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This paper presents the Fast Adaptive Composite Grid Method for Time Dependent Problems (TDFAC), a local grid refinement scheme for nonlinear, time dependent partial differential equations. The method is presented in a one dimensional setting to illustrate the basic features and components. Computational results which show the effectiveness of TDFAC as a computational tool are also included.
An analytical model was constructed to predict the magnitude of stresses produced by rubbing a turbine blade against its tip seal. This model used a linearized approach to the problem, after a parametric study, found that the nonlinear effects were of insignificant magnitude. The important input parameters to the model were: the arc through which rubbing occurs, the turbine rotor speed, normal force exerted on the blade, and the rubbing coefficient of friction. Since it is not possible to exactly specify some of these parameters, values were entered into the model which bracket likely values. The form of the forcing function was another variable which was impossible to specify precisely, but the assumption of a half-sine wave with a period equal to the duration of the rub was taken as a realistic assumption. The analytical model predicted resonances between harmonics of the forcing function decomposition and known harmonics of the blade. Thus, it seemed probable that blade tip rubbing could be at least a contributor to the blade-cracking phenomenon. A full-scale, full-speed test conducted on the space shuttle main engine high pressure fuel turbopump Whirligig tester was conducted at speeds between 33,000 and 28,000 RPM to confirm analytical predictions.
An adaptive control approach is investigated for the Space Station. The main components of the adaptive controller are the parameter identification scheme, the control gain calculation, and the control law. The control law is the Space Station baseline control law. The control gain calculation is based on linear quadratic regulator theory with eigenvalue placement in a vertical strip. The parameter identification scheme is a real-time recursive extended Kalman filter which estimates the inertias and also provides an estimate of the unmodeled disturbances due to the aerodynamic torques and to the nonlinear effects. An analysis of the inertia estimation problem suggests that it is possible to compute accurate estimates of the Space Station inertias during nominal CMG (control moment gyro) operations. The closed-loop adaptive control law is shown to be capable of stabilizing the Space Station after large inertia changes. Results are presented for the pitch axis.
An adaptive approach to Space Station attitude control is investigated. The main components of the controller are the parameter identification scheme, the control gain calculation, and the control law. The control law is a full-state feedback space station baseline control law. The control gain calculation is based on linear-quadratic regulator theory with eigenvalues placement in a vertical strip. The parameter identification scheme is a recursive extended Kalman filter that estimates the inertias and also provides an estimate of the unmodeled disturbances due to the aerodynamic torques and to the nonlinear effects. An analysis of the inertia estimation problem suggests that it is possible to estimate Space Station inertias accurately during nominal control moment gyro operations. The closed-loop adaptive control law is shown to be capable of stabilizing the Space Station after large inertia changes. Results are presented for the pitch axis.
The application of the explicit nonlinear finite element analysis code DYNA3D to crashworthiness problems is discussed. Emphasized in the first part of this work are the most important capabilities of an explicit code for crashworthiness analyses. The areas with significant research promise for the computational simulation of crash events are then addressed.
With stochastic robustness we are creating tools to design robust compensators for practical systems. During this year, the stochastic robustness research achieved the following results: refined the search tools needed for synthesis; successfully designed robust compensators for the American Controls Conference benchmark problem; and successfully designed robust compensators for a nonlinear hypersonic aircraft model with uncertainties in 28 parameters.
Stable and accurate interface conditions are derived for the linear advection-diffusion equation. The conditions are functionally independent of the spatial order of accuracy and rely only on the form of the discrete operator. We focus on high-order finite-difference operators that satisfy the summation-by-parts (SBP) property. We prove that stability is a natural consequence of the SBP operators used in conjunction with the new boundary conditions. In addition, we show that the interface treatments are conservative. New finite-difference operators of spatial accuracy up to sixth order are constructed: these operators satisfy the SBP property. Finite-difference operators are shown to admit design accuracy (p(sup th)-order global accuracy) when (p - 1)(sup th)-order stencil closures are used near the boundaries if the physical boundary conditions are implemented to at least p(sup th)-order accuracy. Stability and accuracy are demonstrated on the nonlinear Burgers' equation for an twelve-subdomain problem with randomly distributed interfaces.
