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1.
The nodal domain integration method is used to develop a numerical model of the linear diffusion equation. The nodal domain integration approach is shown to represent an infinity of finite element mass matrix lumping schemes including the Galerkin and subdomain integration versions of the weighted residual method and an integrated finite difference method. Neumann, Dirichlet and mixed boundary conditions are accommodated analogous to the Galerkin finite element method. In order to reduce the overall integrated approximation relative error, a mass matrix lumping formulation is developed which is based on the Crank-Nicolson time advancement approximation. The optimum mass lumping factors are found to be strongly related to the model timestep size. 相似文献
2.
《Advances in water resources》1998,21(1):27-46
This work examines variable density flow and corresponding solute transport in groundwater systems. Fluid dynamics of salty solutions with significant density variations are of increasing interest in many problems of subsurface hydrology. The mathematical model comprises a set of non-linear, coupled, partial differential equations to be solved for pressure/hydraulic head and mass fraction/concentration of the solute component. The governing equations and underlying assumptions are developed and discussed. The equation of solute mass conservation is formulated in terms of mass fraction and mass concentration. Different levels of the approximation of density variations in the mass balance equations are used for convection problems (e.g. the Boussinesq approximation and its extension, fully density approximation). The impact of these simplifications is studied by use of numerical modelling.Numerical models for nonlinear problems, such as density-driven convection, must be carefully verified in a particular series of tests. Standard benchmarks for proving variable density flow models are the Henry, Elder, and salt dome (HYDROCOIN level 1 case 5) problems. We studied these benchmarks using two finite element simulators - ROCKFLOW, which was developed at the Institute of Fluid Mechanics and Computer Applications in Civil Engineering and FEFLOW, which was developed at the Institute for Water Resources Planning and Systems Research Ltd. Although both simulators are based on the Galerkin finite element method, they differ in many approximation details such as temporal discretization (Crank-Nicolson vs predictor-corrector schemes), spatial discretization (triangular and quadrilateral elements), finite element basis functions (linear, bilinear, biquadratic), iteration schemes (Newton, Picard) and solvers (direct, iterative). The numerical analysis illustrates discretization effects and defects arising from the different levels of the density of approximation. We contribute new results for the salt dome problem, for which inconsistent findings exist in literature. Applications of the verified numerical models to more complex problems, such as thermohaline and three-dimensional convection systems, will be presented in the second part of this paper. 相似文献
3.
《Advances in water resources》2005,28(7):729-744
In this paper, we discuss the local discontinuous Galerkin (LDG) method applied to elliptic flow problems and give details on its implementation, focusing specifically on the case of piecewise linear approximating functions. The LDG method is one a family of discontinuous Galerkin (DG) methods proposed for diffusion models. These DG methods allow for very general hp finite element meshes, and produce locally conservative fluxes which can be used in coupling flow with transport. The drawback to DG methods, when compared to their continuous counterparts, is the number of degrees of freedom required to compute the solution. This motivates a coupled approach, discussed herein, where the solution is allowed to be continuous or discontinuous on a node-by-node basis. This coupled approximation is locally conservative in regions where the numerical solution is discontinuous. Numerical results for fully discontinuous, continuous and coupled discontinuous/continuous solutions are given, where we compare solution accuracy, matrix condition numbers and mass balance errors for the various approaches. 相似文献
4.
T.V. Hromadka 《Advances in water resources》1984,7(2):76-80
The nodal domain integration method is applied to a two-dimensional advection-diffusion process in an anisotropic inhomogeneous medium. The domain is discretised into the union of irregular triangle finite elements with vertex-located nodal points and a linear trial function is used to approximate the governing flow equation's state variable in each element. Non-linear parameters are assumed quasi-constant for small durations in time in each element. The resulting numerical model represents the Galerkin and subdomain integration weighted residual methods and the integrated finite difference method as special cases. Both Dirichlet and Neumann boundary conditions are accommodated in a manner similar to the Galerkin finite element approach. 相似文献
5.
