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Nonlinear optimization problems that are encountered in science and industry are examined. A method of projecting the gradient vector onto a set of linear contraints is developed, and a program that uses this method is presented. The algorithm that generates this projection matrix is based on the Gram-Schmidt method and overcomes some of the objections to the Rosen projection method.
Performance of the two commonly used numerical procedures, one based on artificial compressibility method and the other pressure projection method, are compared. These formulations are selected primarily because they are designed for three-dimensional applications. The computational procedures are compared by obtaining steady state solutions of a wake vortex and unsteady solutions of a curved duct flow. For steady computations, artificial compressibility was very efficient in terms of computing time and robustness. For an unsteady flow which requires small physical time step, pressure projection method was found to be computationally more efficient than an artificial compressibility method. This comparison is intended to give some basis for selecting a method or a flow solution code for large three-dimensional applications where computing resources become a critical issue.
Projection methods for computing stationary probability distributions for Markov chain models are presented. A general projection method is a method which seeks an approximation from a subspace of small dimension to the original problem. Thus, the original matrix problem of size N is approximated by one of dimension m, typically much smaller than N. A particularly successful class of methods based on this principle is that of Krylov subspace methods which utilize subspaces of the form span(v,av,...,A(exp m-1)v). These methods are effective in solving linear systems and eigenvalue problems (Lanczos, Arnoldi,...) as well as nonlinear equations. They can be combined with more traditional iterative methods such as successive overrelaxation, symmetric successive overrelaxation, or with incomplete factorization methods to enhance convergence.
A method for projecting flight configuration eigenvectors onto the vector space of the test configuration eigenvectors is proposed. The underlying concept of the proposed method is that any flight configuration eigenvectors that can be replicated by a linear combination of test configuration eigenvectors, is correlated, if the constituent test eigenvectors are themselves correlated. Therefore, the modal test target mode shapes should be those test modes that combine to form the important modes of the flight configurations. This approach also recognizes that it is the deformed shapes that dictate what sections of the structure are “highly strained” (in a relative sense, within each mode shape), and that it is these highly strained sections that require an accurate stiffness representation to achieve an accurate model correlation.
A method for projecting flight configuration eigenvectors onto the vector space of the test configuration eigenvectors is proposed. The underlying concept of the proposed method is that any flight configuration eigenvectors (that can be replicated by a linear combination of test configuration eigenvectors) is correlated if the constituent test eigenvectors are themselves correlated. Therefore, the modal test target mode shapes should be those test modes that combine to form the important modes of the flight configurations. This approach also recognizes that it is the deformed shapes that dictate what sections of the structure are “highly strained” (in a relative sense, within each mode shape), and that these “highly strained” sections require an accurate stiffness representation to achieve an accurate model correlation.
There are two main requirements for practical simulation of unsteady flow at high Reynolds number: the algorithm must accurately propagate discontinuous flow fields without excessive artificial viscosity, and it must have some adaptive capability to concentrate computational effort where it is most needed. We satisfy the first of these requirements with a second-order Godunov method similar to those used for high-speed flows with shocks, and the second with a grid-based refinement scheme which avoids some of the drawbacks associated with unstructured meshes. These two features of our algorithm place certain constraints on the projection method used to enforce incompressibility. Velocities are cell-based, leading to a Laplacian stencil for the projection which decouples adjacent grid points. We discuss features of the multigrid and multilevel iteration schemes required for solution of the resulting decoupled problem. Variable-density flows require use of a modified projection operator--we have found a multigrid method for this modified projection that successfully handles density jumps of thousands to one. Numerical results are shown for the 2D adaptive and 3D variable-density algorithms.
Numerical models of solidification including a mushy zone are notoriously inefficient; most of them are based on formulations that require the coupled solution of the velocity components in the momentum equation greatly restricting the range of applicability of the models. There are only two models known to the authors that have used a projection or fractional step formulation, but none of these were used to model problems of any significant size. A third model was only applied to a partial mushy zone with no all-fluid region. Our initial attempts at modeling directional solidification in the presence of a developing mushy zone using a projection formulation encountered very serious difficulties once solidification starts. These difficulties were traced to the inability of the method to deal with large local density differences in the vicinity of the fluid-mush interface. As a result, a modified formulation of the projection method has been developed, that maintains the coupling between the body force and the pressure gradient and is presented in this work. The new formulation is shown to be robust and efficient, and can be applied to problems involving very large meshes. This is illustrated in this work through its application to simulations involving Pb-Sb and Pb-Sn alloys.
