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NASA NTRS1985

A hybrid numerical technique for predicting the aerodynamic and acoustic fields of advanced turboprops

A hybrid numerical procedure is presented for the prediction of the aerodynamic and acoustic performance of advanced turboprops. A hybrid scheme is proposed which in principle leads to a consistent simultaneous prediction of both fields. In the inner flow a finite difference method, the Approximate-Factorization Alternating-Direction-Implicit (ADI) scheme, is used to solve the nonlinear Euler equations. In the outer flow the linearized acoustic equations are solved via a Boundary-Integral Equation (BIE) method. The two solutions are iteratively matched across a fictitious interface in the flow so as to maintain continuity. At convergence the resulting aerodynamic load prediction will automatically satisfy the appropriate free-field boundary conditions at the edge of the finite difference grid, while the acoustic predictions will reflect the back-reaction of the radiated field on the magnitude of the loading source terms, as well as refractive effects in the inner flow. The equations and logic needed to match the two solutions are developed and the computer program implementing the procedure is described. Unfortunately, no converged solutions were obtained, due to unexpectedly large running times. The reasons for this are discussed and several means to alleviate the situation are suggested.

Homicz, G. F.↗

DOE OSTI2022

A Reinforcement Learning Approach to Parameter Selection for Distributed Optimal Power Flow

With the increasing penetration of distributed energy resources, distributed optimization algorithms have attracted significant attention for power systems applications due to their potential for superior scalability, privacy, and robustness to a single point-of-failure. The Alternating Direction Method of Multipliers (ADMM) is a popular distributed optimization algorithm; however, its convergence performance is highly dependent on the selection of penalty parameters, which are usually chosen heuristically. In this work, we use reinforcement learning (RL) to develop an adaptive penalty parameter selection policy for alternating current optimal power flow (ACOPF) problem solved via ADMM with the goal of minimizing the number of iterations until convergence. We train our RL policy using deep Q-learning and show that this policy can result in significantly accelerated convergence (up to a 59% reduction in the number of iterations compared to existing, curvatureinformed penalty parameter selection methods). Furthermore, we show that our RL policy demonstrates promise for generalizability, performing well under unseen loading schemes as well as under unseen losses of lines and generators (up to a 50% reduction in iterations). This work thus provides a proof-of-concept for using RL for parameter selection in ADMM for power systems applications.

alternating current optimal power flow↗

NASA NTRS1992

A mean field neural network for hierarchical module placement

This paper proposes a mean field neural network for the two-dimensional module placement problem. An efficient coding scheme with only O(N log N) neurons is employed where N is the number of modules. The neurons are evolved in groups of N in log N iteration steps such that the circuit is recursively partitioned in alternating vertical and horizontal directions. In our simulations, the network was able to find optimal solutions to all test problems with up to 128 modules.

Unaltuna, M. Kemal↗

DOE OSTI2021

Prime factorization using quantum variational imaginary time evolution

The road to computing on quantum devices has been accelerated by the promises that come from using Shor’s algorithm to reduce the complexity of prime factorization. However, this promise hast not yet been realized due to noisy qubits and lack of robust error correction schemes. Here we explore a promising, alternative method for prime factorization that uses well-established techniques from variational imaginary time evolution. We create a Hamiltonian whose ground state encodes the solution to the problem and use variational techniques to evolve a state iteratively towards these prime factors. We show that the number of circuits evaluated in each iteration scales as \(O(n^{5}d)\) , where n is the bit-length of the number to be factorized and d is the depth of the circuit. We use a single layer of entangling gates to factorize 36 numbers represented using 7, 8, and 9-qubit Hamiltonians. We also verify the method’s performance by implementing it on the IBMQ Lima hardware to factorize 55, 65, 77 and 91 which are greater than the largest number (21) to have been factorized on IBMQ hardware.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1972

A method of limit point calculation in finite element structural analysis

An approach is presented for the calculation of limit points for structures described by discrete coordinates, and whose governing equations derive from finite element concepts. The nonlinear load-displacement path of the imperfect structure is first traced by use of a direct iteration scheme and the determinant of the governing algebraic equations is calculated at each solution point. The limit point is then established by extrapolation and imposition of the condition of zero slope of the plot of load vs. determinant. Three problems are solved in illustration of the approach and in comparison with alternative procedures and test data.

Gallagher, R. H.↗

NASA NTRS1983

Numerical solution of a coupled pair of elliptic equations from solid state electronics

Iterative methods are considered for the solution of a coupled pair of second order elliptic partial differential equations which arise in the field of solid state electronics. A finite difference scheme is used which retains the conservative form of the differential equations. Numerical solutions are obtained in two ways, by multigrid and dynamic alternating direction implicit methods. Numerical results are presented which show the multigrid method to be an efficient way of solving this problem.