Transition to turbulence in swept-wing flows has resisted correlation with linear theory because of its sensitivity to freestream conditions and 3-D roughness and because one of the principal instability modes quickly 'becomes nonlinear. In the face of such a formidable problem, two rather long-term fundamental efforts have been underway at DLR Gottinberg and Arizona State University that address swept-wing transition. These efforts have been recently reviewed by Bippes (1997) and Reibert and Saric (1997). Thus, the present work is a continuation of a series of studies on swept-wing boundary layers which have led to a better understanding of the transition process. In particular, we have taken advantage of the sensitivity to 3-D roughness and the modal nature of the instability in order to propose a particular control strategy. Complementing the two aforementioned reviews, general reviews of the swept-wing transition problem are found in Arnal (1997) and Kachanov (1996). Other recent reviews include Reshotko (t997), Crouch (1997), and Herbert (1997a,b). The failure of linear theory is discussed in Reed et al. (1996). The historical work is found in Reed and Sar-ic (1989). The basic idea is that the combination of sweep and chordwise pressure gradient within the boundary layer creates a velocity component perpendicular to the inviscid streamline. This crossflow profile is inflectional and exhibits both traveling and stationary unstable waves called crossflow vortices that are (approximately) aligned along the inviscid streamlines. Under conditions of low freestream turbulence levels, the dominant crossflow wave is stationary (Reibert and Saric t997) while moderate to high turbulence levels initiate dominant traveling waves (Dehle and Bippes 1996; Bippes 1997). 'Me mechanism is relatively insensitive to sound and 2-D surface roughness (Radeztsky et al. 1993) but very sensitive to 3-D roughness near the attachment line. We concentrate our work on low-turbulence freestream flows and stationary crossflow waves. Although the v' and w' components of the disturbances are very small, by convecting streamwise momentum in the wall-normal direction, they produce 0(l) changes inu'.Thus the mean flow is highly distorted with localized inflection points. Transition is then triggered by a high-frequency secondary instability of the distorted mean profile.
The research work presented in this thesis addresses the problem of robust control of uncertain linear and nonlinear systems using Neural network-based Generalized Predictive Control (NGPC) methodology. A brief overview of predictive control and its comparison with Linear Quadratic (LQ) control is given to emphasize advantages and drawbacks of predictive control methods. It is shown that the Generalized Predictive Control (GPC) methodology overcomes the drawbacks associated with traditional LQ control as well as conventional predictive control methods. It is shown that in spite of the model-based nature of GPC it has good robustness properties being special case of receding horizon control. The conditions for choosing tuning parameters for GPC to ensure closed-loop stability are derived. A neural network-based GPC architecture is proposed for the control of linear and nonlinear uncertain systems. A methodology to account for parametric uncertainty in the system is proposed using on-line training capability of multi-layer neural network. Several simulation examples and results from real-time experiments are given to demonstrate the effectiveness of the proposed methodology.
An adaptive unstructured grid refinement technique has been developed and successfully applied to several three dimensional inviscid flow test cases. The method is based on a combination of surface mesh subdivision and local remeshing of the volume grid Simple functions of flow quantities are employed to detect dominant features of the flowfield The method is designed for modular coupling with various error/feature analyzers and flow solvers. Several steady-state, inviscid flow test cases are presented to demonstrate the applicability of the method for solving practical three-dimensional problems. In all cases, accurate solutions featuring complex, nonlinear flow phenomena such as shock waves and vortices have been generated automatically and efficiently.
