MIXED FINITE ELEMENT METHODS FOR THE SHALLOW WATER EQUATIONS INCLUDING CURRENT AND SILT SEDIMENTATION （Ⅰ）——THE CONTINUOUS-TIME CASE

An initial-boundary value problem for shallow equation system consisting of water dynamics equations, silt transport equation, the equation of bottom topography change, and of some boundary and initial conditions is studied, the existence of its generalized solution and semidiscrete mixed finite element (MFE) solution was discussed, and the error estimates of the semidiscrete MFE solution was derived. The error estimates are optimal.     

MIXED FINITE ELEMENT METHODS FOR THE SHALLOW WATER EQUATIONS INCLUDING CURRENT AND SILT SEDIMENTATION (Ⅰ)-THE CONTINUOUS-TIME CASE

An initial-boundary value problem for shallow equation system consisting of water dynamics equations, silt transport equation, the equation of bottom topography change, and of some boundary and initial conditions is studied, the existence of its generalized solution and semidiscrete mixed finite element (MFE) solution was discussed, and the error estimates of the semidiscrete MFE solution was derived. The error estimates are optimal.     

MIXED FINITE ELEMENT METHODS FOR THE SHALLOW WATER EQUATIONS INCLUDING CURRENT AND SILT SEDIMENTA-TION (Ⅰ)-THE CONTINUOUS-TIME CASE

IntroductionTheshallowwaterequationsareanimportantmathematicalmodelforavarietyofprobleminhydraulicengineering .Inrecentyears,therehasbeeninterestinthenumericalsolutionfortheshallowwaterequations.Thenumericalsimulationsfortheshallowwaterequationsystemcanbeappliedtomanypurposes .First,itcanserveasameansformodelingtidalfluctuationsforthosenterestedincapturingtidalenergyforcommercialpurposes.Secondly ,thesesimulationscanbeusedtocomputetidalrangesandsurgessuchashurricanesandtsunamiscausedbyextreme…     

MIXED FINITE ELEMENT METHODS FOR THE SHALLOW WATER EQUATIONS INCLUDING CURRENT AND SILT SEDIMENTATION ( Ⅱ )-THE DISCRETE-TIME CASE ALONG CHARACTERISTICS

The mixed finite element (MFE) methods for a shallow water equation system consisting of water dynamics equations, silt transport equation, and the equation of bottom topography change were derived. A fully discrete MFE scheme for the discrete-time along characteristics is presented and error estimates are established. The existence and convergence of MFE solution of the discrete current velocity, elevation of the bottom topography, thickness of fluid column, and mass rate of sediment is demonstrated.     

MIXED FINITE ELEMENT METHODS FOR THE SHALLOW WATER EQUATIONS INCLUDING CURRENT AND SILT SEDIMENTATION (Ⅱ)——THE DISCRETE-TIME CASE ALONG CHARACTERISTICS

IntroductionThoughatransport_diffusion (orLagrange_Galerkin)method ,whichisalsocalledachar_acteristicsmethod ,isanoldone[1]andhasbeenextensivelyappliedtodealingwithPDEswithdiffusiontermand/orconvection ,ithadnotbeenmixedthefiniteelementmethodstotreatsuccessfullytheconvergenceofnumericalsolutionfortheNavier_Stokesequationsuntiltheearly198 0s[2 ].Intheearly 1 990s,Berm挷ｄｅｚｅｔａｌ.[3]appliedthismethodtodealingwiththeshallowwaterequationsonlyincludingthecurrentandthedepthofwaterandonlyderi…     

The boundary layer method for the solution of singular perturbation problem for the parabolic partial differential equation

In this paper, we discuss the singular perturbation problem of the parabolic partial differential equation. As usual, we must reduce the mesh size in the neighbourhood of the boundary layer so that typical feature of the boundary layer will not be lost. Then we need very large operational quantity when mesh sizes are getting too small. Now we propose the boundary layer scheme, which need not take very fine mesh size in the neighbourhood of the boundary layer. Numerical examples show that the accuracy can be satisfied with moderate step size.     

Mesh adaptation strategies for shallow water flow

This paper shows how the mesh adaptation technique can be exploited for the numerical simulation of shallow water flow. The shallow water equations are numerically approximated by the Galerkin finite element method, using linear elements for the elevation field and quadratic elements for the unit width discharge field; the time advancing scheme is of a fractional step type. The standard mesh refinement technique is coupled with the numerical solver; movement and elimination of nodes of the initial triangulation is not allowed. Two error indicators are discussed and applied in the numerical examples. The conclusion focuses the relevant advantages obtained by applying this adaptive approach by considering specific test cases of steady and unsteady flows. Copyright © 1999 John Wiley & Sons, Ltd.     