Phillips, T. N.↗

NASA NTRS1984

Numerical solution of a coupled pair of elliptic equations from solid state electronics

Phillips, T. N.↗

NASA NTRS1992

Development of iterative techniques for the solution of unsteady compressible viscous flows

A research effort was initiated at Georgia Tech in February 1991 on the development of efficient techniques for the computation of 2-D and 3-D unsteady compressible flow problems. It was found that in 2-D unsteady viscous flow applications, the generalized minimal residual (GMRES) scheme was able to significantly improve the accuracy and stability characteristics of an existing 2-D ADI (Alternating Direction Implicit) time marching scheme. That is, the GMRES/ADI combination allowed 10 to 20 times larger time steps compared to an ADI scheme. Because the GMRES algorithm requires 5 to 10 times the CPU work compared to the ADI scheme, the combined GMRES/ADI scheme yields a net factor of 2 savings in CPU cost. During the past year, we also experimented with GMRES/multigrid/ADI combination. The purpose of this combination was to compute the low frequency components of the change in the flow properties from one time step to the next on a coarse grid. This strategy reduces the memory requirements of the GMRES method roughly by a factor of 4-8 for steady flow problems.

Sankar, Lakshmi N.↗

NASA NTRS1987

Efficient solutions to the Euler equations for supersonic flow with embedded subsonic regions

A line Gauss-Seidel (LGS) relaxation algorithm in conjunction with a one-parameter family of upwind discretizations of the Euler equations in two dimensions is described. Convergence of the basic algorithm to the steady state is quadratic for fully supersonic flows and is linear for other flows. This is in contrast to the block alternating direction implicit methods (either central or upwind differenced) and the upwind biased relaxation schemes, all of which converge linearly, independent of the flow regime. Moreover, the algorithm presented herein is easily coupled with methods to detect regions of subsonic flow embedded in supersonic flow. This allows marching by lines in the supersonic regions, converging each line quadratically, and iterating in the subsonic regions, and yields a very efficient iteration strategy. Numerical results are presented for two-dimensional supersonic and transonic flows containing oblique and normal shock waves which confirm the efficiency of the iteration strategy.

Walters, Robert W.↗

NASA NTRS1992

A transient FETI methodology for large-scale parallel implicit computations in structural mechanics

Explicit codes are often used to simulate the nonlinear dynamics of large-scale structural systems, even for low frequency response, because the storage and CPU requirements entailed by the repeated factorizations traditionally found in implicit codes rapidly overwhelm the available computing resources. With the advent of parallel processing, this trend is accelerating because explicit schemes are also easier to parallelize than implicit ones. However, the time step restriction imposed by the Courant stability condition on all explicit schemes cannot yet -- and perhaps will never -- be offset by the speed of parallel hardware. Therefore, it is essential to develop efficient and robust alternatives to direct methods that are also amenable to massively parallel processing because implicit codes using unconditionally stable time-integration algorithms are computationally more efficient when simulating low-frequency dynamics. Here we present a domain decomposition method for implicit schemes that requires significantly less storage than factorization algorithms, that is several times faster than other popular direct and iterative methods, that can be easily implemented on both shared and local memory parallel processors, and that is both computationally and communication-wise efficient. The proposed transient domain decomposition method is an extension of the method of Finite Element Tearing and Interconnecting (FETI) developed by Farhat and Roux for the solution of static problems. Serial and parallel performance results on the CRAY Y-MP/8 and the iPSC-860/128 systems are reported and analyzed for realistic structural dynamics problems. These results establish the superiority of the FETI method over both the serial/parallel conjugate gradient algorithm with diagonal scaling and the serial/parallel direct method, and contrast the computational power of the iPSC-860/128 parallel processor with that of the CRAY Y-MP/8 system.

Farhat, Charbel↗

DOE OSTI2024

Multi-Fidelity Scheme for Accelerating 4D Finite Element Analysis for Mircoreactors

