基于精细时程积分的u-p格式饱和两相介质动力问题的显-显式时域算法

时域算法的计算效率是求解实际非线性或大量自由度等工程问题最重要的影响因素,而显式算法是提高计算效率最有效的手段。针对u-p格式饱和两相介质动力方程,实现显-显式算法的必要前提是质量矩阵M和流体压缩矩阵S的对角化处理,且整体算法的计算精度和稳定性同样需要权衡。基于对角化的M和S矩阵,将高精度和稳定性的精细时程积分法应用于求解液相方程,并结合求解固、液两相动力方程的中心差分法,提出了一种新的显-显式的高效时域算法。理论分析和数值试验结果均说明提出方法具有良好的稳定性和较高的计算精度。     

广义Tanh函数法中Riccati方程和sine-Gordon方程的新解及其新应用

非线性偏微分方程的显式解析解,特别是行波解,蕴含了方程的丰富信息,对于描述各种现象的发展规律起着至关重要的作用．本文尝试构造 KdV 方程多种形式的新显式行波解．首先,利用试探函数法和 Matlab计算给出了 Riccat 方程的许多新显式解析解．其次,运用广义 Tanh 函数法以及 Riccati 方程的新解得到了 sine-Gordon 方程的许多新显式解析解．最后,作为新的应用,把三角函数法结合 sine-Gordon 方程的新显式解析解并利用简化的变换形式进一步找到了 KdV 方程的许多新显式行波解．这些结果推广和补充了以往的相关研究成果,特别地,这些方法和新的结果可以用于求解许多非线性偏微分方程的新显式行波解．     

关于两端固定轴向运动梁的横向振动

研究两端固定轴向运动梁的横向振动。导出了系统的频率方程，提出了求解该非显式表达的非线性超越方程的数值算法，得到了固有频率和模态函数。     

非线性振动分析的重心插值配点法

将计算区间采用第二类Chebyshev点离散,利用数值稳定性好、计算精度高的重心Lagrange插值近似未知函数,建立未知函数各阶导数在计算节点上的微分矩阵.利用重心Lagrange插值公式离散非线性振动微分方程为非线性代数方程,采用Newton法求解非线性代数方程.计算得到振动位移后,采用微分矩阵和重心Lagange插值计算非线性振动的速度、加速度和振动周期.采用重心插值配点法计算了Duffing型非线性振动方程和非线性单摆振动方程.数值算例表明本文方法具有计算公式简单、程序实施方便和计算精度高的优点.     

饱和两相介质近场波动问题的一种时域全显式数值计算方法

基于 u -p形式的饱和两相介质弹性波动方程,开展了饱和两相介质近场波动问题时域显式数值计算方法的研究。通过对波动方程中的质量矩阵和孔隙流体压缩矩阵进行对角化处理,消除了方程中的动力耦联,实现了波动方程的解耦。分别应用中心差分法和Newmark常平均加速度法求解固相位移和速度,基于向后差分法求解孔隙流体压力,推导得到了饱和两相介质动力响应的时域显式逐步积分的计算列式,建立了饱和两相介质近场波动问题的一种新的时域全显式数值计算方法。进行了该文方法中矩阵对角化合理性的验证。将该方法的数值解与相应的解析解进行对比,二者符合良好,验证了该方法的正确性。将该文建立的时域数值计算方法与透射人工边界方法相结合,应用于饱和两相介质的近场波动问题,进行了饱和土场地地震响应的计算研究,计算结果符合弹性波动理论的基本规律,表明该方法对于饱和两相介质近场波动问题时域计算求解的适用性。基于该方法中时域递推计算格式的传递矩阵,进行了该方法稳定性特性的研究。该文建立的数值计算方法具有时域全显式算法的基本特征。方法中对动力响应的全部分量均采用递推和迭代的模式进行求解,避免了求解耦联的动力方程组。该方法具有较高的计算效率,是进行饱和两相介质近场波动问题时域计算求解的一种有效的算法。     

