有限弹性变形地壳中的地震波

地壳由半无限大的基岩上一层厚度为H^-的表土层组成，入射地震波为垂直的SH波，产生水平地面运动。当浅源大地震发生时，在极震区以外行波传播产生地面运动将使地壳介质有非线性的有限弹性变形。用小参数摄动法使非线性控制方程为线性化的小参数各阶控制方程，得出头两阶线性控制方程的解析解。     

求解非线性阻尼作用地震破裂过程的一种方法

把所有的参数都无量纲化,然后在证明了无量纲的非线性项的阻尼(与运动速度立方正比)系数B<1为小参数之后,把待求解的振幅函数展成B的收敛幂级数,再代入动力方程或方程组,得到一系列线性化的逼近方程或方程组,用以动坐标表示的广义富氏级数代入各级逼近方程或方程组,对Ⅰ、Ⅱ、Ⅲ型破裂的弹性断裂动力学问题和Voigt模型的粘弹性断裂动力学问题,都化为含时间自变量的Mathieu方程或方程组,用WKBJ方法得出其在稳定区域的渐近解。还用此法解了一个三维问题。     

具有非线性阻尼的圬工粗短柱在水平地震力作用下横截面上的压应力重分布(英文)

当合力作用在横截面核心之外时, 此截面的拉应力对于抗拉强度很低的圬工材料是不能承受的, 故应考虑压应力的重新分布。本文采用能反映激发和衰减的非线性Ｒａｙ／ｅｉｇｈ阻尼, 使控制方程无量纲化, 设振幅函数满足所有边界条件而简化控制方程为非齐次的Ｖａｍ ｄｅｒ Ｐｏｌ 方程, 得出其头两次渐近解, 再计算压应力重分布     

具有非线性阻尼的圬工粗短柱在水平地震力作用下横截面上的压应力重 …

当合力作用在横截面核心之外时,此截面的拉应力对于抗拉强度很低的圬工是不能承受的 ,故应考虑压应力的重新分布。本文采用能反映激发和衰减的非线性Ｒａｙ／ｅｉｇｈ阻尼,使控制方程无量纲化,设振幅函数满足所有边界条件而简化控制方程非齐次的ＶａｍｄｅｒＰｏｌ方程,得了其头两次渐近解,再计算压应力重分布。     

冲击环境下非线性岩土与相邻多个结构的相互作用

本文讨论了冲击波作用下非线性弹性岩土与相邻多个结构的相互作用问题。从文[1]给出的各阶渐近方程出发,利用复变函数构造了逼近这个问题的渐近解的完备的函数序列,将问题归结为对一组线性代数方程的求解。作为例子,文中计算了相邻双圆柱结构在冲击波作用下动力响应的位移和应力的结果。     

非线性多自由度结构地震反应的共振特性与突变分析

由渐近解法，在主坐标中给出了非线性多自由度结构地震反应的渐近解，利用渐近解计算了共振区振幅-频率响应的一系列数值结果，进而分析了系统参数对共振区反应特性的影响及突变性质。地震反应中表现出来的突变性、共振畸变、“路径”效应等复杂的非线性特性，对结构系统的动力特性有重要影响。     

在冲击波下非线性岩土与圆柱结构的相互作用

本文研究了非线性弹性岩土中冲击波与圆柱结构动力相互作用问题。引用一种多参数的非线性弹性岩土模型,应用摄动法把系统的基本方程按入射波的小参数展开为各阶渐近线性方程,并引入位移函数得到了解耦的波动方程。文中应用谐振波叠加法,即把对瞬态波情况的求解转变为对各谐振分量情况的求解和叠加,给出了与拉氏变换法一致的位移和应力结果。     

加肋圆柱曲板房顶的非线性垂直地震响应

因纵横向加肋密度不同, 故把题述房顶当做各向异性弹性体［１］ , 则问题的控制方程组为两个四阶偏微分方程（ 有一个是非线性） , 代满足边界条件的振型函数入内并用富氏级数的正交性并结合逐次逼近法, 化控制方程组为一个非线性ｖａｎ ｄｅｒ Ｐｏｌ 方程, 得出问题的解析解, 并且讨论了解的跳跃即动力稳定丧失出现的情况。     

