A comparison of integral formulations for the analysis of low Reynolds number flows

Integral formulations for the analysis of low Reynolds number flows have been developed over the past 25 years. These formulations can typically be categorized as being either direct or indirect and either velocity integral equations or traction integral equations. Depending on the boundary conditions imposed for a given problem, the resulting integral formulation will result in either a Fredholm integral equation of the first kind, second kind or mixed. In general, Fredholm integral equations of the first kind lead to unstable numerical schemes based upon discretization and can result in low-order accuracy. For most practical problems, the direct velocity integral equation results in a Fredholm equation of the first kind. Nevertheless, many researchers have used this integral formulation with great success and any potential matrix ill-conditioning seems to have little influence on the accuracy of the numerical solutions. In this research, three integral formulations are compared, namely, the direct velocity integral equation, the indirect velocity integral equation, and the direct traction integral equation. Three benchmark problems are chosen for which there are analytic solutions. Although the discretized matrix condition numbers are in some cases orders of magnitude larger for the direct velocity integral formulation, there is little to distinguish between the three methods in terms of accuracy. In fact, the results for the direct velocity integral equation were slightly more accurate in most cases for the three benchmark problems considered in this research.     

An Iterative Scheme of Arbitrary Odd Order and Its Basins of Attraction for Nonlinear Systems

In this paper, we propose a fifth-order scheme for solving systems of nonlinear equations. The convergence analysis of the proposed technique is discussed. The proposed method is generalized and extended to be of any odd order of the form 2n − 1. The scheme is composed of three steps, of which the first two steps are based on the two-step Homeier’s method with cubic convergence, and the last is a Newton step with an appropriate approximation for the derivative. Every iteration of the presented method requires the evaluation of two functions, two Fréchet derivatives, and three matrix inversions. A comparison between the efficiency index and the computational efficiency index of the presented scheme with existing methods is performed. The basins of attraction of the proposed scheme illustrated and compared to other schemes of the same order. Different test problems including large systems of equations are considered to compare the performance of the proposed method according to other methods of the same order. As an application, we apply the new scheme to some real-life problems, including the mixed Hammerstein integral equation and Burgers’ equation. Comparisons and examples show that the presented method is efficient and comparable to the existing techniques of the same order.     

Degenerate scale for the analysis of circular thin plate using the boundary integral equation method and boundary element methods

In this paper, the degenerate scale for plate problem is studied. For the continuous model, we use the null-field integral equation, Fourier series and the series expansion in terms of degenerate kernel for fundamental solutions to examine the solvability of BIEM for circular thin plates. Any two of the four boundary integral equations in the plate formulation may be chosen. For the discrete model, the circulant is employed to determine the rank deficiency of the influence matrix. Both approaches, continuous and discrete models, lead to the same result of degenerate scale. We study the nonunique solution analytically for the circular plate and find degenerate scales. The similar properties of solvability condition between the membrane (Laplace) and plate (biharmonic) problems are also examined. The number of degenerate scales for the six boundary integral formulations is also determined. Tel.: 886-2-2462-2192-ext. 6140 or 6177     

Computational methods for solving static field and eddy current problems via Fredholm integral equations

Two-dimensional static field problems can be solved by a method based on Fredholm integral equations (equations of the second kind). This has numerical advantages over the mote commonly used integral equation of the first kind. The method is applicable to both magnetostatic and electrostatic problems formulated in terms of either vector or scalar potentials. It has been extended to the solution of eddy current problems with sinusoidal driving functions. The application of the classical Fredholm equation has been extended to problems containing boundary conditions: 1) potential value, 2) normal derivative value, and 3) an interface condition, all in the same problem. The solutions to the Fredholm equations are single or double (dipole) layers of sources on the problem boundaries and interfaces. This method has been developed into computer codes which use piecewise quadratic approximations to the solutions to the integral equations. Exact integrations are used to replace the integral equations by a matrix equation. The solution to this matrix equation can then be used to directly calculate the field anywhere.     

