Analytical and numerical investigation for true and spurious eigensolutions of an elliptical membrane using the real-part dual BIEM/BEM

The paper performs analytical and numerical investigation of the true and spurious eigensolutions of an elliptical membrane using the real-part boundary integral equation method (BIEM) following the successful work on a circular case by using the dual boundary element method (BEM) (Kuo et al. in Int. J. Numer. Methods Eng. 48:1401–1422, 2000). We extend to the elliptical case in this paper. To analytically study the eigenproblems of an elliptical membrane, the elliptical coordinates and Mathieu functions are adopted. The fundamental solution is expanded into the degenerate kernel by using the elliptical coordinates and the boundary densities are expanded by using the eigenfunction expansion. The Jacobian terms may exist in the degenerate kernel, boundary density and boundary contour integration but they can cancel each other out. Therefore, the orthogonal relations are reserved in the boundary contour integral. It is interesting to find that the BIEM using the real or the imaginary-part kernel to deal with an elliptical membrane yields spurious eigensolutions. This finding agrees with those corresponding to the circular case. The spurious eigenvalues in the real-part BIEM are found to be the zeros of the mth-order (even or odd) modified Mathieu functions of the second kind or their derivatives. To verify this finding, the BEM is implemented. Furthermore, the commercial finite-element code ABAQUS is also utilized to provide eigensolutions for comparisons. It is found that good agreement is obtained.     

Exact solution for free vibration analysis of an eccentric elliptical plate

The method of separation of variables in elliptical coordinates in conjunction with the translational addition theorems for Mathieu functions is used to investigate the free flexural vibrations of a fully clamped thin elastic panel of elliptical planform containing an elliptical cutout of arbitrary size, location, and orientation. The first five natural frequencies are calculated for various plate/cutout aspect ratios and selected cutout location/orientation parameters. Also, a number of representative vibration mode shapes are depicted in graphical form. The accuracy of solutions is demonstrated through proper convergence studies, and the validity of results is established with the aid of a commercial finite element package as well as by comparison with those in the existing literature.     

Analytical investigation for true and spurious eigensolutions of multiply-connected membranes containing elliptical boundaries using the dual BIEM

It is well known that the boundary element method may induce spurious eigenvalues while solving eigenvalue problems. The finding that spurious eigenvalues depend on the geometry of inner boundary and the approach utilized has been revealed analytically and numerically in the literature. However, all the related efforts were focused on eigenproblems involving circular boundaries. On the other hand, the extension to elliptical boundaries seems not straightforward and lacks of attention. Accordingly, this paper performs an analytical investigation of spurious eigenvalues for a confocal elliptical membrane using boundary integral equation methods (BIEM) in conjunction with separable kernels and eigenfunction expansion. To analytically study this eigenproblem, the elliptic coordinates and Mathieu functions are adopted. The fundamental solution is expanded into the separable kernel by using the elliptic coordinates and the boundary densities are expanded by using the eigenfunction expansion. The Jacobian terms may exist in the separable kernel, boundary density and boundary contour integration and they can cancel each other out. Therefore, the orthogonal relations are reserved in the boundary contour integration. In this way, a similar finding about the mechanism of spurious eigenvalues is found and agrees with those corresponding to the annular case. To verify this finding, the boundary element method and the commercial finite-element code ABAQUS are also utilized to provide eigensolutions, respectively, for comparisons. Good agreement is observed from comparisons. Based on the adaptive observer system, the present approach can deal with eigenproblems containing circular and elliptical boundaries at the same time in a semi-analytical manner. By using the BIEM, it is found that spurious eigenvalues are the zeros of the modified Mathieu functions which depend on the inner elliptical boundary and the integral formulation. Finally, several methods including the CHIEF method, the SVD updating technique and the Burton & Miller method are employed to filter out the spurious eigenvalues, respectively. In addition, the efficiency of the CHIEF method is better than those of the SVD updating technique and the Burton & Miller approach, since not only hypersingularity is avoided but also computation effort is saved.     

Solution of pressure diffusion in radially composite reservoirs

This paper presents a new general method for solving the pressure-diffusion equation in cylindrically radial composite reservoirs, where the rock and fluid properties may change radially as a function ofr. Composite systems, such as formations with wellbore filtrate invasion and reservoirs with peripheral water encroachment, can be encountered as a result of drilling, secondary oil recovery, and water influx.The new solution method utilizes the reflection and transmission concept of electromagnetics to solve fluid flow problems in three-dimensional cylindrically radial reservoirs, where heterogeneity is in only one direction. The Green's function for a point source in a three-dimensional radially composite system is developed by using the reflection and transmission method. The method as well as the point source solution are sufficiently general that they may be applied to similar fluid flow and well testing problems involving single-phase flow.     

