Constrained Hamilton variational principle for shallow water problems and Zu-class symplectic algorithm

In this paper, the shallow water problem is discussed. By treating the incompressible condition as the constraint, a constrained Hamilton variational principle is presented for the shallow water problem. Based on the constrained Hamilton variational principle, a shallow water equation based on displacement and pressure (SWE-DP) is developed. A hybrid numerical method combining the finite element method for spatial discretization and the Zu-class method for time integration is created for the SWEDP. The correctness of the proposed SWE-DP is verified by numerical comparisons with two existing shallow water equations (SWEs). The effectiveness of the hybrid numerical method proposed for the SWE-DP is also verified by numerical experiments. Moreover, the numerical experiments demonstrate that the Zu-class method shows excellent performance with respect to simulating the long time evolution of the shallow water.     

Interval Arithmetic and Static Interval Finite Element Method

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混凝土中化学-热-湿-力耦合过程的数值方法

提出了一个火灾下混凝土中化学-热-湿-力耦合过程分析的两级数学模型. 混凝 土模型化为充满两种非混溶孔隙流体的非饱和变形多孔多相介质. 数学模型基于控制干空 气、湿份及基质溶解物的质量守恒、混凝土介质混合体的动量守恒和焓(能量)守恒的耦合 偏微分方程组. 模型中特别考虑到了高温下的脱盐过程. 构造了一个用于数值模拟 化学-热-湿-力耦合行为的有限元求解过程的混合弱形式. 并且针对其中具有非自伴随算子特性的 双曲线控制方程的空间离散进行了特殊考虑. 数值结果例题显示所发展的数学模型和数值方 法在重现火灾条件下的混凝土中化学-热-湿-力耦合行为的有效性.     

Optimal Control in Navier-Stokes Equations

This paper presents a formulation for optimal control of a forced convection flow. The state equation that governs the forced convection flow can be expressed as the incompressible Navier-Stokes equations and energy equation. The optimal control can be formulated as finding a control force to minimize a performance function that is defined to evaluate a control object. The stabilized finite element method is used for the spatial discretization, while the Crank-Nicolson scheme is used for the temporal discretization. The Sakawa-Shindo method, which is an iterative procedure, is applied for minimizing the performance function.     

Examination for adjoint boundary conditions in initial water elevation estimation problems

I present here a method of generating a distribution of initial water elevation by employing the adjoint equation and finite element methods. A shallow‐water equation is employed to simulate flow behavior. The adjoint equation method is utilized to obtain a distribution of initial water elevation for the observed water elevation. The finite element method, using the stabilized bubble function element, is used for spatial discretization, and the Crank–Nicolson method is used for temporal discretizations. In addition to a method for optimally assimilating water elevation, a method is presented for determining adjoint boundary conditions. An examination using the observation data including noise data is also carried out. Copyright © 2009 John Wiley & Sons, Ltd.     

Multiphase flow in heterogeneous porous media: A classical finite element method versus an implicit pressure–explicit saturation‐based mixed finite element–finite volume approach

Various discretization methods exist for the numerical simulation of multiphase flow in porous media. In this paper, two methods are introduced and analyzed—a full‐upwind Galerkin method which belongs to the classical finite element methods, and a mixed‐hybrid finite element method based on an implicit pressure–explicit saturation (IMPES) approach. Both methods are derived from the governing equations of two‐phase flow. Their discretization concepts are compared in detail. Their efficiency is discussed using several examples. Copyright © 1999 John Wiley & Sons, Ltd.     

Godunov–mixed methods for immiscible displacement

The immiscible displacement problem in reservoir engineering can be formulated as a system of partial differential equations which includes an elliptic pressure–velocity equation and a degenerate parabolic saturation equation. We apply a sequential numerical scheme to this problem where time splitting is used to solve the saturation equation. In this procedure one approximates advection by a higher-order Godunov method and diffusion by a mixed finite element method. Numerical results for this scheme applied to gas–oil centrifuge experiments are given.     

