Elliptic averaging methods using the sum of Jacobian elliptic delta and zeta functions as the generating solution

The present paper describes an improved version of the elliptic averaging method that provides a highly accurate periodic solution of a non-linear system based on the single-degree-of-freedom Duffing oscillator with a snap-through spring. In the proposed method, the sum of the Jacobian elliptic delta and zeta functions is used as the generating solution of the averaging method. The proposed method can be used to obtain the non-odd-order solution, which includes both even- and odd-order harmonic components. The stability analysis for the approximate solution obtained by the present method is also discussed. The stability of the solution is determined from the characteristic multiplier based on Floquet’s theorem. The proposed method is applied to a fundamental oscillator in a non-linear system. The numerical results demonstrate that the proposed method is very effective for analyzing the periodic solution of half-swing mode for systems based on Duffing oscillators with a snap-through spring.     

一种处理弹性动力边界元法中奇异积分的方法

利用边界元法求解瞬态弹性动力学问题时,时域基本解函数的分段连续性和奇异性为该问题的求解带来很大的困难。为了解决时域基本解中的奇异性问题,本文依据柯西主值的定义,对经过时间解析积分之后的时域基本解进行奇异值分解,将其分成奇异和正则积分两部分;其中正则部分可通过采用常规高斯积分方法来计算,而奇异部分具有简单的形式,可以利用解析积分计算。经过上述操作之后,就可以达到直接消除时域基本解中奇异积分的目的。和传统方法相比,本文方法并不依赖静力学基本解来消除奇异性,是一种直接求解方法。最后给定两个数值算例来验证本文提出方法的正确性和可行性,结果表明使用本文算法可以解决弹性动力学边界积分方程中的奇异性问题。     

结构动力方程的样条精细积分法

结合精细积分法和样条函数拟合技术的优点,提出了求解结构动力方程的一种有效方法.首先对非齐次项用三次正规化B样条函数进行拟合,然后利用正规化B样条函数形状相同、仅相差一个平移量的特点,构造了一个高效的特解求解方法.按此方法只需求出一个标准B样条项所对应的特解,然后通过时间坐标的平移并结合叠加原理,即可求出任意时刻的特解值.由于特解计算中采用数值积分的方法,避免了矩阵求逆,因而本方法具有较大的适用范围.算例结果证明了该方法的有效性.     

Boundary element method to solve torsion problem of cracked cylinder

Separating the discontinuous solution by use of the single crack solution, together with the regular solution of harmonic function, the torsion problem of a cracked cylinder is reduced to solving a set of mixed-type integral equations and its numerical technique is then proposed by combining the numerical method of singular integral equation with the boundary element method. Several numerical examples are calculated which will be useful to engineering practice. The method proposed is characterized by its fine accuracy and convenience for using, which can be extended to the cases of multiple crack.The project supported by National Natural Science Foundation of China.     

Helmholtz边界积分方程中奇异积分间接求解方法

提出了间接求解传统Helmholtz边界积分方程CBIE的强奇异积分和自由项系数,以及Burton-Miller边界积分方程BMBIE中的超强奇异积分的特解法。对于声场的内域问题,给出了满足Helmholtz控制方程的特解,间接求出了CBIE中的强奇异积分和自由项系数。对于声场外域对应的BMBIE中的超强奇异积分,按Guiggiani方法计算其柯西主值积分需要进行泰勒级数展开的高阶近似,公式繁复,实施困难。本文给出了满足Helmholtz控制方程和Sommerfeld散射条件的特解,提出了间接求出超强奇异积分的方法。推导了轴对称结构外场问题的强奇异积分中的柯西主值积分表达式,并通过轴对称问题算例证明了本文方法的高效性。数值结果表明,对于内域问题,采用本文特解法的计算结果优于直接求解强奇异积分和自由项系数的结果,且本文的特解法可避免针对具体几何信息计算自由项系数,因而具有更好的适用性。对于外域问题,两者精度相当,但本文的特解法可避免对核函数进行高阶泰勒级数展开,更易于数值实施。     

AN IMPROVED IRS METHOD FOR STRUCTURE MODEL CONDENSATION1)

