频率域电磁法三维有限元正演线性方程组迭代算法

在三维频率域电磁法的正演模拟方法中,有限元方法具有计算精度高、适应性强的优点,近年来来得到了越来越多的关注.在正演过程中,主要的计算量集中在求解由偏微分方程组离散得到的线性方程组上,因此求解线性方程组关系着正演计算速度以及模拟精度.由于由有限元方法离散得到的复系数线性方程组条件数非常大,使用常规的迭代法和预条件很难收敛.目前大多数的研究工作采用直接解法,需要大量的计算机内存,限制了可求解问题的规模.本文研究了线性方程组的迭代解法,通过将复系数线性方程组转化为其实对称形式,构造分块对角预条件.在应用预条件的过程中,需要求解两个较小的实数方程,通过辅助空间解法求解.本文的算法适用于可控源电磁法和大地电磁法,对一系列的数值算例的模拟结果证明了迭代算法的效率,结果表明迭代算法可以在小于20次迭代内收敛,同时迭代次数与模型电阻率、问题规模和频率无关.     

三维频率域随钻电磁波测井数值模拟

在现有的普通计算机的内存和运算速度的条件下,三维频率域电磁波测井数值模拟非常困难,为了研究随钻仪器在复杂测井环境中的响应规律,从Maxwell电磁响应方程出发,针对大斜度井井眼和侵入剖面的几何特点,采用新的网格划分方法,并应用基于交错网格的有限差分得到了三维频率域电磁响应差分计算格式,采用改进的ICCG（不完全乔尔斯基共轭梯度）方法,对一维变带宽存储的大型复稀疏矩阵进行了求解,得到了随钻电磁波测井响应.计算结果表明：一维变带宽存储方法很好地解决了大型稀疏矩阵的存储问题;改进的ICCG方法得到的结果真实可靠并大大提高了求解效率;随钻电阻率曲线随着井斜角度增大会出现“极化角”,低阻侵入会使“极化角”弱化;相位电阻率曲线受围岩影响较小,受侵入影响较大.     

空间域位场延拓新方法研究

在空间域进行位场延拓,需要数值求解第一类Fredholm积分方程,由于所得方程组系数矩阵不是稀疏矩阵,求解该方程组需要的计算机内存大,计算量大,导致延拓算法在一般计算机上难以实现,阻碍了对空间域位场延拓方法的研究.在分析系数矩阵结构特征的基础上,本文证明了方程组系数矩阵是对称的分块Toeplitz型矩阵.利用系数矩阵的对称性和分块Toeplitz型矩阵与向量相乘的快速算法,解决了系数矩阵的存储和计算问题,使得空间域位场延拓成为可能,为研究新的位场延拓方法和分析延拓误差提供了一条新的途径.利用模型数据和实测资料,对空间域位场向上延拓、空间域积分迭代法向下延拓进行了检验,结果证实了空间域位场延拓的可行性和正确性.     

电性各向异性地层频率域电磁响应模拟

在现有计算机内存和运算速度条件下，电性各向异性地层中频率域电磁响应的数值模拟是一个难点. 为探索各向异性介质模型电磁响应的有效算法，同时为进一步研究各向异性地层测量响应的校正方法奠定基础，从Maxwell电磁响应方程出发，使用由三个彼此垂直的发射线圈和三个彼此垂直的接收线圈组成的线圈结构模型，应用交错网格有限差分法推导了三维频率域电磁响应的差分计算格式，采用预条件双共轭梯度法对所形成的复系数矩阵进行了迭代求解，从而得到电性各向异性地层模型下的线圈测量响应. 应用这些方法进行了多种条件下电磁响应的计算，计算结果表明，所形成的计算方法是准确可靠的，预条件双共轭梯度法的运算效率也较高；电性各向异性地层的电磁响应与地层倾角（或井斜角）、仪器测量方位、地层的水平电阻率和垂直电阻率有关；根据水平方向磁场强度分量可以确定地层界面；在电磁响应计算过程中，背景电导率的取值要尽量接近整个计算区域的平均电导率.     

