Surface roughness effects on radiant transfer between surfaces
Radiative heat transfer between opaque interacting surfaces for equal temperature adjoint plates with one dimensional surface roughness profile
Hering, R. G.↗
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Surface roughness effects on radiant energy interchange
Surface roughness effects on thermal radiant energy interchange between opaque interacting surfaces formulated for unequal temperature adjoint plates
Hering, R. G.↗
Surface roughness effects on equilibrium temperature of interacting surfaces
Radiatively interacting adjoint plates in presence of collimated solar flux, considering surface roughness effects on equilibrium temperature distribution
Hering, R. G.↗
Comparison of a discrete steepest ascent method with the continuous steepest ascent method for optimal programing
A discrete steepest ascent method which allows controls which are not piecewise constant (for example, it allows all continuous piecewise linear controls) was derived for the solution of optimal programming problems. This method is based on the continuous steepest ascent method of Bryson and Denham and new concepts introduced by Kelley and Denham in their development of compatible adjoints for taking into account the effects of numerical integration. The method is a generalization of the algorithm suggested by Canon, Cullum, and Polak with the details of the gradient computation given. The discrete method was compared with the continuous method for an aerodynamics problem for which an analytic solution is given by Pontryagin's maximum principle, and numerical results are presented. The discrete method converges more rapidly than the continuous method at first, but then for some undetermined reason, loses its exponential convergence rate. A comparsion was also made for the algorithm of Canon, Cullum, and Polak using piecewise constant controls. This algorithm is very competitive with the continuous algorithm.
Childs, A. G.↗
Variational differential equations for engineering type trajectories close to a planet with an atmosphere
The differential equations for the adjoint variables are derived and coded in FORTRAN. The program is written in a form to either take into account or neglect thrust, aerodynamic forces, planet rotation and oblateness, and altitude dependent winds.
Dickmanns, E. D.↗
A shielding application of perturbation theory to determine changes in neutron and gamma doses due to changes in shield layers
The perturbation theory for fixed sources was applied to radiation shielding problems to determine changes in neutron and gamma ray doses due to changes in various shield layers. For a given source and detector position the perturbation method enables dose derivatives due to all layer changes to be determined from one forward and one inhomogeneous adjoint calculation. The direct approach requires two forward calculations for the derivative due to a single layer change. Hence, the perturbation method for obtaining dose derivatives permits an appreciable savings in computation for a multilayered shield. For an illustrative problem, a comparison was made of the fractional change in the dose per unit change in the thickness of each shield layer as calculated by perturbation theory and by successive direct calculations; excellent agreement was obtained between the two methods.
Fieno, D.↗
An Optimization study on a hybrid computer
The maximum principle is applied to minimum-time optimal-control problems, and an optimization algorithm is presented which can be implemented on a hybrid computer. The state and adjoint equations are set up on ASTRAC 2, a high-speed analog computer capable of 1000 differential equation solutions per second. The optimization algorithm is implemented on a PDP-9, an 18-bit, digital computer. The optimization scheme has global and local search phases and uses a vector optimization criterion. Second and third-order bang-bang control systems are studied as examples.
Gonzalez, R. S.↗
Optimal startup control of a jacketed tubular reactor.
The optimal startup policy of a jacketed tubular reactor, in which a first-order, reversible, exothermic reaction takes place, is presented. A distributed maximum principle is presented for determining weak necessary conditions for optimality of a diffusional distributed parameter system. A numerical technique is developed for practical implementation of the distributed maximum principle. This involves the sequential solution of the state and adjoint equations, in conjunction with a functional gradient technique for iteratively improving the control function.
Hahn, D. R.↗
Surface roughness effects on equilibrium temperature.
An analysis is presented for evaluation of equilibrium temperature distribution on radiatively adiabatic, adjoint planes which are uniformly irradiated by a collimated solar flux. The analysis employs a semigrey spectral model. Radiation properties for surface emitted radiation are obtained from the expressions of electromagnetic theory for smooth surfaces. Rough surface properties for solar radiation are given by the Beckmann bidirectional reflectance model. Numerical solutions to the governing equations yield equilibrium temperature distributions for a range of the influencing parameters. Surface roughness has little influence on equilibrium temperature for materials with high values for solar absorptance. However, for low or intermediate values of solar absorptance, roughness effects on the spatial distribution of reflected solar radiation can significantly alter equilibrium temperature particularly at surface elements where radiant interaction is small.
Houchens, A. F.↗
Some impulsive rendezvous trajectories and their possible optimality.
Two- and three-impulse trajectories are investigated for fixed-time, fixed-angle rendezvous between vacant circular coplanar orbits, for trip angles less than, or equal to 2 pi in magnitude. For two-impulse trajectories, general features of the characteristic velocity function are outlined. Parameters of the intermediate orbit are reviewed. Attention is given to limiting cases. Computation of the adjoint system helps to define the domain of possible optimality foajectories: it is a closed domain in the trip time, trip angle plane. Waiting periods on terminal orbits are considered. The domain of possible optimality is defined using Lawden's primer vrtory. This domain extends to infinity if the radius ratio of terminal orbits is less than 15.6. Three-impulse trajectories are tried in cases where two-impulse trajectories, with or without cost, have been found nonoptimal. Improvements on the characteristic velocity are thus obtained.