The focus of our work has been on developing an insight into the physics that govern the optimum design of thermal insulation for use in thermal protection systems of launch vehicle. Of particular interest was to obtain optimality criteria for designing foam insulations that have density (or porosity) distributions through the thickness for optimum thermal performance. We investigate the optimum design of functionally graded thermal insulation for steady state heat transfer through the foam. We showed that the heat transfer in the foam has competing modes, of radiation and conduction. The problem assumed a fixed inside temperature of 400 K and varied the aerodynamic surface heating on the outside surface from 0.2 to 1.0 MW/sq m. The thermal insulation develops a high temperature gradient through the thickness. Investigation of the model developed for heat conduction in foams showed that at high temperatures (as on outside wall) intracellular radiation dominates the heat transfer in the foam. Minimizing radiation requires reducing the pore size, which increases the density of the foam. At low temperatures (as on the inside wall), intracellular conduction (of the metal and air) dominates the heat transfer. Minimizing conduction requires increasing the pore size. This indicated that for every temperature there was an optimum value of density that minimized the heat transfer coefficient. Two optimization studies were performed. One was to minimize the heat transmitted though a fixed thickness insulation by varying density profiles. The second was to obtain the minimum mass insulation for specified thickness. Analytical optimality criteria were derived for the cases considered. The optimality condition for minimum heat transfer required that at each temperature we find the density that minimizes the heat transfer coefficient. Once a relationship between the optimum heat transfer coefficient and the temperature was found, the design problem reduced to the solution of a simple nonlinear differential equation. Preliminary results of this work were presented at the American Society of Composites meeting, and the final version was submitted for publication in the AIAA Journal. In addition to minimizing the transmitted heat, we investigated the optimum design for minimum weight given an acceptable level of heat transmission through the insulation. The optimality criterion developed was different from that obtained for minimizing beat transfer coefficient. For minimum mass design, we had to find for a given temperature the optimum density, which minimized the logarithmic derivative of the insulation thermal conductivity with respect to its density. The logarithmic derivative is defined as the ratio of relative change in the dependent response (thermal conductivity) to the relative change in the independent variable (density). The results have been documented as a conference paper that will be presented at the upcoming AIAA.
This article explains results developed from the following research: 'The Stability of Plates and Shells beyond the Elastic Limit.' A significant improvement is found in the derivation of the relations between the stress factors and the strains resulting from the instability of plates and shells. In a strict analysis, the problem reduces to the solution of two simultaneous nonlinear partial differential equations of the fourth order in the deflection and stress function, and in the approximate analysis to a single linear equation of the Bryan type. Solutions are given for the special cases of a rectangular plate buckling into a cylindrical form, and of an arbitrarily shaped plate under uniform compression. These solutions indicate that the accuracy obtained by the approximate method is satisfactory.
A significant body of work exists showing that providing a nonlinear programming (NLP) solver with expressions for the problem constraint gradient substantially increases the speed of program execution and can also improve the robustness of convergence, especially for local optimizers. Calculation of these derivatives is often accomplished through the computation of spacecraft's state transition matrix (STM). If the two-body gravitational model is employed as is often done in the context of preliminary design, closed form expressions for these derivatives may be provided. If a high fidelity dynamics model, that might include perturbing forces such as the gravitational effect from multiple third bodies and solar radiation pressure is used then these STM's must be computed numerically. We present a method for the power hardward model and a full ephemeris model. An adaptive-step embedded eight order Dormand-Prince numerical integrator is discussed and a method for the computation of the time of flight derivatives in this framework is presented. The use of these numerically calculated derivatieves offer a substantial improvement over finite differencing in the context of a global optimizer. Specifically the inclusion of these STM's into the low thrust missiondesign tool chain in use at NASA Goddard Spaceflight Center allows for an increased preliminary mission design cadence.
