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Surrogate Modeling of Nonlinear Dynamic Systems: A Comparative Study
Surrogate models play a vital role in overcoming the computational challenge in designing and analyzing nonlinear dynamic systems, especially in the presence of uncertainty. This paper presents a comparative study of different surrogate modeling techniques for nonlinear dynamic systems. Four surrogate modeling methods, namely, Gaussian process (GP) regression, a long short-term memory (LSTM) network, a convolutional neural network (CNN) with LSTM (CNN-LSTM), and a CNN with bidirectional LSTM (CNN-BLSTM), are studied and compared. All these model types can predict the future behavior of dynamic systems over long periods based on training data from relatively short periods. The multi-dimensional inputs of surrogate models are organized in a nonlinear autoregressive exogenous model (NARX) scheme to enable recursive prediction over long periods, where current predictions replace inputs from the previous time window. Three numerical examples, including one mathematical example and two nonlinear engineering analysis models, are used to compare the performance of the four surrogate modeling techniques. The results show that the GP-NARX surrogate model tends to have more stable performance than the other three deep learning (DL)-based methods for the three particular examples studied. The tuning effort of GP-NARX is also much lower than its deep learning-based counterparts.
Ergodic Lagrangian dynamics in a superhero universe
We present a fictional scenario that, while undeniably whimsical, provides the foundation for a unique exercise in extended problem solving, physics analysis, and quantitative model development. Starting with the foundational premise of the Wild Cards shared-world superhero universe, we demonstrate how a variety of concepts appropriate to the advanced undergraduate level—ergodicity, functional analysis, Lagrangian mechanics, and the ever-important simplifying approximation—can be combined into a rich, coherent mathematical model. The goal of this case study is to develop a useful pedagogical exercise in exploring an open-ended research question that presents, at first glance, no clear path forward. Being both eclectic and lengthy, this exercise offers a unique way for students to apply their core physics and mathematics education. It is perhaps best used within a senior honors seminar or within a brief (e.g., January term) elective class.
Computer-aided circuit analysis Annual report, 15 May 1966 - 15 Aug. 1967
Numerical methods, programs, and mathematical models for use in digital computer-aided circuit analysis and design
Numerical integration in a rigid-body trajectory program
Comparison of fourth order Runge-Kutta numerical integration technique with two numerical integration formulas in mathematical models used for computing rigid body trajectories
Cavitation data for hydraulic equipment
Development of cavitation B-factors for helium, parahydrogen, nitrogen, fluorine, oxygen, refrigerant 114, and water is discussed. B-factor is defined and numerical relationships are established. Mathematical models are included and alternative methods of determining B-factor are explained.
Development of a global pollution model for CO, CH4, and CH2O
The current status of a global pollution model for carbon monoxide, methane, and formaldehyde is described. The physico-chemical action is considered of these three pollutants in the troposphere. This geographic restriction is convenient since the tropopause provides a natural boundary across which little transport occurs. The data on sources and sinks for these pollutants is based on available information and assumptions relative to the major man-made and natural contributions. The distributions and concentrations of methane, formaldehyde, and carbon monoxide in the atmosphere are interrelated by the chemical reactions in which they participate. A chemical kinetic model based on the pseudo-steady state approximation for the intermediate species was developed to account for these reactions. The numerical procedure used to mathematically describe the pollution transport is a mass conservative scheme employing an integral flux approach.
Electrophoretic separator for purifying biologicals, part 1
A program to develop an engineering model of an electrophoretic separator for purifying biologicals is summarized. An extensive mathematical modeling study and numerous ground based tests were included. Focus was placed on developing an actual electrophoretic separator of the continuous flow type, configured and suitable for flight testing as a space processing applications rocket payload.
Orbit/attitude estimation with Landsat-1 and -2 landmark data
The accuracy of spacecraft orbit and attitude determination is investigated using Landsat-1 and -2 picture data. The data consists of ground control points, of known geodetic coordinates, which are identified on a series of Landsat pictures. Both sequential (extended Kalman filter) and batch (nonlinear Bayesian weighted least-squares) estimation techniques have been implemented in two software systems. Discussion of these systems includes a system overview, mathematical formulation, and sample numerical results. Particular emphasis is given to the application of a spectral analysis technique to derive spacecraft attitude models.
Control of large spaceborne antenna systems with flexible booms by mechanical decoupling
A simple practical method for designing antenna-feed attitude control systems for large deployable spaceborne antenna systems with long flexible booms is proposed. The basic idea is to mechanically decouple the antenna-feed from the boom so that the feed-attitude control system can be designed without taking the boom dynamics into consideration, thus avoiding a complex control problem involving an infinite-dimensional distributed parameter system. The validity of the proposed method is substantiated by analytical and numerical studies using a mathematical model for the flexible boom which could undergo both bending and torsional vibrations. This approach leads to simple antenna-feed attitude control systems which are amenable to physical implementation.
