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 control of drainage basin considering moving boundary

The purpose of this article is to present a technique to optimally control river flood using a drainage basin considering a moving boundary. The main theme of this article is to obtain outflow discharge from the drainage basin that maintains the water level at a downstream point and empties the drainage basin as soon as possible. The water flow phenomenon inside the drainage basin when a river flood occurs is considered. This phenomenon can be analysed by the finite element method considering a moving boundary. The optimal control theory can be implemented to obtain the optimal control discharge. The finite element analysis with a moving boundary is introduced in the optimal control theory. A new boundary condition on the downstream side of the river is proposed. This condition is formulated by the solitary wave condition based on the basic water level being capable of representing natural water surface. As a numerical study, optimal control of shallow water flow is carried out for the Tsurumi River and its drainage basin model.     

Stability of high‐speed two‐layer film casting of Newtonian fluids

The steady‐state flow and its linear stability are investigated for the isothermal two‐layer film casting process. Newtonian fluids are considered in this study. The continuity of traction is ensured at the interface, and the axial velocity is assumed to be uniform across each film layer separately. The effects of inertia, gravity, fluid parameters and processing conditions on the steady‐state flow and its stability are studied. The results indicate that the fluid properties and the processing conditions have significant influence on the flow. The flow stability is strongly dependent on the layer layout with respect to the take‐up rolling process. The frequency of the (unstable) disturbance is insensitive to flow and processing parameters. Copyright © 2006 John Wiley & Sons, Ltd.     

Optimal control of MHD-flow deceleration

A problem of pulsed control for a three-dimensional magnetohydrodynamic (MHD) model is considered. It is demonstrated that singularities of the solution of MHD equations do not develop with time because they are suppressed by a magnetic field. The existence of an optimal control is proved. An optimality system with the solution regular in time as a whole is constructed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 5, pp. 3–10, September–October, 2008.     

A computational method on process optimal control for finite deformation problems

On the principle of non-incremental algorithm, some basic ideas of process optimal control iterative algorithm, based on the Optimal Control Variational Principle in Mechanics, is proposed in this paper. Then the essential governing equations are presented. This work provides a new method to achieve the numerical solutions of the mechanic of finite deformation. Project supported by the National Natural Science Foundation of China (Grant No. 594305050).     

Multilayer film casting of modified Giesekus fluids Part 2. Linear stability analysis

A linear stability analysis of the multilayer film casting of polymeric fluids has been conducted. A modified Giesekus model was used to characterize the rheological behaviors of the fluids. The critical draw ratio at the onset of draw resonance was found to depend on the elongational and shear viscosities of the fluids. Extensional-thickening has a stabilizing effect, whereas shear-thinning and extensional-thinning have destabilizing effects. The critical draw ratios for bilayer films of various thickness fractions are bounded by those for single layer films of the two fluids. When the two fluids have a comparable elongational viscosity, the critical draw ratio at a given Deborah number varies linearly with thickness fraction. When one fluid has a much larger elongational viscosity, it dominates the flow and the critical draw ratios at most thickness fractions remain close to its critical draw ratio as a single layer film. When the dominating fluid exhibits extensional-thickening, a film with a certain thickness fraction has more than one critical draw ratio at a given Deborah number and may not exhibit draw resonance within some range of the Deborah number.     

Optimal Control of a Vortex Trapped by an Airfoil with a Cavity

An airfoil with a cavity traps a vortex; the lift increases but the vortex shows great receptivity to upstream disturbances. A simple potential flow model confirms that the vortex stability basin is of a reduced extent. In this paper we present a control technique stabilizing the vortex position based on a potential flow model. The actuators are sources/sinks at the wall and the suction/blowing law is obtained by the adjoint optimization method. This revised version was published online in July 2006 with corrections to the Cover Date.     

Numerical Approaches to Optimal Control of a Model Equation for Shear Flow Instabilities

We investigate two different discretization approaches of a model optimal-control problem, chosen to be relevant for control of instabilities in shear flows. In the first method, a fully discrete approach has been used, together with a finite-element spatial discretization, to obtain the objective function gradient in terms of a discretely-derived adjoint equation. In the second method, Chebyshev collocation is used for spatial discretization, and the gradient is approximated by discretizing the continuously-derived adjoint equation. The discrete approach always results in a faster convergence of the conjugate-gradient optimization algorithm. Due to the shear in the convective velocity, a low diffusivity in the problem complicates the structure of the computed optimal control, resulting in particularly noticeable differences in convergence rate between the methods. When the diffusivity is higher, the control becomes less complicated, and the difference in convergence rate reduces. The use of approximate gradients results in a higher sensitivity to the degrees of freedom in time. When the system contains a strong instability, it only takes a few iteration to obtain an effective control for both methods,even if there are differences in the formal convergence rate. This indicates that it is possible to use the approximative gradients of the objective function in cases where the control problem mainly consists of controlling strong instabilities. This revised version was published online in July 2006 with corrections to the Cover Date.     

