducing the coordinate transformation technique. The concept of the present method is different from the conventional discretization methods. Unlike the conventional numerical methods where the discrete form of the differential equation only involves mesh points inside the solution domain, the new discretization method reduces the differential equation into a discrete form which may involve some points outside the solution domain. The functional values at these points are computed by the approximate form of the solution along a vertical or horizontal line. This process is called extrapolation. The form of the solution along a line can be approximated by Lagrange interpolated polynomial using all the points on the line or by low order polynomial using 3 local points. In this paper, the proposed new discretization method is first validated by its application to solve sample linear and nonlinear differential equations. It is demonstrated that the present method can easily treat different solution domains without any additional programming work. Then the method is applied to simulate incompressible flows in a smooth expansion channel by solving Navier–Stokes equations. The numerical results obtained by the new discretization method agree very well with available data in the literature. All the numerical examples showed that the present method is very efficient, which is suitable for solving irregular domain problems. Received 19 July 2000     

矩形悬臂厚板的解析解

首先把胡海昌提出的弹性厚板弯曲问题的简化方程表示成为Hamilton正则方程,然后利用辛几何方法对全状态相变量进行分离变量,求出其本征值后,再按本征函数展开的方法求出矩形悬臂厚板的解析解。由于在求解过程中不需要事先人为地选取挠度函数,而是从厚板弯曲的基本方程出发,直接利用数学的方法求出可以满足其边界条件的这类问题的解析解,使得问题的求解更加合理论化。最后还给出了计算实例来验证所采用的方法以及所推导出的公式的正确性。     

The normal shock waves of real gases and the generalized isentropic exponents

When a supersonic gas flow is choked or otherwise disturbed, a shock wave appears. When this shock wave is perpendicular to the direction of the one-dimensional flow, a normal shock wave transforms the flow to a subsonic one. The thermodynamic variables involved are five, i.e. the pressure, the specific volume, the temperature, the enthalpy and the velocity of the flow. To determine their values after the normal Shock, five relations are used namely the equation of state, the enthalpy equation, the laws of conservation of impulse and energy, and the continuity of mass. For the simple case of a perfect gas, these relations are explicit and allow a straightforward solution of the system of the five equations. In the case of a real gas flow the system is solved numerically. The present work considers two other possibilities, i.e. the use of perfect gas like equations describing the phenomenon or the use of the Redlich-Kwong equation of state. Perfect gas like equations may be used now, to describe the phenomenon, after it has been shown that there exist three isentropic exponents, instead of the one used until now, and after it has been observed that the mathematical form of the perfect gas equations fit the real gas isentropic expansion when suitable numerical values of the corresponding constants and exponents are used. The other possibility, the use of the Redlich-Kwong equation of state, allows the calculation of the state variables after the normal shock when only the two constants of this equation and the specific heat of the corresponding perfect gas state (pressure approaching zero) are available. Both methods presented here require the numerical solution of a onevariable equation. In the first method the variable is the Mach number while in the second the specific volume.