This paper presents and compares several numerical solutions of the coupled system of Navier–Stokes and Darcy equations. The schemes are based on combinations of the finite element method and the discontinuous Galerkin method. Accuracy and robustness of the methods are investigated for heterogeneous porous media. The importance of local mass conservation for filtration problems is also discussed. 相似文献
6.
The multiscale finite element method is developed for solving the coupling problems of consolidation of heterogeneous saturated porous media under external loading conditions. Two sets of multiscale base functions are constructed, respectively, for the pressure field of fluid flow and the displacement field of solid skeleton. The coupling problems are then solved with a multiscale numerical procedure in space and time domain. The heterogeneities induced by permeabilities and mechanical parameters of the saturated porous media are both taken into account. Numerical experiments are carried out for different cases in comparison with the standard finite element method. The numerical results show that the coupling multiscale finite element method can be successfully used for solving the complicated coupling problems. It reduces greatly the computing effort in both memory and time for transient problems. 相似文献
7.
A two-dimensional Galerkin finite element model for water flow in variably saturated soil is presented. A fourth-order Runge-Kutta time integration method is employed which allows use of time steps at least 2 times greater than for a traditional finite difference approximation of time derivatives. For short total simulation times computer execution costs for the Runge-Kutta method are greater than for the finite difference approximation due to the start up cost of the Runge-Kutta method, but for longer simulation times the Runge-Kutta method requires considerably less computational effort even when automatic time-step adjustment is used with the finite difference procedure. A comparison of the method of influence coefficients and 2 × 2 Gaussian integration to compute element matrices indicates that the influence coefficient method reduces total execution time to 60% of that required for numerical quadrature. Computed pressure heads using the influence coefficient method and numerical integration are found to be in close agreement with each other even under conditions of highly non-linear soil properties in a heterogeneous domain. Fluxes computed by the two methods are also generally in close agreement except under extremely non-linear conditions when some deviations were observed at short simulation times. 相似文献
8.
《Advances in water resources》2002,25(1):67-84
A discontinuous Galerkin (DG) finite element method is described for the two-dimensional, depth-integrated shallow water equations (SWEs). This method is based on formulating the SWEs as a system of conservation laws, or advection–diffusion equations. A weak formulation is obtained by integrating the equations over a single element, and approximating the unknowns by piecewise, possibly discontinuous, polynomials. Because of its local nature, the DG method easily allows for varying the polynomial order of approximation. It is also “locally conservative”, and incorporates upwinded numerical fluxes for modeling problems with high flow gradients. Numerical results are presented for several test cases, including supercritical flow, river inflow and standard tidal flow in complex domains, and a contaminant transport scenario where we have coupled the shallow water flow equations with a transport equation for a chemical species. 相似文献
9.
本文对声波与弹性波方程进行有限元法离散,构造有限元法频散关系的一般特征值问题,分析了时间离散格式为中心差分的三角网格有限元法声波与弹性波模拟的频散特性. 比较了三种质量矩阵即分布式质量矩阵、集中质量矩阵和混合质量矩阵对有限元法频散的影响;选取四种典型三角网格,分析了混合质量矩阵有限元（MFEM）频散的方向各向异性;数值频散、方向各向异性随插值阶数的增加逐渐减弱,当空间为三阶插值时,频散主要表现为随采样率的变化而几乎无明显方向各向异性, 其频散幅值也较小. 控制其他影响因素不变的情况下,研究了不同波速比介质中弹性波的数值频散. 最后给出了三角网格MFEM的数值耗散性. 相似文献
10.
This paper presents recent results of application of the finite element models to wave overtopping and wave run-up problems in ocean dynamics. Open boundaries are prescribed as natural boundary condition obtained from the continuity equation of the Galerkin finite element formulation. The numerical results are, in general, reasonably good agreements with the histrical field data. 相似文献
11.