模型缩聚法在结构的静力和动力特性分析中有着广泛的应用。应用模型缩聚法,可以有效降低结构的有限元计算规模,节省计算时间和成本,并能获得和实验测量自由度相匹配的有限元模型。本文在改进模型缩聚方法(improved reduced system, IRS)的基础上,提出一种考虑二阶惯性量的改进IRS方法,有效改进了IRS方法的计算精度,和模型缩聚迭代法(iterated IRS, IIRS)相比,此方法计算量更小且计算精度更高。以桁架结构和框架结构为例对所提二阶IRS方法进行了验证,并将计算结果与精确值、Guyan缩聚解、IRS缩聚解和IIRS缩聚解进行了比较,结果表明了所提方法计算精度最好,具有良好的工程应用前景。     

随机有限元分析的二级区域分解并行求解算法

采用蒙特卡罗模拟(MCS)和加权积分法对二维问题进行随机有限元分析。尽管MCS方法对任何有确定解的问题都具有求解精度高的优点,但由于求解所需的计算量巨大使其应用受到限制。利用并行求解技术可有效地处理这种密集型计算问题。基于有限元分裂对接法(FETI)的并行特性并利用预处理共轭梯度法(PCG)的求解高效性,结合整体子区域实现(GSI-PCG)和FETI法,提出二级求解算法,并在工作站机群上实现了数值算例。算例计算结果表明本文GSI(PCG)-FETI算法具有较高的并行加速比和并行效率,具有良好的性能,可有效地进行二维问题的随机有限元分析。     

Longitudinal vibrations of arbitrary non-uniform rods

Free and steady-state forced longitudinal vibrations of non-uniform rods are investigated by an iteration method, which results in a series solution. The series obtained are convergent and linearly independent. Its convergence is verified by convergence tests, its linear independence confirmed by the nonzero value of the corresponding Wronski determinant. Then, the solution obtained is an exact one reducible to a classical solution for the case of uniform rods. In order to verify the method, two examples are presented as an application of the proposed method. The results obtained are equivalent to the method in literature. In contrast to the proposed method capable of dealing with arbitrary non-uniform rods in principle, the method in literature is confined to work on special cases.     

Transverse vibrations of arbitrary non-uniform beams

Free and steady state forced transverse vibrations of non-uniform beams are investigated with a proposed method, leading to a series solution. The obtained series is verified to be convergent and linearly independent in a convergence test and by the non-zero value of the corresponding Wronski determinant, respectively. The obtained solution is rigorous, which can be reduced to a classical solution for uniform beams. The proposed method can deal with arbitrary non-uniform Euler-Bernoulli beams in principle, but the methods in terms of special functions or elementary functions can only work in some special cases.     

一类指数矩阵函数及其应用

研究了一阶常微分方程组特解的精细积分方法. 针对非齐次项为多项式、指数函数以及二者的乘积的情况,在Duhamel积分形式特解的基础上,引入了一类指数矩阵函数. 通过该类函数的线性组合即可表达出非齐次方程的特解. 建立了该类指数矩阵函数的一种高效递推算法,并在此基础上实现了特解的精细积分. 由于特解的积分过程能充分利用通解精细积分过程的中间量,因此两个精细积分过程能有机地结合起来,形成了一种高效、统一的广义精细积分法. 对上述递推算法做了进一步优化,并给出了通用的计算公式.算例结果证明了该方法的有效性.      

Iterative harmonic balance for period-one rotating solution of parametric pendulum

In this study, an iterative method based on harmonic balance for the period-one rotation of parametrically excited pendulum is proposed. Based on the definition of the period-one rotating orbit, the exact form of the solution can be obtained using the Fourier series. An iterative harmonic balance process is proposed to estimate the coefficients in the exact solution form. The general formula for each iteration step is presented. The method is evaluated using two criteria, which are the system energy error and the global residual error. The performance of the proposed method is compared with the results from multiscale method and perturbation method. The numerical results obtained with the Dormand?CPrince method (ODE45 in MATLAB?) are used as the baseline of the evaluation.     

