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At least 883 records · Page 49

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NASA NTRS1984

On the implementation of modal filters for control of structures

The most common technique for the control of structures is modal control. In modal control, the differential equations in terms of actual coordinates are replaced by a set of ordinary differential equations in terms of the modal coordinates known as modal equations. In designing feedback controls in conjunction with the modal equations, one must know the modal states for the modes targeted for control. The sensors measure actual states, however. The modal states can be estimated by means of a Luenberger observer or modal filters. The modal filters produce estimates of the modal states from distributed measurements of the states. If distributed measurements are not available, then they can be reconstructed from measurements at discrete points via interpolation. This paper examines various questions associated with the implementation of modal filters, such as the effect of choice of interpolation functions and sensors locations, as well as of measurement errors, on the state estimation process. The method is demonstrated by means of two numerical examples.

Meirovitch, L.↗

NASA NTRS1985

Mixed models and reduction method for dynamic analysis of anisotropic shells

A time-domain computational procedure is presented for predicting the dynamic response of laminated anisotropic shells. The two key elements of the procedure are: (1) use of mixed finite element models having independent interpolation (shape) functions for stress resultants and generalized displacements for the spatial discretization of the shell, with the stress resultants allowed to be discontinuous at interelement boundaries; and (2) use of a dynamic reduction method, with the global approximation vectors consisting of the static solution and an orthogonal set of Lanczos vectors. The dynamic reduction is accomplished by means of successive application of the finite element method and the classical Rayleigh-Ritz technique. The finite element method is first used to generate the global approximation vectors. Then the Rayleigh-Ritz technique is used to generate a reduced system of ordinary differential equations in the amplitudes of these modes. The temporal integration of the reduced differential equations is performed by using an explicit half-station central difference scheme (Leap-frog method). The effectiveness of the proposed procedure is demonstrated by means of a numerical example and its advantages over reduction methods used with the displacement formulation are discussed.

Noor, A. K.↗

NASA NTRS1987

Time-dependent approximation schemes for some problems of parameter estimation in distributed systems

A parameter estimation method that can be used to estimate functional parameters in delay differential equations and moving boundary problems is discussed. In either problem, the original model equation (which is infinite-dimensional) is approximated by a system of ordinary differential equations that can be solved numerically in an efficient way. The approximation scheme is based on time-dependent spline elements. For the delay equation with time-varying delay, convergence results are presented that indicate the estimates obtained using the approximating system. Numerical test examples converge in some sense to a best-fit parameter for the original system are included by means of which time-varying and state-dependent delays and a time-varying diffusion coefficient in a one-phase, one-dimensional Stefan problem are estimated.

Murphy, K. A.↗

NASA NTRS2019

A Fast Monte Carlo Method for Model-Based Prognostics Based on Stochastic Calculus

This work proposes a fast Monte Carlo method to solve differential equations utilized in model-based prognostics. The methodology is derived from the theory of stochastic calculus, and the goal of such a method is to speed up the estimation of the probability density functions describing the independent variable evolution over time. In the prognostic scenarios presented in this paper, the stochastic differential equations describe variables directly or indirectly related to the degradation of a monitored system. The method allows the estimation of the probability density functions by solving the deterministic equation and approximating the stochastic integrals using samples of the model noise. By so doing, the prognostic problem is solved without the Monte Carlo simulation based on Euler's forward method, which is typically the most time consuming task of the prediction stage. Three different prognostic scenarios are presented as proof of concept: (i) life prediction of electrolytic capacitors, (ii) remaining time to discharge of Lithium-ion batteries, and (iii) prognostic of cracked structures under fatigue loading. The paper shows how the method produces probability density functions that are statistically indistinguishable from the distributions estimated with Euler's forward Monte Carlo simulation. However, the proposed solution is orders of magnitude faster when computing the time-to-failure distribution of the monitored system. The approach may enable complex real-time prognostics and health management solutions with limited computing power.

stochastic calculus↗

NASA NTRS1978

Fracture and crack growth in orthotropic laminates

Goree, J. G.↗

NASA NTRS1980

Stresses in a three-dimensional unidirectional composite containing broken fibers