We present advances in compositional modeling of two-phase multi-component flow through highly complex porous media. Higher-order methods are used to approximate both mass transport and the velocity and pressure fields. We employ the Mixed Hybrid Finite Element (MHFE) method to simultaneously solve, to the same order, the pressure equation and Darcy's law for the velocity. The species balance equation is approximated by the discontinuous Galerkin (DG) approach, combined with a slope limiter. In this work we present an improved DG scheme where phase splitting is analyzed at all element vertices in the two-phase regions, rather than only as element averages. This approximation is higher-order than the commonly employed finite volume method and earlier DG approximations. The method reduces numerical dispersion, allowing for an accurate capture of shock fronts and lower dependence on mesh quality and orientation. Further new features are the extension to unstructured grids and support for arbitrary permeability tensors (allowing for both scalar heterogeneity, and shear anisotropy). The most important advancement in this work is the self-consistent modeling of two-phase multi-component Fickian diffusion. We present several numerical examples to illustrate the powerful features of our combined MHFE–dg method with respect to lower-order calculations, ranging from simple two component fluids to more challenging real problems regarding CO2 injection into a vertical domain saturated with a multi-component petroleum fluid. 相似文献
12.
An efficient and accurate numerical model for multicomponent compressible single-phase flow in fractured media is presented. The discrete-fracture approach is used to model the fractures where the fracture entities are described explicitly in the computational domain. We use the concept of cross flow equilibrium in the fractures. This will allow large matrix elements in the neighborhood of the fractures and considerable speed up of the algorithm. We use an implicit finite volume (FV) scheme to solve the species mass balance equation in the fractures. This step avoids the use of Courant–Freidricks–Levy (CFL) condition and contributes to significant speed up of the code. The hybrid mixed finite element method (MFE) is used to solve for the velocity in both the matrix and the fractures coupled with the discontinuous Galerkin (DG) method to solve the species transport equations in the matrix. Four numerical examples are presented to demonstrate the robustness and efficiency of the proposed model. We show that the combination of the fracture cross-flow equilibrium and the implicit composition calculation in the fractures increase the computational speed 20–130 times in 2D. In 3D, one may expect even a higher computational efficiency. 相似文献
13.
The nodal domain integration method is applied to a one-dimensional advection—diffusion mathematical model without a source term. Comparison of the resulting numerical model to the well known Galerkin finite element, subdomain, and finite difference domain models indicates that a single numerical statement can be developed which includes the Galerkin finite element, subdomain, and finite difference models as special cases. 相似文献
14.
P.W. France 《Journal of Hydrology》1974,21(4):381-398
A numerical method is presented for analysing either steady state or transient three-dimensional groundwater flow problems. The governing equation is formulated in terms of the finite element process using the Galerkin approach, and cubic isoparametric elements are used to simulate the flow domain as these permit accurate modelling of curved boundaries. Particular attention is paid to the time dependent movement of the phreatic surface where an iterative technique based on the replacement of the original transient problem by a discrete number of steady state problems is used to effect a solution. Furthermore, in tracing the movement of the surface use is made of the element formulation theory in order to compute the normal to the boundary.The validity of the technique is first established by analysing a radially symmetrical problem for which an alternative analytical solution is available. Finally, a general three-dimensional flow system is studied for which there is no known analytical solution. It is shown that relatively few elements are required to yield practical solutions. 相似文献
15.
The turbulent advection-diffusion mathematical model in three-dimensional space is solved by a mixed finite element finite difference method. Linear finite elements in the vertical direction and central finite differences in the horizontal directions are used coupled with the Galerkin error minimization procedure. The integration in time is performed in fractional steps (one explicit one implicit) by splitting the differential operator. The method is illustrated by application to the three-dimensional movement of suspended sediment. Its accuracy is checked by comparison to analytical solutions and its efficiency is gauged relative to finite elements and implicit finite difference solutions for two-dimensional suspended sediment transport over a dredged channel. 相似文献
16.