利用直达P波测定地方震震源深度的探讨

震源深度是地震时空参数中的基本参数,是目前最难测准的参数之一。可以使用直达P波测定震源深度。要确保测定结果的精度,必须加密台网观测点,使台间距的大小与震源深度相当。利用四川省西昌和冕宁大桥水库两个遥测地震台网的直达P波资料,测定发生在其监控区域内137个地震的震源深度。结果发现,该区域地震的平均深度和深度下限存在随震级增大而增加的现象。其中有部分地震的震源深度,利用本研究方法测定的结果与使用遥测地震台网计算机定位程序输出的结果相差较大。作者认为,这部分地震使用计算机定位程序得出的震源深度有较大的误差。     

On the solution of transient free-surface flow problems in porous media by the finite element method

A numerical procedure is presented to deal with solution of transient free-surface flows in porous media. The governing boundary-value problem for the piezometric potential is solved by the finite element method. The initial-value problem which describes the transient motion of the free-surface is solved by the method of quasi-linearization. The numerical scheme has been applied to isotropic and anisotropic earth dam problem and also to a ditch drainage problem. Excellent agreements have been reached when compared with known solutions. This computational procedure is shown to be stable and suitable for this class of problems with the aid of a digital computer.     

Efficient mode superposition algorithm for seismic analysis of non-linear structures

A step-by-step integration method is proposed to compute within the framework of the conventional mode superposition technique the response of bilinear hysteretic structures subjected to earthquake ground motions. The method is computationally efficient because only a few modes are needed to obtain an accurate estimate of such a response, and because it does not require the use of excessively small time steps to avoid problems of accuracy or stability. It is developed on the basis that the non-linear terms in the equations of motion for non-linear systems may be considered as additional external forces, and the fact that by doing so such equations of motion can be interpreted as the equations of motion of an equivalent linear system, excited by a modified ground motion. These linear equations are then subjected to a conventional modal decomposition and transformed, as with linear systems, into a set of independent differential equations, each representing the system's response in one of its modes of vibration. To increase the efficiency of the method and account properly for the participation of higher modes, these independent equations are solved using the Nigam-Jennings technique in conjunction with the so-called mode acceleration method. The accuracy and efficiency of the method is verified by means of a comparative study with solutions obtained with a conventional direct integration method. In this comparative study, including only a few modes, the proposed method accurately predicts the seismic response of three two-dimensional frame structures, but requiring only, on an average, about 47 per cent less computer time than when the direct integration method is used.     

基于并行化直接解法的频率域可控源电磁三维正演

电磁法的三维数值模拟是一个对数值算法和计算机硬件要求都非常高的问题.对常用的微分类方法如有限单元法和有限差分法而言,求解最后所得的大型线性方程组是至关重要的一步,直接影响到正演算法的实用性.如何高效、稳定且准确地解线性方程长期以来一直是被探讨的问题.本文实现了基于线性系统直接求解技术的频率域可控源电磁(CSEM)三维正演.使用交错网格有限体积法(FV)来离散化关于二次电场的Helmholtz方程;使用直接解法取代传统的迭代解法来求解离散线性系统,即对系统矩阵进行完全LU分解,具体通过调用大规模并行矩阵直接求解器(MUMPS)来实现.基于理论模型做了一系列数值实验,首先证明了直接解法的高精度和稳定性,并考察了其内存需求、计算时间和并行可伸缩性等主要计算性能,最后检验了所开发的算法快速模拟多场源CSEM问题的能力以及对常规海洋和陆地CSEM模拟的有效性.     

电流线追踪电位电阻率层析成像方法初探

电阻率层析成像技术尽管已有了一些比较好的结果，但从国内外发表的文章可以看出，基本上采用的都是有限元方法，而电阻率层析成像的核心问题也就是雅可比矩阵的求取问题。有限元方法能够很好地实现该问题的求解，但需要的计算机内存及计算时间相当的大，为此我们类比地震学中走时射线追踪技术，开展了电流线追踪电位电阻率层析成像方法研究。     

地震叠前逆时偏移中的去噪与存储

地震叠前逆时偏移是当前公认的地震成像的有效途径,然而它面临着计算量甚巨,低频成像噪音以及存储量大等问题,因此,业内科研工作者对其研究乐此不疲.借助GPU/CPU协同计算可以有效解决计算量的难点,笔者已在另文中阐述,本文着重探讨成像噪音抑制以及存储问题.文中分析了叠前逆时偏移产生成像噪音的机制,据此提出在叠前地震资料中先对数据进行相位与振幅校正,进而在成像后运用拉普拉斯算子滤波法消除成像噪音,从而有效去除成像所产生的低频噪音;针对存储量,采用随机边界,用计算换存储,并借助GPU实现,节省了GPU与CPU之间的数据通讯,数值实验结果表明,采用随机边界方法的逆时偏移结果与直接存储波场的方法得到的结果差别甚小.     