Next-generation microreactors are currently being designed to be operated terrestrial and extraterrestrial for remote surface power production. These systems will provide an alternative source of carbon-free energy that is versatile and can be utilized for various applications. These applications of microreactors have prompted the usage of high-fidelity unstructured finite element (FE) based approaches to provide time-dependent solutions for multiple physics fields. Computing these high-fidelity (4-D) solutions for multiple design iterations, physics fields, and transient events requires an immense computational cost. These high-fidelity solutions are computational expensive and the cost can be reduced through the implementation of less accurate low-fidelity solutions. Unlike the current fleet of commercial nuclear reactors, these next-gen systems present challenges due to the material and physical limitations required. Due to these constraints, the brute force technique of parameterizing important system characteristics determines whether a design meets the project objective. A system such as a nuclear reactor could have thousands of design parameters that affect system performance. In order to analyze the entire parameter space of such a complex system would require millions of CPU hours and countless design iterations. This costly approach is not practical due to regulatory and budget limitations. In this proposal, an approach that utilizes hybrid high-fidelity and low-fidelity FE models to reduce the computational cost of evaluating these applications in 4-D will be presented. The accuracy of the high-fidelity model and the computational efficiency of the low-fidelity model are taken advantage of to produce a solution that closely resembles the full order high-fidelity solution. By utilizing an unconverged coarse FE mesh, operation limits such as temperature, structural loading, etc., can be evaluated in an accelerated fashion and then can be spatially interpolated onto a finer mesh. The resulting error arising from the coarse mesh can be mitigated with a discrepancy function that actively quantifies and corrects the error in the coarse solution. This discrepancy function can be periodically updated with high-fidelity calculations across the temporal domain thus requiring less iterations on the fine mesh. As a result, the computational cost can be reduced for the evaluation and design iteration of microreactors. The proposal for this research contains three sections: Section 2 provides a literature review, Section 3 outlines the methodology for the proposed multi-fidelity scheme, and Section 4 displays preliminary results of the proposed research.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

NASA NTRS1983

Multi-grid calculation of three-dimensional transonic potential flows

A multi-grid algorithm has been developed to speed the iterative convergence of calculations for the transonic potential flow past swept wings and wing-fuselage combinations. The method is based upon a fully-conservative, finite-volume approximation to the steady potential equation which is second-order accurate everywhere in the flow field except near shock waves. The multi-grid scheme is incorporated within the framework of an alternating successive-line-overrelaxation (SLOR) solver of the difference equations. Computed results confirm the second-order accuracy of the scheme, and demonstrate the effectiveness of the multi-grid procedure.

Caughey, D. A.↗

NASA NTRS1995

Accurate orbit determination strategies for the tracking and data relay satellites

The National Aeronautics and Space Administration (NASA) has developed the Tracking and Data Relay Satellite (TDRS) System (TDRSS) for tracking and communications support of low Earth-orbiting satellites. TDRSS has the operational capability of providing 85% coverage for TDRSS-user spacecraft. TDRSS currently consists of five geosynchronous spacecraft and the White Sands Complex (WSC) at White Sands, New Mexico. The Bilateration Ranging Transponder System (BRTS) provides range and Doppler measurements for each TDRS. The ground-based BRTS transponders are tracked as if they were TDRSS-user spacecraft. Since the positions of the BRTS transponders are known, their radiometric tracking measurements can be used to provide a well-determined ephemeris for the TDRS spacecraft. For high-accuracy orbit determination of a TDRSS user, such as the Ocean Topography Experiment (TOPEX)/Poseidon spacecraft, high-accuracy TDRS orbits are required. This paper reports on successive refinements in improved techniques and procedures leading to more accurate TDRS orbit determination strategies using the Goddard Trajectory Determination System (GTDS). These strategies range from the standard operational solution using only the BRTS tracking measurements to a sophisticated iterative process involving several successive simultaneous solutions for multiple TDRSs and a TDRSS-user spacecraft. Results are presented for GTDS-generated TDRS ephemerides produced in simultaneous solutions with the TOPEX/Poseidon spacecraft. Strategies with different user spacecraft, as well as schemes for recovering accurate TDRS orbits following a TDRS maneuver, are also presented. In addition, a comprehensive assessment and evaluation of alternative strategies for TDRS orbit determination, excluding BRTS tracking measurements, are presented.

Oza, D. H.↗

DOE OSTI2021

ANTS

The ANTS code (Alternate Non-Linear Two-phase Solver) is based on a novel non-linear solution algorithm for the solution of the two-phase, subchannel fluid equations. It achieves its performance through decoupling of the two-phase momentum equations (axial and transverse) from the axial phasic mass and energy equations which allows for a nested non-linear iteration scheme. This enables a plane-by-plane solution which the inner iteration focuses on a reduced non-linear equation set for the primitives in phasic mass flow rate, enthalpy and void for each node edge. Single node edges are coupled as part of the outer iteration via surface mass fluxes which appear as source terms in the inner iteration scheme. The outer iteration readily accommodates two-phase flow phenomena closure relationships for subchannel mixing and void drift. A primary feature is the use of a non-staggered mesh computational mesh and steady-state iterative solver in contrast to all existing subchannel codes.

