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NASA NTRS2004

Application of Simulated Annealing and Related Algorithms to TWTA Design

Simulated Annealing (SA) is a stochastic optimization algorithm used to search for global minima in complex design surfaces where exhaustive searches are not computationally feasible. The algorithm is derived by simulating the annealing process, whereby a solid is heated to a liquid state and then cooled slowly to reach thermodynamic equilibrium at each temperature. The idea is that atoms in the solid continually bond and re-bond at various quantum energy levels, and with sufficient cooling time they will rearrange at the minimum energy state to form a perfect crystal. The distribution of energy levels is given by the Boltzmann distribution: as temperature drops, the probability of the presence of high-energy bonds decreases. In searching for an optimal design, local minima and discontinuities are often present in a design surface. SA presents a distinct advantage over other optimization algorithms in its ability to escape from these local minima. Just as high-energy atomic configurations are visited in the actual annealing process in order to eventually reach the minimum energy state, in SA highly non-optimal configurations are visited in order to find otherwise inaccessible global minima. The SA algorithm produces a Markov chain of points in the design space at each temperature, with a monotonically decreasing temperature. A random point is started upon, and the objective function is evaluated at that point. A stochastic perturbation is then made to the parameters of the point to arrive at a proposed new point in the design space, at which the objection function is evaluated as well. If the change in objective function values (Delta)E is negative, the proposed new point is accepted. If (Delta)E is positive, the proposed new point is accepted according to the Metropolis criterion: rho((Delta)f) = exp((-Delta)E/T), where T is the temperature for the current Markov chain. The process then repeats for the remainder of the Markov chain, after which the temperature is decremented and the process repeats. Eventually (and hopefully), a near-globally optimal solution is attained as T approaches zero. Several exciting variants of SA have recently emerged, including Discrete-State Simulated Annealing (DSSA) and Simulated Tempering (ST). The DSSA algorithm takes the thermodynamic analogy one step further by categorizing objective function evaluations into discrete states. In doing so, many of the case-specific problems associated with fine-tuning the SA algorithm can be avoided; for example, theoretical approximations for the initial and final temperature can be derived independently of the case. In this manner, DSSA provides a scheme that is more robust with respect to widely differing design surfaces. ST differs from SA in that the temperature T becomes an additional random variable in the optimization. The system is also kept in equilibrium as the temperature changes, as opposed to the system being driven out of equilibrium as temperature changes in SA. ST is designed to overcome obstacles in design surfaces where numerous local minima are separated by high barriers. These algorithms are incorporated into the optimal design of the traveling-wave tube amplifier (TWTA). The area under scrutiny is the collector, in which it would be ideal to use negative potential to decelerate the spent electron beam to zero kinetic energy just as it reaches the collector surface. In reality this is not plausible due to a number of physical limitations, including repulsion and differing levels of kinetic energy among individual electrons. Instead, the collector is designed with multiple stages depressed below ground potential. The design of this multiple-stage collector is the optimization problem of interest. One remaining problem in SA and DSSA is the difficulty in determining when equilibrium has been reached so that the current Markov chain can be terminated. It has been suggested in recent literature that simulating the thermodynamic properties opecific heat, entropy, and internal energy from the Boltzmann distribution can provide good indicators of having reached equilibrium at a certain temperature. These properties are tested for their efficacy and implemented in SA and DSSA code with respect to TWTA collector optimization.

Radke, Eric M.↗

NASA NTRS1977

Network functions and facilities

Objectives, functions, and organization of the Deep Space Network are summarized. A deep space station, ground communication, and network operations control capabilities are described. Data system configurations for testing, training, and network operations control functions are outlined.