A parallel finite–infinite element model for two-dimensional soil–structure interaction problems

A parallel soil–structure interaction (SSI) model is presented for applications on distributed computer systems. Substructring method is applied to the SSI system and a coupled finite–infinite element based parallel computer program is developed. In the SSI system, infinite elements are used to represent the soil which extends to infinity. In this case, a large finite element mesh is required to define the near field for reliable predictions. The resulting large-scale problems are solved on distributed computer systems in this study. The domain is represented by separated substructures and an interface. The number of substructures are determined by the available processors in the parallel platform. To avoid the formation of large interface equations, smaller interface equations are distributed to processors while substructure contributions are performed. This saves a lot of memory storage and computational effort. Direct solution techniques are used for the solution of interface and substructure equation systems. The program is investigated through some example problems. The example problems exposed the need for solving large-scale problems in order to reach better results. The results of the example problems demonstrated the benefits of the parallel SSI algorithm.     

A finite element-finite difference alternating direction algorithm for three-dimensional groundwater transport

While technically feasible, three-dimensional finite element groundwater transport simulation has not found widespread application because of the considerable computational burden inherent in the approach. An operator splitting algorithm which treats the horizontal plane using finite elements in the first step and the vertical dimension using finite differences in the second step provides considerable savings in both computer memory requirements and computational effort. The algorithm reduces the problem of an (N × M) array of nodes from one in which NM equations must be solved simultaneously, to one of solving N equations M times in the first step and M equations N times in the second step. Applications to type problems and field situations indicate that the method is robust and accurate.     

快速拟线性近似方法及三维频谱激电反演研究

Zhdanov提出的拟线性(QL)近似方法有效解决了积分方程计算速度慢、占用内存多的缺陷,但因为三维频谱激电(SIP)换源频繁,每次换源都需要重新计算剖分单元的并矢格林函数和一次场,所以用该方法模拟三维SIP的速度仍然缓慢.本文根据一次场及格林函数的空间对称性,提出了一种适用于多源电磁法的快速QL近似正演方法.在此基础上,实现了Cole-Cole参数范围约束的三维SIP共轭梯度反演方法.理论试算结果表明:快速QL近似方法适用于三维SIP正演模拟,计算速度较QL近似方法有了很大提高;Cole-Cole参数范围约束的三维SIP共轭梯度反演方法对零频电阻率、极化率等参数的反演效果良好,将异常区域剖分成2000块时迭代一次只需约0.2s的时间,占用内存大约为140 MB.     

用改进的光滑约束最小二乘正交分解法实现电阻率三维反演

对三维电阻率反演问题进行了深入研究,提供了一种利用地表观测数据实现三维反演的实用算法.该方法应用有限差分求正演解,并通过对粗糙度矩阵元素进行适当改进,使之适用于各种情况下粗糙度矩阵的求取,进而建立在模型的总粗糙度极小条件下的反演方程.对反演方程采用收敛速度快且稳定的最小二乘正交分解（LSQR）法进行迭代求解,在迭代求解过程中只需利用偏导数矩阵和其转置矩阵乘以一个向量的结果,回避了直接求偏导数矩阵的繁琐计算,节省了内存,加快了反演的计算速度.不同的计算实例表明上述方法是求解大规模三维电阻率反演问题的有效方法.     

Relation between the extreme angular and areal distortion in cartographic representation

Summary The paper presents the method of deriving representations, optimal from the point of view of global characteristics, for areal and extreme angular distortion (3), (7), if one is optional and the other characteristic is minimized at the same time. If a special minimization method (Section 5) is used for computer processing, the problem can be solved in the region of representation equations in the form of polynomials of the 3rd or 5th degree, while assuming symmetry relative to the equator and the central meridian. Thus, the method enables both the fundamental global characteristics to be regulated freely. Their dependence is illustrated in graphs (Figs 2 and 6) depending on the selected examples of representations of the whole globe. The latter shows that method is a contribution for cartographic practice as well.     