Homogenization of very rough interfaces separating two piezoelectric solids

The main aim of this paper is to derive homogenized equations in explicit form of the linear piezoelectricity in two-dimensional domains separated by an interface which highly oscillates between two parallel straight lines. First, the basic equations of the linear theory of piezoelectricity are written down in matrix form. Then, following the techniques presented recently by these authors, the explicit homogenized equation and the associate continuity condition, for generally anisotropic piezoelectric materials, are derived. They are then written down in component form for some specific cases. Since the obtained equations are totally explicit, they are significant in practical applications.     

Three co-planar moving Griffith cracks in an infinite elastic medium

The dynamic in-plane problem of determining the stress and displacement due to three co-planar Griffith cracks moving steadily at a subsonic speed in a fixed direction in an infinite, isotropic, homogeneous medium under normal stress has been treated. The static problem of determining the stress and displacement around three co-planar Griffith cracks in an infinite isotropic elastic medium has also been considered. In both the cases, employing Fourier integral transform, the problems have been reduced to solving a set of four integral equations. These integral equations have been solved using finite Hilbert transform technique and Cook's result [16] to obtain the exact form of crack opening displacement and stress intensity factors which are presented in the form of graphs.     

Derivation of COM equations using the surface impedance method

The surface impedance method is used for the consistent derivation of coupling of modes equations which describe the interaction of SAW with a periodical system of electrodes of finite thickness. The exact analytic solution of the electrostatic problem in the presence of an arbitrary external electric field for a plane system of electrodes is applied to the calculation of the charge and electric field distributions. Mechanical perturbations are taken into account to first order of the thickness of the electrodes. As a result the scalar self-consistent equation for the electric potential of acoustic waves in the gratings is obtained. For the periodic structure this equation is reduced to the form of COM equations for slowly varying amplitudes. Analytical expressions for all coefficients of the COM equations connecting them with geometrical and material parameters are found. The NSPUDT effect can be considered. Dissipation and energy storage terms can be introduced empirically. The solution of the COM equations is represented in the form of a P matrix with elements written in a convenient form. A simple formula for calculating the location of maximum transducer frequency response is proposed, The balance of energy is considered. Some new relations among the elements of P matrix are found     

Singular bifurcations in higher index differential-algebraic equations

The present paper extends the singularity induced bifurcation theorem (SIBT) to higher index differential-algebraic equations (DAEs) in Hessenberg form. The SIBT arises in power system theory, and is also significant within the context of electrical circuits. This phenomenon is due to the presence of singularities in parameter-dependent problems, and it was originally proved for semiexplicit index-1 DAEs. We introduce a new proof of a matrix pencil-based version of this result, relying on the spectral features of the linearization of the underlying ordinary differential equation. This approach is then applied to higher index DAEs in Hessenberg form. For these structures, the SIBT is shown to follow from a minimal index change at the singularity. Additionally, we show that a Kronecker index change is also necessary for the existence of a singularity induced bifurcation point in semi-explicit index-1 and Hessenberg DAEs.     

Numerical manifold method based on the method of weighted residuals

Usually, the governing equations of the numerical manifold method (NMM) are derived from the minimum potential energy principle. For many applied problems it is difficult to derive in general outset the functional forms of the governing equations. This obviously strongly restricts the implementation of the minimum potential energy principle or other variational principles in NMM. In fact, the governing equations of NMM can be derived from a more general method of weighted residuals. By choosing suitable weight functions, the derivation of the governing equations of the NMM from the weighted residual method leads to the same result as that derived from the minimum potential energy principle. This is demonstrated in the paper by deriving the governing equations of the NMM for linear elasticity problems, and also for Laplaces equation for which the governing equations of the NMM cannot be derived from the minimum potential energy principle. The performance of the method is illustrated by three numerical examples.     