Kropaczek, David J↗

NASA NTRS1993

A transient FETI methodology for large-scale parallel implicit computations in structural mechanics, part 2

Explicit codes are often used to simulate the nonlinear dynamics of large-scale structural systems, even for low frequency response, because the storage and CPU requirements entailed by the repeated factorizations traditionally found in implicit codes rapidly overwhelm the available computing resources. With the advent of parallel processing, this trend is accelerating because explicit schemes are also easier to parallellize than implicit ones. However, the time step restriction imposed by the Courant stability condition on all explicit schemes cannot yet and perhaps will never be offset by the speed of parallel hardware. Therefore, it is essential to develop efficient and robust alternatives to direct methods that are also amenable to massively parallel processing because implicit codes using unconditionally stable time-integration algorithms are computationally more efficient than explicit codes when simulating low-frequency dynamics. Here we present a domain decomposition method for implicit schemes that requires significantly less storage than factorization algorithms, that is several times faster than other popular direct and iterative methods, that can be easily implemented on both shared and local memory parallel processors, and that is both computationally and communication-wise efficient. The proposed transient domain decomposition method is an extension of the method of Finite Element Tearing and Interconnecting (FETI) developed by Farhat and Roux for the solution of static problems. Serial and parallel performance results on the CRAY Y-MP/8 and the iPSC-860/128 systems are reported and analyzed for realistic structural dynamics problems. These results establish the superiority of the FETI method over both the serial/parallel conjugate gradient algorithm with diagonal scaling and the serial/parallel direct method, and contrast the computational power of the iPSC-860/128 parallel processor with that of the CRAY Y-MP/8 system.

Farhat, Charbel↗

NASA NTRS1972

A finite element procedure for nonlinear prebuckling and initial postbuckling analysis

A procedure cast in a form appropriate to the finite element method is presented for geometrically nonlinear prebuckling and postbuckling structural analysis, including the identification of snap-through type of buckling. The principal features of this procedure are the use of direct iteration for solution of the nonlinear algebraic equations in the prebuckling range, an interpolation scheme for determination of the initial bifurcation point, a perturbation method in definition of the load-displacement behavior through the postbuckling regime, and extrapolation in determination of the limit point for snap-through buckling. Three numerical examples are presented in illustration of the procedure and in comparison with alternative approaches.

Mau, S. T.↗

DOE OSTI2023

Technical Report on Waveform Fit Metrics for Global Models

The new WAVEFORMS Initiative in the Ground-based Nuclear Detonation Detection (GNDD) program includes an increased emphasis on the development of Earth models and methods to predict entire seismic and acoustic waveforms more accurately. In general, this increased emphasis is predicated on the need to better characterize seismic events and provide improved model-based discrimination between event types including earthquakes and explosions. More specifically, while current moment tensor inversion methods tend to work well for larger events (M>~4) using tuned 1-D Earth models, the development of state-of-the-art 3-D models and methods is required for the prediction of shorter period waves over large areas for discrimination of smaller events. There is no standard metric for model-based waveform prediction accuracy used in the waveform modeling/inversion community. However, there are several popular waveform misfit definitions; and minimizing the corresponding objective functions is the goal of waveform inversion. Some example misfit definitions employed for adjoint waveform tomography include measures of simple travel time differences (e.g. Tape et al., 2010), cross-correlation travel time differences (e.g. Luo and Schuster, 1991), multi-taper frequency dependent methods (e.g. Lei et al., 2020), time-frequency phase misfit functions (e.g. Fichtner 2010; Rodgers et al., 2022), normalized cross-correlation methods (e.g. Tao et al., 2018), and others. In some cases, these misfit definitions also involve complicated weighting schemes and summations over multiple frequency bands making it difficult to duplicate the misfit measurement with alternative models and datasets. Although each of the misfit definitions mentioned above are useful for developing waveform models, the actual misfit values are not usually meaningful outside of a given project, model, and/or dataset. Therefore, it is difficult to understand and communicate model performance for predicting waveforms and comparing to other models and/or new model iterations with a different dataset. Therefore, there is a need for a generalized method for evaluating overall model performance that is independent from the specific misfit chosen to develop the waveform models that is also intuitive and meaningful. In this report, we describe a new metric we refer to as ‘Percent of Correlated Signal’. The following sections describe and demonstrate the metric with a case study event and a more rigorous test using a random selection of globally distributed events. While the focus here is on global tomography models, the metric is meant to applicable to regional ‘wiggle-for-wiggle’ waveform models/studies as well.

58 GEOSCIENCES↗

NASA NTRS1975

A triangular thin shell finite element: Nonlinear analysis

Aspects of the formulation of a triangular thin shell finite element which pertain to geometrically nonlinear (small strain, finite displacement) behavior are described. The procedure for solution of the resulting nonlinear algebraic equations combines a one-step incremental (tangent stiffness) approach with one iteration in the Newton-Raphson mode. A method is presented which permits a rational estimation of step size in this procedure. Limit points are calculated by means of a superposition scheme coupled to the incremental side of the solution procedure while bifurcation points are calculated through a process of interpolation of the determinants of the tangent-stiffness matrix. Numerical results are obtained for a flat plate and two curved shell problems and are compared with alternative solutions.

Thomas, G. R.↗