Shape determination of wind-resistant wings attached to an oscillating bridge using adjoint equation method

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.

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.

The purpose of this paper is to present a parameter identification method to determine the force of a blast and the elastic modulus of the ground using the measurements of a dynamic elastic wave, the adjoint equation method of optimal control theory, and the finite element method. Before the excavation of rocky ground, it is important to estimate the ground properties. In this paper, the elastic modulus is determined as the performance function is minimized using a technique based on the first‐order adjoint method. The performance function is a square sum of the discrepancies between the computed and the observed values of the velocities. After the determination of the magnitude of the blasting force, we can determine the elastic modulus of the rock. As the basic equation to calculate the velocities of dynamic elastic body, elastic equilibrium equations with linear viscosity are employed. The adjoint equation method has been utilized in order to calculate the gradient of the performance function with respect to the parameters. The gradient of the performance function is calculated using the first‐order adjoint equation. The weighted gradient method is applied for minimization. In order to solve the state equations in space and time, the finite element method and the Newmark \documentclass{article}\usepackage{amssymb}\footskip=0pc\pagestyle{empty}\begin{document}$\frac{1}{4}$\end{document} method are used. In this paper, we tested the practical application of our proposed method for determination of the elastic modulus of rock at the Ikawa tunnel located in the Tokushima prefecture, Japan. Copyright © 2009 John Wiley & Sons, Ltd.

This paper presents an optimal control system that includes a time-delay function for application to flood control setups with a retardation area. This system consists of the present and past controls that express flow behaviour in the retardation area. Optimal control theory is used to obtain a control discharge that satisfies the state equation including the time-delay function and minimizes the performance function. The optimal control and the delayed control discharges are obtained by the solution of an adjoint equation. The weighted gradient method is employed as a minimization algorithm. The Galerkin finite element procedure is employed to discretize the state and adjoint equations in the spatial direction. The bubble function interpolation, originated by the authors' group, using a stabilized term, is employed for the discretization in space. The flood flow in the Tsurumi river is presented as a numerical model. We show in this paper that floods can be controlled by means of a time-delay function.

In order to decrease the fluid drag on an underwater robot manipulator, an optimal trajectory method based on the variational method is presented. By introducing the adjoint variables, which are Lagrange multipliers, we formulate a Lagrange function under certain constraints related to the target angle, target angular velocity, and dynamic equation of the robot manipulator. The state equation (the partial differentiation of the Lagrange function with respect to the state variables), adjoint equation (the partial differentiation of the Lagrange function with respect to the adjoint variables), and sensitivity equation (the partial differentiation of the Lagrange function with respect to torques) can be derived from the stationary conditions of the Lagrange function. Using the state equation, we can calculate the state variables (angles, angular velocities, and angular acceleration) at every time step in the forward time direction. These state variables are stored as data at every time step. Next, by using the adjoint equation, we can calculate the adjoint variables by using these state variables at every time step in the backward time direction. These adjoint variables are stored as data at every time step. Third, the sensitivity equation is calculated by using both the state variables and the adjoint variables. Finally, the optimal trajectory of the manipulator is obtained using the sensitivities. The proposed method is applied to the problem of two-link manipulators. It can obtain the optimal drag reduction trajectory of the manipulator under the constraints mentioned above.

*† This paper deals with the use of the adjoint equation for aerodynamic shape optimization of complex configurations with overset grids methods. The use of overset grid eases the grid generation process, but the nontrivial task of ensuring communication between the overlapping grids needs careful attention. This need is effectively addressed by using a practically useful technique known as Implicit Hole Cutting (IHC) method first introduced by Lee and Baeder 1 and further demonstrated by the authors 2 in a previous study. The method depends on the simplicity of the cell selection process based on the main criterion of cell size, and all grid points including hole interior points and hole fringe points are treated indiscriminately in the flow computation. The simplicity of the IHC method is exploited for the adjoint equation solver. Similar to the flow solver, the adjoint equations are solved on the conventional point-matched grids and overlapped grids within a multi-block framework. Parallel computing with MPI is also used to improve the overall efficiency of the design optimization process. The method is successfully applied to several two and threedimensional shape optimization cases for both external and internal flows problems.

Multiphysics numerical modelling of backward flow forming process of AISI 5140 steel

This paper presents a method to control a flow behavior using the first-order adjoint equation method. A flood causes large-scale and extensive damage to the human property. It is expected that the damage can be suppressed to minimum if the water level or velocity of the river can be controlled. Therefore, the optimal control of a water flow is carried out in this study. In the control theory, the performance function which is defined by the square sum of the discrepancy between the computed and the objective water elevation at the target points is used. The extended performance function is given by the performance function and the state equation. The first-order adjoint equation can be derived by the condition that the first variations of the extended performance function and constraint condition are zero. The gradient of the extended performance function is obtained by solving the first-order adjoint equation. As the minimization technique, the weighted gradient method is applied. The shallow water equation based on the water velocity and elevation is used as a state equation. As the spatial discretization, the stabilized bubble function is employed. As the temporal discretization, the Crank-Nicolson method is applied. In numerical studies, the optimal control of water elevation in the Ikari dam lake is carried out. In this research, the optimal water velocity sent by the pump to minimize the water rise in the Ikari dam lake is computed. The inflow boundary conditions are presupposed as sinusoidal wave water elevation. The results of optimal control is performed effectively.

