Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “numerical algorithms”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 595 records · Page 33
All sourcesNASA NTRSDOE OSTI
NASA NTRS1987

A direct algorithm for solution of incompressible three-dimensional unsteady Navier-Stokes equations

A direct, implicit, numerical solution algorithm, with second-order accuracy in space and time, is constructed for the three-dimensional unsteady incompressible Navier-Stokes equations formulated in terms of velocity and vorticity, using generalized orthogonal coordinates to achieve the accurate solution of complex viscous flow configurations. A numerically stable, efficient, direct inversion procedure is developed for the computationally intensive divergence-curl elliptic velocity problem. This overdetermined partial differential operator is first formulated as a uniquely determined, nonsingular matrix-vector problem; this aspect of the procedure is a unique feature of the present analysis. The three-dimensional vorticity-transport equation is solved by a modified factorization technique which completely eliminates the need for any block-matrix inversions and only scalar tridiagonal matrices need to be inverted. The method is applied to the test problem of the three-dimensional flow within a shear-driven cubical box. Coherent streamwise vortex structures are observed within the steady-state flow at Re = 100.

Osswald, G. A.↗
NASA NTRS1996

Computational aeroacoustics and numerical simulation of supersonic jets

The research project has been a computational study of computational aeroacoustics algorithms and numerical simulations of the flow and noise of supersonic jets. During this study a new method for the implementation of solid wall boundary conditions for complex geometries in three dimensions has been developed. In addition, a detailed study of the simulation of the flow in and noise from supersonic circular and rectangular jets has been conducted. Extensive comparisons have been made with experimental measurements. A summary of the results of the research program are attached as the main body of this report in the form of two publications. Also, the report lists the names of the students who were supported by this grant, their degrees, and the titles of their dissertations. In addition, a list of presentations and publications made by the Principal Investigators and the research students is also included.

Morris, Philip J.↗
DOE OSTI2023

Convergence of Hyperbolic Neural Networks Under Riemannian Stochastic Gradient Descent

Abstract We prove, under mild conditions, the convergence of a Riemannian gradient descent method for a hyperbolic neural network regression model, both in batch gradient descent and stochastic gradient descent. We also discuss a Riemannian version of the Adam algorithm. We show numerical simulations of these algorithms on various benchmarks.

Whiting, Wes (ORCID:0000000247505060)↗
DOE OSTI2021

Explicit physics-informed neural networks for nonlinear closure: The case of transport in tissues

In upscaling methods, closures for nonlinear problems present a well-known challenge. While a number of theoretical methods have been proposed for handling such closures, nonlinearities still remain a significant obstacle for many problems. In this work, we use a combination of formal upscaling and data-driven machine learning for explicitly closing a nonlinear transport and reaction process in multiscale tissues. The classical effectiveness factor model is used to formulate the macroscale reaction kinetics. We train a multilayer perceptron network using training data generated by direct numerical simulations over microscale examples. Once trained, the network is used in an algorithm for numerically solving the upscaled (coarse-grained) differential equation describing mass transport and reaction in two example tissues. The network is described as being explicit in the sense that the network is trained using macroscale concentrations and gradients of concentration as components of the feature space rather than incorporating them as part of a constraint in the optimization process. Network training and solutions to the macroscale transport equations were computed for two different tissues. The two tissue types (brain and liver) exhibit markedly different geometrical complexity and spatial scale (cell size and sample size). The upscaled solutions for the average concentration are compared with numerical solutions derived from the microscale concentration fields by a posteriori averaging. There are three outcomes of this work of particular note. 1) Our overall approach results in an upscaled nonlinear PDE. The PDE is closed using a neural network, and our approach results in the definition of the classical effectiveness factor for effecting closure. 2) We identify particular source terms for the closure problem that are important for representing the structure of the closure. These source terms involve macroscale concentrations and their gradients. We adopt these source terms to use as explicit features in the learning algorithm. We find the trained networks that include the macroscale source terms generate models that are able to predict the correction factor with increased fidelity over those that do not. 3) We find that the trained network exhibits good generalizability, and it is able to predict the effectiveness factor with high fidelity for realistically-structured tissues despite the significantly different scale and geometrical complexity of the two example tissue types. This latter result emphasizes our purposeful connection between conventional averaging methods with the use of machine learning for closure; this contrasts with some machine learning methods for upscaling where the exact form of the macroscale equation remains unknown.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

