Adaptive finite element method for analysis of pollutant dispersion in shallow water

A finite element method for analysis of pollutant dispersion in shallow water is presented. The analysis is divided into two parts ： （ 1 ） computation of the velocity flow field and water surface elevation, and （2） computation of the pollutant concentration field from the dispersion model. The method was combined with an adaptive meshing technique to increase the solution accuracy, as well as to reduce the computational time and computer memory. The finite element formulation and the computer programs were validated by several examples that have known solutions. In addition, the capability of the combined method was demonstrated by analyzing pollutant dispersion in Chao Phraya River near the gulf of Thailand.     

Transient solutions for three-dimensional lid-driven cavity flows by a least-squares finite element method

A time-accurate least-squares finite element method is used to simulate three-dimensional flows in a cubic cavity with a uniform moving top. The time- accurate solutions are obtained by the Crank-Nicolson method for time integration and Newton linearization for the convective terms with extensive linearization steps. A matrix-free algorithm of the Jacobi conjugate gradient method is used to solve the symmetric, positive definite linear system of equations. To show that the least-squares finite element method with the Jacobi conjugate gradient technique has promising potential to provide implicit, fully coupled and time-accurate solutions to large-scale three-dimensional fluid flows, we present results for three-dimensional lid-driven flows in a cubic cavity for Reynolds numbers up to 3200.     

A control-volume based finite element method for three-dimensional incompressible turbulent fluid flow,heat transfer,and related phenomena

A control-volume based finite element method of equal-order type for three-dimensional incompressible turbulent fluid flow, heat transfer, and related phenomena is presented. The discretization equations are based mainly on the physics of the phenomena under consideration, more than on mathematical arguments. Special emphasis is devoted to the discretization of the convective terms and the continuity equation, and to the treatment of the boundary conditions imposed by the use of a high Reynolds k-?, type turbulence model. The pressure-velocity coupling in the fluid flow calculation is made from a derivative of the original SIMPLER method, without pressure correction. The discretized equations are solved in a sequential, rather than a coupled, form with significant advantage in the required computer time and storage. The method is an extension of a former version proposed by us for two-dimensional, laminar problems, and is here successfully applied to the following situations: three-dimensional deflected turbulent jet, and flows in 90° and 45° junctions of ducts with rectangular cross sections. The calculated results are in very good agreement with the experimental and numerical (obtained with the well established finite difference method) data available in the literature.     

Concurrent use of finite element and finite volume methods for shallow water flows in locally 1‐D channel networks

A numerical model is developed for shallow water equation in locally 1‐D channel networks. The model concurrently uses the standard Galerkin finite element method for the continuity equation and the finite volume method with an upwind scheme for the momentum equation. The surface gradient method is consistently employed. A minimum treatment is given for channel junctions so that application to multiply connected channels do not require any special consideration The model is capable of computing different types of transcritical flows, wet and dry flows, and flows with complex source terms. Standardized test problems and laboratory experimental data are used for verifying the model. Applicability of the models is validated in a multiply connected channel network draining hydromorphic farmlands located in a West African savanna, and Manning's roughness coefficient is identified, so that the steady solution is consistent with field observations. Unsteady simulation demonstrates that the model is capable of stably reproducing shifts of hydraulic jumps in flows of sub‐millimeter water depths. Copyright © 2011 John Wiley & Sons, Ltd.     

Surface waves propagation in shallow water: A finite element model

A two-dimensional (in-plane) numerical model for surface waves propagation based on the non-linear dispersive wave approach described by Boussinesq-type equations, which provide an attractive theory for predicting the depth-averaged velocity field resulting from that wave-type propagation in shallow water, is presented. The numerical solution of the corresponding partial differential equations by finite-difference methods has been the subject of several scientific works. In the present work we propose a new approach to the problem: the spatial discretization of the system composed by the Boussinesq equations is made by a finite element method, making use of the weighted residual technique for the solution approach within each element. The model is validated by comparing numerical results with theoretical solutions and with results obtained experimentally.     

