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NASA NTRS1981

Spectral multigrid methods for elliptic equations

An alternative approach which employs multigrid concepts in the iterative solution of spectral equations was examined. Spectral multigrid methods are described for self adjoint elliptic equations with either periodic or Dirichlet boundary conditions. For realistic fluid calculations the relevant boundary conditions are periodic in at least one (angular) coordinate and Dirichlet (or Neumann) in the remaining coordinates. Spectral methods are always effective for flows in strictly rectangular geometries since corners generally introduce singularities into the solution. If the boundary is smooth, then mapping techniques are used to transform the problem into one with a combination of periodic and Dirichlet boundary conditions. It is suggested that spectral multigrid methods in these geometries can be devised by combining the techniques.

Zang, T. A.↗

NASA NTRS1981

An adaptive, multi-level method for elliptic boundary value problems

Subroutine PLTMG is a FORTRAN program for solving self-adjoint elliptic boundary value problems in general regions of R-squared. It is based on a piecewise linear triangle finite element method, an adaptive grid refinement procedure, and a multi-level iterative method to solve the resulting sets of linear equations. This paper describes the method and presents some numerical results and comparisons.

Bank, R. E.↗

NASA NTRS1981

Optimal transfer between close near-circular orbits

A nonlinear analysis is carried out for the optimal transfer between close near-circular noncoplanar orbits with no limit on time. The degeneracy of the linear analyses of Marec and Edelbaum is, thereby, removed and with the aid of a generalized Jacobi test, it is established that the minimum-impulse transfer requires, at most, three impulses. A field of extremals is generated in an orbit element space starting with the initial state and anticipated adjoint variables and integrating these variables by using the optimal thrust control law that maximizes the variational Hamiltonian. The nonoptimality of any four-impulse strategy is determined numerically by observing the reflection of every extremal from either an envelope (conjugate point test) or a switching surface on or before the application of the fourth impulse. This test is carried out in Regions I, II, and III of Marec, as well as the special transition region of Breakwell near the boundary of Region III.

Kechichian, J.↗

NASA NTRS1981

HADY-I, a FORTRAN program for the compressible stability analysis of three-dimensional boundary layers

A computer program HADY-I for calculating the linear incompressible or compressible stability characteristics of the laminar boundary layer on swept and tapered wings is described. The eigenvalue problem and its adjoint arising from the linearized disturbance equations with the appropriate boundary conditions are solved numerically using a combination of Newton-Raphson interative scheme and a variable step size integrator based on the Runge-Kutta-Fehlburh fifth-order formulas. The integrator is used in conjunction with a modified Gram-Schmidt orthonormalization procedure. The computer program HADY-I calculates the growth rates of crossflow or streamwise Tollmien-Schlichting instabilities. It also calculates the group velocities of these disturbances. It is restricted to parallel stability calculations, where the boundary layer (meanflow) is assumed to be parallel. The meanflow solution is an input to the program.

El-Hady, N. M.↗

NASA NTRS1981

Shape determination and control for large space structures

An integral operator approach is used to derive solutions to static shape determination and control problems associated with large space structures. Problem assumptions include a linear self-adjoint system model, observations and control forces at discrete points, and performance criteria for the comparison of estimates or control forms. Results are illustrated by simulations in the one dimensional case with a flexible beam model, and in the multidimensional case with a finite model of a large space antenna. Modal expansions for terms in the solution algorithms are presented, using modes from the static or associated dynamic mode. These expansions provide approximated solutions in the event that a used form analytical solution to the system boundary value problem is not available.

Weeks, C. J.↗

NASA NTRS1981

Sturm-Liouville eigenproblems with an interior pole

The eigenvalues and eigenfunctions of self-adjoint Sturm-Liouville problems with a simple pole on the interior of an interval are investigated. Three general theorems are proved, and it is shown that as n approaches infinity, the eigenfunctions more and more closely resemble those of an ordinary Sturm-Liouville problem. The low-order modes differ significantly from those of a nonsingular eigenproblem in that both eigenvalues and eigenfunctions are complex, and the eigenvalues for all small n may cluster about a common value in contrast to the widely separated eigenvalues of the corresponding nonsingular problem. In addition, the WKB is shown to be accurate for all n, and all eigenvalues of a normal one-dimensional Sturm-Liouville equation with nonperiodic boundary conditions are well separated.