A simple introduction to the SiMPL method for density-based topology optimization

We introduce a novel method for solving density-based topology optimization problems: Sigmoidal Mirror descent with a Projected Latent variable (SiMPL). The SiMPL method (pronounced as “the simple method”) optimizes a design using only first-order derivative information of the objective function. The bound constraints on the density field are enforced with the help of the (negative) Fermi–Dirac entropy, which is also used to define a non-symmetric distance function called a Bregman divergence on the set of admissible designs. This Bregman divergence leads to a simple update rule that is further simplified with the help of a so-called latent variable. Because the SiMPL method involves discretizing the latent variable, it produces a sequence of pointwise-feasible iterates, even when high-order finite elements are used in the discretization. Numerical experiments demonstrate that the method outperforms other popular first-order optimization algorithms. In conclusion, to outline the general applicability of the technique, we include examples with (self-load) compliance minimization and compliant mechanism optimization problems.

Calculus of Variations and Optimization↗
NASA NTRS2009

Addition of Improved Shock-Capturing Schemes to OVERFLOW 2.1

Existing approximate Riemann solvers do not perform well when the grid is not aligned with strong shocks in the flow field. Three new approximate Riemann algorithms are investigated to improve solution accuracy and stability in the vicinity of strong shocks. The new algorithms are compared to the existing upwind algorithms in OVERFLOW 2.1. The new algorithms use a multidimensional pressure gradient based switch to transition to a more numerically dissipative algorithm in the vicinity of strong shocks. One new algorithm also attempts to artificially thicken captured shocks in order to alleviate the errors in the solution introduced by "stair-stepping" of the shock resulting from the approximate Riemann solver. This algorithm performed well for all the example cases and produced results that were almost insensitive to the alignment of the grid and the shock.

Burning, Pieter G.↗
NASA NTRS1974

Nonlinear initial-boundary value solutions by the finite element method

The finite-element method is used to establish a numerical solution algorithm for the Navier-Stokes equations for two-dimensional flows of a viscous compressible fluid. Numerical experiments confirm the advection property for the finite-element equivalent of the nonlinear convection term for both unidirectional and recirculating flowfields. For linear functionals, the algorithm demonstrates good accuracy using coarse discretizations and h squared convergence with discretization refinement.

Baker, A. J.↗
NASA NTRS1991

A parallel-vector Lanczos eigen-solver for structural vibration problems

The Lanczos algorithm for the solution of generalized eigen-problem has been receiving a lot of attention in recent years due to its computational efficiency. The focus of this paper is to develop a Lanczos algorithm which can exploit both the parallel and vector capabilities provided by modern high-performance computers. A partial restoring orthogonality scheme is also developed and incorporated into the basic Lanczos algorithm. The numerical performance in terms of accuracy and efficiency of the proposed parallel-vector Lanczos algorithm is demonstrated by solving for the frequencies and mode shapes of structural problems on multiprocessor supercomputers.

Qin, Jiangning↗
NASA NTRS1994

A combined algorithm for minimum time slewing of flexible spacecraft

The use of Pontryagin's Maximum Principle for the large-angle slewing of large flexible structures usually results in the so-called two-point boundary-value problem (TPBVP), in which many requirements (e.g., minimum time, small flexible amplitude, and limited control powers, etc.) must be satisfied simultaneously. The successful solution of this problem depends largely on the use of an efficient numerical computational algorithm. There are many candidate algorithms available for his problem (e.g., quasilinearization, gradient, and shooting, etc.). In this paper, a proposed algorithm, which combines the quasilinearization method with a time shortening technique and a shooting method, is applied to the minimum-time, three-dimensional, and large-angle maneuver of flexible spacecraft, particularly the orbiting Spacecraft Control Laboratory Experiment (SCOLE) configuration. Theoretically, the nonlinear TPBVP can be solved only through the shooting method to find the 'exact' switching times for the bang-bang controls. However, computationally, a suitable guess for the missing initial costates is crucial because the convergence range of the unknown initial costates is usually narrow, especially for systems with high dimensions and when a multi-bang-bang control strategy is needed. On the other hand, the problems of near minimum time attitude maneuver of general rigid spacecraft and fast slewing of flexible spacecraft have been examined by the authors through a numerical approach based on the quasilinearization algorithm with a time shortening technique. Computational results have demonstrated its broad convergence range and insensitivity to initial costate choices. Consequently, a combined approach is naturally suggested here to solve the minimum time slewing problem. That is, in the computational process, the quasilinearization method is used first to obtain a near minimum time solution. Then, the acquired converged initial costates from the quasilinearization approach are transformed (tailored) to and used as the initial costate guess for starting the shooting method. Finally, the shooting method takes over the remaining calculations until the minimum-time solution converges. The nonlinear equations of motion of the SCOLE are formulated by using Lagrange's equations, with the mast modeled as a continuous beam subject to three-dimensional deformations. The numerical results will be presented and some related computational issues will also be discussed.