An approximate solution is developed for the determination of the interlaminar normal and shear stresses in the vicinity of a crack in a three dimensional composite containing unidirectional linearly elastic fibers in an infinite linearly elastic matrix. In order to reduce the complexity of the formulation, certain assumptions are made as to the physically significant stresses to be retained. These simplifications reduce the partial differential equations of elasticity to differential-difference equations which are tractable using Fourier transform techniques. This 'material modeling' approach is in contrast with solutions developed by considering each lamina as a homogeneous, orthotropic layer. The resulting solution does not contain the classical singular stress field for the fibers and the influence of broken fibers on unbroken fibers is felt by a change in stress concentration factors. The matrix stresses however, are unbounded as the fiber spacing vanishes and an equivalent fiber-matrix geometry is proposed which gives the correct singular behavior. The numerical solution is considered in detail and several specific examples are presented. The potential for damaged or debonded zones to be generated by an embedded crack is discussed, and stress concentration factors for fibers near the crack are given. Detailed comparisons are made between the present solution, the analogous two-dimensional problem, and corresponding shear-lag models.

Goree, J. G.↗

NASA NTRSDate not supplied

On the Implementation and Further Validation of a Time Domain Boundary Element Method Broadband Impedance Boundary Condition

A time domain boundary integral equation with Burton-Miller reformulation is presented for acoustic scattering by surfaces with liners in a uniform mean flow. The Ingard-Myers impedance boundary condition is implemented using a broadband multipole impedance model and converted into time domain differential equations to augment the boundary integral equation. The coupled integral-differential equations are solved numerically by a March-On-in-Time (MOT) scheme. While the Ingard-Myers condition is known to support Kelvin-Helmholtz instability due to its use of a vortex sheet interface between the flow and the liner surface, it is found that by neglecting a second derivative term in the current time domain impedance boundary condition formulation, the instability an be effectively suppressed in computation. The proposed formulation and implementation are validated using a NASA Langley Research Center Grazing Flow Impedance Tube (GFIT) experimental dataset with satisfactory results. Moreover, a minimization procedure for finding the poles and coefficients of the broadband multiple impedance model is formulated in this paper by which, unlike the commonly used vector-fitting method, passivity of the model is ensured. Numerical tests show the proposed minimization approach is effective for modeling liners that are commonly used in aeroacoustic applications.

Time Domain Impedance Boundary Condition↗

NASA NTRS1973

Decomposition-aggregation stability analysis

This report presents the development and description of the decomposition aggregation approach to stability investigations of high dimension mathematical models of dynamic systems. The high dimension vector differential equation describing a large dynamic system is decomposed into a number of lower dimension vector differential equations which represent interconnected subsystems. Then a method is described by which the stability properties of each subsystem are aggregated into a single vector Liapunov function, representing the aggregate system model, consisting of subsystem Liapunov functions as components. A linear vector differential inequality is then formed in terms of the vector Liapunov function. The matrix of the model, which reflects the stability properties of the subsystems and the nature of their interconnections, is analyzed to conclude over-all system stability characteristics. The technique is applied in detail to investigate the stability characteristics of a dynamic model of a hypothetical spinning Skylab.

Siljak, D. D.↗

NASA NTRS1989

Semidiscrete Galerkin modelling of compressible viscous flow past a circular cone at incidence

A numerical study of the laminar and compressible boundary layer, about a circular cone in a supersonic free stream, is presented. It is thought that if accurate and efficient numerical schemes can be produced to solve the boundary layer equations, they can be joined to numerical codes that solve the inviscid outer flow. The combination of these numerical codes is competitive with the accurate, but computationally expensive, Navier-Stokes schemes. The primary goal is to develop a finite element method for the calculation of 3-D compressible laminar boundary layer about a yawed cone. The proposed method can, in principle, be extended to apply to the 3-D boundary layer of pointed bodies of arbitrary cross section. The 3-D boundary layer equations governing supersonic free stream flow about a cone are examined. The 3-D partial differential equations are reduced to 2-D integral equations by applying the Howarth, Mangler, Crocco transformations, a linear relation between viscosity, and a Blasius-type of similarity variable. This is equivalent to a Dorodnitsyn-type formulation. The reduced equations are independent of density and curvature effects, and resemble the weak form of the 2-D incompressible boundary layer equations in Cartesian coordinates. In addition the coordinate normal to the wall has been stretched, which reduces the gradients across the layer and provides high resolution near the surface. Utilizing the parabolic nature of the boundary layer equations, a finite element method is applied to the Dorodnitsyn formulation. The formulation is presented in a Petrov-Galerkin finite element form and discretized across the layer using linear interpolation functions. The finite element discretization yields a system of ordinary differential equations in the circumferential direction. The circumferential derivatives are solved by an implicit and noniterative finite difference marching scheme. Solutions are presented for a 15 deg half angle cone at angles of attack of 5 and 10 deg. The numerical solutions assume a laminar boundary layer with free stream Mach number of 7. Results include circumferential distribution of skin friction and surface heat transfer, and cross flow velocity distributions across the layer.