Computational fluid dynamics (CFD) is increasingly becoming the tool of choice for researchers and engineers facing complex fluid flow problems. One area of CFD that has attracted significant attention in recent years is the development of meshless or mesh-free methods[1]. These methods have so far been used in many applications like fracture mechanics, astrophysics and fluid flow. The key advantage of meshless methods over mesh-based methods like finite volume (FV) and finite element (FE) methods is its non-requirement of pre-specified connectivity between nodes in order to derive approximation and interpolation. This leads to the ability of finite difference (FD) methods to treat irregular domains and boundaries. A direct result of this is another advantage of mesh-free methods having the adaptability for problems involving large movement or wholesale motion of boundaries or embedded bodies. In this area, it has the edge over composite or overset grid methods in being able to use only one coordinate frame.A convecting meshless scheme for boundary driven fluid flow is described. This scheme can be applied to self-propulsion problems, or, in general, flows involving motion of rigid or deformable bodies embedded within the flow.The generalized finite difference method[2] (GFD) was chosen this scheme. A meshless cloud of nodes is embedded around the body surface along with the structured Cartesian background nodes for the entire flow field. By incorporating a subdomain for each node and its neighbours, the use of Taylor series expansion enables the derivatives to be found. Two forms of incompressible momentum equations are used: the standard Eulerian form in primitive variables for the structured nodes and the Arbitrary Lagrangian-Eulerian (ALE) form for the meshless cloud of nodes. GFD with moving least squares approximation is used for spatial discretization of the convecting meshless nodes. The projection method with Crank-Nicolson discretization is used to achieve second order accurate time discretization. Following prescribed body motion or fluid structure interaction, the meshless cloud acquire new locations in the next time step and the momentum equations in ALE form can then be solved.Test cases were performed to verify code usability and method accuracy and convergence. The simulation of the decaying vortex has been a popular benchmark for the gauging of accuracy. Basically the exact solutions of the time-dependent function values are known and both residues and absolute errors were calculated. Solving using different grid sizes proved the present method to be of second order spatial accuracy. The meshless method was then tried on the driven cavity flow at Reynolds number 1000. The results were well-compared with popular benchmark results[4]. In considering the effects of implementing the convecting meshless nodes across a normal flow field, the decaying vortex was again simulated, with a moving patch of nodes within the Cartesian field. The residues were well controlled and the errors were not significantly increased. Finally, a couple of external flows cases were simulated, basically on a flow past a stationary bluff body and a flapping ellipse inside an enclosed body of fluid.Future works in the pipeline include writing a parallel code for larger scale simulation and simulating oscillating cylinders and flapping of elliptic aerofoil(s).

This work presents a numerical method to automatically identify the friction coefficient which governs natural free surface flows. The mathematical formulation is based on the optimal control theory. A Lagrangian operator, which partial derivatives provide respectively the state equations, the adjoint equations and the optimality condition, is introduced. A finite element method is used for space discretization. Two solution algorithms are proposed and tested on some steady flows. The first one is very robust and converges for any initial distribution of the friction coefficient. The second one converges for realistic initial trials, uses less memory and could be well extended to unsteady flows.

This paper presents a complete derivation and implementation of the Arbitrary Lagrangian Eulerian (ALE) formulation for the simulation of large deformation quasi-static and dynamic problems. While most of the previous work done on ALE for dynamic applications was mainly based on operator split and explicit calculations, this work derives the quasi-static and dynamic ALE equations using a fully coupled implicit approach. Full expression for the ALE virtual work equations and finite element matrices are given. Time integration relations for the dynamic equations are also derived. Several quasi-static and dynamic large deformation applications are solved and presented.