The relative performance of two iterative schemes, based on projection techniques, is compared on a shared memory multiprocessor - VAX 6240. The CG accelerated Block-SSOR method and the CG accelerated Symmetric-Kaczmarz method are considered for the solution of large sparse nonsymmetric systems of linear equations. It is shown that the regular structure of many matrices can be exploited by the CG-accelerated Block-SSOR method to provide good speedup in a multiprocessing environment. However, the CG accelerated Symmetric-Kaczmarz method, while being a viable alternative on a scalar machine, is unable to benefit from multiprocessing.
A methodology for robust eigenstructure assignment for multivariable feedback systems is presented. The algorithm is based upon a pole placement technique using projections onto subspaces of admissible eigenvectors. New ideas are introduced to generate target (desired) sets of unitary eigenvectors and determine optimal feasible eigenvectors in a least-square sense. Useful connections are established between the pole-placement by independent modal space control and the method introduced in this paper. A multicriterion optimization algorithm is also presented, which takes efficient advantage of the present eigenstructure assignment method. These developments show significant improvement over an earlier version of this algorithm in both computational cost and accuracy. This optimization process appears to be numerically robust and suitable for high-dimensional multicriterion optimizations; it is especially attractive for computer-aided design of control systems.
In this paper a new image processing technique for flash LIDAR data is presented as a potential tool to enable safe and precise spacecraft landings in future robotic or crewed lunar and planetary missions. Flash LIDARs can generate, in real-time, range data that can be interpreted as a 3-dimensional (3-D) image and transformed into a corresponding digital elevation map (DEM). The NASA Autonomous Landing and Hazard Avoidance (ALHAT) project is capitalizing on this new technology by developing, testing and analyzing flash LIDARs to detect hazardous terrain features such as craters, rocks, and slopes during the descent phase of spacecraft landings. Using a flash LIDAR for this application looks very promising, however through theoretical and simulation analysis the ALHAT team has determined that a single frame, or mosaic, of flash LIDAR data may not be sufficient to build a landing site DEM with acceptable spatial resolution, precision, size, or for a mosaic, in time, to meet current system requirements. One way to overcome this potential limitation is by enhancing the flash LIDAR output images. We propose a new super-resolution algorithm applicable to flash LIDAR range data that will create a DEM with sufficient accuracy, precision and size to meet current ALHAT requirements. The performance of our super-resolution algorithm is analyzed by processing data generated during a series of simulation runs by a high fidelity model of a flash LIDAR imaging a high resolution synthetic lunar elevation map. The flash LIDAR model is attached to a simulated spacecraft by a gimbal that points the LIDAR to a target landing site. For each simulation run, a sequence of flash LIDAR frames is recorded and processed as the spacecraft descends toward the landing site. Each run has a different trajectory profile with varying LIDAR look angles of the terrain. We process the output LIDAR frames using our SR algorithm and the results show that the achieved level of accuracy and precision of the SR generated landing site DEM is more than adequate for detecting hazardous terrain features and identifying safe areas.
New ideas which lead to feedback control laws for large flexible structures which are insensitive to model uncertainty are presented. A pole placement method is presented which leads to near-unitary closed loop eigenvectors, and a new method is introduced to design the control while simultaneously considering three competing measures of optimality. Robustness versus Integral algorithms are applicable to at least moderately high-dimensioned systems. In the present discussion, controls for two coupled flexible bodies are considered. A 6x24 gain matrix is designed to control a 12 modes system using 6 actuators. Researchers also developed control laws for the R2P2 simulator at Martin Marietta; in this case 3 actuators are used to control a 12th order system. Simulation studies indicate that researchers indeed achieved robust designs without significant difficulties associated with spillover into the uncontrolled modes. Here, several key ideas and numerical results are given. In the references, details of the formulation, discussions of salient features, and connection to the availiable literature are given.
Within the framework of the second phase of the Tropospheric Ozone Assessment Report (TOAR-II), it was recognized that an essential first step for deriving accurate trends from the ground-based networks that monitor ozone in the free troposphere is putting the measurements on the same basis with respect to absolute references and processing methods. The relevant procedures are referred to as “harmonization” or “homogenization”. The TOAR II working group, “HEGIFTOM” (Harmonization and Evaluation of Ground-based Instruments for Free-Tropospheric Ozone Measurements), has carried out harmonization for five types of network (Figure below) instruments (mid-1990s to 2020): ozonesondes, commercial aircraft IAGOS landing/takeoff profiles, Fourier-Transform Infrared spectrometer (FTIR), tropospheric Lidar, and Brewer/Dobson Umkehr. First, we summarize the homogenization effort for each network that provides new quality-assessed ozone profile or segment (partial column) data sets, including uncertainty estimates and quality flags. Second, with the HEGIFTOM datasets forming the basis for a global assessment of tropospheric ozone column trends, results derived with various trend detection algorithms will be presented. The ultimate goal is evaluation of the consistency of the calculated trends among different techniques at selected stations and/or regions.