三维非均匀不稳定渗流方程的二维不均匀粗化算法

在无源汇条件下,根据流过某一个横截面的流体流量等于流过这一横截面内所有精细网格的流体流量之和这一特点提出了粗化网格等效渗透率的计算方法。在粗化区内,利用直接解法求解二维渗流方程,再用这些解合成粗化网格的三维合成解,并由合成解计算粗化网格的等效渗透率。根据精度的要求采用了不均匀网格粗化,在流体流速大的区域采用精细网格。利用所得等效渗透率计算了粗化网格的某三维非均匀不稳定渗流场的压降解,结果表明三维非均匀不稳定渗流方程的二维不均匀粗化解非常逼近采用精细网格的解,但计算的速度比采用精细网格提高了80倍。     

Computations of wall distances by solving a transport equation

Computations of wall distances still play a key role in modern turbulence modeling. Motivated by the expense involved in the computation, an approach solving partial differential equations is considered. An Euler-like transport equation is proposed based on the Eikonal equation. Thus, the efficient algorithms and code components developed for solving transport equations such as Euler and Navier-Stokes equations can be reused. This article provides a detailed implementation of the transport equation in the Cartesian coordinates based on the code of computational fluid dynamics for missiles (MICFD) of Beihang University. The transport equation is robust and rapidly convergent by the implicit lower-upper symmetric Gauss-Seidel (LUSGS) time advancement and upwind spatial discretization. Geometric derivatives must also be upwind determined to ensure accuracy. Special treatments on initial and boundary conditions are discussed. This distance solving approach is successfully applied on several complex geometries with 1-1 blocking or overset grids.     

Computations of wall distances by solving a transport equation

Computations of wall distances still play a key role in modern turbulence modeling. Motivated by the expense involved in the computation, an approach solving partial differential equations is considered. An Euler-like transport equation is proposed based on the Eikonal equation. Thus, the efficient algorithms and code components developed for solving transport equations such as Euler and Navier-Stokes equations can be reused. This article provides a detailed implementation of the transport equation in the Cartesian coordinates based on the code of computational fluid dynamics for missiles (MICFD) of Beihang University. The transport equation is robust and rapidly convergent by the implicit lower-upper symmetric Gauss-Seidel (LUSGS) time advancement and upwind spatial discretization. Geometric derivatives must also be upwind determined to ensure accuracy. Special treatments on initial and boundary conditions are discussed. This distance solving approach is successfully applied on several complex geometries with 1–1 blocking or overset grids.     

Transient forced convection heat transfer in a channel

A numerical analysis is made of incompressible transient turbulent flow heat transfer between two parallel plates when there is a step jump in space along the channel in wall heat flux or wall temperature. The variation of the fluid velocity and effective diffusivity over the channel cross section are accounted for. The fluid is assumed to have a fully-developed turbulent velocity profile throughout the length of the channel. The thermal responses of the system are obtained by solving energy equation for air by a digital computer. The results are presented in graphical forms. The stability of the finite difference solution is studied and condition for the stability of the difference solution is derived. A method is given to obtain velocity distributions from the distribution of turbulent eddy diffusivity of momentum. Variations of Nusselt numbers are obtained as a function of time and space. Steady-state values are also given and compared with the published results.     

Renormalization group methods for a Mathieu equation with delayed feedback

This paper presents the application of the renormalization group (RG) methods to the delayed differential equation. By analyzing the Mathieu equation with time delay feedback, we get the amplitude and phase equations, and then obtain the approximate solutions by solving the corresponding RG equations. It shows that the approximate solutions obtained from the RG method are superior to those from the conventionally perturbation methods.     

A Study on Elliptical Gas Flow in Tri-porosity Gas Reservoirs

Elliptical flow is common in the near vertical fracture area and in anisotropic reservoirs. However, the classical radial flow models cannot provide a complete analysis for elliptical flow. This article presents a new mathematic model for gas elliptical flow in tri-porosity gas reservoirs. The differential equation of the new model is written in Mathieu equation, so that the solution can also be expressed by Mathieu functions. The numerical solution of the corresponding Mathieu functions ce _2n (ξ, −q), Ke _2n (ξ, −q) and their derivatives are obtained to derive the dimensionless pseudo pressure drop in Laplace space. The sensitivities of tri-porosity systems, including the parameters related to anisotropies C _De^2S and ξ _w, the storativity ratios ω _f and ω _m, and the interporosity flow coefficients λ_vf and λ_mf, are studied using Laplace numerical inversion. The new solution includes not only the factors considered in classic solutions in previous articles, but also incorporates the effect of reservoir anisotropy.     