Use of non-spherical gravity harmonics for relative motion GN/C (category 2)
Detailed analysis of the Automatic Rendezvous and Capture problem indicate a need for three different regions of mathematical description for the GN&C algorithms: (1) multi-vehicle orbital mechanics to the rendezvous interface point, i.e., within 100 nm; (2) relative motion solutions (such as Clohessy-Wiltshire type) from the far-field to the near-field interface, i.e., within 1 nm and; (3) close proximity motion - the near-field motion where the relative differences in the gravitational and orbit inertial accelerations can be neglected from the equations of motion. Limit boundaries to these regions can be precisely defined by further analysis and will be functions of the tracking measurement accuracies and the computer resources available for the solution of the algorithms. This paper analyzes the relative motion in Regions 2 and 3 above and present the derivation and discussion of the general case of non-spherical gravitational perturbed relative motion. Mathematical deviations from the numerically integrated spherical gravity case and solutions from the Clohessy-Wiltshire equations are presented in the analysis. Based upon this preliminary analysis, it is recommended that further efforts be used to assess the relative position and velocity differences in Region 2 due to non-spherical gravity harmonics and that viable GN&C algorithms be developed to include these gravity perturbations (especially the effects of the first gravity harmonic, J2).
The Determination of Forces and Moments on a Gimballed SRM Nozzle Using a Cold Flow Model
The Solid Rocket Motor Air Flow Facility (SAF) at NASA Marshall Space Flight Center was used to characterize the flow in the critical aft end and nozzle of a solid propellant rocket motor (SRM) as part of the design phase of development. The SAF is a high pressure, blowdown facility which supplies a controlled flow of air to a subscale model of the internal port and nozzle of a SRM to enable measurement and evaluation of the flow field and surface pressure distributions. The ASRM Aft Section/Nozzle Model is an 8 percent scale model of the 19 second burn time aft port geometry and nozzle of the Advanced Solid Rocket Motor, the now canceled new generation space Shuttle Booster. It has the capability to simulate fixed nozzle gimbal angles of 0, 4, and 8 degrees. The model was tested at full scale motor Reynolds Numbers with extensive surface pressure instrumentation to enable detailed mapping of the surface pressure distributions over the nozzle interior surface, the exterior surface of the nozzle nose and the surface of the simulated propellant grain in the aft motor port. A mathematical analysis and associated numerical procedure were developed to integrate the measured surface pressure distributions to determine the lateral and axial forces on the moveable section of the nozzle, the effective model thrust and the effective aerodynamic thrust vector (as opposed to the geometric nozzle gimbal angle). The nozzle lateral and axial aerodynamic loads and moments about the pivot point are required for design purposes and require complex, three dimensional flow analyses. The alignment of the thrust vector with the nozzle geometric centerline is also a design requirement requiring three dimensional analyses which were supported by this experimental program. The model was tested with all three gimbal angles at three pressure levels to determine Reynolds number effects and reproducibility. This program was successful in demonstrating that a measured surface pressure distribution could be integrated to determine the lateral and axial loads, moments and thrust vector alignment for the scaled model of a large space booster nozzle. Numerical results were provided which are scaleable to the full scale rocket motor and can be used as benchmark data for 3-D CFD analyses.
Investigation of Coronal Large Scale Structures Utilizing Spartan 201 Data
Two telescopes aboard Spartan 201, a small satellite has been launched from the Space Shuttles, on April 8th, 1993, September 8th, 1994, September 7th, 1995 and November 20th, 1997. The main objective of the mission was to answer some of the most fundamental unanswered questions of solar physics-What accelerates the solar wind and what heats the corona? The two telescopes are 1) Ultraviolet Coronal Spectrometer (UVCS) provided by the Smithsonian Astrophysical Observatory which uses ultraviolet emissions from neutral hydrogen and ions in the corona to determine velocities of the coronal plasma within the solar wind source region, and the temperature and density distributions of protons and 2) White Light Coronagraph (WLC) provided by NASA's Goddard Space Flight Center which measures visible light to determine the density distribution of coronal electrons within the same region. The PI has had the primary responsibility in the development and application of computer codes necessary for scientific data analysis activities, end instrument calibration for the white-light coronagraph for the entire Spartan mission. The PI was responsible for the science output from the WLC instrument. PI has also been involved in the investigation of coronal density distributions in large-scale structures by use of numerical models which are (mathematically) sufficient to reproduce the details of the observed brightness and polarized brightness distributions found in SPARTAN 201 data.
Genetically Engineered Microelectronic Infrared Filters
A genetic algorithm is used for design of infrared filters and in the understanding of the material structure of a resonant tunneling diode. These two components are examples of microdevices and nanodevices that can be numerically simulated using fundamental mathematical and physical models. Because the number of parameters that can be used in the design of one of these devices is large, and because experimental exploration of the design space is unfeasible, reliable software models integrated with global optimization methods are examined The genetic algorithm and engineering design codes have been implemented on massively parallel computers to exploit their high performance. Design results are presented for the infrared filter showing new and optimized device design. Results for nanodevices are presented in a companion paper at this workshop.
