A finite analytic method for solving the 2-D time-dependent advection–diffusion equation with time-invariant coefficients

Difficulty in solving the transient advection–diffusion equation (ADE) stems from the relationship between the advection derivatives and the time derivative. For a solution method to be viable, it must account for this relationship by being accurate in both space and time. This research presents a unique method for solving the time-dependent ADE that does not discretize the derivative terms but rather solves the equation analytically in the space–time domain. The method is computationally efficient and numerically accurate and addresses the common limitations of numerical dispersion and spurious oscillations that can be prevalent in other solution methods. The method is based on the improved finite analytic (IFA) solution method [Lowry TS, Li S-G. A characteristic based finite analytic method for solving the two-dimensional steady-state advection–diffusion equation. Water Resour Res 38 (7), 10.1029/2001WR000518] in space coupled with a Laplace transformation in time. In this way, the method has no Courant condition and maintains accuracy in space and time, performing well even at high Peclet numbers. The method is compared to a hybrid method of characteristics, a random walk particle tracking method, and an Eulerian–Lagrangian Localized Adjoint Method using various degrees of flow-field heterogeneity across multiple Peclet numbers. Results show the IFALT method to be computationally more efficient while producing similar or better accuracy than the other methods.     

流体饱和多孔隙介质二维弹性波方程正演模拟的小波有限元法

本文将小波有限元法引入到流体饱和多孔隙介质二维波动方程的正演模拟中,以二维Daubechies小波的尺度函数代替多项式函数作为插值函数,构造二维张量积小波单元.引入一类特征函数解决了Daubechies小波没有显式解析表达式所带来的基函数积分值计算问题,并推导出计算分数节点上Daubechies小波函数值的递推公式,从而构造出由小波系数空间到波场位移空间的快速小波变换.数值模拟结果表明该方法是有效的.     

A finite element solution for the fractional advection–dispersion equation

The fractional advection–dispersion equation (FADE) known as its non-local dispersion, has been proven to be a promising tool to simulate anomalous solute transport in groundwater. We present an unconditionally stable finite element (FEM) approach to solve the one-dimensional FADE based on the Caputo definition of the fractional derivative with considering its singularity at the boundaries. The stability and accuracy of the FEM solution is verified against the analytical solution, and the sensitivity of the FEM solution to the fractional order α and the skewness parameter β is analyzed. We find that the proposed numerical approach converge to the numerical solution of the advection–dispersion equation (ADE) as the fractional order α equals 2. The problem caused by using the first- or third-kind boundary with an integral-order derivative at the inlet is remedied by using the third-kind boundary with a fractional-order derivative there. The problems for concentration estimation at boundaries caused by the singularity of the fractional derivative can be solved by using the concept of transition probability conservation. The FEM solution of this study has smaller numerical dispersion than that of the FD solution by Meerschaert and Tadjeran (J Comput Appl Math 2004). For a given α, the spatial distribution of concentration exhibits a symmetric non-Fickian behavior when β = 0. The spatial distribution of concentration shows a Fickian behavior on the left-hand side of the spatial domain and a notable non-Fickian behavior on the right-hand side of the spatial domain when β = 1, whereas when β = −1 the spatial distribution of concentration is the opposite of that of β = 1. Finally, the numerical approach is applied to simulate the atrazine transport in a saturated soil column and the results indicat that the FEM solution of the FADE could better simulate the atrazine transport process than that of the ADE, especially at the tail of the breakthrough curves.     

On the solution of the time-dependent convection-diffusion equation by the finite element method

A Petrov-Galerkin finite element method is presented for the time-dependent convection-diffusion equation. The scheme is based on bilinear time-space trial and quadratic in time-linear in space test functions, the latter being nonconforming. Second order in time and third order in space accuracy is obtained, and the schemes are free of numerical diffusion and disperson effects. Numerical results are presented which show excellent approximation properties.     

A finite element model for the diffusion—convection equation with applications to air pollution problems

A method of solution for the diffusion-convection equation in one spatial dimension has been developed previously. The generalization of this method into two-spatial dimensions will be presented. The method employs space-time volume elements with edges joining the nodes at subsequent time levels oriented along the characteristics of the associated pure convection problem. The accuracy and utility of the method are demonstrated by solving several examples and results are compared with the exact solutions.     