Peltier, J. P.↗
Eigenvalues of singular differential operators by finite difference methods. I.
Approximation of the eigenvalues of certain self-adjoint operators defined by a formal differential operator in a Hilbert space. In general, two problems are studied. The first is the problem of defining a suitable Hilbert space operator that has eigenvalues. The second problem concerns the finite difference operators to be used.
Baxley, J. V.↗
Eigenvalues of singular differential operators by finite difference methods. II.
Note is made of an earlier paper which defined finite difference operators for the Hilbert space L2(m), and gave the eigenvalues for these operators. The present work examines eigenvalues for higher order singular differential operators by using finite difference methods. The two self-adjoint operators investigated are defined by a particular value in the same Hilbert space, L2(m), and are strictly positive with compact inverses. A class of finite difference operators is considered, with the idea of application to the theory of Toeplitz matrices. The approximating operators consist of a good approximation plus a perturbing operator.
Baxley, J. V.↗
Shielding application of perturbation theory to determine changes in neutron and gamma doses due to changes in shield layers
Perturbation theory formulas were derived and applied to determine changes in neutron and gamma-ray doses due to changes in various radiation shield layers for fixed sources. For a given source and detector position, the perturbation method enables dose derivatives with respect to density, or equivalently thickness, for every layer to be determined from one forward and one inhomogeneous adjoint calculation. A direct determination without the perturbation approach would require two forward calculations to evaluate the dose derivative due to a change in a single layer. Hence, the perturbation method for obtaining dose derivatives requires fewer computations for design studies of multilayer shields. For an illustrative problem, a comparison was made of the fractional change in the dose per unit change in the thickness of each shield layer in a two-layer spherical configuration as calculated by perturbation theory and by successive direct calculations; excellent agreement was obtained between the two methods.
Fieno, D.↗
A mesh gradient technique for numerical optimization
A class of successive-improvement optimization methods in which directions of descent are defined in the state space along each trial trajectory are considered. The given problem is first decomposed into two discrete levels by imposing mesh points. Level 1 consists of running optimal subarcs between each successive pair of mesh points. For normal systems, these optimal two-point boundary value problems can be solved by following a routine prescription if the mesh spacing is sufficiently close. A spacing criterion is given. Under appropriate conditions, the criterion value depends only on the coordinates of the mesh points, and its gradient with respect to those coordinates may be defined by interpreting the adjoint variables as partial derivatives of the criterion value function. In level 2, the gradient data is used to generate improvement steps or search directions in the state space which satisfy the boundary values and constraints of the given problem.
Willis, E. A., Jr.↗
A shielding application of perturbation theory to determine changes in neutron and gamma doses due to changes in shield layers
Perturbation theory for fixed sources was applied to radiation shielding problems to determine changes in neutron and gamma ray doses due to changes in various shield layers. For a given source and detector position, the perturbation method enables dose derivatives due to all layer changes to be determined from one forward and one inhomogeneous adjoint calculation. The direct approach requires two forward calculations for the derivative due to a single layer change. Hence, the perturbation method for a obtaining dose derivatives permits an appreciable savings in computation for a multilayered shield. A comparison was made of the fractional change in the dose per unit change in shield layer thickness as calculated by perturbation theory and by successive direct calculations; excellent agreement was obtained between the two methods.
Fieno, D.↗
Topics in structural dynamics: Nonlinear unsteady transonic flows and Monte Carlo methods in acoustics
The results are reported of two unrelated studies. The first was an investigation of the formulation of the equations for non-uniform unsteady flows, by perturbation of an irrotational flow to obtain the linear Green's equation. The resulting integral equation was found to contain a kernel which could be expressed as the solution of the adjoint flow equation, a linear equation for small perturbations, but with non-constant coefficients determined by the steady flow conditions. It is believed that the non-uniform flow effects may prove important in transonic flutter, and that in such cases, the use of doublet type solutions of the wave equation would then prove to be erroneous. The second task covered an initial investigation into the use of the Monte Carlo method for solution of acoustical field problems. Computed results are given for a rectangular room problem, and for a problem involving a circular duct with a source located at the closed end.
Haviland, J. K.↗
Rotation in vibration, optimization, and aeroelastic stability problems
The effects of rotation in the areas of vibrations, dynamic stability, optimization, and aeroelasticity were studied. The governing equations of motion for the study of vibration and dynamic stability of a rapidly rotating deformable body were developed starting from the nonlinear theory of elasticity. Some common features such as the limitations of the classical theory of elasticity, the choice of axis system, the property of self-adjointness, the phenomenon of frequency splitting, shortcomings of stability methods as applied to gyroscopic systems, and the effect of internal and external damping on stability in gyroscopic systems are identified and discussed, and are then applied to three specific problems.
Kaza, K. R. V.↗
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.↗