Least-squares mixed finite element method for a class of stokes equation

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功能梯度材料板件三维分析的半解析梯度有限元法

将半解析有限元与梯度有限元相结合,形成一种半解析梯度有限元来求解功能梯度材料板件问题。该方法兼有有限元法的适应性强、程序统一,半解析有限元法的节省单元与计算工作量,梯度有限元法的适应构件内部材料性能任意梯度分布等特点,并实现用一维数值计算给出构件三维分析结果。算例分析表明了方法的精度、功能与上述特点,充分揭示了功能梯度材料板件力学响应的三维形态。半解析梯度有限元法可推广应用到其他功能梯度材料面结构的各类分析中。     

Orthogonality of modal bases in hp finite element models

In this paper, we exploit orthogonality of modal bases (SIAM J. Sci. Comput. 1999; 20 :1671–1695) used in hp finite element models. We calculate entries of coefficient matrix analytically without using any numerical integration, which can be computationally very expensive. We use properties of Jacobi polynomials and recast the entries of the coefficient matrix so that they can be evaluated analytically. We implement this in the context of the least‐squares finite element model although this procedure can be used in other finite element formulations. In this paper, we only develop analytical expressions for rectangular elements. Spectral convergence of the L₂ least‐squares functional is verified using exact solution of Kovasznay flow. Numerical results for transient flow over a backward‐facing step are also presented. We also solve steady flow past a circular cylinder and show the reduction in computational cost using expressions developed herein. Copyright © 2007 John Wiley & Sons, Ltd.     

Compatibility between finite volumes and finite elements using solutions of shallow water equations for substance transport

This paper formulates a finite volume analogue of a finite element schematization of three‐dimensional shallow water equations. The resulting finite volume schematization, when applied to the continuity equation, exactly reproduces the set of matrix equations that is obtained by the application of the corresponding finite element schematization to the continuity equation. The procedure allows the consistent and mass conserving coupling of the finite element Telemac model for three‐dimensional flow with the finite volume Delft3D‐WAQ model for water quality. The work has been carried out as part of a joint development by LNHE and WL∣Delft Hydraulics to explore the mutual interaction of their software. Copyright © 2006 John Wiley & Sons, Ltd.     

Improvements in mass conservation using alternative boundary implementations for a quasi‐bubble finite element shallow water model

Finite element approaches generally do not guarantee exact satisfaction of conservation laws especially when Dirichlet‐type boundary conditions are imposed. This article discusses improvement of the global mass conservation property of quasi‐bubble finite element solutions for the shallow water equations, focusing on implementations of the surface‐elevation boundary conditions. We propose two alternative implementations, which are shown by numerical verification to be effective in improving the smoothness of solutions near the boundary and in reducing the mass conservation error. The improvement of the mass conservation property contributes to augmenting the reliability and robustness of long‐term time integrations. Copyright © 2006 John Wiley & Sons, Ltd.     

Two‐dimensional dispersion analyses of finite element approximations to the shallow water equations

Dispersion analysis of discrete solutions to the shallow water equations has been extensively used as a tool to define the relationships between frequency and wave number and to determine if an algorithm leads to a dual wave number response and near 2Δx oscillations. In this paper, we explore the application of two‐dimensional dispersion analysis to cluster based and Galerkin finite element‐based discretizations of the primitive shallow water equations and the generalized wave continuity equation (GWCE) reformulation of the harmonic shallow water equations on a number of grid configurations. It is demonstrated that for various algorithms and grid configurations, contradictions exist between the results of one‐dimensional and two‐dimensional dispersion analysis as a result of subtle changes in the mass matrix. Numerical experiments indicate that the two‐dimensional dispersion analysis correctly predicts the existence and onset of near 2Δx noise in the solution. Copyright © 2004 John Wiley & Sons, Ltd.     

平面非协调动态有限元法

本文通过引入高阶非协调位移模式，从Ｈｅｌｉｎｇｅｒ－Ｒｅｉｓｓｎｅｒ泛函出发，给出了非协调动态有限元的一般列式，得到了相应的收敛准则。算例表明本文方法简单、有效     

Application of Taylor-least squares finite element to three-dimensional advection-diffusion equation

The Taylor-least squares (TLS) scheme, developed to solve the unsteady advection-diffusion equation for advection-dominated cases in one and two dimensions, is extended to three dimensions and applied to some 3D examples to demonstrate its accuracy. The serendipity Hermite element is selected as an interpolation function on a linear hexagonal element. As a validation of the code and as a simple sensitivity analysis of the scheme on the different types of shape functions, the 2D example problem of the previous study is solved again. Four 3D problems, two with advection and two with advection-diffusion, are also solved. The first two examples are advection of a steep 3D Gaussian hill in rotational flow fields. For the advection-diffusion problems with fairly general flow fields and diffusion tensors, analytical solutions are obtained using the ray method. Despite the steepness of the initial conditions, very good agreement is observed between the analytical and TLS solutions.     