Galerkin finite element method and the groundwater flow equation: 1. Convergence of the method

This paper is concerned with the convergence of the Galerkin finite element method applied to a groundwater flow problem containing a borehole, with special reference to quadrature effects and the accuracy of the solution. It is shown that there exists an optimal quadrature rule for every choice of piecewise polynomial basis functions. Another interesting result proved here is that, in a direct application of the method the accuracy is very nearly independent of the degree of the polynomial basis functions, but strongly dependent on the distance of the borehole from the boundary if this is small.     

用于弹性波方程数值模拟的有限差分系数确定方法

有限差分方法是波场数值模拟的一个重要方法,交错网格差分格式比规则网格差分格式稳定性更好,但方法本身都存在因网格化而形成的数值频散效应,这会降低波场模拟的精度与分辨率.为了缓解有限差分算子的数值频散效应,精确求解空间偏导数,本文把求解波动方程的线性化方法推广到用于求解弹性波方程交错网格有限差分系数;同时应用最大最小准则作为模拟退火(SA)优化算法求解差分系数的数值频散误差判定标准来求解有限差分系数.通过上述两种方法,分别利用均匀各向同性介质和复杂构造模型进行了数值正演模拟和数值频散分析,并与传统泰勒展开算法、最小二乘算法进行比较,验证了线性化方法和模拟退火方法都能有效压制数值频散,并比较了各个算法的特点.     

The w-plane finite element method for the solution of the steady-state scalar wave equation in two dimensions

A novel finite element method has been described in this paper for the solution of the steady-state scalar wave equation in two dimensions. In this method the physical domain of the problem is mapped into an image domain; the governing equation and the prescribed boundary conditions are also appropriately transformed. Because logarithmic mapping functions are used, the physical domain is logarithmically condensed in the image plane. The method is therefore particularly suitable for the efficient and economical solution of large or very large aspect ratio problems. The high degree of accuracy which the method is capable of is demonstrated by means of two typical examples. Possible extension of the method to complex problems including non-linearity, multiplicity, etc. is also discussed.     

Mixing cell method for solving the solute transport equation with spatially variable coefficients

The advection–dispersion equation with spatially variable coefficients does not have an exact analytical solution and is therefore solved numerically. However, solutions obtained with several of the traditional finite difference or finite element techniques typically exhibit spurious oscillation or numerical dispersion when advection is dominant. The mixing cell and semi-analytical solution methods proposed in this study avoid such oscillation or numerical dispersion when advection dominates. Both the mixing cell and semi-analytical solution methods calculate the spatial step size by equating numerical dispersion to physical dispersion. Because of the spatial variability of the coefficients the spatial step size varies in space. When the time step size Δt→0, the mixing cell method reduces to the semi-analytical solution method. The results of application to two cases show that the mixing cell and semi-analytical solution methods are better than a finite difference method used in the study. © 1998 John Wiley & Sons, Ltd.     

比例边界有限元子结构法研究

比例边界有限元法最初应用于土-结构的相互作用分析，经过近几年的完善和发展，如今已经能够应用到其他很多领域。但是因为比例边界有限元理论是基于相似性要求的，使得其在处理几何形状复杂的结构时，会有很大的局限性，从而在某些领域的应用仍旧受到限制。同时由于其全时空耦合，导致大量计算量和工作量，也是其应用受限的一个原因。采用子结构法，打破这些局限性，并且分别针对有限域、无限域的问题，对比例边界有限元子结构法进行了研究，得出了有利于比例边界有限元法在工程实践中应用的结论，为其在实际工程应用中提供了可靠的依据和规律。     