Bainum, P. M.↗
NASA NTRS2016

Comparison of NACA 0012 Laminar Flow Solutions: Structured and Unstructured Grid Methods

In this paper we consider the solution of the compressible Navier-Stokes equations for a class of laminar airfoil flows. The principal objective of this paper is to demonstrate that members of this class of laminar flows have steady-state solutions. These laminar airfoil flow cases are often used to evaluate accuracy, stability and convergence of numerical solution algorithms for the Navier-Stokes equations. In recent years, such flows have also been used as test cases for high-order numerical schemes. While generally consistent steady-state solutions have been obtained for these flows using higher order schemes, a number of results have been published with various solutions, including unsteady ones. We demonstrate with two different numerical methods and a range of meshes with a maximum density that exceeds 8 × 106 grid points that steady-state solutions are obtained. Furthermore, numerical evidence is presented that even when solving the equations with an unsteady algorithm, one obtains steady-state solutions.

Swanson, R. C.↗
NASA NTRS1988

Investigation of practical applications of H infinity control theory to the design of control systems for large space structures

The applicability of H infinity control theory to the problems of large space structures (LSS) control was investigated. A complete evaluation to any technique as a candidate for large space structure control involves analytical evaluation, algorithmic evaluation, evaluation via simulation studies, and experimental evaluation. The results of analytical and algorithmic evaluations are documented. The analytical evaluation involves the determination of the appropriateness of the underlying assumptions inherent in the H infinity theory, the determination of the capability of the H infinity theory to achieve the design goals likely to be imposed on an LSS control design, and the identification of any LSS specific simplifications or complications of the theory. The resuls of the analytical evaluation are presented in the form of a tutorial on the subject of H infinity control theory with the LSS control designer in mind. The algorthmic evaluation of H infinity for LSS control pertains to the identification of general, high level algorithms for effecting the application of H infinity to LSS control problems, the identification of specific, numerically reliable algorithms necessary for a computer implementation of the general algorithms, the recommendation of a flexible software system for implementing the H infinity design steps, and ultimately the actual development of the necessary computer codes. Finally, the state of the art in H infinity applications is summarized with a brief outline of the most promising areas of current research.

Irwin, R. Dennis↗
NASA NTRS1992

Antenna analysis using neural networks

Conventional computing schemes have long been used to analyze problems in electromagnetics (EM). The vast majority of EM applications require computationally intensive algorithms involving numerical integration and solutions to large systems of equations. The feasibility of using neural network computing algorithms for antenna analysis is investigated. The ultimate goal is to use a trained neural network algorithm to reduce the computational demands of existing reflector surface error compensation techniques. Neural networks are computational algorithms based on neurobiological systems. Neural nets consist of massively parallel interconnected nonlinear computational elements. They are often employed in pattern recognition and image processing problems. Recently, neural network analysis has been applied in the electromagnetics area for the design of frequency selective surfaces and beam forming networks. The backpropagation training algorithm was employed to simulate classical antenna array synthesis techniques. The Woodward-Lawson (W-L) and Dolph-Chebyshev (D-C) array pattern synthesis techniques were used to train the neural network. The inputs to the network were samples of the desired synthesis pattern. The outputs are the array element excitations required to synthesize the desired pattern. Once trained, the network is used to simulate the W-L or D-C techniques. Various sector patterns and cosecant-type patterns (27 total) generated using W-L synthesis were used to train the network. Desired pattern samples were then fed to the neural network. The outputs of the network were the simulated W-L excitations. A 20 element linear array was used. There were 41 input pattern samples with 40 output excitations (20 real parts, 20 imaginary). A comparison between the simulated and actual W-L techniques is shown for a triangular-shaped pattern. Dolph-Chebyshev is a different class of synthesis technique in that D-C is used for side lobe control as opposed to pattern shaping. The interesting thing about D-C synthesis is that the side lobes have the same amplitude. Five-element arrays were used. Again, 41 pattern samples were used for the input. Nine actual D-C patterns ranging from -10 dB to -30 dB side lobe levels were used to train the network. A comparison between simulated and actual D-C techniques for a pattern with -22 dB side lobe level is shown. The goal for this research was to evaluate the performance of neural network computing with antennas. Future applications will employ the backpropagation training algorithm to drastically reduce the computational complexity involved in performing EM compensation for surface errors in large space reflector antennas.

