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.     

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.     

Shape optimisation of an oscillating body in fluid flow by adjoint equation and ALE finite element methods

The purpose of this paper is to determine the shape of an oscillating body by minimising drag and lift forces, located in a transient incompressible viscous fluid flow by means of the Arbitrary Lagrangian Eulerian finite element method and an optimal control theory. A performance function is expressed by the drag and lift forces. The performance function should be minimised satisfying the state equation and the constant volume condition. Therefore, this problem can be transformed into a minimisation problem without constraint by the Lagrange multiplier method. The adjoint equation and the gradient of the performance function are used to update the shape of the body. In this study, as a minimisation technique, the weighted gradient method is applied. The final shape is obtained of which drag and lift forces are reduced by 66.2% and 92.8%, respectively. The final shape obtained by this study is compared with the final shape of the non-oscillating body. The obtained final shape of the oscillating body is significantly different from the non-oscillating body.     

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.     

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.     

Computational water purification controls using second‐order adjoint equations with stability confirmation

This paper presents a computational method for water purification using second‐order adjoint equations. In Japan, the waters of polluted rivers are purified by conveying the waters from other rivers into the main rivers or by using outflows from sewage plants. The shallow water flow equation based on the water velocity and elevation and the advection diffusion equation of COD concentration are governing equations. The control problem involves finding a flow velocity into the main river that can reduce the COD concentration as close to the target value as possible. In other words, the problem is to find a water velocity to minimize the performance function, which is the square sum of the discrepancy between the computed and the observed COD concentrations. The present research was motivated by the need to apply water purification controls to practical projects. We have found that the controls occasionally tend to be unstable, and the stability of control must be ensured. By expanding the extended performance function into the Taylor series, the necessary condition for the stationary state is derived. Based on this condition, the first‐and second‐order adjoint equations can be obtained. The backward solution of the adjoint equation leads to the gradient and the Hessian product; these serve as the basis of the quasi‐Newton method. From the condition that the performance function must be minimum, the stability confirmation index can be determined. Using this index, we have derived the trust region method, the computation of which confirms the stability of control. Verification was carried out using a simple channel model. By varying the peak value of the inflow velocity, the outlet velocity has been determined such that the water elevation at the target point is zero. Depending on the peak value of the inflow, unstable control arises; this is determined by the stability confirmation index presented in this paper. The trust region method with the stability confirmation index is shown to be adaptable to judge the stability of control. The present method was applied to the water purification of Teganuma river in Japan. The steady fundamental state was computed with the inflow, outflow, and COD concentration at the inlet being specified. The control velocity at the control point can be determined for a fixed control duration with and without the stability confirmation index. The inflow, outflow, and COD concentration are specified as functions of time. It is shown that this method is suitable for practical use because control stability can be ensured. Moreover, it is also noted that the maximum flow velocity for stable control depending on the given control duration can be obtained. 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.     

A method to carry out shape optimization with a large number of design variables

A new method for shape optimization with relatively large number of design variables is proposed. It is well known that gradient‐based methods converge to a local optimum. As a result, utilization of a richer design space does not necessarily lead to a better design. This is demonstrated via the design of an airfoil for maximum lift for Re = 1000 and α = 4° flow. The airfoil is represented by fourth‐order non‐uniform rational B‐splines, and the control points are used as design variables. Starting with a NACA0012 airfoil, it is found that the optimal airfoil obtained with 13 control points has far superior aerodynamic performance than the ones obtained with 39 and 61 control points. For effective utilization of a richer design space, it is proposed that the number of design variables be increased gradually. The method is demonstrated by designing high lift airfoils for Re = 1000 and 1 × 10⁴. The objective function is the maximization of the time‐averaged lift coefficient for α = 4°. The optimization cycle with 27 control points is initiated with the optimal airfoil obtained with 13 control points. The process is continued with gradual increase in the number of design variables. Beyond a certain number of control points, the optimization leads to a spontaneous appearance of corrugations on the upper surface of the airfoil. The corrugations are responsible for the generation of small vortices that add to the suction on the upper surface of the airfoil and lead to enhanced lift. A stabilized finite element method is used to solve the unsteady flow and adjoint equations. Copyright © 2012 John Wiley & Sons, Ltd.     

