A boundary element solution to the soap bubble problem

The boundary element method (BEM) is applied to the soap bubble problem, that is to the problem of determining the surface that a soap bubble constrained by bounding contours assumes under the action of molecular forces. This is also the shape of a uniformly stretched membrane bounded by one or more non-intersecting curves. As the slopes of the membrane surface are finite, their square can not be neglected and the resulting governing equation is non-linear. The problem is solved using the analogue equation method (AEM). According to this method the non-linear membrane is substituted by a linear one subjected to a fictitious transverse load. The fictitious load is established using the BEM. Numerical examples are presented which illustrate the method and demonstrate its accuracy. This application of the BEM to non-linear problems shows that BEM is a versatile computational method for all-purpose use in engineering analysis. The solution of the problem at hand is very important in engineering, since the soap bubble surface can be used as the best initial form for membrane roofs.     

简谐激励力作用下悬垂缆线的谐波共振

本文研究在简谐激励力作用下的悬垂缆线的谐波共振。用Hamilton原理导出悬垂缆线面内运动的非线性偏微分方程。通过假设悬垂缆线的挠度曲线，运用Galerkin方法将偏微分方程转化为常微分方程。用多尺度法研究悬垂缆线的超谐波共振和次谐波共振，得到了系统的定常周期解，平均方程和幅频曲线。研究了非线性对幅频曲线的影响和定常运动的稳定区域。     

Self-similar evolution of a twin boundary in anti-plane shear

This paper studies the transient motion of a twin boundary in two dimensions. The twinning deformation is described as an anti-plane shear deformation with discontinuous strains. The material is assumed to be compressible and hyperelastic with a stored energy function consisting of multiple potential wells. The quasi-steady-state evolution of a twinning step is studied. The model includes an anisotropic kinetic relation that governs the twin boundary motion in two dimensions under applied stress. A self-similar solution for the motion of the twinning step is found with a specific initial shape. General solutions to the linearized evolution equation are established in the form of an infinite series for arbitrary initial shapes. Stability of the self-similar solution is discussed.     

A quasi‐interpolation method for solving stiff ordinary differential equations

Based on the idea of quasi‐interpolation and radial basis functions approximation, a numerical method is developed to quasi‐interpolate the forcing term of differential equations by using radial basis functions. A highly accurate approximation for the solution can then be obtained by solving the corresponding fundamental equation and a small size system of equations related to the initial or boundary conditions. This overcomes the ill‐conditioning problem resulting from using the radial basis functions as a global interpolant. Error estimation is given for a particular second‐order stiff differential equation with boundary layer. The result of computations indicates that the method can be applied to solve very stiff problems. With the use of multiquadric, a special class of radial basis functions, it has been shown that a reasonable choice for the optimal shape parameter is obtained by taking the same value of the shape parameter as the perturbed parameter contained in the stiff equation. Copyright © 2000 John Wiley & Sons, Ltd.     

Characterization of magnetohydrostatic axial separation system

The use of the magnetohydrostatic axial separation system for separation and characterization of particulates is considered. It is shown that one reference curve determined with a given paramagnetic fluid at a given current level is sufficient for the calculation and construction of a set of calibration curves for different fluids and current levels. In cases where a solution of the magnetic field is available, calculation of reference curves are straightforward. In other cases they must be determined experimentally. Procedures for calculations of the density and susceptibility of particles by use of the calibration curves are outlined, and expected axial separation patterns of a mixture of particles are evaluated. It is shown that lower resolution is associated with high levitational forces. Increase of resolution may be obtained at higher current levels or by smaller rate of tapering of the poles in the region of small air gap.     

Flow through variably saturated soils

The equation of flow through variably saturated porous media is discretized via the Galerkin finite element formulation. The discretization is coupled with an approach for mesh generation and optimization of the node numbering scheme. Sensitivity analysis showed that the solution behavior is controlled by dimensionless quantities equivalent to Peclet and Courant numbers. For the form of equation investigated, no universal limiting values of Pe and Cr can be established because the values of these parameters depend on both the constitutive relations used and on initial conditions. For more efficient solution of the problem, a deformation scheme of the computational mesh is proposed, which accounts for the limiting Peclet and Courant numbers and for the shape of the deformed elements. Comparisons with other solutions showed that the numerical scheme performs very well.     

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.     