基于黏弹性的微注射成型流动模拟

应用有限元方法研究了微注射成型中瞬态、可压缩、非牛顿熔体流动的黏弹性对流动前沿及流动平衡的影响。基于Phan-Thien-Tanner模型建立了熔体流动的本构方程,利用Hele-Shaw假设和简化建立了瞬态、可压缩、非牛顿熔体流动的连续性方程、动量方程、能量方程;为了有效地描述微注射成型的尺寸效应,采用了边界滑移和表面张力边界条件。通过分部积分和待定系数法导出了带有边界信息的变分方程和求解应力分量的半解析公式,构造了有限元离散求解及超松驰迭代算法。模拟结果表明:熔体的黏弹性对浇口附近的压力和后续的熔体流动前沿有重要影响;与黏性模型相比,黏弹性模型可以控制模拟压力的快速增长,减少不同型腔之间的充填差异,与短射实验结果也更吻合。     

Free vibration analysis of an elliptical plate with eccentric elliptical cut-outs

The elaborated collocation multipole method is employed to obtain a semi-analytical solution, involving proper products of angular and radial Mathieu functions, for the free flexural vibrations of a fully clamped thin elastic plate of elliptical planform containing multiple elliptical cutouts of arbitrary size, location, and orientation. The problem boundary conditions are satisfied by uniformly collocating points on the boundaries, and exactly calculating the normal derivative of plate displacement at the collocation points through use of appropriate directional derivative in each coordinate system. The multipole expansion is truncated to yield a coupled algebraic linear system of equations that is then solved for the nontrivial eigensolutions. Extensive numerical simulations present the first three calculated natural frequencies and the associated deformed mode shapes of an elliptical plate with elliptical/circular cutouts, for a wide range of plate/cutout aspect ratios, and cutout location/orientation parameters. The accuracy of solutions is checked through appropriate convergence studies, and the validity of results is established with the aid of a commercial finite element package as well as by comparison with the data available in the existing literature.     

Estimates of azimuthal numbers associated with elementary elliptic cylinder wave functions

The paper deals with issues related to the construction of solutions, 2 π-periodic in the angular variable, of the Mathieu differential equation for the circular elliptic cylinder harmonics, the associated characteristic values, and the azimuthal numbers needed to form the elementary elliptic cylinder wave functions. A superposition of the latter is one possible form for representing the analytic solution of the thermoelastic wave propagation problem in long waveguides with elliptic cross-section contour. The classical Sturm-Liouville problem for the Mathieu equation is reduced to a spectral problem for a linear self-adjoint operator in the Hilbert space of infinite square summable two-sided sequences. An approach is proposed that permits one to derive rather simple algorithms for computing the characteristic values of the angular Mathieu equation with real parameters and the corresponding eigenfunctions. Priority is given to the application of the most symmetric forms and equations that have not yet been used in the theory of the Mathieu equation. These algorithms amount to constructing a matrix diagonalizing an infinite symmetric pentadiagonal matrix. The problem of generalizing the notion of azimuthal number of a wave propagating in a cylindrical waveguide to the case of elliptic geometry is considered. Two-sided mutually refining estimates are constructed for the spectral values of the Mathieu differential operator with periodic and half-periodic (antiperiodic) boundary conditions.     

A Quasilinear Based Procedure for Saturated/Unsaturated Water Movement in Soils

The quasilinear form of Richards equation for one-dimensional unsaturated flow in soils can be readily solved for a wide variety of conditions. However, it cannot explain saturated/unsaturated flow and the constant diffusivity assumption, used to linearise the transient quasilinear equation, can introduce significant error. This paper presents a quasi-analytical solution to transient saturated/unsaturated flow based on the quasilinear equation, with saturated flow explained by a transformed Darcy's equation. The procedure presented is based on the modified finite analytic method. With this approach, the problem domain is divided into elements, with the element equations being solutions to a constant coefficient form of the governing partial differential equation. While the element equations are based on a constant diffusivity assumption, transient diffusivity behaviour is incorporated by time stepping. Profile heterogeneity can be incorporated into the procedure by allowing flow properties to vary from element to element. Two procedures are presented for the temporal solution; a Laplace transform procedure and a finite difference scheme. An advantage of the Laplace transform procedure is the ability to incorporate transient boundary condition behaviour directly into the analytical solutions. The scheme is shown to work well for two different flow problems, for three soil types. The technique presented can yield results of high accuracy if the spatial discretisation is sufficient, or alternatively can produce approximate solutions with low computational overheads by using large sized elements. Error was shown to be stable, linearly related to element size.     

