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NASA NTRS2016

A Nonlinear Modal Aeroelastic Solver for FUN3D

A nonlinear structural solver has been implemented internally within the NASA FUN3D computational fluid dynamics code, allowing for some new aeroelastic capabilities. Using a modal representation of the structure, a set of differential or differential-algebraic equations are derived for general thin structures with geometric nonlinearities. ODEPACK and LAPACK routines are linked with FUN3D, and the nonlinear equations are solved at each CFD time step. The existing predictor-corrector method is retained, whereby the structural solution is updated after mesh deformation. The nonlinear solver is validated using a test case for a flexible aeroshell at transonic, supersonic, and hypersonic flow conditions. Agreement with linear theory is seen for the static aeroelastic solutions at relatively low dynamic pressures, but structural nonlinearities limit deformation amplitudes at high dynamic pressures. No flutter was found at any of the tested trajectory points, though LCO may be possible in the transonic regime.

Goldman, Benjamin D.↗
NASA NTRS1993

Control of nonlinear systems with applications to constrained robots and spacecraft attitude stabilization

This thesis is organized in two parts. In Part 1, control systems described by a class of nonlinear differential and algebraic equations are introduced. A procedure for local stabilization based on a local state realization is developed. An alternative approach to local stabilization is developed based on a classical linearization of the nonlinear differential-algebraic equations. A theoretical framework is established for solving a tracking problem associated with the differential-algebraic system. First, a simple procedure is developed for the design of a feedback control law which ensures, at least locally, that the tracking error in the closed loop system lies within any given bound if the reference inputs are sufficiently slowly varying. Next, by imposing additional assumptions, a procedure is developed for the design of a feedback control law which ensures that the tracking error in the closed loop system approaches zero exponentially for reference inputs which are not necessarily slowly varying. The control design methodologies are used for simultaneous force and position control in constrained robot systems. The differential-algebraic equations are shown to characterize the slow dynamics of a certain nonlinear control system in nonstandard singularly perturbed form. In Part 2, the attitude stabilization (reorientation) of a rigid spacecraft using only two control torques is considered. First, the case of momentum wheel actuators is considered. The complete spacecraft dynamics are not controllable. However, the spacecraft dynamics are small time locally controllable in a reduced sense. The reduced spacecraft dynamics cannot be asymptotically stabilized using continuous feedback, but a discontinuous feedback control strategy is constructed. Next, the case of gas jet actuators is considered. If the uncontrolled principal axis is not an axis of symmetry, the complete spacecraft dynamics are small time locally controllable. However, the spacecraft attitude cannot be asymptotically stabilized using continuous feedback, but a discontinuous stabilizing feedback control strategy is constructed. If the uncontrolled principal axis is an axis of symmetry, the complete spacecraft dynamics cannot be stabilized. However, the spacecraft dynamics are small time locally controllable in a reduced sense. The reduced spacecraft dynamics cannot be asymptotically stabilized using continuous feedback, but again a discontinuous feedback control strategy is constructed.

Krishnan, Hariharan↗
NASA NTRS1972

Lie algebras and linear differential equations.

Certain symmetry properties possessed by the solutions of linear differential equations are examined. For this purpose, some basic ideas from the theory of finite dimensional linear systems are used together with the work of Wei and Norman on the use of Lie algebraic methods in differential equation theory.