近年来,随着地震波数值模拟对计算精度和效率的要求越来越高,间断有限元方法开始受到越来越多的关注.本文中,针对具有吸收边界条件的二维地震声波波动方程,作者提出了一种基于局部间断有限元方法的数值模拟算法.该算法在空间上使用局部间断有限元方法进行离散,在时间上采用了显式蛙跳格式.在这种时空离散的组合方式下,每个时间步上,此算法在空间剖分的每个单元上的求解计算是相互独立的,因而具有极高的并行性.通过数值算例,我们将该算法与连续有限元方法进行了比较.结果表明,本算法不仅具有对起伏构造的良好适应性,而且在计算效率和计算精度等方面,都具有优越性. 相似文献
17.
The behaviour of numerical solutions of the one-dimensional advection-dispersion equation is investigated and comparisons between the consistent and the lumped formulations of Galerkin finite element schemes are made. Well-known criteria for the control of accuracy in the lumped (finite difference) formulation are reviewed. It is found that, because the numerical error produced by the consistent formulation is generally less than that produced by the lumped formulation, these criteria can also be used for the control of numerical dispersion in the consistent formulation. However, because the error in both types of solutions decreases in time when the discretization is invariant, the criteria can be relaxed with advancing simulation time. For the consistent formulation it is found that beyond some initial time period, the numerical error depends only on the temporal discretization. This suggests that constant accuracy can be maintained throughout the simulation period while allowing the time step length to grow. 相似文献
18.
间断Galerkin有限元法（DG-FEM）作为一种有效的高阶有限元法受到了国内外学者的广泛关注.本文基于任意高阶间断Galerkin有限元法对弹性波方程进行空间离散,并将离散后所得的非齐次线性常微分方程系统齐次化,最后结合针对齐次问题的强稳定性保持龙格库塔（SSP Runge-Kutta）算法,将DG-FEM推广至时间任意高阶精度.另外,借鉴近最佳匹配层（NPML）的思想,基于复频移（CFS）拉伸坐标变换推导了一种新的PML吸收边界条件（简称为CFS-NPML）,该CFS-NPML能够与DG-FEM算法很好地结合,形成有效的起伏地表地震波传播数值模拟技术.数值试验结果表明,DG-FEM具有高阶精度,可以适应任意复杂起伏地表和复杂构造情况下的弹性波传播数值模拟.同时,CFS-NPML对包括面波等震相的人为边界反射都具有良好的吸收效果. 相似文献
19.
Daniel R. Lynch 《Advances in water resources》1984,7(2):67-75
Numerical mass balance relations are derived for common formulations of the hydraulic and species transport equations, by summing the Galerkin equations. Precise mass balance is demonstrated, provided the Galerkin equation is retained at all boundaries. The effects of quadrature, variable coefficients, transients and irregular geometry are addressed, and numerical experiments verify the algebra. 相似文献
20.
Local transmitting boundaries for transient elastic analysis 总被引：1，自引：0，他引：1
The aim of this paper is to investigate and develop alternative methods of analyzing problems in dynamic soil–structure-interaction (SSI). The interaction means that the amplitude of structural response is effected by additional energy dissipation through radiation and material damping in the soil. The surrounding soft soil behaves as a natural damper for a massive and stiff structure supported or embedded in it. The main focus is the major difficulty posed by such an analysis — the phenomena of waves that radiate outward from the excited structures towards infinity. In numerical calculations only a finite region of the foundation medium is analyzed and something is done to prevent the outgoing radiation waves from reflecting at the boundary region.Development of a simple and efficient finite element (FE) procedure for the solution directly in the time domain of transient SSI problems is the main concern. The central feature of the procedure is local absorbing boundaries used to render the computational domain finite. These boundaries are local in both time and space and are completely defined by a pair of symmetric stiffness and damping matrices. As the effort for implementing them is the same as for the impedance boundary condition (BC) considering the angle of incidence, standard assembly procedure can be used. Due to the local nature they also preserve the overall structure of the global equations of motion. Even though the focus is in the time domain the same equations of motions can be used to determine the solution under time-harmonic excitation directly in the frequency domain. Explicit formulae for the element matrices are included in the paper and numerical examples for transient radiation model problems to illustrate the validity and accuracy of the new procedures, are given. 相似文献