Finite element method for shallow water equation including open boundary condition

A method to deal with an open boundary condition in the analysis of water surface waves, the tide, etc. by means of the finite element method is proposed in this paper. The present method has two important features relating to the treatment of the open boundary condition. The first feature is to consider the non-reflective virtual boundary condition which has been developed in the numerical wave analysis method. The incident wave conditions without spurious reflected waves can be imposed at the open boundary. The second feature is to identify the amplitude of the components of incident waves in terms of observed water elevations in the field of standing waves. This can be done as a parameter identification based on an optimization problem by applying the conjugate gradient method. The applicability of this method to wave propagation problems is verified by several numerical computations.     

A finite element/volume method model of the depth‐averaged horizontally 2D shallow water equations

Analysis of surface water flows is of central importance in understanding and predicting a wide range of water engineering issues. Dynamics of surface water is reasonably well described using the shallow water equations (SWEs) with the hydrostatic pressure assumption. The SWEs are nonlinear hyperbolic partial differential equations that are in general required to be solved numerically. Application of a simple and efficient numerical model is desirable for solving the SWEs in practical problems. This study develops a new numerical model of the depth‐averaged horizontally 2D SWEs referred to as 2D finite element/volume method (2D FEVM) model. The continuity equation is solved with the conforming, standard Galerkin FEM scheme and momentum equations with an upwind, cell‐centered finite volume method scheme, utilizing the water surface elevation and the line discharges as unknowns aligned in a staggered manner. The 2D FEVM model relies on neither Riemann solvers nor high‐resolution algorithms in order to serve as a simple numerical model. Water at a rest state is exactly preserved in the model. A fully explicit temporal integration is achieved in the model using an efficient approximate matrix inversion method. A series of test problems, containing three benchmark problems and three experiments of transcritical flows, are carried out to assess accuracy and versatility of the model. Copyright © 2014 John Wiley & Sons, Ltd.     

应用三维有限元法探讨RC梁柱节点的破坏机理

为了探究节点加强后RC梁柱节点的抗剪承载力、延性提高以及节点破坏模式由剪切破坏转变为梁弯曲破坏的实质,本文在RC节点的三维有限元分析的基础上探讨RC梁柱节点加强后的破坏机理。通过详细考察加固前后节点内部不同位置混凝土的应力-应变发展规律,探讨加强前后节点混凝土的宏观损伤发展过程;同时,通过考察核心混凝土的应力-应变关系及发展过程定量探讨加强钢板、梁柱主筋及箍筋对核心混凝土的约束作用。基于上述混凝土损伤过程的宏观分析,可以得出由于梁主筋的粘结加强、加强钢板以及箍筋对混凝土约束作用,使节点核心混凝土的实际强度增大、损伤延迟,RC节点由加强前的节点剪切破坏模式转变成梁端的弯曲破坏模式,从而提高RC梁柱节点的抗剪承载力和延性。     

On Taylor weak statement finite element methods for computational fluid dynamics

A Taylor series augmentation of a weak statement (a ‘Taylor weak statement’ or ‘Taylor-Galerkin’ method) is used to systematically reduce the dispersion error in a finite element approximation of the one-dimensional transient advection equation. A frequency analysis is applied to determine the phase velocity of semi-implicit linear, quadratic and cubic basis one-dimensional finite element methods and of several comparative finite difference/finite volume algorithms. The finite element methods analysed include both Galerkin and Taylor weak statements. The frequency analysis is used to obtain an improved linear basis Taylor weak statement finite element algorithm. Solutions are reported for verification problems in one and two dimensions and are compared with finite volume solutions. The improved finite element algorithms have sufficient phase accuracy to achieve highly accurate linear transient solutions with little or no artificial diffusion.     

随机结构有限元分析的递推求解方法

提出了一种新的谱随机有限元分析方法——递推求解方法。该方法将随机结构的随机响应表示成非正交多项式展式,建立了和摄动法类似的一系列确定的递推方程,并通过确定性有限元方法对这些递推方程进行静力问题求解。算例表明,当随机量出现较大涨落时,计算结果相对于传统摄动法有不小的改进。     

Criteria for finite element algorithm of generalized heat conduction equation

To eliminate oscillation and overbounding of finite element solutions of classical heat conduction equation, the author and Xiao have put forward two new concepts of monotonies and have derived and proved several criteria. This idea is borrowed here to deal with generalized heat conduction equation and finite element criteria for eliminating oscillation and overbounding are also presented. Some new and useful conclusions are drawn.