有限元法2.5维CSAMT数值模拟

目前国内大多数CSAMT资料处理都用一维反演方法,即假定地球介质是水平层状地电结构,这对于横向变化较大的介质结构的反演则是不真实的．为了合理的模拟可控信号源的三维特征应该使用三维有限元方法．受计算机内存和计算速度的限制,三维方法的实用化受到约束．在许多情况下,地电结构沿走向变化很小,只沿倾向发生变化．这种地电结构是二维的,而人工源是三维的,因此CSAMT资料的观测可用2．5维有限元方法进行数值模拟．本文从麦克斯韦方程组出发,建立了2．5维有限元CSAMT数值模拟方法,其核心是把地电参数变化小的走向方向转化成波数域,用一系列波数模拟三维源的特征．用一个横向均匀的三层地电结构模型展示了2．5维数值模拟的特征,并和一维模拟结果进行了比较,证实了有限元方法2．5维CSAMT数值模拟的可靠性．在此基础上对一个地电结构已知的实际模型进行了2．5维数值模拟并和该剖面的野外实测剖面数据进行了比较,进一步有效地说明了本文介绍的2．5维CSAMT数值模拟方法是仿真和可靠的,为在此基础上的2．5维反演打下了良好的基础．     

多重网格在二维泊松方程有限元分析中的应用

本文简要介绍多重网格(MG)算法的基本原理及基本步骤,然后将多重网格算法引入有限单元中,对二维泊松方程进行求解.单元数尺度从8×8逐次增加至1024×1024,并与单重网格中高斯-赛德尔迭代法(GS)、共轭梯度法(CG) 在程序运行时间以及迭代次数方面进行比较.结果表明MG在计算速度和迭代次数都明显优于GS、CG方法.在1024×1024网格中,MG不仅比GS快500多倍,比CG快60多倍,而且与理论解的误差更小.     

Hillslope hydrographs by the finite element method

Simulation models may be used to explore the implications of making specific assumptions about the nature of a real world system, and then to make predictions of the behaviour of that system under a set of naturally occurring conditions. It is important that understanding generated by the former should be gained before predictive use of the system model. This paper describes and uses a finite-element model of transient, partially saturated water flow within a hillslope soil mantle overlying an impermeable bedrock, to make an investigation into the effects of parameter variations and initial conditions on the hillslope hydrograph. The results clearly demonstrate that the response of the hillslope system to rainfall is highly non-linear and that the initial conditions, particularly in the unsaturated zone, are of paramount importance in governing the timing and magnitude of the hydrograph peak. Hillslope convergence appears as the dominant topographic parameter but the non-linearity of the response and the complex interdependence between the soil and topographic parameters restrict the possibility of further definite conclusions about the relative sensitivity of the simulated hillslope hydrograph to changes in these parameters.     

修正辛格式有限元法的地震波场模拟

三角网格有限元法具有网格剖分的灵活性,能有效模拟地震波在复杂介质中的传播.但传统有限元法用于地震波场模拟时计算效率较低,消耗较大计算资源.本文采用改进的核矩阵存储（IKMS）策略以提高有限元法的计算效率,该方法不用组合总体刚度矩阵,且相比于常规有限元法节省成倍的内存.对于时间离散,将有限元离散后的地震波运动方程变换至Hamilton体系,在显式二阶辛Runge-Kutta-Nystr m（RKN）格式的基础之上加入额外空间离散算子构造修正辛差分格式,通过Taylor展开式得到具有四阶时间精度时间格式,且辛系数全为正数.本文从理论上分析了时空改进方法相比传统辛-有限元方法在频散压制、稳定性提升等方面的优势.数值算例进一步证实本方法具有内存消耗少、稳定性强和数值频散弱等优点.

     

A novel spherical polar finite element for the solution of the steady-state scalar wave equation in three-dimensions

A novel method has been proposed for the finite element solution of the steady-state scalar wave equation in three-dimensions. In this the governing equation and the prescribed boundary conditions in the physical space are transformed into a spherical polar space in which the radial direction is logarithmically condensed; the physical problem domain is also mapped into the new space. The transformed equation is then solved in the mapped domain using conventional finite elements. Because physical dimensions of the problem are logarithmically condensed in the proposed spherical polar space, the method is particularly suitable for solving truly three-dimensional problems in which the aspect ratio(s) is large or very large. A number of illustrative examples considered show that the proposed method is capable of a high degree of accuracy, achieved efficiently and economically. A hybrid scheme has also been proposed for dealing with awkward-shaped domains.     