Smith, William T.↗
NASA NTRS1987

A Taylor weak-statement algorithm for hyperbolic conservation laws

Finite element analysis, applied to computational fluid dynamics (CFD) problem classes, presents a formal procedure for establishing the ingredients of a discrete approximation numerical solution algorithm. A classical Galerkin weak-statement formulation, formed on a Taylor series extension of the conservation law system, is developed herein that embeds a set of parameters eligible for constraint according to specification of suitable norms. The derived family of Taylor weak statements is shown to contain, as special cases, over one dozen independently derived CFD algorithms published over the past several decades for the high speed flow problem class. A theoretical analysis is completed that facilitates direct qualitative comparisons. Numerical results for definitive linear and nonlinear test problems permit direct quantitative performance comparisons.

Baker, A. J.↗
NASA NTRS1995

Robust integration schemes for generalized viscoplasticity with internal-state variables. Part 2: Algorithmic developments and implementation

This two-part report is concerned with the development of a general framework for the implicit time-stepping integrators for the flow and evolution equations in generalized viscoplastic models. The primary goal is to present a complete theoretical formulation, and to address in detail the algorithmic and numerical analysis aspects involved in its finite element implementation, as well as to critically assess the numerical performance of the developed schemes in a comprehensive set of test cases. On the theoretical side, the general framework is developed on the basis of the unconditionally-stable, backward-Euler difference scheme as a starting point. Its mathematical structure is of sufficient generality to allow a unified treatment of different classes of viscoplastic models with internal variables. In particular, two specific models of this type, which are representative of the present start-of-art in metal viscoplasticity, are considered in applications reported here; i.e., fully associative (GVIPS) and non-associative (NAV) models. The matrix forms developed for both these models are directly applicable for both initially isotropic and anisotropic materials, in general (three-dimensional) situations as well as subspace applications (i.e., plane stress/strain, axisymmetric, generalized plane stress in shells). On the computational side, issues related to efficiency and robustness are emphasized in developing the (local) interative algorithm. In particular, closed-form expressions for residual vectors and (consistent) material tangent stiffness arrays are given explicitly for both GVIPS and NAV models, with their maximum sizes 'optimized' to depend only on the number of independent stress components (but independent of the number of viscoplastic internal state parameters). Significant robustness of the local iterative solution is provided by complementing the basic Newton-Raphson scheme with a line-search strategy for convergence. In the present second part of the report, we focus on the specific details of the numerical schemes, and associated computer algorithms, for the finite-element implementation of GVIPS and NAV models.

Li, Wei↗
NASA NTRS1982

A penalty finite element algorithm for parabolic flow problems

The thin-layer simplification of the two-dimensional Navier-Stokes equations for steady viscous flow are developed using an order of magnitude analysis. A space marching finite element solution algorithm is developed, wherein the first order continuity effects are enforced as a penalty function differential constraint. Numerical results are presented to document accuracy and convergence features of the numerical solution algorithm.

Baker, A. J.↗
NASA NTRS1988

Material parameter measurements at high temperatures

Alternate fixtures of techniques for the measurement of the constitutive material parameters at elevated temperatures are presented. The technique utilizes scattered field data from material coated cylinders between parallel plates or material coated hemispheres over a finite size groundplane. The data acquisition is centered around the HP 8510B Network Analyzer. The parameters are then found from a numerical search algorithm using the Newton-Ralphson technique with the measured and calculated fields from these canonical scatters. Numerical and experimental results are shown.

Dominek, A.↗