A hybrid analytical–numerical solution for 3D notched plates

This study used a hybrid analytical and numerical method to analyze three-dimensional (3D) elastic bodies with sharp-V notches. The proposed method separates the 3D equilibrium equation into primary and shadow parts, where the solution of the primary part is the analytical solution under the generalized plane-strain theory, and the shadow part is solved numerically using a weak form based on the finite element theory. A least-squares method is then used to find the multiplication factors of these primary and shadow modes using 3D finite element results. Numerical simulations indicate that the proposed method can accurately simulate the singularities near a sharp V-notch. The major advantage of this method is that a 3D whole displacement field with the singular effect based on the theoretical solution near the notch can be obtained for anisotropic materials under arbitrary loads.     

风与柔性结构流固耦合作用的预处理方法研究

针对强耦合方法求解风与柔性结构流固耦合作用时,大量计算资源都耗费在对强耦合方程求解中这一弊端,本文研究了强耦合方程的预处理求解方法。在风与柔性结构流固耦合作用的强耦合整体方程的基础上,将时空离散和线性化后的类似结构方程看成是鞍点问题,首先推导得到了类似结构方程的预处理矩阵;再基于此推导出了强耦合整体方程的预处理矩阵。首先采用预处理方法对经典二维流固耦合问题进行了计算,验证了提出的预处理矩阵的正确性;然后对风与三维膜结构的流固耦合作用进行了分析,评估了所提出预处理方法的相关计算参数。计算结果表明,所提出的预处理方法可使强耦合整体方程的求解在计算精度和计算效率上都得到较大提升,证明本文提出的预处理方法适用于风与柔性结构的流固耦合分析。     

随机激励的耗散的Hamilton系统理论的研究进展

近几年中,利用Ｈａｍｉｌｔｏｎ系统的可积性与共振性概念及Ｐｏｉｓｓｏｎ括号性质等,提出了高斯白噪声激励下多自由度非线性随机系统的精确平稳解的泛函构造与求解方法,并在此基础上提出了等效非线性系统法,提出了拟Ｈａｍｉｌｔｏｎ系统的随机平均法,并在该法基础上研究了拟Ｈａｍｉｌｔｏｎ系统随机稳定性、随机分岔、可靠性及最优非线性随机控制,从而基本上形成了一个非线性随机动力学与控制的Ｈａｍｉｌｔｏｎ理论框架．本文简要介绍了这方面的进展．     

Drift response of a bilinear hysteretic system to periodic excitation under sustained load effects

This paper presents an investigation of response characteristics for hysteretic systems idealized as a bilinear hysteretic model subjected to period excitations composed of a harmonic function and a sustained load. It is shown that the displacement solution can exhibit a drift sequence persistently repeated at a frequency identical to the excitation frequency in the case of zero post-yielding stiffness. The periodic-like drift sequence is further classified into three major types according to their different hysteretic looping behaviors. An approximate solution approach based on the method of weighted residuals is proposed to analyze the drift amplitude per response cycle. The assumed response shape is composed of two concatenated harmonic functions each with a frequency slightly detuned from the excitation frequency. The method is accompanied with a subsequent first-order analysis to obtain a closed-form approximation for the drift response. Good response predictions of the proposed solution method are demonstrated through both undamped and damped drift-frequency analyses.     

Hybrid graded element model for transient heat conduction in functionally graded materials

This paper presents a hybrid graded element model for the transient heat conduction problem in functionally graded materials (FGMs). First, a Laplace transform approach is used to handle the time variable. Then, a fundamental solution in Laplace space for FGMs is constructed. Next, a hybrid graded element is formulated based on the obtained fundamental solution and a frame field. As a result, the graded properties of FGMs are naturally reflected by using the fundamental solution to interpolate the intra-element field. Further, Stefest’s algorithm is employed to convert the results in Laplace space back into the time-space domain. Finally, the performance of the proposed method is assessed by several benchmark examples. The results demonstrate well the efficiency and accuracy of the proposed method.     