Hybrid seismic modeling based on discrete-wave number and finite-difference methods

Any calculation of seismic wave propagation comprising the seismic source, the travel path, and the receiver site in a single finite-difference (FD) model requires a considerable amount of computer time and memory. Moreover, the methods currently available for including point sources in the 2D FD calculations are far-field approximations only. Therefore we have developed a new hybrid method for treating the seismic wave fields at localized 2D near-surface structures embedded in a 1D background medium, and excited by a point source. The source radiation and propagation in the background model is solved by the discrete-wave number (DW) method, while the propagation in the local 2D structure is calculated by the FD method. The coupling between the two sets of calculations is performed on a rectangular excitation box surrounding the local structure. We show the usefulness of the method in ground-motion studies where both near-field source effects and local site effects are important. Technical problems connected with the inconsistency between the 3D source radiation and the 2D FD calculation are minor for the relatively distant in-plane point explosive sources, but are more serious for the in-plane dislocation sources.     

可控源音频大地电磁数据的反演方法

从反演方程、构造目标函数和求解三方面对用于可控源音频大地电磁法(CSAMT)的实用反演方法中的四种进行了描述．水平层状地层CSAMT法资料的直接反演法首次尝试了一维空间的全资料CSAMT反演，效果较好，但该方法尚难应用于2D、3D复杂介质中；奥克姆反演方法既考虑了横向的光滑函数，又考虑了纵向的光滑函数，得到比较光滑的横向、纵向变化的背景电性结果，但有可能把一些小构造光滑掉．快速松驰反演算法和共轭梯度算法由于计算速度快，占内存少而被用于三维反演中，二者相比，快速松驰算法在求解雅可比矩阵时只做一次正演计算，在更新模型时解小型方程组，所以在速度上更胜一筹．在后三种算法中，由于复杂电性结构无解析解，正演计算都采用数值计算．数值计算的可靠性、速度影响着反演算法的有效性，这方面的研究也将是2D、3D复杂电性结构反演的研究方向之一.     

稀疏存储的显式有限元三角网格地震波数值模拟及其PML吸收边界条件

有限元法是复杂介质地震模拟的有力工具,它能比较客观地反映地震波的传播,比较细致地再现地震图像.但是,为了获得较精确的结果,有限元法模拟地震波的传播需要的网格点数多,具有计算量大和消耗内存多的缺点.针对上述缺点,本文对刚度矩阵采用压缩存储行(CSR)格式,以减少计算量并节省内存;采用集中质量矩阵得到对角的质量矩阵以提高有限元法(显式有限元)的计算效率;时间离散采用保能量的Newmark算法以提高有限元法的计算精度;采用变分形式(弱形式)的PML吸收边界条件对人工截断边界进行处理.通过与高精度的数值方法--谱元法的数值试验的对比表明,上述方法的引入可使有限元法在计算精度和计算效率方面均可取得比较显著的改进.为了获得相当的计算精度,相比于7阶谱元法,显式有限元法需要更精细的网格.然而,显式有限元法的计算速度比前者快近2倍,而内存需求仅为谱元法的1/4~1/6.     

模拟震源动力学破裂过程的数值方法研究

为模拟震源的动力学破裂过程,本文讨论一种求解剪切裂纹动力学扩展问题的有限差分法。在研究二维反平面破裂的基础上,我们把所用方法推广到三维问题。研究了许多简单震源模型,通过对比同一问题的解析解和其它数值解,对方法的正确性和计算精度进行了检验,结果表明我们的数值方法是可行的。最后,作为初步应用的例子,我们研究了非均匀断层的自发破裂问题。     

Application of Biot's poroelasticity to some soil dynamics problems in civil engineering

This review type of paper shows how the poroelastodynamic theory of Biot can be applied to some soil dynamics problems encountered in transportation engineering, which have been solved by the present authors. These problems involve rigid walls retaining poroelastic soil and subjected to harmonic seismic waves and moving loads on poroelastic soil. Both classes of problems involve a soil layer over bedrock, are of the plane strain type and are solved analytically by two methods: a direct (almost exact and exact for the above two classes of problems) method and an approximate method. The effects of shear modulus, porosity, permeability and hysteretic damping of the soil medium as well as the seismic frequency for retaining walls and velocity for moving loads on the dynamic response are numerically evaluated in order to assess their relative importance on that response.