焊管冷弯回弹曲线方程构建及工艺优化

目的 研究316L奥氏体不锈钢板材JCOE弯曲卸载回弹和应力分布特征,对预弯曲量实施相应补偿,以提高弯曲成形精度。方法 基于弹塑性变形理论,利用有限元模拟研究各关键成形参数对板材弯曲卸载回弹的影响规律,将影响指标线性/非线性拟合叠加,构建回弹曲线方程。结果 板料回弹量与上模下压量、下模开口量呈线性关系,与上模下行速度、摩擦因数呈指数关系。结论 优化后的最佳冷弯成形工艺参数如下:上模下压量为12 mm,下模开口量为150 mm,上模下行速度为4 mm/s,摩擦因数为0.15。对于径壁比值≤10的板料成形,理论计算得到的弯曲回弹量与实测平均值吻合较好,构建的模型可以为实际生产奠定理论基础。     

一种半隐式的气动弹性时域求解方法

针对结构小幅运动、流动强非线性的气动弹性问题,基于Adams 隐式方法发展了一种半隐式的线性多步法(SILMS)。将隐式的广义气动力项采用显式插值求解,结构项仍采用隐式格式计算,该方法融合了显式方法耦合简单和隐式方法稳定性好的特点。结构运动采用模态坐标描述,流动方程应用计算流体力学(CFD)技术求解,二者采用松耦合方法进行时域推进。算例计算了一个标准气动弹性模型(Isogai wing)的颤振结果,并与几种经典的时域模拟方法进行了比较,证明该方法具有效率高、稳定性好、精度高的优点。     

多点激励下隔震桥梁非线性随机振动的时域显式迭代模拟法

由于地震动的随机性与空间变异性、隔震支座的非线性特征及其安装数目少的特点,隔震桥梁地震响应分析是一典型的多输入局部非线性随机振动问题。采用Bouc-Wen模型描述隔震支座的非线性恢复力,建立多点地震激励下隔震桥梁结构的运动方程,并将得到的非线性运动方程改写为关于状态变量的拟线性方程,基于精细时程积分法和龙格-库塔法构造拟线性方程及隔震支座滞回位移方程的时域显式降维迭代求解格式。再以此高效算法为基础,应用随机模拟法计算多点激励下隔震桥梁随机地震响应的统计量。数值算例表明,所建立的方法计算速度快、精度高,可用于实际桥梁结构的抗震性能验算。     

大规模非线性系统随机振动显式迭代Monte Carlo模拟法

探讨了非平稳随机激励下大规模非线性系统随机振动Monte Carlo模拟法。引入等效激励的概念,把非线性系统动力方程写成状态方程的形式,并采用精细积分法对状态方程进行数值求解,导出非线性系统振动分析的显式迭代法。基于所导出的显式迭代公式,可以有效提高单次确定性非线性振动分析的计算效率,从而可以通过Monte Carlo模拟获得非平稳随机激励下非线性系统随机响应的统计信息,同时还可以获取非线性系统随机响应的演化概率密度函数。数值算例表明,所提出的方法迭代收敛速度快,计算精度高,适用于求解大规模非线性系统随机振动问题。     

任意阶次多项式最小二乘拟合不确定度计算方法与最佳拟合阶次分析

阐述了一种采用矩阵方程表示的任意阶次多项式最小二乘拟合的不确定度计算方法,同时通过仿真研究了该方法的特性。通过该方法,可以得到拟合不确定度最小的拟合阶次,也就是最佳的拟合阶次。仿真得到的最佳拟合阶次与仿真模型的原函数阶次都为5阶,从而验证了该方法的有效性。     

最优微分方程本征值集的唯一性

进一步讨论了最优微分方程，证明了存在本征解的最优微分方程的本征值集与本征解集是唯一的。     

A new method for the numerical solution of integral equation approximations

A new numerical technique for solving the Ornstein-Zernike equation is described. It is particularly useful in solving the Ornstein-Zernike equation for approximations and pair potentials (such as the Percus-Yevick and mean spherical approximations for finite ranged potentials) which imply a finiteranged direct correlation function since for such approximations the numerical technique is essentially exact. The only approximation involved in such cases is the discretization of direct and total correlation functions over the finite range on which the direct correlation function is nonzero. Thus, the new method avoids truncation of the total correlation function and should permit the critical point and spinodal curve to be mapped out with greater accuracy than is permitted by existing methods. Preliminary explorations on the stability and accuracy of the method are described.Paper presented at the Tenth Symposium on Thermophysical Properties, June 20–23, 1988, Gaithersburg, Maryland, U.S.A.     