Brockett, R. W.↗
NASA NTRS1993

Structured adaptive grid generation using algebraic methods

The accuracy of the numerical algorithm depends not only on the formal order of approximation but also on the distribution of grid points in the computational domain. Grid adaptation is a procedure which allows optimal grid redistribution as the solution progresses. It offers the prospect of accurate flow field simulations without the use of an excessively timely, computationally expensive, grid. Grid adaptive schemes are divided into two basic categories: differential and algebraic. The differential method is based on a variational approach where a function which contains a measure of grid smoothness, orthogonality and volume variation is minimized by using a variational principle. This approach provided a solid mathematical basis for the adaptive method, but the Euler-Lagrange equations must be solved in addition to the original governing equations. On the other hand, the algebraic method requires much less computational effort, but the grid may not be smooth. The algebraic techniques are based on devising an algorithm where the grid movement is governed by estimates of the local error in the numerical solution. This is achieved by requiring the points in the large error regions to attract other points and points in the low error region to repel other points. The development of a fast, efficient, and robust algebraic adaptive algorithm for structured flow simulation applications is presented. This development is accomplished in a three step process. The first step is to define an adaptive weighting mesh (distribution mesh) on the basis of the equidistribution law applied to the flow field solution. The second, and probably the most crucial step, is to redistribute grid points in the computational domain according to the aforementioned weighting mesh. The third and the last step is to reevaluate the flow property by an appropriate search/interpolate scheme at the new grid locations. The adaptive weighting mesh provides the information on the desired concentration of points to the grid redistribution scheme. The evaluation of the weighting mesh is accomplished by utilizing the weight function representing the solution variation and the equidistribution law. The selection of the weight function plays a key role in grid adaptation. A new weight function utilizing a properly weighted boolean sum of various flowfield characteristics is defined. The redistribution scheme is developed utilizing Non-Uniform Rational B-Splines (NURBS) representation. The application of NURBS representation results in a well distributed smooth grid by maintaining the fidelity of the geometry associated with boundary curves. Several algebraic methods are applied to smooth and/or nearly orthogonalize the grid lines. An elliptic solver is utilized to smooth the grid lines if there are grid crossings. Various computational examples of practical interest are presented to demonstrate the success of these methods.

Yang, Jiann-Cherng↗
NASA NTRS1986

Quasi-generalized variables

The numerical solution of a system of differential and algebraic equations is difficult, due to the appearance of numerical instabilities. A method is presented here which permits numerical solutions of such a system to be obtained which satisfy the algebraic constraint equations exactly without reducing the order of the differential equations. The method is demonstrated using examples from mechanics.

Baumgarten, J.↗
NASA NTRS1990

Calculation of turbulent flow in complex geometries with a second-moment closure model

A full Reynolds stress turbulence model is applied to calculate the flow in various channels with complex cross-section. The model is shown to be more robust and to produce more consistent results than algebraic stress models. The relationship between various versions of the latter is provided. It is shown that differential or algebraic stress models derived from a direct application of a popular linear approximation for the pressure-strain terms in the Reynolds stress equations will always underpredict the secondary motion, and hence the accompanying distortion of other flow properties. A simple approximation which leads to better predictions is presented and related to previous practice.

Demuren, A. O.↗
NASA NTRS1989

Labeled trees and the efficient computation of derivations

The effective parallel symbolic computation of operators under composition is discussed. Examples include differential operators under composition and vector fields under the Lie bracket. Data structures consisting of formal linear combinations of rooted labeled trees are discussed. A multiplication on rooted labeled trees is defined, thereby making the set of these data structures into an associative algebra. An algebra homomorphism is defined from the original algebra of operators into this algebra of trees. An algebra homomorphism from the algebra of trees into the algebra of differential operators is then described. The cancellation which occurs when noncommuting operators are expressed in terms of commuting ones occurs naturally when the operators are represented using this data structure. This leads to an algorithm which, for operators which are derivations, speeds up the computation exponentially in the degree of the operator. It is shown that the algebra of trees leads naturally to a parallel version of the algorithm.

Grossman, Robert↗
NASA NTRS1965

Universal variables.

Universal variable parameters and formulas used for two-body problem, differential correction and various orbits and perturbations

ORBIT CALCULATION↗
NASA NTRS1972

Equations of motion in the linear approximation.

Attempt to develop a gauge-invariant theory of the motion of singularities in an n-dimensional Riemannian space. A gauge-invariant theory is developed for a nongeodetic particle represented by a time-like world line in the linearized field theory. The solution of the basic algebraic and differential equations is to be constructed from the source line, the future null cones emanating from it, a scale factor, and nothing else, and the leading term of the solution is the Schwarzschild field. A class of possible solutions is examined, and it is shown that this class contains just one acceptable member - namely, a particle in constant acceleration.

Robinson, I.↗
NASA NTRS1977

Some comments on series solutions

The use of power series and truncated power series in the MACSYMA system for algebraic manipulation is illustrated. Algebraic and differential equations are solved using Taylor series or asymptotic series. Deficiencies of the current scheme are noted, and remedies suggested.

Fateman, R. J.↗
NASA NTRS1982

Development and application of the GIM code for the Cyber 203 computer

The GIM computer code for fluid dynamics research was developed. Enhancement of the computer code, implicit algorithm development, turbulence model implementation, chemistry model development, interactive input module coding and wing/body flowfield computation are described. The GIM quasi-parabolic code development was completed, and the code used to compute a number of example cases. Turbulence models, algebraic and differential equations, were added to the basic viscous code. An equilibrium reacting chemistry model and implicit finite difference scheme were also added. Development was completed on the interactive module for generating the input data for GIM. Solutions for inviscid hypersonic flow over a wing/body configuration are also presented.