A high-order characteristics/finite element method for the incompressible Navier-Stokes equations

In this paper we consider a discretization of the incompressible Navier-Stokes equations involving a second-order time scheme based on the characteristics method and a spatial discretization of finite element type. Theoretical and numerical analyses are detailed and we obtain stability results abnd optimal eror estimates on the velocity and pressure under a time step restriction less stringent than the standard Courant-Freidrichs-Levy condition. Finally, some numerical results obtained wiht the code N3S are shown which justify the interest of this scheme and its advantages with respect to an analogous first-order time scheme. © 1997 John Wiley & Sons, Ltd.     

广义对流扩散方程的隐式特征线混合元方法

提出了一个广义对流扩散方程的混合有限元方法,方程的基本变量及其空间梯度和流量在单 元内均作为独立变量分别插值. 基于胡海昌-Washizu三变量广义变分原理结合特征线法给 出了控制方程的单元弱形式. 混合元方法采用基于一点积分方案并结合可以滤掉虚假的 数值震荡的隐式特征线法. 数值结果证明了所提出的方法可以提供和四点积分同样的数 值计算结果,并能够提高计算效率.     

饱和土一维动力响应分析的弱式微分求积元法

基于伽辽金加权残值法，本文首先建立一维饱和土动力学控制微分方程的弱形式，而后分别采用微分求积法和有限元法将其空间坐标离散，得到以土体骨架位移、流体-土骨架相对位移和孔隙流体压力为自由度的单元离散方程，从而采用Crank-Nicolson 法求解．数值算例一方面通过与解析解的对比，验证了离散方程和数值程序的正确性．另一方面，通过地表位移和基底孔隙压力的收敛性分析，检验了求积元和有限元法的收敛效率．数值结果表明：所建立的弱式微分求积法在饱和土动力分析中不仅具有显著优于常规有限元法的收敛效率，而且还具有可变阶的收敛性能，为今后高效率分析提供了一种可能．     

A coupled continuous and discontinuous finite element method for the incompressible flows

In this paper, we develop a coupled continuous Galerkin and discontinuous Galerkin finite element method based on a split scheme to solve the incompressible Navier–Stokes equations. In order to use the equal order interpolation functions for velocity and pressure, we decouple the original Navier–Stokes equations and obtain three distinct equations through the split method, which are nonlinear hyperbolic, elliptic, and Helmholtz equations, respectively. The hybrid method combines the merits of discontinuous Galerkin (DG) and finite element method (FEM). Therefore, DG is concerned to accomplish the spatial discretization of the nonlinear hyperbolic equation to avoid using the stabilization approaches that appeared in FEM. Moreover, FEM is utilized to deal with the Poisson and Helmholtz equations to reduce the computational cost compared with DG. As for the temporal discretization, a second‐order stiffly stable approach is employed. Several typical benchmarks, namely, the Poiseuille flow, the backward‐facing step flow, and the flow around the cylinder with a wide range of Reynolds numbers, are considered to demonstrate and validate the feasibility, accuracy, and efficiency of this coupled method. Copyright © 2016 John Wiley & Sons, Ltd.     

A finite element method for the two‐dimensional extended Boussinesq equations

A new numerical method for Nwogu's (ASCE Journal of Waterway, Port, Coastal and Ocean Engineering 1993; 119 :618)two‐dimensional extended Boussinesq equations is presented using a linear triangular finite element spatial discretization coupled with a sophisticated adaptive time integration package. The authors have previously presented a finite element method for the one‐dimensional form of these equations (M. Walkley and M. Berzins (International Journal for Numerical Methods in Fluids 1999; 29 (2):143)) and this paper describes the extension of these ideas to the two‐dimensional equations and the application of the method to complex geometries using unstructured triangular grids. Computational results are presented for two standard test problems and a realistic harbour model. Copyright © 2002 John Wiley & Sons, Ltd.     