Renzetti, N. A.↗

NASA NTRS1998

Stochastic Methods for Aircraft Design

The global stochastic optimization method, simulated annealing (SA), was adapted and applied to various problems in aircraft design. The research was aimed at overcoming the problem of finding an optimal design in a space with multiple minima and roughness ubiquitous to numerically generated nonlinear objective functions. SA was modified to reduce the number of objective function evaluations for an optimal design, historically the main criticism of stochastic methods. SA was applied to many CFD/MDO problems including: low sonic-boom bodies, minimum drag on supersonic fore-bodies, minimum drag on supersonic aeroelastic fore-bodies, minimum drag on HSCT aeroelastic wings, FLOPS preliminary design code, another preliminary aircraft design study with vortex lattice aerodynamics, HSR complete aircraft aerodynamics. In every case, SA provided a simple, robust and reliable optimization method which found optimal designs in order 100 objective function evaluations. Perhaps most importantly, from this academic/industrial project, technology has been successfully transferred; this method is the method of choice for optimization problems at Northrop Grumman.

Pelz, Richard B.↗

NASA NTRS1986

Closed-loop, pilot/vehicle analysis of the approach and landing task

In the case of approach and landing, it is universally accepted that the pilot uses more than one vehicle response, or output, to close his control loops. Therefore, to model this task, a multi-loop analysis technique is required. The analysis problem has been in obtaining reasonable analytic estimates of the describing functions representing the pilot's loop compensation. Once these pilot describing functions are obtained, appropriate performance and workload metrics must then be developed for the landing task. The optimal control approach provides a powerful technique for obtaining the necessary describing functions, once the appropriate task objective is defined in terms of a quadratic objective function. An approach is presented through the use of a simple, reasonable objective function and model-based metrics to evaluate loop performance and pilot workload. The results of an analysis of the LAHOS (Landing and Approach of Higher Order Systems) study performed by R.E. Smith is also presented.

Anderson, M. R.↗

NASA NTRS2019

Numerical Investigation and Optimization of a Flushwall Injector for Scramjet Applications at Hypervelocity Flow Conditions

An investigation utilizing Reynolds-averaged simulations (RAS) was performed in order to find optimal designs for an interdigitated flushwall injector suitable for scramjet applications at hypervelocity conditions. The flight Mach number, duct height, spanwise width, and injection angle were the design variables selected to maximize two objective functions: the thrust potential and combustion efficiency. A Latin hypercube sampling design-of-experiments method was used to select design points for RAS. A methodology was developed that automated building geometries and generating grids for each design. The ensuing RAS analysis generated the performance database from which the two objective functions of interest were computed using a one-dimensional performance utility. The data were fitted using four surrogate models: an artificial neural network (ANN) model, a cubic polynomial, a quadratic polynomial, and a Kriging model. Variance-based decomposition showed that both objective functions were primarily driven by changes in the duct height. Multiobjective design optimization was performed for all four surrogate models via a genetic algorithm method. Optimal solutions were obtained at the upper and lower bounds of the flight Mach number range. The Kriging model obtained an optimal solution set that predicted high values for both objective functions. Additionally, three challenge points were selected to assess the designs on the Pareto fronts. Further sampling among the designs of the Pareto fronts are required in order to lower the errors and perform more accurate surrogate-based optimization. sed optimization.

Shenoy, Rajiv R.↗

NASA NTRS1988

Kinematic calibration of manipulators with closed loop actuated joints

A method for performing forward kinematic calibration of manipulators having one or more closed-loop-actuated joints is presented. The technique is an extension of algorithms designed for open-loop jointed manipulators. The calibration is equivalent to minimizing an objective function subject to constraints. The objective function is taken as the integral of end-effector position and orientation error. The constraints arise from the closed-loop mechanisms present in the manipulator.