Application of finite element methods to thermohydrodynamic lubrication

A finite element formulation is presented for the equations governing the steady thermohydrodynamic behaviour of liquid lubricated bearings. This formulation permits application of the iterative solution scheme to bearings of arbitrary geometry. A generalized Reynolds equation resulting from the combination of the mass and momentum conservation equations is cast into variational form and used to derive general finite element equations. The method of weighted residuals with Galerkin's criterion is used to generate finite element matrix equations for the thermal energy equation. In addition to the finite element formulation, a discussion of appropriate finite difference techniques is also given for problems without complex geometry. As an example, the formulations are applied to obtain numerical solutions for a three-dimensional sector thrust bearing operating in the thermohydrodynamic regime. Pressure, velocity and temperature distributions are give, and the thermohydrodynamic solutions are compared with the results of classical isothermal theory.     

Construction of the exact solution of the ring-plate problem by solving functional equations

This paper presents a method of constructing the exact solution of the ring-plate problem. The method is based on the general solution formula (nonseries form) of the biharmonic equation. The method changes solving the boundary-value problems of the ring plate into solving three functional equations and computing the coefficients of a simple Fourier series, or only solving four functional equations. The method is believed to be new. The simpler formulas of the solutions of all cases of the ring-plate boundary-value problems without any free boundary are obtained. Several examples are given.     

On least squares approximations to indefinite problems of the mixed type

A least squares method is presented for computing approximate solutions of indefinite partial differential equations of the mixed type such as those that arise in connection with transonic flutter analysis. The mehod retains the advantages of finite difference schemes namely simplicity and sparsity of the resulting matrix system. However, it offers some great advantages over finite difference schemes. First, the method is insensitive to the value of the forcing frequency i.e., the resulting matrix system is always symmetric and positive definite,. As a result, iterative methods may be successfully employed to solve the matrix system, thus taking full advantage of the sparsity. Furthermore, the method is insensitive to the type of the partial differential equation, i.e., the computational algorithm is the same in elliptic and hyperbolic regions. In this work the method is formulated and numerical results for model problems are presented. Some theoretical aspects of least squares approximations are also discussed.     

Numerical Simulations and Moments of the Field from a Point Source in a Random Medium

Abstract

Numerical simulations of the field emitted by a point source in a medium with weak random variations of refractive index are often made using the paraxial form of the wave equation in Cartesian coordinates. In this case the representation of the point source can lead to problems. The present paper shows how these difficulties can be avoided by the appropriate use of circular coordinates. The form of the second and fourth moment equations in circular coordinates is also discussed.     

Free vibration analysis of sandwich beams using improved dynamic stiffness method

In this paper, free vibration of three-layered symmetric sandwich beam is investigated using dynamic stiffness and finite element methods. To determine the governing equations of motion by the present theory, the core density has been taken into consideration. The governing partial differential equations of motion for one element contained three layers are derived using Hamilton’s principle. This formulation leads to two partial differential equations which are coupled in axial and bending deformations. For the harmonic motion, these equations are combined to form one ordinary differential equation. Closed form analytical solution for this equation is determined. By applying the boundary conditions, the element dynamic stiffness matrix is developed. They are assembled and the boundary conditions of the beam are applied, so that the dynamic stiffness matrix of the beam is derived. Natural frequencies and mode shapes are computed by the use of numerical techniques and the known Wittrick–Williams algorithm. After validation of the present model, the effect of various parameters such as density, thickness and shear modulus of the core for various boundary conditions on the first natural frequency is studied.     

Numerical solutions for a class of multi-part mixed boundary value problems

This paper proposes a procedure for the extension of a widely used numerical method^1?4 for the solution of systems of singular integral equations, from mixed boundary value problems with two or three parts to those having many parts. The procedure is applied to three previously solved mixed boundary value problems and the result are presented in graphical form. The effectiveness of the given procedure is observed through the results for some limiting and special cases found in the literature.     