Three Program Architecture for Design Optimization
In this presentation, I would like to review historical perspective on the program architecture used to build design optimization capabilities based on mathematical programming and other numerical search techniques. It is rather straightforward to classify the program architecture in three categories as shown above. However, the relative importance of each of the three approaches has not been static, instead dynamically changing as the capabilities of available computational resource increases. For example, we considered that the direct coupling architecture would never be used for practical problems, but availability of such computer systems as multi-processor. In this presentation, I would like to review the roles of three architecture from historical as well as current and future perspective. There may also be some possibility for emergence of hybrid architecture. I hope to provide some seeds for active discussion where we are heading to in the very dynamic environment for high speed computing and communication.
Interface Configuration Experiments (ICE) Explore the Effects of Microgravity on Fluids
The Interface Configuration Experiment (ICE) is actually a series of experiments that explore the striking behavior of liquid-vapor interfaces (i.e., fluid surfaces) in a low gravity environment under which major shifts in liquid position can arise from small changes in container shape or contact angle. Although these experiments are designed to test current mathematical theory, there are numerous practical applications that could result from these studies. When designing fluid management systems for space-based operations, it is important to be able to predict the locations and configurations that fluids will assume in containers under low-gravity conditions. The increased ability to predict, and hence control, fluid interfaces is vital to systems and/or processes where capillary forces play a significant role both in space and on the Earth. Some of these applications are in general coating processes (paints, pesticides, printing, etc.), fluid transport in porous media (ground water flows, oil recovery, etc.), liquid propellant systems in space (liquid fuel and oxygen), capillary-pumped loops and heat pipes, and space-based life-support systems. In space, almost every fluid system is affected, if not dominated, by capillarity. Knowledge of the liquid-vapor interface behavior, and in particular the interface shape from which any analysis must begin, is required as a foundation to predict how these fluids will react in microgravity and on Earth. With such knowledge, system designs can be optimized, thereby decreasing costs and complexity, while increasing performance and reliability. ICE has increased, and will continue to increase this knowledge, as it probes the specific peculiarities of current theory upon which our current understanding of these effects is based. Several versions of ICE were conducted in NASA Lewis Research Center's drop towers and on the space shuttle during the first and second United States Microgravity Laboratory missions (USML-1 and USML-2). Additional tests are planned for the space shuttle and for the Russian Mir space station. These studies will focus on interfacial problems concerning surface existence, uniqueness, configuration, stability, and flow characteristics.
Numerical Experiments in Error Control for Sound Propagation Using a Damping Layer Boundary Treatment
This paper presents results from numerical experiments for controlling the error caused by a damping layer boundary treatment when simulating the propagation of an acoustic signal from a continuous pressure source. The computations are with the 2D Linearized Euler Equations (LEE) for both a uniform mean flow and a steady parallel jet. The numerical experiments are with algorithms that are third, fifth, seventh and ninth order accurate in space and time. The numerical domain is enclosed in a damping layer boundary treatment. The damping is implemented in a time accurate manner, with simple polynomial damping profiles of second, fourth, sixth and eighth power. At the outer boundaries of the damping layer the propagating solution is uniformly set to zero. The complete boundary treatment is remarkably simple and intrinsically independant from the dimension of the spatial domain. The reported results show the relative effect on the error from the boundary treatment by varying the damping layer width, damping profile power, damping amplitude, propagtion time, grid resolution and algorithm order. The issue that is being addressed is not the accuracy of the numerical solution when compared to a mathematical solution, but the effect of the complete boundary treatment on the numerical solution, and to what degree the error in the numerical solution from the complete boundary treatment can be controlled. We report maximum relative absolute errors from just the boundary treatment that range from O[10-2] to O[10-7].
Numerical Experiments in Error Control for Sound Propagation Using a Damping Layer Boundary Treatment
This paper presents results from numerical experiments for controlling the error caused by a damping layer boundary treatment when simulating the propagation of an acoustic signal from a continuous pressure source. The computations are with the 2D Linearized Euler Equations (LEE) for both a uniform mean flow and a steady parallel jet. The numerical experiments are with algorithms that are third, fifth, seventh and ninth order accurate in space and time. The numerical domain is enclosed in a damping layer boundary treatment. The damping is implemented in a time accurate manner, with simple polynomial damping profiles of second, fourth, sixth and eighth power. At the outer boundaries of the damping layer the propagating solution is uniformly set to zero. The complete boundary treatment is remarkably simple and intrinsically independant from the dimension of the spatial domain. The reported results show the relative effect on the error from the boundary treatment by varying the damping layer width, damping profile power, damping amplitude, propagtion time, grid resolution and algorithm order. The issue that is being addressed is not the accuracy of the numerical solution when compared to a mathematical solution, but the effect of the complete boundary treatment on the numerical solution, and to what degree the error in the numerical solution from the complete boundary treatment can be controlled. We report maximum relative absolute errors from just the boundary treatment that range from O[10-2] to O[10-7].