The Lagrange–Galerkin method for the two‐dimensional shallow water equations on adaptive grids

The weak Lagrange–Galerkin finite element method for the two‐dimensional shallow water equations on adaptive unstructured grids is presented. The equations are written in conservation form and the domains are discretized using triangular elements. Lagrangian methods integrate the governing equations along the characteristic curves, thus being well suited for resolving the non‐linearities introduced by the advection operator of the fluid dynamics equations. An additional fortuitous consequence of using Lagrangian methods is that the resulting spatial operator is self‐adjoint, thereby justifying the use of a Galerkin formulation; this formulation has been proven to be optimal for such differential operators. The weak Lagrange–Galerkin method automatically takes into account the dilation of the control volume, thereby resulting in a conservative scheme. The use of linear triangular elements permits the construction of accurate (by virtue of the second‐order spatial and temporal accuracies of the scheme) and efficient (by virtue of the less stringent Courant–Friedrich–Lewy (CFL) condition of Lagrangian methods) schemes on adaptive unstructured triangular grids. Lagrangian methods are natural candidates for use with adaptive unstructured grids because the resolution of the grid can be increased without having to decrease the time step in order to satisfy stability. An advancing front adaptive unstructured triangular mesh generator is presented. The highlight of this algorithm is that the weak Lagrange–Galerkin method is used to project the conservation variables from the old mesh onto the newly adapted mesh. In addition, two new schemes for computing the characteristic curves are presented: a composite mid‐point rule and a general family of Runge–Kutta schemes. Results for the two‐dimensional advection equation with and without time‐dependent velocity fields are illustrated to confirm the accuracy of the particle trajectories. Results for the two‐dimensional shallow water equations on a non‐linear soliton wave are presented to illustrate the power and flexibility of this strategy. Copyright © 2000 John Wiley & Sons, Ltd.     

An efficient finite difference scheme for free‐surface flows in narrow rivers and estuaries

This paper presents a free‐surface correction (FSC) method for solving laterally averaged, 2‐D momentum and continuity equations. The FSC method is a predictor–corrector scheme, in which an intermediate free surface elevation is first calculated from the vertically integrated continuity equation after an intermediate, longitudinal velocity distribution is determined from the momentum equation. In the finite difference equation for the intermediate velocity, the vertical eddy viscosity term and the bottom‐ and sidewall friction terms are discretized implicitly, while the pressure gradient term, convection terms, and the horizontal eddy viscosity term are discretized explicitly. The intermediate free surface elevation is then adjusted by solving a FSC equation before the intermediate velocity field is corrected. The finite difference scheme is simple and can be easily implemented in existing laterally averaged 2‐D models. It is unconditionally stable with respect to gravitational waves, shear stresses on the bottom and side walls, and the vertical eddy viscosity term. It has been tested and validated with analytical solutions and field data measured in a narrow, riverine estuary in southwest Florida. Model simulations show that this numerical scheme is very efficient and normally can be run with a Courant number larger than 10. It can be used for rivers where the upstream bed elevation is higher than the downstream water surface elevation without any problem. Copyright © 2003 John Wiley & Sons, Ltd.     

A shape optimisation method of a body located in adiabatic flows

The purpose of this study is to derive an optimal shape of a body located in adiabatic flow. In this study, we use the equation of motion, the equation of continuity and the pressure–density relation derived from the Poisson’s law as the governing equation. The formulation is based on an optimal control theory in which a performance function of fluid force is taken into consideration. The performance function should be minimised satisfying the governing equations. This problem can be solved without constraints by using the adjoint equation with adjoint variables corresponding to the state equation. The performance function is defined by the drag and lift forces acting on the body. The weighted gradient method is applied as a minimisation technique, the Galerkin finite element method is used as a spatial discretisation and the implicit scheme is used as a temporal discretisation to solve the state equations. The mixed interpolation, the bubble function for velocity and the linear function for density, is employed as the interpolation. The optimal shape is obtained for a body in adiabatic flows.     