Everett, Louis J.↗

NASA NTRS2019

Numerical Investigation and Optimization of a Flushwall Injector for Scramjet Applications at Hypervelocity Flow Conditions

An investigation utilizing Reynolds-averaged simulations (RAS) was performed in order to demonstrate the use of design and analysis of computer experiments (DACE) methods in Sandia’s DAKOTA software package for surrogate modeling and optimization. These methods were applied to a flow- path fueled with an interdigitated flushwall injector suitable for scramjet applications at hyper- velocity conditions and ascending along a constant dynamic pressure flight trajectory. The flight Mach number, duct height, spanwise width, and injection angle were the design variables selected to maximize two objective functions: the thrust potential and combustion efficiency. Because the RAS of this case are computationally expensive, surrogate models are used for optimization. To build a surrogate model a RAS database is created. The sequence of the design variables comprising the database were generated using a Latin hypercube sampling (LHS) method. A methodology was also developed to automatically build geometries and generate structured grids for each design point. The ensuing RAS analysis generated the simulation database from which the two objective functions were computed using a one-dimensionalization (1D) of the three-dimensional simulation data. The data were fitted using four surrogate models: an artificial neural network (ANN), a cubic polynomial, a quadratic polynomial, and a Kriging model. Variance-based decomposition showed that both objective functions were primarily driven by changes in the duct height. Multiobjective design optimization was performed for all four surrogate models via a genetic algorithm method. Optimal solutions were obtained at the upper and lower bounds of the flight Mach number range. The Kriging model predicted an optimal solution set that exhibited high values for both objective functions. Additionally, three challenge points were selected to assess the designs on the Pareto fronts. Further sampling among the designs of the Pareto fronts may be required to lower the surrogate model errors and perform more accurate surrogate-model-based optimization.

Shenoy, Rajiv R.↗

NASA NTRS2012

Unsteady Adjoint Approach for Design Optimization of Flapping Airfoils

This paper describes the work for optimizing the propulsive efficiency of flapping airfoils, i.e., improving the thrust under constraining aerodynamic work during the flapping flights by changing their shape and trajectory of motion with the unsteady discrete adjoint approach. For unsteady problems, it is essential to properly resolving time scales of motion under consideration and it must be compatible with the objective sought after. We include both the instantaneous and time-averaged (periodic) formulations in this study. For the design optimization with shape parameters or motion parameters, the time-averaged objective function is found to be more useful, while the instantaneous one is more suitable for flow control. The instantaneous objective function is operationally straightforward. On the other hand, the time-averaged objective function requires additional steps in the adjoint approach; the unsteady discrete adjoint equations for a periodic flow must be reformulated and the corresponding system of equations solved iteratively. We compare the design results from shape and trajectory optimizations and investigate the physical relevance of design variables to the flapping motion at on- and off-design conditions.

Lee, Byung Joon↗

NASA NTRS1988

Structural optimization of rotor blades with integrated dynamics and aerodynamics

The problem of structural optimization of helicopter rotor blades with integrated dynamic and aerodynamic design considerations is addressed. Results of recent optimization work on rotor blades for minimum weight with constraints on multiple coupled natural flap-lag frequencies, blade autorotational inertia and centrifugal stress has been reviewed. A strategy has been defined for the ongoing activities in the integrated dynamic/aerodynamic optimization of rotor blades. As a first step, the integrated dynamic/airload optimization problem has been formulated. To calculate system sensitivity derivatives necessary for the optimization recently developed, Global Sensitivity Equations (GSE) are being investigated. A need for multiple objective functions for the integrated optimization problem has been demonstrated and various techniques for solving the multiple objective function optimization are being investigated. The method called the Global Criteria Approach has been applied to a test problem with the blade in vacuum and the blade weight and the centrifugal stress as the multiple objectives. The results indicate that the method is quite effective in solving optimization problems with conflicting objective functions.

