Optimal Trajectory of Underwater Manipulator Using Adjoint Variable Method for Reducing Drag

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.

Rate optimization of fractional flow reservoir model based on the continuous adjoint method

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

Viscous attenuation can have a strong impact on seismic wave propagation, but it is rarely taken into account in full waveform inversion (FWI). When viscoelasticity is considered in time domain FWI, the displacement formulation of the wave equation is usually used instead of the popular velocity–stress formulation. However, inversion schemes rely on the adjoint equations, which are quite different for the velocity–stress formulation than for the displacement formulation. In this paper, we apply the adjoint state method to the isotropic viscoelastic wave equation in the velocity–stress formulation based on the generalized standard linear solid rheology. By applying linear transformations to the wave equation before deriving the adjoint state equations, we obtain two symmetric sets of partial differential equations for the forward and adjoint variables. The resulting sets of equations only differ by a sign change and can be solved by the same numerical implementation. We also investigate the crosstalk between parameter classes (velocity and attenuation) of the viscoelastic equation. More specifically, we show that the attenuation levels can be used to recover the quality factors of P and S waves, but that they are very sensitive to velocity errors. Finally, we present a synthetic example of viscoelastic FWI in the context of monitoring CO₂ geological sequestration. We show that FWI based on our formulation can indeed recover P- and S-wave velocities and their attenuation levels when attenuation is high enough. Both changes in velocity and attenuation levels recovered with FWI can be used to track the CO₂ plume during and after injection. Further studies are required to evaluate the performance of viscoelastic FWI on real data.

The Pontryagin maximum principle is used to prove a theorem concerning optimal control in regional macroeconomics. A boundary value problem for optimal trajectories of the state and adjoint variables is formulated, and optimal curves are analyzed. An algorithm is proposed for solving the boundary value problem of optimal control. The performance of the algorithm is demonstrated by computing an optimal control and the corresponding optimal trajectories.

ABSTRACTThis paper deals with the optimal control of constrained mechanical systems, with potential additional kinematic constraints at the final time. Correspondingly, the equations of motion of the underlying mechanical system assume the form of differential‐algebraic equations with end constraints. The proposed discretisation of the optimality conditions yields a scheme which is capable of preserving control angular momentum maps resulting from the rotational symmetry of the underlying optimal control problem. The numerical solution of the discretised system is first tested with two solution strategies: Monolithic and staggered approaches, and then also solved with hybrid approaches, which combine salient features of each individual strategy. The monolithic strategy solves all the optimality conditions for all time steps as a single system of non‐linear equations and relies on a Newton‐Raphson scheme, which guarantees quadratic rates of convergence in the vicinity of the optimal solution trajectory. The staggered strategy is based on the Forward‐Backward Sweep Method (FBSM), where state and adjoint equations are solved separately, and the control equations provide an update of the control variables, which we here achieve with also a Newton‐Raphson scheme. The proposed hybrid strategies combine the advantages of a conventional gradient‐based FBSM with the individual Newton‐based solution procedures once the solution is close to the optimal trajectory. The strategies are developed and compared through three representative numerical examples, which show that all schemes yield very similar solutions. However, the hybrid approaches become more advantageous in the computation time when the time‐step decreases or the size of the problem increases.

The problem of calculating gradients for discretized optimal control problems is addressed. These gradients are needed by a sequential quadratic programming (SQP) method, which is used to solve the discretized optimal control problem. Two approaches for gradient calculation are discussed and compared numerically. The first approach uses sensitivity equations, the second uses adjoint equations. It turns out that it depends on the problem dimensions which approach is more preferable compared to the other. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Superconvergence and a posteriori error estimates of splitting positive definite mixed finite element methods for elliptic optimal control problems

Open-loop optimal control of batch chromatographic separation processes using direct collocation

A new generalization of boundary condition iteration (BCI) methods is developed, based on a suggestion of Denn and Aris. This simplifies to Horn's equation for up to two state equations and to previous boundary iteration methods when is explicitly soluble for as a function of the constants, the state and the adjoint variables. The new general method is also applicable when this latter function is unobtainable.Distinct improvement in the convergence rates of the existing BCI methods has been obtained through the introduction of a correction specific to each state variable.A convergence procedure for use with Horn's equation is proposed and the resultant algorithm has desirable properties.