第一类算子方程的吉洪诺夫正则解的收敛性和渐近阶估计

本文对于第一类算子方程当右端近似给定和当算子与右端皆近似分别给定时,讨论如何确定正则参数。并给出吉洪诺夫正则解的收敛性和渐近阶估计。     

粘弹性拉张型地震破裂过程的解析解

本文应用包括地壳破裂发震过程中具有激发及衰减的非线性Rayleigh阻尼,用Voigt粘弹性模型表示地壳,它能更全面地反映地壳介质分子之间的内摩擦造成的粘滞性阻尼,在数学方法上,用解非线性问题解析法的摄动理论结合动坐标的富氏级数,把问题的非线性控制方程组化为各阶线性化的控制方程组后,再简化为标准的Mathieu方程构成的耦连方程组,再用WKBJ方法,给出其在稳定区域的近似解,从而得出了问题的解析解。     

利用阵列感应测井仪信号反演非均匀分布地层电导率

基于非线性积分方程,利用井轴上阵列感应成像测井仪（AIT）的单频测量信号,将变形玻恩（Bom）迭代方法用于地球物理测井反演,用正则化方法求出地层电导率分布。文中首先给出对于未知电导率分布的非线性积分方程,然后用玻恩近似使非线性积分方程线性化,求出电导率分布。在迭代过程中,数值模式匹配法用于求解阵列感应的正演问题。数值计算表明,用井轴上 AIT 的单频测量信号得到的成像结果与真电导率剖面分布吻合较好。     

Nonlinear waves on a coupled density front

Abstract

It is shown that the inclusion of the nonlinear terms in the equations of motion of a coupled density front of zero potential vorticity results in wave solutions which merely propagate with time. The linear theory, on the other hand, predicts an exponential temporal growth. The nonlinear equation admits steady solutions representing standing waves whereas if the nonlinear terms are omitted no steady solutions exist. The general initial value problem is difficult to solve numerically since the linear problem is ill posed.

In addition we prove that the general similarity solution of the nonlinear equation tends to zero for large times, at any point in space, regardless of the initial condition.     

用变分玻恩迭代方法重建二维非均匀介质结构

提出了用于二维轴对称非均匀介质结构的反演和成像的一种新的反演迭代方法──变分玻恩迭代方法（ＶＢＩＭ）．首先利用玻恩近似将非线性积分方程线性化,然后应用变分方法导出用于反演的电场积分方程．正演数据则利用高效的数值模式匹配方法获得．数值结果表明,ＶＢＩＭ与ＢＩＭ相比,其收敛速度、成像质量等均得到较大的改善。     

电极型电阻率测井反演算法研究

低频电极型测井仪的响应总是受井眼、围岩和侵入的影响，而传统的测井综合解释无法完全有效地校正这些影响．本文将收敛速度快而效果好的变形波恩迭代（DBIM）方法用于处理轴对称二维非均匀介质分布的电导率反演问题．在每次迭代过程中采用了快速的半解析的模式匹配的正演方法，半解析地表达出反演中所需计算的格林函数的偏导数；并在此基础上半解析地求出了反演的非线性积分方程中的积分运算，大大提高了反演速度和精度．     

Use of Picard and Newton iteration for solving nonlinear ground water flow equations

This study examines the use of Picard and Newton iteration to solve the nonlinear, saturated ground water flow equation. Here, a simple three-node problem is used to demonstrate the convergence difficulties that can arise when solving the nonlinear, saturated ground water flow equation in both homogeneous and heterogeneous systems with and without nonlinear boundary conditions. For these cases, the characteristic types of convergence patterns are examined. Viewing these convergence patterns as orbits of an attractor in a dynamical system provides further insight. It is shown that the nonlinearity that arises from nonlinear head-dependent boundary conditions can cause more convergence difficulties than the nonlinearity that arises from flow in an unconfined aquifer. Furthermore, the effects of damping on both convergence and convergence rate are investigated. It is shown that no single strategy is effective for all problems and how understanding pitfalls and merits of several methods can be helpful in overcoming convergence difficulties. Results show that Picard iterations can be a simple and effective method for the solution of nonlinear, saturated ground water flow problems.     