The Space Shuttle program is cooperating in a pilot project to apply formal methods to live requirements analysis activities. As one of the larger ongoing shuttle Change Requests (CR's), the Global Positioning System (GPS) CR involves a significant upgrade to the Shuttle's navigation capability. Shuttles are to be outfitted with GPS receivers and the primary avionics software will be enhanced to accept GPS-provided positions and integrate them into navigation calculations. Prior to implementing the CR, requirements analysts at Loral Space Information Systems, the Shuttle software contractor, must scrutinize the CR to identify and resolve any requirements issues. We describe an ongoing task of the Formal Methods Demonstration Project for Space Applications whose goal is to find an effective way to use formal methods in the GPS CR requirements analysis phase. This phase is currently under way and a small team from NASA Langley, ViGYAN Inc. and Loral is now engaged in this task. Background on the GPS CR is provided and an overview of the hardware/software architecture is presented. We outline the approach being taken to formalize the requirements, only a subset of which is being attempted. The approach features the use of the PVS specification language to model 'principal functions', which are major units of Shuttle software. Conventional state machine techniques form the basis of our approach. Given this background, we present interim results based on a snapshot of work in progress. Samples of requirements specifications rendered in PVS are offered to illustration. We walk through a specification sketch for the principal function known as GPS Receiver State processing. Results to date are summarized and feedback from Loral requirements analysts is highlighted. Preliminary data is shown comparing issues detected by the formal methods team versus those detected using existing requirements analysis methods. We conclude by discussing our plan to complete the remaining activities of this task.
The method described as the method of nearcharacteristics by Nakagawa (1980) is renamed the method of projected characteristics. Making full use of properties of the projected characteristics, a new and simpler formulation is developed. As a result, the formulation for the examination of the general three-dimensional problems is presented. It is noted that since in practice numerical solutions must be obtained, the final formulation is given in the form of difference equations. The possibility of including effects of viscous and ohmic dissipations in the formulation is considered, and the physical interpretation is discussed. A systematic manner is then presented for deriving physically self-consistent, time-dependent boundary equations for MHD initial boundary problems. It is demonstrated that the full use of the compatibility equations (differential equations relating variations at two spatial locations and times) is required in determining the time-dependent boundary conditions. In order to provide a clear physical picture as an example, the evolution of axisymmetric global magnetic field by photospheric differential rotation is considered.
A comprehensive optimization program has been developed for computing fuel-optimal trajectories between the earth and a point in the sun-earth-moon system. It presents methods for generating fuel optimal two-impulse trajectories which may originate at the earth or a point in space and fuel optimal three-impulse trajectories between two points in space. The extrapolation of the state vector and the computation of the state transition matrix are accomplished by the Stumpff-Weiss method. The cost and constraint gradients are computed analytically in terms of the terminal state and the state transition matrix. The 4-body Lambert problem is solved by using the Newton-Raphson method. An accelerated gradient projection method is used to optimize a 2-impulse trajectory with terminal constraint. The Davidon's Variance Method is used both in the accelerated gradient projection method and the outer loop of a 3-impulse trajectory optimization problem.
A numerical method of solving fully nonlinear MHD equation is described. In particular, the formulation based on the newly developed method of projected characteristics (Nakagawa, 1981) suitable to study the evolution of magnetic arches due to motions of their foot-points is presented. The final formulation is given in the form of difference equations; therefore, the analysis of numerical stability is also presented. Further, the most important derivation of physically self-consistent, time-dependent boundary conditions (i.e. the evolving boundary equations) is given in detail, and some results obtained with such boundary equations are reported.
Methods are explored for economically generating electrical power to meet future requirements. The Low-Cost Silicon Solar Array Project (LSSA) was established to reduce the price of solar arrays by improving manufacturing technology, adapting mass production techniques, and promoting user acceptance. The new manufacturing technology includes the consideration of new silicon refinement processes, silicon sheet growth techniques, encapsulants, and automated assembly production being developed under contract by industries and universities.