A parallel between Keplerian integrals and integrals of the adjoint equations
Keplerian integrals analogy to integrals of adjoint equations suggested by optimum space navigation trajectories considerations
Burns, I. F.↗
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Admitting the Inadmissible: Adjoint Formulation for Incomplete Cost Functionals in Aerodynamic Optimization
We derive the adjoint equations for problems in aerodynamic optimization which are improperly considered as "inadmissible." For example, a cost functional which depends on the density, rather than on the pressure, is considered "inadmissible" for an optimization problem governed by the Euler equations. We show that for such problems additional terms should be included in the Lagrangian functional when deriving the adjoint equations. These terms are obtained from the restriction of the interior PDE to the control surface. Demonstrations of the explicit derivation of the adjoint equations for "inadmissible" cost functionals are given for the potential, Euler, and Navier-Stokes equations.
Arian, Eyal↗
Application of adjoint operators to neural learning
A technique for the efficient analytical computation of such parameters of the neural architecture as synaptic weights and neural gain is presented as a single solution of a set of adjoint equations. The learning model discussed concentrates on the adiabatic approximation only. A problem of interest is represented by a system of N coupled equations, and then adjoint operators are introduced. A neural network is formalized as an adaptive dynamical system whose temporal evolution is governed by a set of coupled nonlinear differential equations. An approach based on the minimization of a constrained neuromorphic energylike function is applied, and the complete learning dynamics are obtained as a result of the calculations.
Barhen, J.↗
Learning a trajectory using adjoint functions and teacher forcing
A new methodology for faster supervised temporal learning in nonlinear neural networks is presented which builds upon the concept of adjoint operators to allow fast computation of the gradients of an error functional with respect to all parameters of the neural architecture, and exploits the concept of teacher forcing to incorporate information on the desired output into the activation dynamics. The importance of the initial or final time conditions for the adjoint equations is discussed. A new algorithm is presented in which the adjoint equations are solved simultaneously (i.e., forward in time) with the activation dynamics of the neural network. We also indicate how teacher forcing can be modulated in time as learning proceeds. The results obtained show that the learning time is reduced by one to two orders of magnitude with respect to previously published results, while trajectory tracking is significantly improved. The proposed methodology makes hardware implementation of temporal learning attractive for real-time applications.
Toomarian, Nikzad B.↗
Low-thrust trajectory analysis for the geosynchronous mission
Methodology employed in development of a computer program designed to analyze optimal low-thrust trajectories is described, and application of the program to a Solar Electric Propulsion Stage (SEPS) geosynchronous mission is discussed. To avoid the zero inclination and eccentricity singularities which plague many small-force perturbation techniques, a special set of state variables (equinoctial) is used. Adjoint equations are derived for the minimum time problem and are also free from the singularities. Solutions to the state and adjoint equations are obtained by both orbit averaging and precision numerical integration; an evaluation of these approaches is made.
Jasper, T. P.↗
Modeling and adjoints for continuous systems.
Trajectory differential equations in state- variable form, presenting compatible difference equation adjoint scheme for trajectory integration
Denham, W. F.↗
Performance Trades Study for Robust Airfoil Shape Optimization
From time to time, existing aircraft need to be redesigned for new missions with modified operating conditions such as required lift or cruise speed. This research is motivated by the needs of conceptual and preliminary design teams for smooth airfoil shapes that are similar to the baseline design but have improved drag performance over a range of flight conditions. The proposed modified profile optimization method (MPOM) modifies a large number of design variables to search for nonintuitive performance improvements, while avoiding off-design performance degradation. Given a good initial design, the MPOM generates fairly smooth airfoils that are better than the baseline without making drastic shape changes. Moreover, the MPOM allows users to gain valuable information by exploring performance trades over various design conditions. Four simulation cases of airfoil optimization in transonic viscous ow are included to demonstrate the usefulness of the MPOM as a performance trades study tool. Simulation results are obtained by solving fully turbulent Navier-Stokes equations and the corresponding discrete adjoint equations using an unstructured grid computational fluid dynamics code FUN2D.