基于网格退化和重组技术的类桁架拓扑优化方法

发展了一种基于有限元网格退化和重组技术的类桁架拓扑优化方法,可在不改变设计域包络的情况下(如维持流型、艺术外观和附加功能等)对结构进行减重。首先,分别对二维问题和三维问题发展相应的有限元网格退化算法,并在刚度等效的意义上对网格进行重组,建立了具有杆元拓扑特征的有限元模型。其次,以全局种子网格的长度尺寸和杆元横截面积为优化变量,构造了域内双层驱动拓扑优化问题,得到具有最优体分比的杆元拓扑结构。数值算例表明,所提方法可获得新型式的结构拓扑优化方案,并可将结构拓扑优化理论推向工程化应用。     

The improvement of numerical modeling in the solution of incompressible viscous flow problems using finite element method based on spherical Hankel shape functions

In this paper, the finite element method with new spherical Hankel shape functions is developed for simulating 2‐dimensional incompressible viscous fluid problems. In order to approximate the hydrodynamic variables, the finite element method based on new shape functions is reformulated. The governing equations are the Navier‐Stokes equations solved by the finite element method with the classic Lagrange and spherical Hankel shape functions. The new shape functions are derived using the first and second kinds of Bessel functions. In addition, these functions have properties such as piecewise continuity. For the enrichment of Hankel radial basis functions, polynomial terms are added to the functional expansion that only employs spherical Hankel radial basis functions in the approximation. In addition, the participation of spherical Bessel function fields has enhanced the robustness and efficiency of the interpolation. To demonstrate the efficiency and accuracy of these shape functions, 4 benchmark tests in fluid mechanics are considered. Then, the present model results are compared with the classic finite element results and available analytical and numerical solutions. The results show that the proposed method, even with less number of elements, is more accurate than the classic finite element method.     

Adaptive non‐conformal mesh refinement and extended finite element method for viscous flow inside complex moving geometries

Two different techniques to analyze non‐Newtonian viscous flow in complex geometries with internal moving parts and narrow gaps are compared. The first technique is a non‐conforming mesh refinement approach based on the fictitious domain method (FDM), and the second one is the extended finite element method (XFEM). The refinement technique uses one fixed reference mesh, and to impose continuity across non‐conforming regions, constraints using Lagrangian multipliers are used. The size of elements locally in the high shear rate regions is reduced to increase accuracy. FDM is shown to have limitations; therefore, XFEM is applied to decouple the fluid from the internal moving rigid bodies. In XFEM, the discontinuous field variables are captured by using virtual degrees of freedom that serve as enrichment and by applying special integration over the intersected elements. The accuracy of the two methods is demonstrated by direct comparison with results of a boundary‐fitted mesh applied to a two‐dimensional cross section of a twin‐screw extruder. Compared with non‐conforming FDM, XFEM shows a considerable improvement in accuracy around the rigid body, especially in the narrow gap regions. Copyright © 2011 John Wiley & Sons, Ltd.     

Characteristic Galerkin method for convection-diffusion equations and implicit algorithm using precise integration

This paper presents a finite element procedure for solving transient, multidimensional convection-diffusion equations. The procedure is based on the characteristic Galerkin method with an implicit algorithm using precise integration method. With the operator splitting procedure, the precise integration method is introduced to determine the material derivative in the convection-diffusion equation, consequently, the physical quantities of material points. An implicit algorithm with a combination of both the precise and the traditional numerical integration procedures in time domain in the Lagrange coordinates for the characteristic Galerkin method is formulated. The stability analysis of the algorithm shows that the unconditional stability of present implicit algorithm is enhanced as compared with that of the traditional implicit numerical integration procedure. The numerical results validate the presented method in solving convection-diffusion equations. As compared with SUPG method and explicit characteristic Galerkin method, the present method gives the results with higher accuracy and better stability. The project sponsored by the State Scientific and Technological Commission of China through “China State Key Project: the Theory and Methodology for Scientific and Engineering Computations with Large Scale”, the National Natural Science Foundation of China and the European Commission Research Project CI1^*CT94-0014.     

