On the computation and approximation of ultra-high-degree spherical harmonic series

Spherical harmonic series, commonly used to represent the Earth’s gravitational field, are now routinely expanded to ultra-high degree (> 2,000), where the computations of the associated Legendre functions exhibit extremely large ranges (thousands of orders) of magnitudes with varying latitude. We show that in the degree-and-order domain, (ℓ,m), of these functions (with full ortho-normalization), their rather stable oscillatory behavior is distinctly separated from a region of very strong attenuation by a simple linear relationship: , where θ is the polar angle. Derivatives and integrals of associated Legendre functions have these same characteristics. This leads to an operational approach to the computation of spherical harmonic series, including derivatives and integrals of such series, that neglects the numerically insignificant functions on the basis of the above empirical relationship and obviates any concern about their broad range of magnitudes in the recursion formulas that are used to compute them. Tests with a simulated gravitational field show that the errors in so doing can be made less than the data noise at all latitudes and up to expansion degree of at least 10,800. Neglecting numerically insignificant terms in the spherical harmonic series also offers a computational savings of at least one third.     

Numerical study on attenuation of bubble pulse through tuning the air-gun array with the particle swarm optimization method

A numerical method for improving the quality of the far-field signal generated by a marine seismic air-gun array is presented, in which particle swarm optimization is used in conjunction with the theory of the oscillating spherical bubble to tune the array. Two sets of objective functions, including the primary-to-bubble ratio and the variance of the normalized amplitude spectrum, are built to screen the array. With this method, attempts are made to improve the far-field signals generated by four typical arrays, including the planar array, spatial array, irregular array and delayed array. It is concluded that the far-field signal can be largely improved just through the simple adjustment of the chamber volume, the firing depth, the horizontal location or the firing time of each individual gun in the array. We believe that this method will be useful in the air-gun array design and helpful to find a desirable array from a population of air-gun array distributions for a brief time.     

动态映射函数对GPS基线解算质量的影响

介绍尼尔映射函数(NMF)、维也纳映射函数(VMF1)和全球映射函数(GMF)3种常用映射函数,讨论评估GPS基线质量的2个指标--基线重复性和平均值的标准偏差。根据中国境内IGS站的观测数据,基于GAMIT软件的数据处理结果,研究3种映射函数对GPS基线重复性和平均值标准偏差的影响。结果表明,在进行GPS基线处理时,动态映射函数的解算精度比经验映射函数要高,其中VMF1模型在解算基线重复性时结果更好,而GMF模型在计算平均值的标准偏差时效果更佳。     

基于C2阶连续函数的广义Hoek-Brown强度准则屈服面与塑性势面棱角圆化方法

Hoek-Brown强度准则的屈服面与塑性势面在棱角处导数无定义,具有数值奇异性,采用圆角进行光滑过渡只能满足一阶导数连续性,而二阶导数仍然无定义,使得棱边上一致切线模量矩阵无法正确计算,导致有限元总体平衡方程组Newton-Raphson隐式迭代二阶收敛性丧失。提出基于C2阶连续函数的广义Hoek-Brown准则屈服面与塑性势面棱角圆化方法,使得棱角处函数曲面二阶连续可导,棱边上一致切线模量矩阵可精确计算。基于ABAQUS数值开发平台,采用FORTRAN语言编制Hoek-Brown准则理想弹塑性UMAT用户子程序,通过数值算例验证所提方法的正确性。     

Theoretical models and recorded response in the estimation of cumulative seismic damage on non‐linear structures

A damage‐function model is proposed for the inelastic response of structures in terms of initial damage and of displacement amplitudes and secant stiffnesses of response cycles. The model is used to obtain an analytical closed‐form solution for the probability distribution of cumulative damage after an earthquake ground motion given the distribution prior to such excitation and information on the inelastic structural response. The formulation is applied to a reinforced concrete frame and the results show the capabilities of the method to yield updated distributions of damage. Copyright © 2000 John Wiley & Sons, Ltd.