Li, Wu↗
Adjoint-based Sensitivities of Flutter Predictions based on the Linearized Frequency-domain Approach
Flutter is a critical factor in designing and certifying aircraft. The linearized frequency-domain method offers a lower cost alternative to time-marching computational fluid dynamics for high-fidelity flutter analysis. In this work, adjoint-based sensitivities are added to a flutter analysis based on the linearized frequency-domain method to efficiently compute derivatives of flutter cost functions with respect to design variables or uncertain parameters. The derivation of the adjoint equations, which involve complications such as derivatives of a nonlinear generalized eigenvalue problem with complex-valued inputs and derivatives of the linearized Navier-Stokes equations, is provided. The implemented adjoint terms and derivatives are verified before demonstrating the approach for derivatives of flutter dynamic pressure with respect to Mach number for the AGARD 445.6 wing.
Aeroelasticity↗
Adjoint-Based Methodology for Time-Dependent Optimization
This paper presents a discrete adjoint method for a broad class of time-dependent optimization problems. The time-dependent adjoint equations are derived in terms of the discrete residual of an arbitrary finite volume scheme which approximates unsteady conservation law equations. Although only the 2-D unsteady Euler equations are considered in the present analysis, this time-dependent adjoint method is applicable to the 3-D unsteady Reynolds-averaged Navier-Stokes equations with minor modifications. The discrete adjoint operators involving the derivatives of the discrete residual and the cost functional with respect to the flow variables are computed using a complex-variable approach, which provides discrete consistency and drastically reduces the implementation and debugging cycle. The implementation of the time-dependent adjoint method is validated by comparing the sensitivity derivative with that obtained by forward mode differentiation. Our numerical results show that O(10) optimization iterations of the steepest descent method are needed to reduce the objective functional by 3-6 orders of magnitude for test problems considered.
Yamaleev, N. K.↗
Recent Enhancements to Modeling Sonic Boom Propagation using Augmented Burgers’ Equation
Sonic boom propagation through the atmosphere is modeled with an augmented Burgers’ equation which includes nonlinearity and loss mechanisms. This work details an updated discretization of the governing equations which is fully conservative and duality preserving. Adjoint equations, for all the mechanisms involved, are re-derived and implemented using adjoint consistent discretizations. Computation of loudness metrics is performed using digital filters. The updated implementation is demonstrated and compared against the previous formulation for selected cases, and the differences are documented and discussed. The improved discretization results in faster mesh convergence of the loudness metrics and substantially de-creases runtime. In addition, the adjoint solutions provide mesh-converged gradients which are free from spurious oscillations.
Sonic Boom↗
Martian Aerocapture Terminal Point Guidance: A Reference Path Optimization Study
An effective method of terminal point guidance is to employ influence coefficients, which are solved from a set of differential equations adjoint to the linearized perturbations of the equations of motion about a reference trajectory. Hence, to optimize this type of guidance, one must first optimize the reference trajectory that the guidance is based upon. This study concentrates on various methods to optimize a reference trajectory for a Martian aerocapture maneuver, including a parametric analysis and first order gradient method. Resulting reference trajectories were tested in separate 2000 6-DOF Monte Carlo runs, using the Atmospheric Guidance Algorithm Testbed for the Mars Surveyor Program 2001 (MSP '01) Orbiter. These results were compared to an August 1998 study using the same terminal point control guidance algorithm and simulation testbed. Satisfactory improvements over the 1998 study are amply demonstrated.
Ro, Theodore U.↗
Variational Methods in Design Optimization and Sensitivity Analysis for Two-Dimensional Euler Equations
Variational methods (VM) sensitivity analysis employed to derive the costate (adjoint) equations, the transversality conditions, and the functional sensitivity derivatives. In the derivation of the sensitivity equations, the variational methods use the generalized calculus of variations, in which the variable boundary is considered as the design function. The converged solution of the state equations together with the converged solution of the costate equations are integrated along the domain boundary to uniquely determine the functional sensitivity derivatives with respect to the design function. The application of the variational methods to aerodynamic shape optimization problems is demonstrated for internal flow problems at supersonic Mach number range. The study shows, that while maintaining the accuracy of the functional sensitivity derivatives within the reasonable range for engineering prediction purposes, the variational methods show a substantial gain in computational efficiency, i.e., computer time and memory, when compared with the finite difference sensitivity analysis.