Optimal control applied to water flow using second order adjoint method

This paper presents an optimal control applied to water flow using the first and second order adjoint equations. The gradient of the performance function with respect to control variables is analytically obtained by the first order adjoint equation. It is not necessary to compute the Hessian matrix directly using the second order adjoint equation. Two numerical studies have been performed to show the adaptability of the present method. The performance of the second order adjoint method is compared with that of the weighted gradient method, Broyden–Fletcher–Goldfarb–Shanno method and Lanczos method. The precise forms of the adjoint equations and the gradient to use for the minimisation algorithm are derived. The computation by the Lanczos method is shown as superior to those of the other methods discussed in this paper. The message passing interface library is used for the communication of parallel computing.     

Optimal flow control for Navier–Stokes equations: drag minimization

Optimal control and shape optimization techniques have an increasing role in Fluid Dynamics problems governed by partial differential equations (PDEs). In this paper, we consider the problem of drag minimization for a body in relative motion in a fluid by controlling the velocity through the body boundary. With this aim, we handle with an optimal control approach applied to the steady incompressible Navier–Stokes equations. We use the Lagrangian functional approach and we consider the Lagrangian multiplier method for the treatment of the Dirichlet boundary conditions, which include the control function itself. Moreover, we express the drag coefficient, which is the functional to be minimized, through the variational form of the Navier–Stokes equations. In this way, we can derive, in a straightforward manner, the adjoint and sensitivity equations associated with the optimal control problem, even in the presence of Dirichlet control functions. The problem is solved numerically by an iterative optimization procedure applied to state and adjoint PDEs which we approximate by the finite element method. Copyright © 2007 John Wiley & Sons, Ltd.     

热传导问题灵敏度分析的伴随法

在热传导灵敏度分析的直接法的研究基础上，进一步探讨了稳态和瞬态热传导问题灵敏度分析的伴随法.推导了伴随法的计算列式，对于瞬态热传导问题，研究了瞬态约束处理的关键点方法，并提出伴随方程的精细积分解法。算例表明，稳态问题灵敏度计算，伴随法与直接法的结果是一致的；瞬态问题灵敏度计算，两种方法的精度相当。     

Optimization of unsteady incompressible Navier–Stokes flows using variational level set method

This paper presents the optimization of unsteady Navier–Stokes flows using the variational level set method. The solid–liquid interface is expressed by the level set function implicitly, and the fluid velocity is constrained to be zero in the solid domain. An optimization problem, which is constrained by the Navier–Stokes equations and a fluid volume constraint, is analyzed by the Lagrangian multiplier based adjoint approach. The corresponding continuous adjoint equations and the shape sensitivity are derived. The level set function is evolved by solving the Hamilton–Jacobian equation with the upwind finite difference method. The optimization method can be used to design channels for flows with or without body forces. The numerical examples demonstrate the feasibility and robustness of this optimization method for unsteady Navier–Stokes flows.Copyright © 2012 John Wiley & Sons, Ltd.     

Calculation of curved open channel flow using physical curvilinear non‐orthogonal co‐ordinates

There are two main difficulties in numerical simulation calculations using FD/FV method for the flows in real rivers. Firstly, the boundaries are very complex and secondly, the generated grid is usually very non‐uniform locally. Some numerical models in this field solve the first difficulty by the use of physical curvilinear orthogonal co‐ordinates. However, it is very difficult to generate an orthogonal grid for real rivers and the orthogonal restriction often forces the grid to be over concentrated where high resolution is not required. Recently, more and more models solve the first difficulty by the use of generalized curvilinear co‐ordinates (ξ,η). The governing equations are expressed in a covariant or contra‐variant form in terms of generalized curvilinearco‐ordinates (ξ,η). However, some studies in real rivers indicate that this kind of method has some undesirable mesh sensitivities. Sharp differences in adjacent mesh size may easily lead to a calculation stability problem oreven a false simulation result. Both approaches used presently have their own disadvantages in solving the two difficulties that exist in real rivers. In this paper, the authors present a method for two‐dimensional shallow water flow calculations to solve both of the main difficulties, by formulating the governing equations in a physical form in terms of physical curvilinear non‐orthogonal co‐ordinates (s,n). Derivation of the governing equations is explained, and two numerical examples are employed to demonstrate that the presented method is applicable to non‐orthogonal and significantly non‐uniform grids. Copyright © 2004 John Wiley & Sons, Ltd.     