In this paper, we develop second- and third-order non-oscillatory shock-capturing hyperbolic residual distribution schemes for irregular triangular grids, extending our second- and third-order schemes to discontinuous problems. We present extended first-order N- and Rusanov-scheme formulations for hyperbolic advection-diffusion system, and demonstrate that the hyperbolic diffusion term does not affect the solution of inviscid problems for vanishingly small viscous coefficient. We then propose second- and third-order blended hyperbolic residual-distribution schemes with the extended first-order Rusanov-scheme. We show that these proposed schemes are extremely accurate in predicting non-oscillatory solutions for discontinuous problems. We also propose a characteristics-based nonlinear wave sensor for accurately detecting shocks, compression, and expansion regions. Using this proposed sensor, we demonstrate that the developed hyperbolic blended schemes do not produce entropy-violating solutions (unphysical stocks). We then verify the design order of accuracy of these blended schemes on irregular triangular grids.
The Collatz Conjecture is a famous mathematics problem that is simple to state and understand but remains unsolved. In this paper, the problem is recast as a discrete-time nonlinear system. and viewed in a new light from the perspective of nonlinear systems and feedback control theory. In particular, connections are made between the Collatz sequence of numbers and the behavior of closed-loop dynamical systems designed using a feedback control method called sliding mode control. Trajectories of such systems are characterized by a reaching phase and a sliding phase, the latter of which exhibits exponential convergence. As sliding mode control design is rooted in Lyapunov stability theory, the analogy suggests a new direction for proving or disproving the Collatz Conjecture. Although several possible approaches are discussed, no formal proof is given in this paper.
This paper presents a Dynamic Internal Predictive Power Scheduling (DIPPS) approach for optimizing power management in microgrids, particularly focusing on external power exchanges among diverse prosumers. DIPPS utilizes a dynamic objective function with a time-varying binary parameter to control the timing of power transfers to the external grid, facilitated by efficient usage of energy storage for surplus renewable power. The microgrid power scheduling problem is modeled as a mixed-integer nonlinear programming (MINLP-PS) and subsequently transformed into a mixed-integer linear programming (MILPPS) optimization through McCormick's relaxation to reduce computational complexity. A predictive window window with 6 data points is solved at an average of 0.92s, a 97.6% improvement over the 38.27s required for the MINLP-PS formulation, implying the numerical feasibility of the DIPPS approach for real-time implementation. Finally, the approach is validated against a static objective using real-world load data across three case studies with different time-varying parameters, demonstrating the ability of DIPPS to optimize power exchanges and efficiently utilize distributed resources while shifting the external power transfers to specified time durations.
The accurate calculation of the aerodynamic forces in unsteady transonic flow requires the solution of the nonlinear flow equations. The aeroelastician, on the other hand, seeks to treat his problems (flutter, for example) by means of linear equations whenever possible. He may do this, even when the underlying flow is nonlinear, if the perturbation forces are linear over some (perhaps small) range of unsteady amplitude of motion. This paper assesses the range of parameters over which linear behavior occurs. In particular calculations are made for an NACA 64A006 airfoil oscillating in pitch over a range of amplitudes, frequencies, and Mach numbers. The primary aerodynamic method used is the well known LTRAN2 code of Ballhaus and Goorjian that provides a finite-difference solution to the low frequency, small disturbance, two-dimensional potential flow equation. Comparisons are made with linear subsonic theory, local linearization, and, for steady flow, with the full potential equation code of Bauer, Garabedian, and Korn.
The theoretical basis and programming strategy utilized in the construction of the computer program BEST3D (boundary element solution technology - three dimensional) and detailed input instructions are provided for the use of the program. An extensive set of test cases and sample problems is included in the manual and is also available for distribution with the program. The BEST3D program was developed under the 3-D Inelastic Analysis Methods for Hot Section Components contract (NAS3-23697). The overall objective of this program was the development of new computer programs allowing more accurate and efficient three-dimensional thermal and stress analysis of hot section components, i.e., combustor liners, turbine blades, and turbine vanes. The BEST3D program allows both linear and nonlinear analysis of static and quasi-static elastic problems and transient dynamic analysis for elastic problems. Calculation of elastic natural frequencies and mode shapes is also provided.