Meade, Andrew James, Jr.↗

NASA NTRS1996

Multigrid Methods for Fully Implicit Oil Reservoir Simulation

In this paper we consider the simultaneous flow of oil and water in reservoir rock. This displacement process is modeled by two basic equations: the material balance or continuity equations and the equation of motion (Darcy's law). For the numerical solution of this system of nonlinear partial differential equations there are two approaches: the fully implicit or simultaneous solution method and the sequential solution method. In the sequential solution method the system of partial differential equations is manipulated to give an elliptic pressure equation and a hyperbolic (or parabolic) saturation equation. In the IMPES approach the pressure equation is first solved, using values for the saturation from the previous time level. Next the saturations are updated by some explicit time stepping method; this implies that the method is only conditionally stable. For the numerical solution of the linear, elliptic pressure equation multigrid methods have become an accepted technique. On the other hand, the fully implicit method is unconditionally stable, but it has the disadvantage that in every time step a large system of nonlinear algebraic equations has to be solved. The most time-consuming part of any fully implicit reservoir simulator is the solution of this large system of equations. Usually this is done by Newton's method. The resulting systems of linear equations are then either solved by a direct method or by some conjugate gradient type method. In this paper we consider the possibility of applying multigrid methods for the iterative solution of the systems of nonlinear equations. There are two ways of using multigrid for this job: either we use a nonlinear multigrid method or we use a linear multigrid method to deal with the linear systems that arise in Newton's method. So far only a few authors have reported on the use of multigrid methods for fully implicit simulations. Two-level FAS algorithm is presented for the black-oil equations, and linear multigrid for two-phase flow problems with strong heterogeneities and anisotropies is studied. Here we consider both possibilities. Moreover we present a novel way for constructing the coarse grid correction operator in linear multigrid algorithms. This approach has the advantage in that it preserves the sparsity pattern of the fine grid matrix and it can be extended to systems of equations in a straightforward manner. We compare the linear and nonlinear multigrid algorithms by means of a numerical experiment.

Molenaar, J.↗

DOE OSTI2022

Crowd Dynamics: Modeling and Control of Multiagent Systems

This review aims to present recent developments in modeling and control of multiagent systems. A particular focus is set on crowd dynamics characterized by complex interactions among agents, also called social interactions, and large-scale systems. Specifically, in a crowd each individual agent interacts with a field generated by the other agents and the environment. These systems can be modeled at the microscopic scale by ordinary differential equations, while an alternative description at the mesoscopic scale is given by a partial differential equation for the propagation of the probability density of the agents. Control actions can be applied at the individual level as well as at the level of the corresponding fields. This article presents and compares different control types, and the specific application to multilane, multiclass traffic is developed in some detail, showing the main tools at work in a hybrid setting with relevant impacts on autonomous driving.

33 ADVANCED PROPULSION SYSTEMS↗

DOE OSTI2025

The Method of Finite Averages

The Method of Finite Averages (MoFA) is a rigorous multiscale modeling methodology for efficiently modeling multi-physical phenomena in heterogeneous porous media. The code developed in this project aims to perform the numerical calculations required to formulate, implement, and verify MoFA models for Earth and Energy systems (i.e., model verification refers to performing fully-resolved simulations of the systems and comparing their results to those of the models). In general, MoFA transforms partial differential equations (PDEs) describing the fine-scale physics of a system into coupled ordinary differential equations (ODEs)---in time---that describe the coarse-scale---or "average"---physical behaviors of the system. This transformation significantly expedites system simulation, as the coarse-scale ODEs involve vastly fewer degrees of freedom than the fine-scale PDEs. The code developed under this project will allow users to 1.) generate system geometries and numerical meshes, 2.) solve the PDE and ODE systems required for MoFA model formulation and implementation, 3.) solve the PDE systems required to obtain fully-resolved simulation results for model verification, and 4.) compare and plot results (e.g., the model and fully-resolved simulation solutions, the error between the solutions, etc.).