Practical techniques for a three‐dimensional FEM analysis of incompressible fluid flow contained with slip walls and a downstream tube bundle

Two practical techniques are proposed in this paper to simulate a flow contained in a plenum with a downstream tube bundle under a PC environment. First, a technique to impose slip wall conditions on smooth‐faced planes and sharp edges is proposed to compensate for the mesh coarseness relative to boundary layer thickness. In particular, a new type of Poisson equation is formulated to simultaneously satisfy both such velocity boundary conditions on walls and the incompressibility constraint. Second, a numerical model for a downstream tube bundle is proposed, where hydraulic resistance in a tube is imposed as a traction boundary condition on a fluid surface contacting the tube bundle end. The effectiveness of the techniques is numerically demonstrated in the application to a flow in a condenser water box. Copyright © 2001 John Wiley & Sons, Ltd.     

Control of hydrothermal waves in a thermocapillary flow using a gradient‐based control strategy

The problem of controlling the hydrothermal waves in a thermocapillary flow is addressed using a gradient‐based control strategy. The state equations are the two‐dimensional unsteady incompressible Navier–Stokes and energy equations under the Boussinesq approximation. The modeled problem is the ‘open boat’ process of crystal growth, the flow which is driven by Marangoni and buoyancy effects. The control is a spatially and temporally varying heat flux boundary condition at the free surface. The control that minimizes the hydrothermal waves is found using a conjugate gradient method, where the gradient of the objective function with respect to the control variables is obtained from solving a set of adjoint equations. The effectiveness of choices of the parameters governing the control algorithm is examined. Almost complete suppression of the hydrothermal waves is obtained for certain choices of the parameters governing the control algorithm. The numerical issues involved with finding the control using the optimizer are discussed, and the features of the resulting control are analyzed with the goal of understanding how it affects the flow.Copyright © 2012 John Wiley & Sons, Ltd.     

Water quality control by bank placement based on optimal control and finite element method

This paper presents a method for quality control by bank placement based on an optimal control theory and the finite element method. The shallow water equation is employed for the analysis of the flow condition and the advection‐diffusion equation is used for the analysis of pollutant concentration. The optimal control theory is utilized to obtain a control value for the objective state value. The shear‐slip mesh update method which is suitable for the rotational problem of body is employed. To solve the optimization problem, the time domain decomposition method is applied as a technique of storage requirements reduction. The Sakawa–Shindo method is employed as a minimization technique. The Crank–Nicolson method is applied to the temporal discretization. A method for optimal control of bank placement has been presented. Copyright © 2003 John Wiley & Sons, Ltd.     

Finite element analysis of filling stage in die‐casting process using marker surface method and adaptive grid refinement technique

The marker surface method and the adaptive grid refinement technique have been applied to the three‐dimensional (3‐D) finite element analysis of the filling stage in the die‐casting process. Especially, the marker surface plugging technique and the marker surface regeneration technique incorporated in the marker surface method have been proposed for the efficient analysis of 3‐D practical problems. Through the marker surface plugging technique, new parts of marker surface are effective lycreated in order to eliminate the gaps between the parts of marker surface or between the edge of marker surface and cavity wall. By using the marker surface regeneration technique, the marker surface including a great number of marker elements is recreated on the basis of its original shape in order to decrease the number of marker elements and computational time. A3‐D example used as the benchmark test and a typical industrial problem of the die‐casting process have been analysed. The numerical results have been in good agreement with the experimental results and the efficiency of the adaptive grid refinement technique has been verified. It has been shown that the proposed techniques incorporated in the marker surface method and the adaptive grid refinement technique can be effectively applied to general industrial problems. Copyright © 2004 John Wiley & Sons, Ltd.     