A streamline diffusion nonconforming finite element method for the time-dependent linearized Navier-Stokes equations

A nonconforming finite element method of finite difference streamline diffusion type is proposed to solve the time-dependent linearized Navier-Stokes equations. The backward Euler scheme is used for time discretization. Crouzeix-Raviart nonconforming finite element approximation, namely, nonconforming （P1）2 - P0 element, is used for the velocity and pressure fields with the streamline diffusion technique to cope with usual instabilities caused by the convection and time terms. Stability and error estimates are derived with suitable norms.     

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.     

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.     

PEGASE: A NAVIER–STOKES SOLVER FOR DIRECT NUMERICAL SIMULATION OF INCOMPRESSIBLE FLOWS

A hybrid conservative finite difference/finite element scheme is proposed for the solution of the unsteady incompressible Navier–Stokes equations. Using velocity–pressure variables on a non-staggeredgrid system, the solution is obtained with a projection method basedon the resolution of a pressure Poisson equation. The new proposed scheme is derived from the finite element spatial discretization using the Galerkin method with piecewise bilinear polynomial basis functions defined on quadrilateral elements. It is applied to the pressure gradient term and to the non-linear convection term as in the so-called group finite element method. It ensures strong coupling between spatial directions, inhibiting the development of oscillations during long-term computations, as demonstrated by the validation studies. Two- and three-dimensional unsteady separated flows with open boundaries have been simulated with the proposed method using Cartesian uniform mesh grids. Several examples of calculations on the backward-facing step configuration are reported and the results obtained are compared with those given by other methods. © 1997 by John Wiley & Sons, Ltd. Int. j. numer. methods fluids 24: 833–861, 1997.     

A discrete element model for simulating saturated granular soil

A numerical model is developed to simulate saturated granular soil, based on the discrete element method. Soil particles are represented by Lagrangian discrete elements, and pore fluid, by appropriate discrete elements which represent alternately Lagrangian mass of water and Eulerian volume of space. Macro-scale behavior of the model is verified by simulating undrained biaxial compression tests. Micro-scale behavior is compared to previous literature through pore pressure pattern visualization during shear tests. It is demonstrated that dynamic pore pressure patterns are generated by superposed stress waves. These pore-pressure patterns travel much faster than average drainage rate of the pore fluid and may initiate soil fabric change, ultimately leading to liquefaction in loose sands. Thus, this work demonstrates a tool to roughly link dynamic stress wave patterns to initiation of liquefaction phenomena.     

A PREDICTIVE CONTROL METHOD TO OPERATE WATER GATE OF DAM

A new predictive control method for operating water regulating gate of dams is presented based on a hydraulic model. To consider the hydrodynamic behavior of surface waves through a reservoir, the shallow water equation is used with the discretization by the finite element method. This method provides the appropriate solution of outflow discharge which prevents the overflow of dam by the operation of water gate assuming that the inflow discharge is known as a function of some moment in the future. To show the applicability of this method, one dimensional channels with single and with multiple dams and Moriyoshizan dam reservoir have been computed as the numerical examples. It is shown that the water surface elevation of a reservoir is sufficiently controlled by the present method.     

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

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Numerical modelling of shear-dependent mass transfer in large arteries

A numerical scheme for the simulation of blood flow and transport processes in large arteries is presented. Blood flow is described by the unsteady 3D incompressible Navier–Stokes equations for Newtonian fluids; solute transport is modelled by the advection–diffusion equation. The resistance of the arterial wall to transmural transport is described by a shear-dependent wall permeability model. The finite element formulation of the Navier–Stokes equations is based on an operator-splitting method and implicit time discretization. The streamline upwind/Petrov–Galerkin (SUPG) method is applied for stabilization of the advective terms in the transport equation and in the flow equations. A numerical simulation is carried out for pulsatile mass transport in a 3D arterial bend to demonstrate the influence of arterial flow patterns on wall permeability characteristics and transmural mass transfer. The main result is a substantial wall flux reduction at the inner side of the curved region. © 1997 John Wiley & Sons, Ltd.