复合材料蠕变本构关系及实验测定Ⅰ.本构关系

蠕变是复合材料最重要的力学性能之一,实验表明:复合材料在蠕变条件下的变形可以分为弹性变形、粘弹性变形和粘塑性变形.应用不可逆过程的热力学和广义变量的概念可以分析材料的蠕变变形.本文首先回顾了热力学的基本方程;基于Schapery本构关系的假设和思路推导了蠕变本构关系的一般形式,其中包括弹性变形、粘弹性变形和粘塑性变形;考虑到广义力选取的不唯一性,本文提出了广义力选取的原则以使得到的本构关系尽可能地简单;由此本文给出了复合材料的一维蠕变,各向同性复合材料的二维蠕变和纤维增强复合材料平面内的蠕变的本构关系.     

An improved form of the hypersingular boundary integral equation for exterior acoustic problems

An improved form of the hypersingular boundary integral equation (BIE) for acoustic problems is developed in this paper. One popular method for overcoming non-unique problems that occur at characteristic frequencies is the well-known Burton and Miller (1971) method [7], which consists of a linear combination of the Helmholtz equation and its normal derivative equation. The crucial part in implementing this formulation is dealing with the hypersingular integrals. This paper proposes an improved reformulation of the Burton–Miller method and is used to regularize the hypersingular integrals using a new singularity subtraction technique and properties from the associated Laplace equations. It contains only weakly singular integrals and is directly valid for acoustic problems with arbitrary boundary conditions. This work is expected to lead to considerable progress in subsequent developments of the fast multipole boundary element method (FMBEM) for acoustic problems. Numerical examples of both radiation and scattering problems clearly demonstrate that the improved BIE can provide efficient, accurate, and reliable results for 3-D acoustics.     

基于二阶矩阵微分方程的机械振动系统线性二次型调节器设计

本文讨论机械振动系统线性二次型状态调节器（LQR）问题，直接针对系统二阶运动微分方程，性能指标为一个依赖于二阶导数的泛函。由欧拉-拉格朗日方程得出一个系统矩阵增广的二阶线性微分方程，指出该方程稳定的特征对就是最优控制振动系统闭环特征对，并给出求解最优控制状态反馈矩阵的方法，另外，由本文方法还可得出基于速度和加速度反馈的最优控制反馈矩阵。这里不涉及求解代数矩阵Riccati方程。     

ON THE CHOICE OF A DERIVATIVE BOUNDARY ELEMENT FORMULATION USING HERMITE INTERPOLATION

This paper reports on some problems that can arise with the use of regularized derivative boundary integral equations. It concentrates on developing a formulation for the simple Laplace equation using a cubic Hermite interpolation and shows how certain combinations of derivative and conventional boundary integral equations can result in a solution scheme severely lacking in stability. With some simple two- and three-dimensional geometries, the derivative equations on their own do not provide enough information to solve a Dirichlet problem. Even combinations of the conventional and derivative equations fail for some simple geometries. We conclude that the only consistently successful combination is that of the conventional equation with the tangential derivative equation, which showed cubic convergence of results with mesh refinement. Numerical results are presented for this scheme in both two and three dimensions.     

A general integral equation formulatin for homogeneous orthotropic potential problems

In this paper an integral equation formulation is proposed for the analysis of orthotropic potential problems. The two primary integral equations of the method are derived from the original governing differential equation firstly by rewriting it in a slightly different form and then applying the direct boundary element method formulation. The solution procedure is based on the use of the fundamental solutions for the isotropic potential case and special attention is given to the differentiation of a singular integral which yields an additional term as well as to the evaluation of the resulting Cauchy principal value integral. A simple discretization for the boundary and its interior domain is adopted in order to express the primary integral equations of the method in matrix form. Three examples are presented, the results of which illustrate the satisfactory accuracy of the method. The main feature of the proposed formulation is its generality, which makes possible its direct extension to solve such as heat conduction or subsurface flow in anisotropic media and, foremost, to orthotropic and anisotropic elasticity or elastoplasticity.