Stainaker, J. F.↗
NASA NTRS1986

Design of Linear Quadratic Regulators and Kalman Filters

AESOP solves problems associated with design of controls and state estimators for linear time-invariant systems. Systems considered are modeled in state-variable form by set of linear differential and algebraic equations with constant coefficients. Two key problems solved by AESOP are linear quadratic regulator (LQR) design problem and steady-state Kalman filter design problem. AESOP is interactive. User solves design problems and analyzes solutions in single interactive session. Both numerical and graphical information available to user during the session.

Lehtinen, B.↗
NASA NTRS1988

A multivariate variational objective analysis-assimilation method. Part 1: Development of the basic model

The variational method of undetermined multipliers is used to derive a multivariate model for objective analysis. The model is intended for the assimilation of 3-D fields of rawinsonde height, temperature and wind, and mean level temperature observed by satellite into a dynamically consistent data set. Relative measurement errors are taken into account. The dynamic equations are the two nonlinear horizontal momentum equations, the hydrostatic equation, and an integrated continuity equation. The model Euler-Lagrange equations are eleven linear and/or nonlinear partial differential and/or algebraic equations. A cyclical solution sequence is described. Other model features include a nonlinear terrain-following vertical coordinate that eliminates truncation error in the pressure gradient terms of the horizontal momentum equations and easily accommodates satellite observed mean layer temperatures in the middle and upper troposphere. A projection of the pressure gradient onto equivalent pressure surfaces removes most of the adverse impacts of the lower coordinate surface on the variational adjustment.

Achtemeier, Gary L.↗
NASA NTRS1988

A general-purpose approach to computer-aided dynamic analysis of a flexible helicopter

A general purpose mathematical formulation is described for dynamic analysis of a helicopter consisting of flexible and/or rigid bodies that undergo large translations and rotations. Rigid body and elastic sets of generalized coordinates are used. The rigid body coordinates define the location and the orientation of a body coordinate frame (global frame) with respect to an inertial frame. The elastic coordinates are introduced using a finite element approach in order to model flexible components. The compatibility conditions between two adjacent elements in a flexible body are imposed using a Boolean matrix, whereas the compatibility conditions between two adjacent bodies are imposed using the Lagrange multiplier approach. Since the form of the constraint equations depends upon the type of kinematic joint and involves only the generalized coordinates of the two participating elements, then a library of constraint elements can be developed to impose the kinematic constraint in an automated fashion. For the body constraints, the Lagrange multipliers yield the reaction forces and torques of the bodies at the joints. The virtual work approach is used to derive the equations of motion, which are a system of differential and algebraic equations that are highly nonlinear. The formulation presented is general and is compared with hard-wired formulations commonly used in helicopter analysis.

Agrawal, Om P.↗
NASA NTRS1988

Development of the general interpolants method for the CYBER 200 series of supercomputers

The General Interpolants Method (GIM) is a 3-D, time-dependent, hybrid procedure for generating numerical analogs of the conservation laws. This study is directed toward the development and application of the GIM computer code for fluid dynamic research applications as implemented for the Cyber 200 series of supercomputers. An elliptic and quasi-parabolic version of the GIM code are discussed. Turbulence models, algebraic and differential equations, were added to the basic viscous code. An equilibrium reacting chemistry model and an implicit finite difference scheme are also included.

Stalnaker, J. F.↗
NASA NTRS1989

A nearly-linear computational-cost scheme for the forward dynamics of an N-body pendulum

The dynamic equations of motion of an n-body pendulum with spherical joints are derived to be a mixed system of differential and algebraic equations (DAE's). The DAE's are kept in implicit form to save arithmetic and preserve the sparsity of the system and are solved by the robust implicit integration method. At each solution point, the predicted solution is corrected to its exact solution within given tolerance using Newton's iterative method. For each iteration, a linear system of the form J delta X = E has to be solved. The computational cost for solving this linear system directly by LU factorization is O(n exp 3), and it can be reduced significantly by exploring the structure of J. It is shown that by recognizing the recursive patterns and exploiting the sparsity of the system the multiplicative and additive computational costs for solving J delta X = E are O(n) and O(n exp 2), respectively. The formulation and solution method for an n-body pendulum is presented. The computational cost is shown to be nearly linearly proportional to the number of bodies.

Chou, Jack C. K.↗