Ibrahim, A. H.↗
Distributed Blowing and Suction for the Purpose of Streak Control in a Boundary Layer Subjected to a Favorable Pressure Gradient
An analysis of the optimal control by blowing and suction in order to generate stream- wise velocity streaks is presented. The problem is examined using an iterative process that employs the Parabolized Stability Equations for an incompressible uid along with its adjoint equations. In particular, distributions of blowing and suction are computed for both the normal and tangential velocity perturbations for various choices of parameters.
Forgoston, Eric↗
Inclusion of known integrals in the optimal trajectory problem
The classical problem of determination of the rocket trajectory which minimizes mass expenditure during motion between two points in the field of a single gravitating body is analyzed. The known integrals of the system are incorporated into the adjoint equations resulting in a reduction from a seventh-order adjoint system to a third-order adjoint system. The first case which is treated in that of planar motion under specific end conditions. In this case a regularization of the recently derived equations is achieved. The general three-dimensional case is also reduced from a seventh-order adjoint system to a third-order adjoint system. In this case a regularization has not been found.
Burns, R. E.↗
Maneuvering of flexible spacecraft with application to SCOLE
This paper is concerned with the derivation of the equations of motion for the Spacecraft Control Laboratory Experiment (SCOLE). For future reference, the equations of motion of a similar structure orbiting the earth are also derived. The structure is assumed to undergo large rigid-body maneuvers and small elastic deformations. A perturbation approach is presented where the quantities defining the rigid-body maneuver are assumed to be relatively large, with the elastic deformations and deviations from the rigid-body maneuver being relatively small. The perturbation equations have the form of linear, non-self-adjoint equations with time-dependent coefficients. An active control technique can then be formulated to permit maneuvering of the spacecraft and simultaneously suppressing the elastic vibration.
Meirovitch, L.↗
Computing Sensitivities in Evolutionary Systems: A Real-time Reduced Order Modeling Strategy
We present a new methodology for computing sensitivities in evolutionary systems using a model-driven low-rank approximation. To this end, we formulate a variational principle that seeks to minimize the distance between the time derivative of the reduced approximation and sensitivity dynamics. The first order optimality condition of the variational principle leads to a system of closed form evolution equations for an orthonormal basis and corresponding sensitivity coefficients. This approach allows for the computation of sensitivities with respect to a large number of parameters in an accurate and tractable manner by extracting correlations between different sensitivities on the fly. The presented method requires solving forward evolution equations, sidestepping the restrictions imposed by the forward/backward workflow of adjoint sensitivities. For example, the presented method, unlike the adjoint equation, does not impose any input/output load and can be used in applications in which real-time sensitivities are of interest. We demonstrate the utility of the method for three test cases: (1) computing sensitivity with respect to model parameters in the Rössler system, (2) computing sensitivity with respect to an infinite-dimensional forcing parameter in the chaotic Kuramoto--Sivashinsky equation, and (3) computing sensitivity with respect to reaction parameters for species transport in a turbulent reacting flow.
Reduced order model↗
A solution of Mx(double dot) + Cx(dot) + Kx = 0 applicable to the design of active dampers
A solution is presented for the equations of motion for the damped linear oscillator, Mx(double dot) + Cx(dot) + Kx = 0. The algorithm solves a transformed set of equations in terms of the modal variables of the undamped system and, at the same time, solves the adjoint equation of the transformed problem. The adjoint solution is normalized to give the inverse of the solution matrix of the transformed problem. The normalized inverse is useful in design for direct computation of sensitivity derivatives of damping ratios with respect to damping rates. The algorithm is programmed to reduce storage requirements by a factor of three-fourths compared to standard complex eigenvalue subroutines. A numerical example is included.
Thurston, G. A.↗
The Dirichlet problem for the two-dimensional Helmholtz equation for an open boundary
Development of a complete theory of the two-dimensional Dirichlet problem for an open boundary. It is shown that the solution of the Dirichlet problem for an open boundary requires the solution of a Fredholm integral equation of the first kind. Although a Fredholm integral equation of the first kind usually has no solution if the kernel is continuous, owing to the logarithmic singularity of the kernel, the equation in this case is converted to a singular integral equation with a Cauchy kernel. It is proven that the homogeneous adjoint equation of the singular integral equation has no nonzero solution. By virtue of this result, and with the aid of an existence theorem known in the theory of singular integral equations, the existence of solutions of the singular integral equation, and then of the unique solution of the Fredholm integral equation of the first kind is proved.
Hayashi, Y.↗