Effect of a hard artificial asperity on the crack closure behavior in an annealed SAE 1015 steel

The load-crack opening displacement (COD) curves and deformation characteristics in the vicinity of a hard artificial asperity in an annealed SAE 1015 steel were studied. The artificial asperity was found to have a significant effect on the trend of the load-COD curves. The lower portion of the load-COD curves in the unloading phase exhibited a convex shape without the asperity, but a concave shape with the asperity. The concave shape, signifying the acceleration in the COD decrease, was further verified by varying the size of the asperity, conducting special compression tests and elastic-plastic load-COD tests. The plastic deformation in the vicinity of both asperity and crack tip was studied via microhardness tests, etching techniques, and finite element analysis. Based on the experimental observations, a modified crack closure process model was proposed, where three stages of the unloading curve was defined: (i) the asperity does not contact the upper crack face, (ii) a process where both the asperity and the specimen material deform elastically, and the elastic-wedge model is applicable, and (iii) the plastic deformation of the specimen material adjacent to the asperity occurs, thus resulting in the concavely shaped load-COD curves. An equation was proposed to estimate the COD values, in which the plastic deformation both at the crack tip and at the asperity was considered. The residual COD calculated from the proposed equation was found to be consistent with the experimental results.     

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.     

扬声器低频谐波失真的数值分析

扬声器作为一种非线性振动系统，在低频段仅考虑力学恢复力非线性可用经典的Duffing方程来描述；同时考虑恢复力和磁场非线性可用广义的Duffing方程来表征。Yoshinisa研究了仅含恢复力非线性扬声器低频非线性现象中的低频谐波失真现象，但对恢复力和磁场非线性同时存在的扬声器低频谐波失真问题未作研究。西方利用Matlab软件求解扬声器非线性振动系统的广义Duffing方程的数值解，又利用Spectra Plus频谱分析软件得到扬声器低频谐波失真与频率的关系曲线，通过分析低频谐波失真与频率的关系曲线，并着重讨论磁场的非线性对扬声器低频谐波失真的影响，得出一些有价值的结论。     

Application of a pseudo-viscous method to the calculation of the steady supersonic flow past a waisted body

A finite difference method for solving mixed initial and boundary value problems governed by hyperbolic partial differential equations is described. The method has been developed specifically to calculate the flow field associated with any arbitrary two-dimensional or axi-symmetric body placed in a uniform supersonic stream. In order to preserve accuracy in the neighborhood of the body surface, the equations governing the motion are formulated so that

• 1 Streamlines form one system of co-ordinate curves.
• 2 Two of the equations of motion correspond to characteristic relations.

The boundary condition at the body is applied by omitting the equation associated with the characteristic curve reaching the body surface from inside. The method presented is a pseudo-viscous method and flow fields which include shock waves can be treated with ease. The method has been used to calculate the flow past a particular axi-symmetric waisted body. The shape of this body is such that a secondary internal shock wave is generated by coalescence of Mach lines behind the bow shock wave. The results obtained are compared with theoretical and experimental results obtained by other workers. The investigation has been carried out as part of a programme of work needed to validate the pseudo-viscous method used.     

Non-linear vibration of frames by the incremental dynamic stiffness method

The dynamic stiffness method is extended to large amplitude free and forced vibrations of frames. When the steady state vibration is concerned, the time variable is replaced by the frequency parameter in the Fourier series sense and the governing partial differential equations are replaced by a set of ordinary differential equations in the spatial variables alone. The frequency-dependent shape functons are generated approximately for the spatial discretization. These shape functions are the exact solutions of a beam element subjected to mono-frequency excitation and constant axial force to minimize the spatial discretization errors. The system of ordinary differential equations is replaced by a system of non-linear algebraic equations with the Fourier coefficients of the nodal displacements as unknowns. The Fourier nodal coefficients are solved by the Newtonian algorithm in an incremental manner. When an approximate solution is available, an improved solution is obtained by solving a system of linear equations with the Fourier nodal increments as unknowns. The method is very suitable for parametric studies. When the excitation frequency is taken as a parameter, the free vibration response of various resonances can be obtained without actually computing the linear natural modes. For regular points along the response curves, the accuracy of the gradient matrix (Jacobian or tangential stiffness matrix) is secondary (cf. the modified Newtonian method). However, at the critical positions such as the turning points at resonances and the branching points at bifurcations, the gradient matrix becomes important. The minimum number of harmonic terms required is governed by the conditions of completeness and balanceability for predicting physically realistic response curves. The evaluations of the newly introduced mixed geometric matrices and their derivatives are given explicitly for the computation of the gradient matrix.     

On the settling of a bidisperse suspension with particles having different sizes and densities

The settling of a bidisperse suspension with small particles having different sizes and densities can be described by an initial value problem for a system of two non-linear, first-order conservation laws. Solutions to this problem are in general discontinuous and exhibit kinematic shocks that separate areas of different composition. The solution requires the construction of so-called elementary curves in phase space, which are determined from eigenvector fields of the Jacobian of the flux function. Differences in solution behavior to the previously analyzed equal-density case are due to an umbilic point, which appears for different densities only. The initial value problem is eventually solved by the front tracking method, which generates a series of Riemann problems. It turns out that the solution of the problem predicts a fairly complex process of sediment formation, and that the stationary solution can consist of non-constant smooth transitions. This observation is of interest for manufacturing of functionally graded materials.     