An immersed boundary method for incompressible flows using volume of body function

A simple and effective immersed boundary method using volume of body (VOB) function is implemented on unstructured Cartesian meshes. The flow solver is a second‐order accurate implicit pressure‐correction method for the incompressible Navier–Stokes equations. The domain inside the immersed body is viewed as being occupied by the same fluid as outside with a prescribed divergence‐free velocity field. Under this view a fluid–body interface is similar to a fluid–fluid interface encountered in the volume of fluid (VOF) method for the two‐fluid flow problems. The body can thus be identified by the VOB function similar to the VOF function. In fluid–body interface cells the velocity is obtained by a volume‐averaged mixture of body and fluid velocities. The pressure inside the immersed body satisfies the same pressure Poisson equation as outside. To enhance stability and convergence, multigrid methods are developed to solve the difference equations for both pressure and velocity. Various steady and unsteady flows with stationary and moving bodies are computed to validate and to demonstrate the capability of the current method. Copyright © 2005 John Wiley & Sons, Ltd.     

多体系统指标2运动方程HHT方法违约校正1）

采用Cartesian绝对坐标建模方法,完整约束多体系统运动方程是指标3的微分--代数方程（differentialalgebraic equations,DAEs）,数值求解指标3的DAEs属于高指标问题,通过对位置约束方程求导,可使运动方程的指标降为2.位置约束方程求导得到的是速度约束方程.直接求解指标3的运动方程,速度约束方程得不到满足,而且高指标DAEs的数值求解存在一些问题.论文首先采用HHT（Hilber--Hughes--Taylor）直接积分方法求解降指标得到的指标2运动方程,此时速度约束方程参与离散计算,从机器精度上讲速度约束自然得到满足,而位置约束方程没有参与计算,存在“违约”.针对违约问题,采用基于Moore--Penrose广义逆理论的违约校正方法,消除位置约束方程的违约.指标2运动方程HHT方法违约校正,将HHT方法和违约校正方法很好地结合,在数值求解指标2运动方程的过程中,位置约束方程和速度约束方程都不存在违约问题,而且新方法没有引入新的未知数向量,离散得到的非线性方程组的方程数量与原指标2运动方程的方程数量相同,求解规模没有扩大.新方法的实用和有效性通过算例的数值实验得到验证,数值实验也说明新方法保持了HHT方法本身具有的数值阻尼可以控制和二阶精度的特性.最后从非线性方程组的求解规模和计算速度上与其他方法进行了比较分析,说明新方法的优势所在.     

Aerodynamic design using the truncated Newton algorithm and the continuous adjoint approach

In this paper, the so‐called ‘continuous adjoint‐direct approach’ is used within the truncated Newton algorithm for the optimization of aerodynamic shapes, using the Euler equations. It is known that the direct differentiation (DD) of the flow equations with respect to the design variables, followed by the adjoint approach, is the best way to compute the exact matrix, for use along with the Newton optimization method. In contrast to this, in this paper, the adjoint approach followed by the DD of both the flow and adjoint equations (i.e. the other way round) is proved to be the most efficient way to compute the product of the Hessian matrix with any vector required by the truncated Newton algorithm, in which the Newton equations are solved iteratively by means of the conjugate gradient (CG) method. Using numerical experiments, it is demonstrated that just a few CG steps per Newton iteration are enough. Considering that the cost of solving either the adjoint or the DD equations is approximately equal to that of solving the flow equations, the cost per Newton iteration scales linearly with the (small) number of CG steps, rather than the (much higher, in large‐scale problems) number of design variables. By doing so, the curse of dimensionality is alleviated, as shown in a number of applications related to the inverse design of ducts or cascade airfoils for inviscid flows. Copyright © 2011 John Wiley & Sons, Ltd.     