The paper focuses on the study of a semi-activated system, based on a combination of two movements of forced pitching and free-heaving motion. Therefore, quantifying with accuracy the hydrodynamic forces applied on the hydrofoil seems to be crucial. This is investigated throughout a numerical analysis of the hydrofoil dynamics. The deformable structure is oscillating in a low-Reynolds number flow. In this study, a hydrofoil animated by a combined forced pitching and heaving movements is considered. Various materials of the hydrofoil structure are studied, from the rigid material to a more flexible one. A partitioned implicit coupling approach is applied in order to consider the Fluid-Structure Interaction (FSI) effects, while the Navier–Stokes equations are solved using the Arbitrary Lagrangian–Eulerian (ALE) method. Both the viscous incompressible Navier–Stokes equations and the elasticity equation are solved using finite volume method. The study is based on the analysis of the hydrodynamic loads acting on the structure. Therefore, the induced dynamics and the power coefficient of the structure are investigated. It is shown that the flexibility of the hydrofoil has an effect on its hydrodynamic behavior. Indeed it increases the load fluctuations and the horizontal mean force component. Furthermore, the unsteady vortices around the hydrofoil are highly impacted by its deformations. Finally, the structure deformations mostly improve the device energy efficiency.

Abstract. This work proposes a new FE model to predict dynamic crack propagation mechanisms in quasi-brittle materials. The numerical strategy uses a Moving Mesh (MM) technique consistent with the Arbitrary Lagrangian-Eulerian (ALE) formulation to reproduce the variation of the geometry of the computational domain caused by dynamically growing cracks. Specifically, the motion of the mesh nodes takes place consistently with conditions dictated by classic Fracture Mechanics, which provide conditions concerning the direction of propagation and the velocity of advancing cracks. As a remarkable key novelty, the proposed method introduces the ALE formulation of the M-integral for extracting Dynamic Stress Intensity Factors (DSIFs) at a moving crack front. This formulation allows extracting fracture variables on deforming elements without losing accuracy. Comparisons with analytical and numerical data are proposed to assess the validity and efficiency of the proposed strategy.

The unsteady, three-dimensional Navier-Stokes equations coupled with the Euler equations of rigid-body dynamics are sequentially solved to simulate and analyze the aerodynamic response of a high angle of attack delta wing undergoing oscillatory motion. The governing equations of fluid flow and dynamics of the multidisciplinary problem are solved using a time-accurate solution of the laminar, unsteady, compressible, full Navier-Stokes equations with the implicit, upwind, Roe flux-difference splitting, finite-volume scheme and a four-state Runge-Kutta scheme, respectively. The primary model under consideration consists of a 65° swept, sharp-edged, cropped delta wing of zero thickness at 20° angle of attack. In a freestream of Mach 0.85 and Reynolds number of 3.23 x 106, the flow over the upper surface of the wing develops a complex shock system which interacts with the leading-edge primary vortices producing vortex breakdown.\nThe effect of the oscillatory motion of the wing on the vortex breakdown and overall aerodynamic response is detailed to provide insight to the complicated physics associated with unsteady flows and the phenomenon of wing rock. Forced sinusoidal single and coupled mode rolling and pitching motion is presented for the wing in a transonic freestream. The Reynolds number, frequency of oscillation, and the phase angle are varied.\nComparison between the single and coupled mode forced rolling and pitching oscillation cases illustrate the effects of coupling the motion. This investigation shows that even when coupled, forced rolling oscillation at a reduced frequency of 2 π eliminates the vortex breakdown which results in an increase in lift. The coupling effect for in phase forced oscillations show that the lift coefficient of the pitching-alone case and the rolling-moment coefficient of the rolling-alone case dominate the resulting response. However, with a phase lead in the pitching motion, the coupled motion results in a non-periodic response of the rolling moment.\nThe second class of problems involve releasing the wing in roll to respond to the flowfield. Two models of sharp-edged delta wings, the previous 65° swept model and an 80° swept, sharp-edged delta wing, are used to observe the aerodynamic response of a wing free to roll in a transonic and subsonic freestream, respectively. These cases demonstrate damped oscillations, self-sustained limit cycle oscillations, and divergent rolling oscillations. Ultimately, an active control model using a mass injection system was applied on the surface of the wing to suppress the self-sustained limit cycle oscillation known as wing rock.\nComparisons with experimental investigations complete this study, validating the analysis and illustrating the complex details afforded by computational investigations.

An arbitrary Lagrangian-Eulerian (ALE) approach is used to model the orthogonal metal cutting in a steady state situation. The thermomechanical model includes the effects of elasticity, plasticity, strain rate, large strains and friction with heat generated between the tool and the chip. The ALE formulation can combine the advantages of both the Eulerian and Lagrangian approaches in a single description. Particularly, problems linked to the free surface in a Eulerian description and those linked to severe mesh distortions in a Lagrangian one can be solved by this formulation. The ALE governing equations are briefly reviewed in this paper; finite element and finite volume methods are used for the discretization of the conservation equations and an explicit time integration is adopted. Only the steady state solution is required; the ALE formulation is exploited to update the free and the contact surfaces. The model predicts the thermomechanical quantities, the chip geometry and the cutting forces from the cutting data and the material and friction parameters. Cutting experiments were performed with 42CD4 steel and comparisons of experimental tool forces and chip geometry with the numerical results are presented.