Optimal semiactive control of structures with isolated base

The paper investigates the benefits of implementing the semiactive control systems in relation to passive viscous damping in the context of seismically isolated structures. Frequency response functions are compiled from the computed time history response to pulse-like seismic excitation. A simple semiactive control policy is evaluated in regard to passive linear viscous damping and an optimal non-causal semiactive control strategy. The optimal control strategy minimizes the integral of the squared absolute accelerations subject to the constraint that the nonlinear equations of motion are satisfied. The optimization procedure involves a numerical solution to the Euler-Lagrange equations Published in Prikladnaya Mekhanika, Vol. 42, No. 2, pp. 129–135, February 2006.     

A reduced‐order approach for optimal control of fluids using proper orthogonal decomposition

In this article, a reduced‐order modeling approach, suitable for active control of fluid dynamical systems, based on proper orthogonal decomposition (POD) is presented. The rationale behind the reduced‐order modeling is that numerical simulation of Navier–Stokes equations is still too costly for the purpose of optimization and control of unsteady flows. The possibility of obtaining reduced‐order models that reduce the computational complexity associated with the Navier–Stokes equations is examined while capturing the essential dynamics by using the POD. The POD allows the extraction of a reduced set of basis functions, perhaps just a few, from a computational or experimental database through an eigenvalue analysis. The solution is then obtained as a linear combination of this reduced set of basis functions by means of Galerkin projection. This makes it attractive for optimal control and estimation of systems governed by partial differential equations (PDEs). It is used here in active control of fluid flows governed by the Navier–Stokes equations. In particular, flow over a backward‐facing step is considered. Reduced‐order models/low‐dimensional dynamical models for this system are obtained using POD basis functions (global) from the finite element discretizations of the Navier–Stokes equations. Their effectiveness in flow control applications is shown on a recirculation control problem using blowing on the channel boundary. Implementational issues are discussed and numerical experiments are presented. Copyright © 2000 John Wiley & Sons, Ltd.     

Optimal bounded control for maximizing reliability of Duhem hysteretic systems

The optimal bounded control of stochastic-excited systems with Duhem hysteretic components for maximizing system reliability is investigated. The Duhem hysteretic force is transformed to energy-depending damping and stiffness by the energy dissipation balance technique. The controlled system is transformed to the equivalent nonhysteretic system. Stochastic averaging is then implemented to obtain the Itô stochastic equation associated with the total energy of the vibrating system, appropriate for evaluating system responses. Dynamical programming equations for maximizing system reliability are formulated by the dynamical programming principle. The optimal bounded control is derived from the maximization condition in the dynamical programming equation. Finally, the conditional reliability function and mean time of first-passage failure of the optimal Duhem systems are numerically solved from the Kolmogorov equations. The proposed procedure is illustrated with a representative example.     

Optimal bounded control for minimizing the response of quasi-integrable Hamiltonian systems

A procedure for designing optimal bounded control to minimize the response of quasi-integrable Hamiltonian systems is proposed based on the stochastic averaging method for quasi-integrable Hamiltonian systems and the stochastic dynamical programming principle. The equations of motion of a controlled quasi-integrable Hamiltonian system are first reduced to a set of partially completed averaged Itô stochastic differential equations by using the stochastic averaging method for quasi-integrable Hamiltonian systems. Then, the dynamical programming equation for the control problems of minimizing the response of the averaged system is formulated based on the dynamical programming principle. The optimal control law is derived from the dynamical programming equation and control constraints without solving the dynamical programming equation. The response of optimally controlled systems is predicted through solving the Fokker-Planck-Kolmogrov equation associated with fully completed averaged Itô equations. Finally, two examples are worked out in detail to illustrate the application and effectiveness of the proposed control strategy.     

有限变形问题过程最优控制算法

在非增量算法的基础上,提出了用最优控制变分原理形成过程最优控制迭代求解的基本思路,并给出求解的基本控制方程。这一工作为有限变形力学问题的数值求解提供了一个新的处理方法。     

High‐order filtering for control volume flow simulation

A general methodology is presented in order to obtain a hierarchy of high‐order filter functions, starting from the standard top‐hat filter, naturally linked to control volumes flow simulations. The goal is to have a new filtered variable better represented in its high resolved wavenumber components by using a suitable deconvolution. The proposed formulation is applied to the integral momentum equation, that is the evolution equation for the top‐hat filtered variable, by performing a spatial reconstruction based on the approximate inversion of the averaging operator. A theoretical analysis for the Burgers' model equation is presented, demonstrating that the local de‐averaging is an effective tool to obtain a higher‐order accuracy. It is also shown that the subgrid‐scale term, to be modeled in the deconvolved balance equation, has a smaller absolute importance in the resolved wavenumber range for increasing deconvolution order. A numerical analysis of the procedure is presented, based on high‐order upwind and central fluxes reconstruction, leading to congruent control volume schemes. Finally, the features of the present high‐order conservative formulation are tested in the numerical simulation of a sample turbulent flow: the flow behind a backward‐facing step. Copyright © 2001 John Wiley & Sons, Ltd.     