A stability analysis of an oscillating body located in fluid flow using automatic differentiation

This paper presents a stability analysis of an oscillating body subjected to fluid forces located in a transient incompressible viscous flow. If the body is supported by elastic springs, oscillation will begin. If the characteristic period of the body and the excited oscillating period due to fluid forces match each other, resonance can occur. Stability analysis is therefore needed to determine the nonlinear behavior of the body. This paper presents an analysis of the changing stability of bodies by the numerical computation. To implement the computation, the motion of fluid around a body is expressed by the Navier–Stokes equation described in the arbitrary Lagrangian–Eulerian form. The fluid influence on the body is discretized by the finite element method based on a mixed interpolation by the bubble function in space. The motion of the body is assumed to be expressed by the equations of motion. To evaluate stability, stability function is defined by the total energy of the oscillating body. The stability is judged according to a stability index, obtained by the use of the automatic differentiation (AD) of the stability function. AD is a derivative computation method that gives high accuracy. By the use of AD, the second‐order derivative matrix, which is needed to compute the stability index, can be obtained exactly. For the numerical studies, analyses of one degree of freedom and two degrees of freedom (2DOF) for a circular cylinder and 2DOF for a rectangular cylinder are carried out. A combination of a cylinder and supporting elastic spring can produce stable, neutral and unstable states. It is shown that the stability of the cylinder can be determined by the stability index. This paper shows new possibilities for stability analysis of bodies located in a fluid flow. Copyright © 2008 John Wiley & Sons, Ltd.     

Parameter estimation for a three‐dimensional numerical barotropic tidal model with adjoint method

The parameters of a three‐dimensional (3‐D) barotropic tidal model are estimated using the adjoint method. The mode splitting technique is employed in both forward and adjoint models. In the external mode, the alternating direction implicit method is used to discretize the two‐dimensional depth‐averaged equations and a semi‐implicit scheme is used for the 3‐D internal mode computations. In this model the bottom friction is expressed in terms of bottom velocity which is different from the previous works. Besides, the bottom friction coefficients (BFCs) are supposed to be spatially varying, i.e. the BFC at some grid points are selected as the independent BFC, while the BFC at the other grid points can be obtained through linear interpolation with these independent BFCs. On the basis of the simulation of M₂ tide in the Bohai and North Yellow Seas (BNYS), twin experiments are carried out to invert the prescribed distributions of model parameters. The parameters inverted are the Fourier coefficients of open boundary conditions (OBCs), the BFC and the vertical eddy viscosity profiles. In these twin experiments, the real topography of BNYS is installed. The ‘observations’ are produced by the tidal model and recorded at the position of TOPEX/Poseidon altimeter data, tidal gauge data and current data. The experiments discuss the influence of initial guesses, model errors and data number. The inversion has obtained satisfactory results and the prescribed distributions have been successfully inverted. The results indicate that the inversion of BFC is more sensitive to data error than that of OBC and the vertical eddy viscosity profiles. Copyright © 2007 John Wiley & Sons, Ltd.     

High‐resolution,monotone solution of the adjoint shallow‐water equations

A monotone, second‐order accurate numerical scheme is presented for solving the differential form of the adjoint shallow‐water equations in generalized two‐dimensional coordinates. Fluctuation‐splitting is utilized to achieve a high‐resolution solution of the equations in primitive form. One‐step and two‐step schemes are presented and shown to achieve solutions of similarly high accuracy in one dimension. However, the two‐step method is shown to yield more accurate solutions to problems in which unsteady wave speeds are present. In two dimensions, the two‐step scheme is tested in the context of two parameter identification problems, and it is shown to accurately transmit the information needed to identify unknown forcing parameters based on measurements of the system response. The first problem involves the identification of an upstream flood hydrograph based on downstream depth measurements. The second problem involves the identification of a long wave state in the far‐field based on near‐field depth measurements. Copyright © 2002 John Wiley & Sons, Ltd.     