The effects of price- and output-stabilising policies in an interdependent world economy

Three methods for design sensitivity analysis such as finite difference method(FDM), direct differentiation method(DDM) and adjoint variable method(AVM) are well known. FDM and DDM for design sensitivity analysis cost too much when the number of design variables is too large. An AVM is required to compute adjoint variables from the simultaneous linear system equation, the so-called adjoint equation. Because the adjoint equation is independent of the number of design variables, an AVM is efficient for when number of design variables is too large. In this study, AVM has been extended to the eigenproblem of damped systems whose eigenvlaues and eigenvectors are complex numbers. Moreover, this method is implemented into a commercial finite element analysis program by means of the semi-analytical method to show applicability of the developed method into practical structural problems. The proposed method is compared with FDM and verified its accuracy for analytical and practical cases.

Estimation of inlet temperature of a developing fluid flow in a parallel plate channel

Thecompletesetofthestateand adjoint differentialequationsforthenonsingularequinoctialorbitelementsexpressed in polar coordinatesand written in terms of the truelongitudeispresented. Previous formulations adopted the equinoctial frame as the orbital frame for component resolution of the various perturbation accelerations. The consideration of the position dependency of the J2 acceleration in the context of precision integrated orbital transfer trajectories led us to adopt the more convenient polar frame coupled with the use of the true longitude as the accessory variable needed in the description of the variational equations, inasmuch as it is more dife cult to generate the contribution of the J2 perturbation to the adjoint equations if the variational equations are left in terms of the eccentric longitude. I. Introduction T HEconsideration of thenonsingularequinoctialorbitelements has been of great benee t in trajectory propagation and optimization.References1 ‐7 madeuseofthecorresponding variational equations resolved in the equinoctial orbital coordinate system. In Refs. 8‐12, modern numerical methods based on the techniques of collocation and nonlinear programming are applied to the solutionoflow-thrustEarth-boundandinterplanetarytrajectories.These direct methods have introduced robustness characteristics to the optimization process and were e rst used with great success by launch vehicle trajectory optimization and performance analysts. In particular, Betts 9,10 uses a variety of equinoctial element sets within the framework of the direct method. Following the more traditional approach based on the calculus of variations and the indirect shooting method,thispaper developsthe differentialequationsforthe adjoint or multiplier variables to be numerically integrated simultaneously withthestatedifferentialequationstosolvethetwo-point-boundaryvalue orbit transfer problem. As noted by Betts, the elements of the matrix of partial derivatives of the Hamiltonian with respect to the equinoctial elements derived here in analytic form can be used to developacertainanalyticJacobianmatrixinconnectionwithnonlinear programming (NLP) solvers used with direct methods, thereby saving considerable computation time with respect to the adoption ofa numerically determined Jacobian. The variational equations are usually expressed in terms of the eccentric longitude, which can either be left as an accessory variable or be adopted as the fastor sixth orbital element. In Ref. 3, the averaged J2 perturbation effect was accounted for in the design of the optimal transfer trajectory. However, when the exact or rather position-dependent J2 effect must be considered in the context of precision integrated transfer solutions, the J2 perturbation acceleration components become dependent on the eccentric longitude. Because these components are given in the rotating Euler‐Hill or polar frame, it is much simpler to use this frame as the orbital frame instead of the equinoctial framebecause otherwise the transformed expressions for these components become more complicated. These expressions are still quite complicated if expressed in terms of the eccentric longitude even if the polar frame is used. This is important because we must take the partial derivatives of these expressions with respect to the elements to generate the adjoint differential equations. This task is made easier by expressing the variational equations for the J2 perturbation in terms of the true

A fixed-interval smoothing algorithm for linear systems with Markovian switching coefficients is developed in this paper. This smoothing algorithm uses two multiple model-filters, where one of the filters propagates in the forward-time direction and the other one propagates in the backward-time direction. Simulation results of target trajectory reconstruction for a manoeuvring target will be given to compare the performance of this smoothing algorithm to a multiple-model filter and a smoother designed using a single model. >