An adaptive finite element method for the modeling of the equilibrium of red blood cells

This contribution is concerned with the numerical modeling of an isolated red blood cell (RBC), and more generally of phospholipid membranes. We propose an adaptive Eulerian finite element approximation, based on the level set method, of a shape optimization problem arising in the study of RBCs. We simulate the equilibrium shapes that minimize the elastic bending energy under prescribed constraints of fixed volume and surface area. An anisotropic mesh adaptation technique is used in the vicinity of the cell membrane to enhance the robustness of the method. Efficient time and spatial discretizations are considered and implemented. We address in detail the main features of the proposed method, and finally we report several numerical experiments in the two‐dimensional and the three‐dimensional axisymmetric cases. The effectiveness of the numerical method is further demonstrated through numerical comparisons with semi‐analytical solutions provided by a reduced order model. Copyright © 2015 John Wiley & Sons, Ltd.     

A finite element method for free surface flows of incompressible fluids in three dimensions. Part I. Boundary fitted mesh motion

Computational fluid mechanics techniques for examining free surface problems in two‐dimensional form are now well established. Extending these methods to three dimensions requires a reconsideration of some of the difficult issues from two‐dimensional problems as well as developing new formulations to handle added geometric complexity. This paper presents a new finite element formulation for handling three‐dimensional free surface problems with a boundary‐fitted mesh and full Newton iteration, which solves for velocity, pressure, and mesh variables simultaneously. A boundary‐fitted, pseudo‐solid approach is used for moving the mesh, which treats the interior of the mesh as a fictitious elastic solid that deforms in response to boundary motion. To minimize mesh distortion near free boundary under large deformations, the mesh motion equations are rotated into normal and tangential components prior to applying boundary conditions. The Navier–Stokes equations are discretized using a Galerkin–least square/pressure stabilization formulation, which provides good convergence properties with iterative solvers. The result is a method that can track large deformations and rotations of free surface boundaries in three dimensions. The method is applied to two sample problems: solid body rotation of a fluid and extrusion from a nozzle with a rectangular cross‐section. The extrusion example exhibits a variety of free surface shapes that arise from changing processing conditions. Copyright © 2000 John Wiley & Sons, Ltd.     

A finite element method for free surface flows of incompressible fluids in three dimensions. Part II. Dynamic wetting lines

To date, few researchers have solved three‐dimensional free surface problems with dynamic wetting lines. This paper extends the free surface finite element method (FEM) described in a companion paper [Cairncross RA, Schunk PR, Baer TA, Sackinger PA, Rao RR. A finite element method for free surface flows of incompressible fluid in three dimensions. Part I. Boundary fitted mesh motion. International Journal for Numerical Methods in Fluids 2000; 33 : 375–403] to handle dynamic wetting. A generalization of the technique used in two‐dimensional modeling to circumvent double‐valued velocities at the wetting line, the so‐called kinematic paradox, is presented for a wetting line in three dimensions. This approach requires the fluid velocity normal to the contact line to be zero, the fluid velocity tangent to the contact line to be equal to the tangential component of web velocity, and the fluid velocity into the web to be zero. In addition, slip is allowed in a narrow strip along the substrate surface near the dynamic contact line. For realistic wetting line motion, a contact angle that varies with wetting speed is required because contact lines in three dimensions typically advance or recede at different rates depending upon location and/or have both advancing and receding portions. The theory is applied to capillary rise of static fluid in a corner, the initial motion of a Newtonian droplet down an inclined plane, and extrusion of a Newtonian fluid from a nozzle onto a moving substrate. The extrusion results are compared with experimental visualization. Copyright © 2000 John Wiley & Sons, Ltd.