Stochastic optimal control of quasi non-integrable Hamiltonian systems with stochastic maximum principle

A new procedure for designing optimal control of quasi non-integrable Hamiltonian systems under stochastic excitations is proposed based on the stochastic averaging method for quasi non-integrable Hamiltonian systems and the stochastic maximum principle. First, the control problem consisting of 2n-dimensional equations governing the controlled quasi non-integrable system and performance index is converted into a partially averaged one consisting of one-dimensional equation of the controlled system and performance index by using the stochastic averaging method. Then, the adjoint equation and the maximum condition of the partially averaged control problem are derived based on the stochastic maximum principle. The optimal control forces are determined from the maximum condition and solving the forward?Cbackward stochastic differential equations (FBSDE). For infinite time-interval ergodic control, the adjoint variable is a stationary process and the FBSDE is reduced to a partial differential equation. Finally, the response statistics of optimally controlled system is predicted by solving the Fokker?CPlank equation (FPE) associated with the fully averaged It? equation of the controlled system. An example of two degree-of-freedom (DOF) quasi non-integrable Hamiltonian system is worked out to illustrate the proposed procedure and its effectiveness.     

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.     

Shape optimization of a body located in low Reynolds number flow

The purpose of this study is to perform a numerical application of the shape optimization formulation of a body located in an incompressible viscous flow field. The formulation is based on an optimal control theory in which a performance function of the fluid force is introduced. The performance function should be minimized satisfying the state equation. This problem can be transformed into the minimization problem without constraint condition by the Lagrange multiplier method and the adjoint equations using adjoint variables corresponding to the state equations. As a numerical study, the drag force minimization problem in the steady Stokes flow, which means approximated equation of the low Reynolds number Navier–Stokes equation is carried out. After that, the unsteady Navier–Stokes flow is analysed. As the minimization algorithm, the steepest descent method is successfully applied. Copyright © 2005 John Wiley & Sons, Ltd.     

Adjoint approach of the spatial sensitivity to disturbances of internal flows with free surface

A local adjoint technique is developed in order to determine the most sensitive location to perturbations of steady states near bifurcation points in the case of confined flows with free‐surface boundary. Transitions to stationary or periodic flows are studied. The method is validated by comparison of its results with those given by a time approach. It is then applied to the stability study and the feedback control of thermocapillary flows in liquid bridge. Copyright © 2005 John Wiley & Sons, Ltd.     

Generation of turbulent inflow and initial conditions based on multi‐correlated random fields

Direct or large eddy simulation of a turbulent flow field is strongly influenced by its initial or inflow boundary condition. This paper presents a new stochastic approach to generate an artificial turbulent velocity field for initial or inflow boundary condition based on digital filtering. Each velocity component of the artificial turbulent velocity field is generated by linear combination of individual uncorrelated random fields. These uncorrelated random fields are obtained by filtering random white‐noise fields. Using common elements in these linear combinations results in multi‐correlation among different velocity components. The generated velocity field reproduces locally desired Reynolds stress components and integral length scales including cross‐integral length scales. The method appears to be simple, flexible and more accurate in comparison with previously developed methods. The accuracy and performance of the method are demonstrated by numerical simulation of a homogeneous turbulent shear flow with high and low shear rates. To assess the accuracy and performance of the method, simulation results are compared with a reference simulation. Copyright © 2007 John Wiley & Sons, Ltd.     

The independent set perturbation adjoint method: A new method of differentiating mesh‐based fluids models

A new scheme for differentiating complex mesh‐based numerical models (e.g. finite element models), the Independent Set Perturbation Adjoint method (ISP‐Adjoint), is presented. Differentiation of the matrices and source terms making up the discrete forward model is realized by a graph coloring approach (forming independent sets of variables) combined with a perturbation method to obtain gradients in numerical discretizations. This information is then convolved with the ‘mathematical adjoint’, which uses the transpose matrix of the discrete forward model. The adjoint code is simple to implement even with complex governing equations, discretization methods and non‐linear parameterizations. Importantly, the adjoint code is independent of the implementation of the forward code. This greatly reduces the effort required to implement the adjoint model and maintain it as the forward model continues to be developed; as compared with more traditional approaches such as applying automatic differentiation tools. The approach can be readily extended to reduced‐order models. The method is applied to a one‐dimensional Burgers' equation problem, with a highly non‐linear high‐resolution discretization method, and to a two‐dimensional, non‐linear, reduced‐order model of an idealized ocean gyre. Copyright © 2010 John Wiley & Sons, Ltd.