Steady‐state solution of incompressible Navier–Stokes equations using discrete least‐squares meshless method

A fractional step method for the solution of the steady state incompressible Navier–Stokes equations is proposed in this paper in conjunction with a meshless method, named discrete least‐squares meshless (DLSM). The proposed fractional step method is a first‐order accurate scheme, named semi‐incremental fractional step method, which is a general form of the previous first‐order fractional step methods, i.e. non‐incremental and incremental schemes. One of the most important advantages of the proposed scheme is its capability to use large time step sizes for the solution of incompressible Navier–Stokes equations. DLSM method uses moving least‐squares shape functions for function approximation and discrete least‐squares technique for discretization of the governing differential equations and their boundary conditions. As there is no need for a background mesh, the DLSM method can be called a truly meshless method and enjoys symmetric and positive‐definite properties. Several numerical examples are used to demonstrate the ability and the efficiency of the proposed scheme and the discrete least‐squares meshless method. The results are shown to compare favorably with those of the previously published works. Copyright © 2010 John Wiley & Sons, Ltd.     

Implicit solution of harmonic balance equation system using the LU-SGS method and one-step Jacobi/Gauss-Seidel iteration

ABSTRACT

This paper presents efficient alternative numerical methods for an implicit solution of the harmonic balance equation system for analysing temporal periodic unsteady flows. The proposed method employs approximate factorisation to decouple the common residual term and the time spectral source term of a harmonic balance equation system when it is discretised implicitly. With this approximate factorisation, the complexity of implicit solution of the discrete system is greatly reduced. The common residual term can be dealt with using a lower-upper symmetric-Gauss-Seidel (LU-SGS) method and the time spectral source term is integrated using a Jacobi iteration (JI) or one step Gauss-Seidel (GS) iteration, leading to the LU-SGS/JI method or LU-SGS/GS method. The NASA stage 35 compressor and the 1.5 stage Aachen turbine were used to demonstrate the effectiveness of the proposed methods in stabilising solution and its advantages in comparison with the existing lower-upper symmetric-Gauss-Seidel/block Jacobi (LU-SGS/BJ) method. The LU-SGS/GS method and the LU-SGS/JI method are more robust than the LU-SGS/BJ method in stabilising solution. The LU-SGS/GS method also has faster and tighter convergence and lower memory consumption in comparison with the LU-SGS/BJ method.     

Constrained parameter-splitting perturbation method for the improved solutions to the nonlinear vibrations of Euler–Bernoulli cantilevers

In this paper, a new method named constrained parameter-splitting perturbation method for improving the solutions obtained from the parameter-splitting perturbation method is proposed for solving the problems in some extremal cases, such as the strongly nonlinear vibration of an Euler–Bernoulli cantilever. The proposed method takes the advantages of both the perturbation method and the harmonic balance method. The idea is that the solution obtained by the parameter-splitting perturbation method is substituted into the equation of motion and then the accumulative error of the equation is minimized for determining the unknown splitting parameters under the constraints constructed under the frame of harmonic balance method. The forced vibration of an oscillator with cubic geometric nonlinearity and inertia nonlinearity and the forced vibration of a planar microcantilever beam with a lumped tip mass are studied as examples to reveal the efficacy of the proposed method. The inspection of the steady-state response including its stability is conducted by means of comparing the frequency-response curves obtained by the proposed method with those obtained by the numerical continuation method and harmonic balance method, respectively, to show the efficacy and the advantages of the proposed method. Meanwhile, the nonlinear ordering effect on the solutions of the proposed method is also studied by comparing the results obtained by using different nonlinear orderings in the systems. In the last, we found through convergence examinations that it is necessary to have corrections to the erroneous solution which are obtained by harmonic balance method and Floquet theory in stability analysis.

     

An Euler solver for three-dimensional turbomachinery flows

A time-marching finite volume numerical procedure is presented for three-dimensional Euler analysis of turbomachinery flows. The proposed scheme is applied to the conservative form of the Euler equations written in general curvilinear co-ordinates. A simple but computationally efficient grid is constructed. Numerical solution results for three 3D turbine cascade flows have been presented and compared with available measurements as well as with another state-of-the-art 3D Euler analysis numerical solution in order to demonstrate the accuracy and computational efficiency of the analysis method. Also, the predicted results are compared with a 3D potential flow solver and comparison is made with the analytical solution. The proposed method is an accurate and reliable technique for solving the compressible flow equations in turbomachinery geometries.