A parametric study of the finite element eigenfunction method for the scattering of SH-waves

The performance of a coupled Finite Element—Analytic method in solving elastic wave scattering in infinite spaces is studied for plane SH-waves. The influence of several parameters on the accuracy of the results is investigated. These parameters are the number of terms in the series solution, the number of compatibility points across an imaginary circular interface (i.e. the points for which the continuity of both stresses and displacements are enforced), and the ratio of element dimension to wavelength in the direction of propagation. It was found that the method converges rapidly with the number of terms. Furthermore, both element dimension to wavelength ratio and the number of compatibility points significantly influence the accuracy of the approximate solution.     

有限元算法在声波方程数值模拟中的频散分析

针对有限元算法在地震波数值模拟中的数值频散问题,利用集中质量矩阵双线性插值有限元算法,推导了二维声波方程的频散函数.在此基础上采用定量分析方法,对比分析了网格纵横长度比变化时的入射方向、空间采样间隔、地震波频率以及地层速度对数值频散的影响.数值算例和模型正演结果表明：当采用集中质量矩阵双线性插值有限元算法时,为了有效地压制数值频散,在所使用震源子波的峰值频率对应的波长内,采样点数目应不少于20个;减小网格长度的纵横比可以有效地抑制入射角(波传播方向与z轴的夹角)较小的地震波的数值频散;地震波频率越高,传播速度越慢,频散越严重,尤其是当相速度与其所对应的频率比值小于2倍空间采样间隔时,不仅会出现严重的数值频散,还会出现假频现象.     

Comparison of finite difference and finite element solutions to the variably saturated flow equation

Numerical solutions to the equation governing variably saturated flow are usually obtained using either the finite difference (FD) method or the finite element (FE) method. A detailed comparison of these methods shows that the main difference between them is in how the numerical schemes spatially average the variation of material properties. Further differences are also observed in the way that flux boundaries are represented in FE and FD methods. A modified finite element (MFE) algorithm is used to explore the significance of these differences. The MFE algorithm enables a direct comparison with a typical FD solution scheme, and explicitly demonstrates the differences between FE and FD methods. The MFE algorithm provides an improved approximation to the partial differential equation over the usual FD approach while being computationally simpler to implement than the standard FE solution. One of the main limitations of the MFE algorithm is that the algorithm was developed by imposing several restrictions upon the more general FE solution; however, the MFE is shown to be preferable over the usual FE and FD solutions for some of the test problems considered in this study. The comparison results show that the FE (or MFE) solution can avoid the erroneous results encountered in the FD solution for coarsely discretized problems. The improvement in the FE solution is attributed to the broader hydraulic conductivity averaging and differences in the representation of flux type boundaries.     

一种基于亥姆霍兹分解的大地电磁测深有限元正演预条件解法

本研究针对大地电磁测深法有限元数值模拟中,迭代法求解线性方程组效率较低的问题,利用亥姆霍兹分解原理,将电场矢量双旋度方程的预条件问题转化为基于矢量位的泊松问题和基于标量位的拉普拉斯问题,并在四面体非结构化棱边元离散的情况下,借助节点元辅助网格离散上述预条件问题,进一步利用代数多重网格方法（AMG）实施求解,最终实现预条件算法.利用经典的COMMEMI理论模型进行试算并与前人的积分方程解进行对比,验证了本文数值模拟程序与预条件方法的正确性和可靠性.此外,利用不同自由度规模的实验模型对这一预条件算法的效率进行了测试.结果表明,这一算法可以有效地提升大地电磁测深法棱边有限元数值模拟迭代法的收敛性,计算效率较通用的不完全LU分解预条件算法明显更高;在较大自由度网格（>1000万）数值模拟计算中,其算法效率及内存占用相对直接解法有较大优势,也使小型工作站上利用较大自由度的有限元网格进行大地电磁测深数值模拟计算成为可能.

     

流体饱和多孔隙介质弹性波方程边界元解法研究

基于流体饱和多孔隙各向同性介质模型,本文首先推导了流体饱和多孔隙介质中弹性波传播的频率域系统动力方程及边界积分方程,然后给出了流体饱和多孔隙介质弹性波方程的基本解,最后,利用本文给出的边界元方法对流体饱和多孔隙各向同性介质中的弹性波传播进行了数值模拟.结果表明：不论是从固相位移,还是液相位移的地震合成记录都能看到明显的慢速P波,本文提出的流体饱和多孔隙介质弹性波边界元法是有效可行的.