矩阵方程AXB=C的中心对称最小二乘解及其最佳逼近的迭代算法

本文建立了求矩阵方程AXB=C的中心对称最小二乘解的迭代算法。在不考虑舍入误差时,对任意给定的初始中心对称矩阵,该算法能够在有限步迭代后得到此方程的中心对称最小二乘解。当选取特殊的初始矩阵时,可得到极小范数中心对称最小二乘解。另外,在上述解集合中也可得到给定矩阵的最佳逼近矩阵的表达式。     

Two-dimensional non-Fourier heat conduction with arbitrary initial and periodic boundary conditions

In this paper, the (2+1)-dimensional hyperbolic heat conduction equation is analytically solved under the influence of arbitrary initial conditions for a rectangular plate with homogeneous boundary conditions of first-kind. The temperature field is obtained as a double Fourier series. The presented solution is valid even for discontinuous but integrable initial conditions. Afterwards, the solution is generalized by means of a transformation to cover problems with inhomogeneous first-kind boundary conditions. Another interesting issue is that the obtained solution can be considered as a solution to the Klein–Gordon equation under the influence of arbitrary initial conditions by means of a simple transformation.     

Compound parabolic tapered fiber for fiber coupling with a highly divergent source

A novel fiber tapering shape, which is based on compound parabolic geometry, is proposed to increase the acceptance angle of a compound parabolic concentrator. The proposed design is described by use of ray optics on a step-index multimode fiber.     

Symbolic derivation of the equations of caustics about a crack tip

Summary For the determination of the shape of the initial curve of a caustic about a crack tip in plane elasticity problems (whence the shape of the caustic itself is automatically obtained) we need to solve a nonlinear algebraic equation with unknown the distance of each point of the initial curve from the crack tip and parameter the polar angle. Here this nonlinear equation is solved (for a particular crack problem) by the classical method of successive substitutions (a one-point iterative method) in numerical analysis, but symbolically and not numerically, with respect to the polar angle. This yields a semianalytical equation for the initial curve (and, further, for the caustic itself) of particular importance for the study of its properties from the analytical point of view. On the other hand, the present results show the usefulness of symbolic computations in crack problems in fracture mechanics.     

Transient creep behavior of a metal matrix composite with a dilute concentration of randomly oriented spheroidal inclusions

The transient creep behavior of a metal matrix composite containing a dilute concentration of randomly oriented spheroidal inclusions is derived explicitly from the constitutive equation of the matrix. This theory can account for the influence of inclusion shape, elastic inhomogeneity between both phases, and the volume fraction of inclusions. The micro-macro transition is carried out by considering the mechanics of incremental creep, which discloses the nature of stress relaxation in the ductile matrix and the connection between the micro and macro creep strains. The transient creep curves of the composite are displayed with several inclusion shapes. Consistent with the known elastic behavior, spherical inclusions are found to provide the weakest reinforcing effect, whereas thin, circular discs possess the most effective strengthening shape. According to this theory and in line with the experimental data, the creep resistance of cobalt at 500°C can improve by more than 80% after adding a mere 5% of rutile particles into it.     

Lamb waves in a thin isotropic layer between two anisotropic layers

Attenuative Lamb wave propagation in adhesively bonded anisotropic composite plates is introduced. The isotropic adhesive exhibits viscous behavior to stimulate the poor curing of the middle layer. Viscosity is assumed to vary linearly with frequency, implying that attenuation per wavelength is constant. Attenuation can be implemented in the analysis through modification of elastic properties of isotropic adhesive. The new properties become complex, but cause no further complications in the analysis. The characteristic equation is the same as that used for the elastic plate case, except that both real and imaginary parts of the wave number (i.e., the attenuation) must be computed. Based on the Lowe's solution in finding the complex roots of characteristic equation, the effect of longitudinal and shear attenuation coefficients of the middle adhesive layer on phase velocity dispersion curves and attenuation dispersion curves of Lamb waves propagating in bonded anisotropic composites is visualized numerically.     

A boundary element method for calculating the shape and velocity of two-dimensional long bubble in stagnant and flowing liquid

A numerical method based on a boundary element method (BEM) has been developed for computing the velocity and the shape of long bubbles moving steadily in stagnant and flowing liquid in 2D case: plane and axisymmetrical. The flow is assumed to be inviscid and incompressible. The method consists in solving simultaneously a Poisson equation characterizing the flow and an equation for bubble shape in the form of a functional-differential equation resulting from both Bernoulli equation and the jump conditions at the interface. The Poisson equation is solved by a BEM with an iterative loop for nonlinear source term while the system of nonlinear algebraic equations obtained by discretizing the equation on the interface is solved by the Powell's hybrid algorithm. The bubble shape and velocity are obtained as a part of the solution. The problem of multiple solutions is investigated numerically and the maximum velocity criterion is used for selecting the physical solution. The results obtained by the simulation are in good agreement with the experimental and numerical results of previous studies.