Chattopadhyay, Aditi↗

NASA NTRS1989

Structural optimization of rotor blades with integrated dynamics and aerodynamics

Chattopadhyay, Aditi↗

NASA NTRS1999

Structural Optimization for Reliability Using Nonlinear Goal Programming

This report details the development of a reliability based multi-objective design tool for solving structural optimization problems. Based on two different optimization techniques, namely sequential unconstrained minimization and nonlinear goal programming, the developed design method has the capability to take into account the effects of variability on the proposed design through a user specified reliability design criterion. In its sequential unconstrained minimization mode, the developed design tool uses a composite objective function, in conjunction with weight ordered design objectives, in order to take into account conflicting and multiple design criteria. Multiple design criteria of interest including structural weight, load induced stress and deflection, and mechanical reliability. The nonlinear goal programming mode, on the other hand, provides for a design method that eliminates the difficulty of having to define an objective function and constraints, while at the same time has the capability of handling rank ordered design objectives or goals. For simulation purposes the design of a pressure vessel cover plate was undertaken as a test bed for the newly developed design tool. The formulation of this structural optimization problem into sequential unconstrained minimization and goal programming form is presented. The resulting optimization problem was solved using: (i) the linear extended interior penalty function method algorithm; and (ii) Powell's conjugate directions method. Both single and multi-objective numerical test cases are included demonstrating the design tool's capabilities as it applies to this design problem.

El-Sayed, Mohamed E.↗

NASA NTRS1985

Optimal design application on the advanced aeroelastic rotor blade

The vibration and performance optimization procedure using regression analysis was successfully applied to an advanced aeroelastic blade design study. The major advantage of this regression technique is that multiple optimizations can be performed to evaluate the effects of various objective functions and constraint functions. The data bases obtained from the rotorcraft flight simulation program C81 and Myklestad mode shape program are analytically determined as a function of each design variable. This approach has been verified for various blade radial ballast weight locations and blade planforms. This method can also be utilized to ascertain the effect of a particular cost function which is composed of several objective functions with different weighting factors for various mission requirements without any additional effort.

Wei, F. S.↗

NASA NTRS2015

Impedance Eduction in Sound Fields With Peripherally Varying Liners and Flow

A two-dimensional impedance eduction theory is extended to three-dimensional sound fields and peripherally varying duct liners. The approach is to first measure the acoustic pressure field at a series of flush-mounted wall microphones located around the periphery of the flow duct. The numerical solution for the acoustic pressure field at these microphones is also obtained by solving the three-dimensional convected Helmholtz equation using the finite element method. A quadratic objective function based on the difference between the measured and finite element solution is constructed and the unknown impedance function is obtained by minimizing this objective function. Impedance spectra educed for two uniform-structure liners (a wire-mesh and a conventional liner) and a hard-soft-hard peripherally varying liner (for which the soft segment is that of the conventional liner) are presented. Results are presented at three mean flow Mach numbers and fourteen sound source frequencies. The impedance spectra of the uniform-structure liners are also computed using a two-dimensional impedance eduction theory. The primary conclusions of the study are: 1) when measured data is used with the uniform-structure liners, the three-dimensional theory reproduces the same impedance spectra as the two-dimensional theory except for frequencies corresponding to very low or very high liner attenuation; and 2) good agreement between the educed impedance spectra of the uniform structure conventional liner and the soft segment of the peripherally varying liner is obtained.

Watson, W. R.↗

NASA NTRS1988

Use of adjoint methods in the probabilistic finite element approach to fracture mechanics

The adjoint method approach to probabilistic finite element methods (PFEM) is presented. When the number of objective functions is small compared to the number of random variables, the adjoint method is far superior to the direct method in evaluating the objective function derivatives with respect to the random variables. The PFEM is extended to probabilistic fracture mechanics (PFM) using an element which has the near crack-tip singular strain field embedded. Since only two objective functions (i.e., mode I and II stress intensity factors) are needed for PFM, the adjoint method is well suited.