Functional equation for the fading correction of imaging plates

Several experiments were performed to investigate the effects of temperature on the fading of the commercial imaging plate, IP (BAS-UR). The fading characteristics were measured under controlled temperatures between 0°C and 60°C after irradiation with ²³⁸U alpha-ray and ⁶⁰Co gamma-ray sources. Applying Arrhenius' equation to the experimental results, we calculated a universal functional equation that includes two variables: elapsed time (t) and temperature (K). The photo-stimulated luminescence (PSL) calculated by this equation showed good agreement with the results of our experimental ones between 0°C and 40°C. From this equation, the activation energies of the fading are estimated to be about 0.9 eV for both ²³⁸U alpha-ray and ⁶⁰Co gamma-ray sources.     

A homotopy analysis method for the nonlinear partial differential equations arising in engineering

In this article, we have established the homotopy analysis method (HAM) for solving a few partial differential equations arising in engineering. This technique provides the solutions in rapid convergence series with computable terms for the problems with high degree of nonlinear terms appearing in the governing differential equations. The convergence analysis of the proposed method is also discussed. Finally, we have given some illustrative examples to demonstrate the validity and applicability of the proposed method.     

Cubic equations of state for transport properties: An equation for the thermal conductivity of oxygen

A scheme for the development of equations for the transport properties in terms of pressure and temperature, so-called transport equations of state, is presented. The surfaces of transport properties and density as a function of pressure and temperature reveal similarities, which become even more evident when the residual transport property as a function of pressure and temperature is considered. Even the spinodals of transport and thermal properties coincide in the p, T plane, as can be shown mathematically and as was already empirically found for water and oxygen. Based on these similarities a cubic transport equation of state is evaluated for the residual thermal conductivity of oxygen. The new equation is only a little less accurate than the already established virial transport equation of state for oxygen. It is, however, much simpler and needs only a few parameters. The accuracy is still good enough for practical applications. The results demonstrate that cubic equations of state can describe transport properties and are a basis for generalized estimation methods for the transport properties of fluids.     

Numerical treatment of a skew-derivative problem for the Laplace equation in the exterior of an open arc

The skew-derivative problem for harmonic functions in the exterior of an open arc in a plane is considered. This problem models the electric current in a semiconductor film from an electrode of arbitrary shape in the presence of a magnetic field. A numerical method for solving the problem is proposed. The method is based on a boundary-integral-equation approach. The proposed numerical method is tested. The numerical simulation is presented for different values of the parameters and different shapes of the electrode. Physical effects found in numerical experiments are discussed.     

DGH方程Cauchy问题解的解析性

Cauchy问题是偏微分方程研究中的重要问题之一，而初值的性质在很大程度上决定了偏微分方程解的性质。本文研究了DGH方程的Cauchy问题在初值解析的情形下解的性质：我们在一个合适度量的Banach空间中利用压缩的思想证明，DGH方程Cauchy问题初值解析时，其解关于空间变量全局解析，而关于时间变量局部解析。     

A Fundamental Equation for Calculation of the Thermodynamic Properties of Ethanol

A formulation for the thermodynamic properties of ethanol (C₂H₅OH) in the liquid, vapor, and saturation states is presented. The formulation is valid for single-phase and saturation states from 250 to 650K at pressures up to 280MPa. The formulation includes a fundamental equation and ancillary functions for the estimation of saturation properties. The experimental data used to determine the fundamental equation include pressure-density-temperature, ideal gas heat capacity, speed of sound, and vapor pressure. Saturation values computed from the ancillary functions were used to ensure thermodynamic consistency at the vapor-liquid phase boundary. Comparisons between experimental data and values computed using the fundamental equation are given to verify the uncertainties in the calculated properties. The formulation presented may be used to compute densities to within ±0.2%, heat capacities to within ±3%, and speed of sound to within ±1%. Saturation values of the vapor pressure and saturation densities are represented to within ±0.5%, except near the critical point.     

论Helmholtz方程的一类边界积分方程的合理性

本文导出了Helmholtz 方程超定边值问题有解的一个充要条件,和用非解析开拓法证明了文[1]中的Helmholtz 方程在外域中的解的边界积分表示式的合理性,并将此类边界积分表示式推广用于带空洞的有限域。这样就比较严密而又浅近地证明了基于该表示式建立起来的间接变量和直接变量边界积分方程的合理性。