On the stress problem of large elliptical cutouts and cracks in circular cylindrical shells

Numerical solutions are presented for stresses around an elliptical hole in a long, thin, circular cylindrical shell subjected to axial tension for both the symmetric orientations of the hole with respect to the shell. The method of analysis involves obtaining a series solution to the governing shell equations in terms of Mathieu functions by the method of separation of variables and satisfying the boundary conditions numerically term by term in a Fourier series formulation. Results are presented in the form of charts from which stress concentration factors can be directly read over a wide range of the two parameters, namely, axis ratio of the ellipse and a curvature parameter defining the hole size with respect to dimensions of the shell.An interesting feature of the investigation is the analysis of limiting cases of circumferential and axial cracks for axial tension and internal pressure loadings respectively. The method developed involves determining the solution completely in elliptic coordinates and then determining the singular stresses by carrying out a transformation to polar coordinates with crack tip as the origin through a Taylor series expansion. Membrane and bending stress intensity factors are computed and plotted over a sufficiently wide range of the curvature parameter extending from small to large sized cracks. As an outcome of the analysis, a “hybrid” technique has been developed by which singularity conditions at the crack tip can be handled effectively in dealing with boundary conditions in crack problems.     

压电材料中椭圆片状裂纹问题精确解的一种方法

在线性压电陶瓷本构关系和裂纹边界绝缘的框架下,用超奇异积分方程的方法对椭圆类片状裂纹问题进行了重新研究．超奇异积分方程中的未知位移间断和电势间断近似地表示为基本密度函数与多项式之积,其中基本密度函数反映了椭圆片状裂纹前沿电弹性场的奇异性,而多项式在均布载荷作用下可用一个常数来表达．引入椭球坐标系后,得到了均布载荷作用下未知位移间断和电势间断的解析解．使用这些解析解和电弹性场强度的定义,得到了裂纹前沿Ⅰ型、Ⅱ型和Ⅲ型应力强度因子以及电位移强度因子的精确表达式．法向均布载荷作用下的结果与现有精确解完全一致,切向均布载荷作用下的结果则尚未见有其它报道．     

Groundwater Fluctuations in Sloping Aquifers Induced by Time-Varying Replenishment and Seepage from a Uniformly Rising Stream

This paper analyses the classical problem of transient surface?Cgroundwater interaction in a stream?Caquifer system under rather realistic conditions. The downward sloping unconfined aquifer is in contact with a constant water level at one end, and a fully penetrating stream at the other end whose water level is rising at a uniform speed. Furthermore, the aquifer is replenished by a vertical time-varying recharge. Closed form analytical expressions for hydraulic head and flow rate in the aquifer are obtained by solving the linearized Boussinesq equation using Laplace transform method. Effects of aquifer parameters on transient water table and flow rate are illustrated with a numerical example. To assess the efficiency of the linearization method, analytical solutions are compared with numerical solutions of the corresponding non-linear equation.     

弹性椭圆夹杂的线性温变问题

研究了线性温变作用下椭圆夹杂的热弹性问题。通过构造辅助函数，将复变函数的分区全纯函数理论，Riemann边值问题和Cauchy型积分相结合，求得各分区之间的解析关系，从而获得了无穷远均匀加载和线性温变共同作用下椭圆夹杂平面热弹性场的封闭形式解。从本文解答的特殊情况可直接得到已有的若干结果，并可得到一些具有实际意义的新结果。本文发展的分析方法，为求解复杂多连通域的平面热弹性问题提供了一条有效途径。     

Transient spherical flow of non-newtonian power-law fluids in porous media

The transient spherical flow behavior of a slightly compressible non-Newtonian, power-law fluids in porous media is studied. A nonlinear partial differential equation of parabolic type is derived. The diffusivity equation for spherical flow is a special case of the new equation. We obtain analytical, asymptotic and approximate solutions by using the methods of Laplace transform and weighted mass conservation. The structures of asymptotic and approximate solutions are similar, which enriches the theory of one-dimensional flow of non-Newtonian fluids through porous media.     

Existence of n-cycles and border-collision bifurcations in piecewise-linear continuous maps with applications to recurrent neural networks

The purpose of these comments is to correct the derivation of perturbed first-order $$\varepsilon ^{1}$$-equation, for modeling the acoustic scattering waves from a composite beam. We have introduced the correct solutions of displacements and potential functions in the first and second orders, although the mathematical treatment in solving the coupled nonlinear PDEs in the time-space domain looks good. So, the revised equation and the correct second-order $$\varepsilon ^2$$ equation are presented in this note.