Liu, Wing Kam↗

NASA NTRS1991

Aerodynamic design optimization using sensitivity analysis and computational fluid dynamics

A new and efficient method is presented for aerodynamic design optimization, which is based on a computational fluid dynamics (CFD)-sensitivity analysis algorithm. The method is applied to design a scramjet-afterbody configuration for an optimized axial thrust. The Euler equations are solved for the inviscid analysis of the flow, which in turn provides the objective function and the constraints. The CFD analysis is then coupled with the optimization procedure that uses a constrained minimization method. The sensitivity coefficients, i.e. gradients of the objective function and the constraints, needed for the optimization are obtained using a quasi-analytical method rather than the traditional brute force method of finite difference approximations. During the one-dimensional search of the optimization procedure, an approximate flow analysis (predicted flow) based on a first-order Taylor series expansion is used to reduce the computational cost. Finally, the sensitivity of the optimum objective function to various design parameters, which are kept constant during the optimization, is computed to predict new optimum solutions. The flow analysis of the demonstrative example are compared with the experimental data. It is shown that the method is more efficient than the traditional methods.

Baysal, Oktay↗

NASA NTRS1991

Towards practical control design using neural computation

The objective is to develop neural network based control design techniques which address the issue of performance/control effort tradeoff. Additionally, the control design needs to address the important issue if achieving adequate performance in the presence of actuator nonlinearities such as position and rate limits. These issues are discussed using the example of aircraft flight control. Given a set of pilot input commands, a feedforward net is trained to control the vehicle within the constraints imposed by the actuators. This is achieved by minimizing an objective function which is the sum of the tracking errors, control input rates and control input deflections. A tradeoff between tracking performance and control smoothness is obtained by varying, adaptively, the weights of the objective function. The neurocontroller performance is evaluated in the presence of actuator dynamics using a simulation of the vehicle. Appropriate selection of the different weights in the objective function resulted in the good tracking of the pilot commands and smooth neurocontrol. An extension of the neurocontroller design approach is proposed to enhance its practicality.

Troudet, Terry↗

NASA NTRS1984

Multiobjective optimization in structural design with uncertain parameters and stochastic processes

The application of multiobjective optimization techniques to structural design problems involving uncertain parameters and random processes is studied. The design of a cantilever beam with a tip mass subjected to a stochastic base excitation is considered for illustration. Several of the problem parameters are assumed to be random variables and the structural mass, fatigue damage, and negative of natural frequency of vibration are considered for minimization. The solution of this three-criteria design problem is found by using global criterion, utility function, game theory, goal programming, goal attainment, bounded objective function, and lexicographic methods. It is observed that the game theory approach is superior in finding a better optimum solution, assuming the proper balance of the various objective functions. The procedures used in the present investigation are expected to be useful in the design of general dynamic systems involving uncertain parameters, stochastic process, and multiple objectives.

Rao, S. S.↗

NASA NTRS2005

A Quasi-3-D Theory for Impedance Eduction in Uniform Grazing Flow

A 2-D impedance eduction methodology is extended to quasi-3-D sound fields in uniform or shearing mean flow. We introduce a nonlocal, nonreflecting boundary condition to terminate the duct and then educe the impedance by minimizing an objective function. The introduction of a parallel, sparse, equation solver significantly reduces the wall clock time for educing the impedance when compared to that of the sequential band solver used in the 2-D methodology. The accuracy, efficiency, and robustness of the methodology is demonstrated using two examples. In the first example, we show that the method reproduces the known impedance of a ceramic tubular test liner. In the second example, we illustrate that the approach educes the impedance of a four-segment liner where the first, second, and fourth segments consist of a perforated face sheet bonded to honeycomb, and the third segment is a cut from the ceramic tubular test liner. The ability of the method to educe the impedances of multisegmented liners has the potential to significantly reduce the amount of time and cost required to determine the impedance of several uniform liners by allowing them to be placed in series in the test section and to educe the impedance of each segment using a single numerical experiment. Finally, we probe the objective function in great detail and show that it contains a single minimum. Thus, our objective function is ideal for use with local, inexpensive, gradient-based optimizers.

Watson, W. R.↗