Optimal control of unsteady compressible viscous flows

The control of complex, unsteady flows is a pacing technology for advances in fluid mechanics. Recently, optimal control theory has become popular as a means of predicting best case controls that can guide the design of practical flow control systems. However, most of the prior work in this area has focused on incompressible flow which precludes many of the important physical flow phenomena that must be controlled in practice including the coupling of fluid dynamics, acoustics, and heat transfer. This paper presents the formulation and numerical solution of a class of optimal boundary control problems governed by the unsteady two‐dimensional compressible Navier–Stokes equations. Fundamental issues including the choice of the control space and the associated regularization term in the objective function, as well as issues in the gradient computation via the adjoint equation method are discussed. Numerical results are presented for a model problem consisting of two counter‐rotating viscous vortices above an infinite wall which, due to the self‐induced velocity field, propagate downward and interact with the wall. The wall boundary control is the temporal and spatial distribution of wall‐normal velocity. Optimal controls for objective functions that target kinetic energy, heat transfer, and wall shear stress are presented along with the influence of control regularization for each case. Copyright © 2002 John Wiley & Sons, Ltd.     

Optimal control of quadratic functionals for affine nonlinear systems

In this paper we analyze the optimal control problem for a class of affine nonlinear systems under the assumption that the associated Lie algebra is nilpotent. The Lie brackets generated by the vector fields which define the nonlinear system represent a remarkable mathematical instrument for the control of affine systems. We determine the optimal control which corresponds to the nilpotent operator of the first order. In particular, we obtain the control that minimizes the energy of the given nonlinear system. Applications of this control to bilinear systems with first order nilpotent operator are considered.     

Optimal control of hyperbolic H-hemivariational inequalities with state constraints

The optimal control problems of hyperbolic H-hemivariational inequalities with the state constraints and nonnornotone multivalued mapping term are considered. The optimal solutions are obtained. In addition, their approximating problems are also studied.     

Optimal control of nonholonomic motion planning for a flee-falling cat

The nonholonomic motion phnning of a free-falling cat is investigated.Nonholonomicity arises in a free-falling cat subject to nonintegrable angle velocity constraints or nonintegrable conservation laws.When the total angular momentum is zero,the motion equation of a free-falling cat is established based on the model of two symmetric rigid bodies and conservation of angular momentum.The control of system can be converted to the problem of nonholonomic motion planning for a free-falling cat.Based on Ritz approximation theory,the Gauss-Newton method for motion planning by a falling cat is proposed.The effectiveness of the numerical algorithm is demonstrated through simulation on model of a free-falling cat.     

Optimal control of nonholonomic motion planning for a free-falling cat

The nonholonomic motion planning of a free-falling cat is investigated. Non-holonomicity arises in a free-falling cat subject to nonintegrable angle velocity constraints or nonintegrable conservation laws. When the total angular momentum is zero, the motion equation of a free-falling cat is established based on the model of two symmetric rigid bodies and conservation of angular momentum. The control of system can be converted to the problem of nonholonomic motion planning for a free-falling cat. Based on Ritz approximation theory, the Gauss-Newton method for motion planning by a falling cat is proposed. The effectiveness of the numerical algorithm is demonstrated through simulation on model of a free-falling cat.     

Numerical solution of steady free‐surface flows by the adjoint optimal shape design method

Numerical solution of flows that are partially bounded by a freely moving boundary is of great importance in practical applications such as ship hydrodynamics. Free‐boundary problems can be reformulated into optimal shape design problems, which can in principle be solved efficiently by the adjoint method. In this work we investigate the suitability of the adjoint shape optimization method for solving steady free‐surface flows. The asymptotic convergence behaviour of the method is determined for free‐surface flows in 2D and 3D. It is shown that the convergence behaviour depends sensitively on the occurrence of critical modes. The convergence behaviour is moreover shown to be mesh‐width independent, provided that proper preconditioning is applied. Numerical results are presented for 2D flow over an obstacle in a channel. The observed convergence behaviour is indeed mesh‐width independent and conform the derived asymptotic estimates. Copyright © 2003 John Wiley & Sons, Ltd.