Boyd, J. P.↗

NASA NTRS1982

Iterative spectral methods and spectral solutions to compressible flows

A spectral multigrid scheme is described which can solve pseudospectral discretizations of self-adjoint elliptic problems in O(N log N) operations. An iterative technique for efficiently implementing semi-implicit time-stepping for pseudospectral discretizations of Navier-Stokes equations is discussed. This approach can handle variable coefficient terms in an effective manner. Pseudospectral solutions of compressible flow problems are presented. These include one dimensional problems and two dimensional Euler solutions. Results are given both for shock-capturing approaches and for shock-fitting ones.

Hussaini, M. Y.↗

NASA NTRS1983

Simultaneous control and optimization for elastic systems

Studies are conducted to determine the dynamic response of beams and plates to loads which are extreme within a certain class of admissible loads. Two approaches to this problem are suggested. In approach one, Pontryagin's maximality principle is regarded as an additional constraint. The optimality of design is then determined by standard numerical techniques. The "adjoint variable approach' to sensitivity of structural design is applied for a given inhomogeneous term. The inhomogeneous term is an extremal element of admissible load vectors, which constitute a closed subspace of a Sobolev space. Again, the maximality principle is invoked. While only beam and plate theory problems are used as examples, generalizations are easy to perceive.

Komkov, V.↗

NASA NTRS1981

Static shape determination and control for a large space antenna

An integral operator approach is used to derive solutions to static shape determination and control problems associated with large space structures. Problem assumptions include a linear self-adjoint system model, observations and control forces at discrete points, and quadratic performance criteria for the comparison of estimates or control forces. Results are illustrated by simulations with a finite element model of a large space antenna. Modal expansions for terms in the solution algorithms are presented, using modes from the static or associated dynamic model. These expansions provide approximate solutions in the event that a closed form analytical solution to the system boundary value problem is not available.

Weeks, C.↗

NASA NTRS1984

A substructure coupling procedure applicable to general linear time-invariant dynamic systems

A substructure synthesis procedure applicable to structural systems containing general nonconservative terms is presented. In their final form, the non-self-adjoint substructure equations of motion are cast in state vector form through the use of a variational principle. A reduced-order model for each substructure is implemented by representing the substructure as a combination of a small number of Ritz vectors. For the method presented, the substructure Ritz vectors are identified as a truncated set of substructure eigenmodes, which are typically complex, along with a set of generalized real attachment modes. The formation of the generalized attachment modes does not require any knowledge of the substructure flexible modes; hence, only the eigenmodes used explicitly as Ritz vectors need to be extracted from the substructure eigenproblem. An example problem is presented to illustrate the method.

Howsman, T. G.↗

NASA NTRS1984

An analytic solution for the orbital perturbations of the Venus Radar Mapper due to gravitational harmonics

Hill's variational equations are solved analytically for the orbital perturbations of a spacecraft nominally in an elliptic orbit around a non-spherical body. The rotation of the central planet about its spin-axis is not considered in the analysis. The perturbations are restricted to the planetary gravitational harmonics only. An extremely simple algorithm is derived to transform the spherical harmonic potentials to the orbital coordinate system, and the resulting accelerations are shown to be simply trigonometric functions of the true anomaly. With the principal matrix solution for the differential equations of the adjoint system given in closed form, the orthogonality of the trigonometric functions makes it possible to obtain an analytic solution for the non-homogeneous problem, at intervals of 2 pi in true anomaly. The solution for orbital perturbations can be extended over several revolutions by applying well-known results from Floquet's theory. The technique is demonstrated with results presented on the spacecraft periapsis altitude for the forthcoming Venus Radar Mapper Mission.

Vijayaraghavan, A.↗

NASA NTRS1984

Sensitivity analysis of discrete structural systems: A survey

Methods for calculating sensitivity derivatives for discrete structural systems are surveyed, primarily covering literature published during the past two decades. Methods are described for calculating derivatives of static displacements and stresses, eigenvalues and eigenvectors, transient structural response, and derivatives of optimum structural designs with respect to problem parameters. The survey is focused on publications addressed to structural analysis, but also includes a number of methods developed in nonstructural fields such as electronics, controls, and physical chemistry which are directly applicable to structural problems. Most notable among the nonstructural-based methods are the adjoint variable technique from control theory, and the Green's function and FAST methods from physical chemistry.