A NONLINEAR GALERKIN MIXED ELEMENT METHOD AND A POSTERIORI ERROR ESTIMATOR FOR THE STATIONARY NAVIER-STOKES EQUATIONS

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｎｏｎｌｉｎｅａｒＧａｌｅｒｋｉｎｍｅｔｈｏｄｉｓａｍｕｌｔｉ＿ｌｅｖｅｌｓｃｈｅｍｅｔｏｆｉｎｄｔｈｅａｐｐｒｏｘｉｍａｔｅｓｏｌｕｔｉｏｎｆｏｒｔｈｅｄｉｓｓｉｐａｔｉｖｅＰＤＥ (ｐａｒｔｉａｌｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎ) .Ｔｈｉｓｍｅｔｈｏｄｃｏｎｓｉｓｔｓｉｎｓｐｌｉｔｔｉｎｇｔｈｅｕｎｋｎｏｗｎｉｎｔｏｔｗｏ (ｏｒｍｏｒｅ)ｔｅｒｍｓ ,ｗｈｉｃｈｂｅｌｏｎｇｔｏｔｈｅｄｉｓｃｒｅｔｅｓｐａｃｅｓｗｉｔｈｄｉｆｆｅｒｅｎｔｍｅｓｈｓｉｚｅ .Ｔｈｅ…     

Galerkin-Petrov least squares mixed element method for stationary incompressible magnetohydrodynamics

The Galerkin-Petrov least squares method is combined with the mixed finite element method to deal with the stationary, incompressible magnetohydrodynamics system of equations with viscosity. A Galerkin-Petrov least squares mixed finite element format for the stationary incompressible magnetohydrodynamics equations is presented. And the existence and error estimates of its solution are derived. Through this method, the combination among the mixed finite element spaces does not demand the discrete Babuska-Brezzi stability conditions so that the mixed finite element spaces could be chosen arbitrartily and the error estimates with optimal order could be obtained.     

A Nonlinear Galerkin/Petrov-Least Squares Mixed Element Method for the Stationary Navier-Stokes Equation

A nonlinear Galerkin/ Petrov- least squares mixed element (NGPLSME) method for the stationary Navier-Stokes equations is presented and analyzed. The scheme is that Petrov-least squares forms of residuals are added to the nonlinear Galerkin mixed element method so that it is stable for any combination of discrete velocity and pressure spaces without requiring the Babuska-Brezzi stability condition. The existence, uniqueness and convergence ( at optimal rate ) of the NGPLSME solution is proved in the case of sufficient viscosity ( or small data).     

A nonlinear Galerkin/Petrov-least squares mixed element method for the stationary Navier-Stokes equations

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｎｏｎｌｉｎｅａｒＧａｌｅｒｋｉｎｍｅｔｈｏｄｉｓａｍｕｌｔｉ＿ｌｅｖｅｌｓｃｈｅｍｅｔｏｆｉｎｄｔｈｅａｐｐｒｏｘｉｍａｔｅｓｏｌｕｔｉｏｎｆｏｒｔｈｅｄｉｓｓｉｐａｔｉｖｅＰＤＥ .ＴｈｉｓｍｅｔｈｏｄｈａｓｆｉｒｓｔｍａｉｎｌｙｂｅｅｎａｄｄｒｅｓｓｅｄｂｙＦｏｉａｓ＿Ｍａｎｌｅｙ＿Ｔｅｍａｍ[1],Ｍａｒｉｏｎ＿Ｔｅｍａｍ[2 ],Ｆｏｉａｓ＿Ｊｏｌｌｙ＿Ｋｅｖｒｅｋｉｄｉｓ＿Ｔｉｔｉ[3]ａｎｄＤｅｖｕｌｄｅｒ＿Ｍａｒｉｏｎ＿Ｔｉｔｉ[4 ]ｉｎｔｈｅｃａｓｅｏｆｓｐｅｃｔ…     

Solution of spline function of elastic plates

In this paper.from four and three-order differential equations defined by cubic andquadratic splines of generaized beam.The beam functions with many boundary conditionsand under various loads are reduced.The approximate solution of deformation surface andstress of elastic thin plate is very accurate.     

An approximate solution considering flow inertia between spherical surfaces

In this paper, the analytical expressions of the pressure distribution, velocity distribution and discharge of the flow between spherical surfaces are found by using the method of iterative approximate solution when the inertia terms of Navier-Stokes equations in spherical coordinates are taken into consideration. Furthermore, using these expressions, we can directly obtain the corresponding analytical expressions of the laminar radial flow between parallel disks, which are fully identical with corresponding results presented by refs. [3,4].     

LIQUID-SOLID COUPLED SYSTEM OF MICROPUMP

This paper employs the integral-averaged method of thickness to approximate the periodical flows in a piezoelectric micropump, with a shallow water equation including nonlinearity and viscous damp presented to characterize the flows in micropump. The finite element method is used to obtain a matrix equation of fluid pressure. The fluid pressure equation is combined with the vibration equation of a silicon diaphragm to construct a liquid-solid coupled equation for reflecting the interaction between solid diaphragm and fluid motion in a micropump. Numerical results of a mode analysis of the coupled system indicate that the natural frequencies of the coupled system are much lower than those of the non-coupled system. The influence of additional mass and viscous damp of fluid on the natural frequencies of the coupled system is more significant as the pump thickness is small. It is found that the vibration shape functions of silicon diaphragm of the coupled system are almost the same as those of the non-coupled system. This paper also gives the first-order amplitude-frequency relationship of the silicon diaphragm, which is necessary for the flow-rate-frequency analysis of a micropump.     