基于地磁场的自主导航研究

研究了基于地磁场的自主导航,建立了以卫星轨道动力学方程为基础的系统状态方程,并详细推导了以地磁场大小和地磁场矢量为观测量时的观测方程.由于系统的非线性,传统的卡尔曼滤波无能为力,为了解决系统的非线性问题把常用来解决非线性问题的EKF引入到基于地磁场的自主导航系统中.最后用Matlab对基于地磁场的自主导航系统进行了仿真,仿真结果表明EKF有很好的收敛性和稳定性,以地磁场矢量为观测量的导航精度要远高于以地磁场大小为观测量的导航精度.     

双侧向电阻率测井反演算法研究

双侧向测井仪因其激励源的复杂性给其测量数据的反演带来了极大的困难．本文利用变形玻恩（Born）迭代法对轴对称二维非均匀介质中双侧向测井仪的位场数据进行了严格反演．在每一次迭代过程中采用了高效数值模式匹配的正演计算方法,它可半解析地求出反演中所需计算的格林（Green）函数及其偏导数,同时利用所推导出的位场非线性积分方程中积分运算的半解析形式,大大提高了计算效率和反演质量．通过反演实例表明,反演结果与电导率原状剖面吻合较好,并且受噪声影响较小．     

Solution of the nonlinear transport equation using modified Picard iteration

The transport and fate of reactive chemicals in groundwater is governed by equations which are often difficult to solve due to the nonlinear relationship between the solute concentrations for the liquid and solid phases. The nonlinearity may cause mass balance errors during the numerical simulation in addition to numerical errors for linear transport system. We have generalized the modified Picard iteration algorithm of Celia et al.⁵ for unsaturated flow to solve the nonlinear transport equation. Written in a ‘mixed-form’ formulation, the total solute concentration is expanded in a Taylor series with respect to the solution concentration to linearize the transport equation, which is then solved with a conventional finite element method. Numerical results of this mixed-form algorithm are compared with those obtained with the concentration-based scheme using conventional Picard iteration. In general, the new solver resulted in negligible mass balance errors (< ∥10⁻⁸∥%) and required less computational time than the conventional iteration scheme for the test examples, including transport involving highly nonlinear adsorption under steady-state as well as transient flow conditions. In contrast, mass balance errors resulting from the conventional Picard iteration method were higher than 10% for some highly nonlinear problems. Application of the modified Picard iteration scheme to solve the nonlinear transport equation may greatly reduce the mass balance errors and increase computational efficiency.     

Potential Symmetry Properties of a Family of Equations Occuring in Ice Sheet Dynamics

In this paper we derive some similarity solutions of a nonlinear equation associated with a free boundary problem arising in the shallow-water approximation in glaciology. In addition we present a classical potential symmetry analysis of this second-order nonlinear degenerate parabolic equation related to non-Newtonian ice sheet dynamics in the isothermal case. After obtaining a general result connecting the thickness function of the ice sheet and the solution of the nonlinear equation (without any unilateral formulation), a particular example of a similarity solution to a problem formulated with Cauchy boundary conditions is described. This allows us to obtain several qualitative properties on the free moving boundary in the presence of an accumulation-ablation function with realistic physical properties.     

Nonlinear aspects of phreatic water filtration

Horizontal gravity filtration of groundwater in soil is considered. Under Boussinesq approximation, the problem is reduced to a one-dimensional nonlinear parabolic equation in phreatic water level. The problem of linearizing the original equation is discussed. The comparison of gravity-filtration problem solutions in the nonlinear and linearized formulations shows considerable discrepancies to exist between the solutions, especially, for boundary problems with mixed boundary conditions, when the value of the function is not fixed on the right boundary. An analytical solution is obtained for steady-state flow from a water body into the soil with subsequent leakage into underlying beds. Two regimes are shown to exist: one with an infinite exponential tail, and another in the form of a finite groundwater mound. A new approach is proposed to the linearization problem—quasilinearization with the use of the Burgers equation.