Adelman, H. M.↗

NASA NTRS1984

A substructure coupling procedure applicable to general linear time-invariant dynamic systems

A substructure synthesis procedure applicable to structural systems containing general nonconservative terms is presented. In their final form, the nonself-adjoint substructure equations of motion are cast in state vector form through the use of a variational principle. A reduced-order mode for each substructure is implemented by representing the substructure as a combination of a small number of Ritz vectors. For the method presented, the substructure Ritz vectors are identified as a truncated set of substructure eigenmodes, which are typically complex, along with a set of generalized real attachment modes. The formation of the generalized attachment modes does not require any knowledge of the substructure flexible modes; hence, only the eigenmodes used explicitly as Ritz vectors need to be extracted from the substructure eigenproblem. An example problem is presented to illustrate the method.

Howsman, T. G.↗

NASA NTRS1985

Sensitivity derivatives for static test loading boundary conditions

Attention is given to the application of the adjoint variable method under the special combination of displacement and stress boundary conditions which occur when static test conditions are simulated. In the two-step problem-solving method employed, calculation of internal stresses, displacements and total reaction force at the boundary is followed by the calculation of the applied load and the total reaction force due to unit displacements.

Gurdal, Z.↗

NASA NTRS1984

The solution of non-linear hyperbolic equation systems by the finite element method

A finite-element method for the solution of nonlinear hyperbolic systems of equations, such as those encountered in non-self-adjoint problems of transient phenomena in convection-diffusion or in the mixed representation of wave problems, is developed and demonstrated. The problem is rewritten in moving coordinates and reinterpolated to the original mesh by a Taylor expansion prior to a standard Galerkin spatial discretization, and it is shown that this procedure is equivalent to the time-discretization approach of Donea (1984). Numerical results for sample problems are presented graphically, including such shallow-water problems as the breaking of a dam, the shoaling of a wave, and the outflow of a river; compressible flows such as the isothermal flow in a nozzle and the Riemann shock-tube problem; and the two-dimensional scalar-advection, nonlinear-shallow-water, and Euler equations.

Loehner, R.↗

NASA NTRS1985

On the design derivatives of eigenvalues and eigenvectors for distributed parameter systems

In this paper, analytic expressions are obtained for the design derivatives of eigenvalues and eigenfunctions of self-adjoint linear distributed parameter systems. Explicit treatment of boundary conditions is avoided by casting the eigenvalue equation into integral form. Results are expressed in terms of the linear operators defining the eigenvalue problem, and are therefore quite general. Sufficiency conditions appropriate to structural optimization of eigenvalues are obtained.

Reiss, R.↗

NASA NTRS1985

Design of stiffened composite panels with a fracture constraint

An automated procedure for designing minimum-weight composite panels subject to a local damage constraint under tensile loading was developed. A finite element program based on linear elastic fracture mechanics for calculating stress intensity factors (SIF) was incorporated in the design cycle. Panel fracture toughness was obtained by using a strain based criterion. A general purpose mathematical optimization algorithm was used for the weight minimization. Analytical sensitivity derivatives of the SIF employing the adjoint variable technique was used to enhance the computational efficiency of the procedure. Design results for both unstiffened and stiffened plates are presented.

Gurdal, Z.↗

NASA NTRS1985

Effect of maximum lift to drag ratio on optimal aeroassisted plane change

The influence of the maximum lift-to-drag ratio on the turning performance of an Orbital Transfer Vehicle is analyzed. Chapman's variables are used to formulate the equations of motion which are valid for both atmospheric flight and flight in a vacuum in a Newtonian gravitational field. Of the six adjoint variables involved in the variational formulation, four exact integrals and two approximate relations are obtained. This leads to an approximate but explicit control law for the lift and bank control. The control law is tested numerically for a whole range of entry speeds, from parabolic entry to near-circular entry with several values of maximum lift-to-drag ratio. The extensive numerical results, which are very accurate as compared to the exact optimal values, show that the maximum plane change for any speed ratio V(entry)/V(final) is simply proportional to the maximum lift-to-drag ratio, and depends solely on this parameter.

Johannesen, J. R.↗