生物芯片压电微流体泵液-固耦合系统模态分析

对压电微流体泵粘性流体周期流动进行厚度积分平均近似,得到包含粘性的,非线性浅水波动方程,并采用有限元法得到微泵液体压强矩阵方程.液体压强矩阵方程和压电硅片振动有限元方程耦合,得到一个包含微泵进出口扩散管的液-固耦合系统振动方程.液-固耦合系统的模态分析结果表明,微泵液-固耦合系统的自然频率比不耦合的硅片振动自然频率低很多.随着微泵厚度的减少,液体附加质量和粘性阻尼对耦合系统自然频率的影响更加明显.同时发现,对应的压电片振型函数在液-固耦合前后没有明显变化.还给出硅片一阶模态的振幅-频率特征曲线.对薄型无阀压电微流体泵,浅水波模型合理地表达了微泵液体流动和压电硅片振动的相互作用,以及液体附加质量和粘性阻尼对微泵液-固耦合系统动力特征的影响.     

变深度浅水域中非定常船波

以Green—Naghdi(G—N)方程为基础，采用波动方程／有限元法计算船舶经过变深度浅水域时非定常波浪特性．把运动船舶对水面的扰动作为移动压强直接加在Green-Naghdi方程里，以描述运动船体和水面的相互作用．以Series60 CB＝0．6船为算例，给出自由面坡高，波浪阻力在船舶经过一个水下凸包时变化规律，并与浅水方程的结果进行了比较．计算结果表明，当船舶经过凸包时，波浪阻力先增加，后减少，并逐渐趋于正常．同时发现，当船速小于临界速度时(Fr＝√gh＜1．0)，G—N方程给出的船后尾波比浅水方程的结果明显，波浪阻力也比浅水方程的结果有所提高，频率散射必须考虑．当船速大于临界速度时(Fr＝√gh＞1．0)，G—N方程的计算结果与浅水方程差别不大，频率散射的影响可以忽略．     

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.     

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

ＩｎｔｒｏｄｕｃｔｉｏｎＡｇｅｎｅｒａｌｔｈｅｏｒｙｏｆｔｈｅｌｅａｓｔ＿ｓｑｕａｒｅｓｍｅｔｈｏｄｈａｓｂｅｅｎｄｅｖｅｌｏｐｅｄｂｙＡＫＡｚｉｚ,ＲＢＫｅｌｌｏｇｇａｎｄＡＢＳｔｅｐｈｅｎｓｉｎ[1].Ｔｈｅｍｏｓｔｉｍｐｏｒｔａｎｔａｄｖａｎｔａｇｅｌｅａｄｓｔｏａｓｙｍｍｅｔｒｉｃｐｏｓｉｔｉｖｅｄｅｆｉｎｉｔｅｐｒｏｂｌｅｍ.ＪＨＢｒａｍｂｌｅａｎｄＪＡＮｉｔｓｈｅｐｒｅｓｅｎｔｅｄａｌｅａｓｔ＿ｓｑｕａｒｅｓｍｅｔｈｏｄｆｏｒＤｉｒｉｃｈｌｅｔｐｒｏｂｌｅｍｓｉｎ[2].Ｔｈｅｍｅｔｈｏｄｇｅ…     

Residual a posteriori error estimate two-grid methods for the steady Navier-Stokes equation with stream function form

Residual based on a posteriori error estimates for conforming element solutions of incompressible Navier-Stokes equations with stream function form which were computed with seven recently proposed two-level method were derived. The posteriori error estimates contained additional terms in comparison to the error estimates for the solution obtained by the standard finite element method. The importance of these additional terms in the error estimates was investigated by studying their asymptotic behavior. For optimal scaled meshes, these bounds are not of higher order than of convergence of discrete solution.     

USE OF EIGENVALUE TECHNIQUE IN FINITE ELEMENT TIDAL COMPUTATIONS

This paper presents the results of some studies on the development and application of a finite element method (FEM) with a closed-form solution technique for time discretization. The closed-form solution is based on the eigenvalues/vectors of a coefficient matrix. The method is first applied to the one-dimensional linearized shallow water equations and then extended to the two-dimensional shallow water equations. An attempt is made to improve its efficiency by incorporating time splitting and using the closed-form solution technique only for linear terms. Some case studies of a rectangular channel and harbour are presented to illustrate the satisfactory working of the method. © 1997 by John Wiley & Sons, Ltd. Int. j. numer. methods fluids 24: 953–963, 1997.