Pietrzyk, KyleM [Lawrence Livermore National Labor↗

DOE OSTI2023

Calculation fo the Reflection and Transmission Coefficients for Gradual Tapered Waveguide Structures

It is required to find the reflection and transmission coefficients for a tapered waveguide. It is assumed that the taper is very long and gradual such that no higher order modes are excited from the discontinuity of the waveguide. Three different methods are presented for finding the reflection coefficient and two methods are presented for finding the transmission coefficient. A traditional differential equation method which finds only the reflection coefficient is presented. An additional set of differential equations are presented which finds both the reflection and transmission coefficients. A FEM (finite element method), which solves for both the reflection and transmission coefficients, is presented with its limitations. An example of a tapered coaxial line and waveguide is presented with the different methods compared and contrasted.

43 PARTICLE ACCELERATORS↗

DOE OSTI2023

Earth Filter Mathematica Toolbox V4 User’s Guide: Simplifying, Visualizing and Quick Modeling of All Fission Products from 60 Nuclear Sources

The Earth Filter Mathematica toolbox is a set of user-friendly routines that accurately calculate the ingrowth and decay of any of the 908 fission products for an infinite combination of sources over a large dynamic range and large time scales. This toolbox was developed by Dr. W. P. Brug at Los Alamos National Laboratory and eliminates the need for the user to define and solve differential equations by automatically scanning the ENDF-349 dataset and creating a complete set of differential equations (including delayed neutron branches to both the ground and metastable states) for every fission product. Additionally, this code allows the user to choose the data source for half-lives from several key references (i.e., TORI, IAEA, ENDF or NUDAT.) The complete set of ENDF-349 independent and cumulative fission yields for all 60 isotopic targets and 4 different neutron spectrums are included. This allows nearly any nuclear fission source to be accurately modeled as a linear combination of endpoint sources. The Earth Filter Toolbox has no limit as to the length of split list, and split lists of 5 or more endpoint sources can be quite common.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

NASA NTRS1966

Matrix methods and automation in structural engineering

The partial differential equations of motion of shell structures subject to arbitrary loads may be extremely difficult or even impossible to solve. Since the digital computer is now an available tool for the structural engineer, current research has been directed toward methods which involve matrix formulation of large systems of algebraic equations and matrix solutions for discrete elements rather than the solution of the partial differential equations. These methods require rapid and accurate computer solutions. Recognizing the accuracy problems inherent in working with large matrices, a comprehensive survey of available computer programs was performed for large matrix inversion and for eigenvalue and eigenvector solutions under a Research Grant from the National Aeronautics and Space Administration and the techniques are developed in this paper for using these programs most efficiently for structural applications. The contents of the paper include: automation of matrix compilation, methods of very large matrix inversion and solution of simultaneous equation, and techniques for finding eigenvalues and vectors. In addition, a finite element stiffness matrix approach developed at the Denver Research Institute for both plates and shells subject to arbitrary dynamic loads is described as it was instrumented with complete automation on the digital computer.

Anita S West↗

NASA NTRS1973

Wind-driven currents in a shallow lake or sea.

A technique is given for obtaining exact analytical solutions for wind-driven currents in a shallow lake or sea with a large class of bottom topographies which may or may not contain an island. Welander's second-order partial differential equation for the lake's surface displacement is used for determining the lake circulation. The technique, together with some results based on conformal mapping, is used to reduce the problem of finding solutions to Welander's equation for a closed body of water to the problem of solving an ordinary differential equation. The method is applied to an elliptically shaped lake and a circular lake containing an eccentrically located circular island.

Goldstein, M.↗

NASA NTRS1997

Mathematical Analysis and Optimization of Infiltration Processes

A variety of infiltration techniques can be used to fabricate solid materials, particularly composites. In general these processes can be described with at least one time dependent partial differential equation describing the evolution of the solid phase, coupled to one or more partial differential equations describing mass transport through a porous structure. This paper presents a detailed mathematical analysis of a relatively simple set of equations which is used to describe chemical vapor infiltration. The results demonstrate that the process is controlled by only two parameters, alpha and beta. The optimization problem associated with minimizing the infiltration time is also considered. Allowing alpha and beta to vary with time leads to significant reductions in the infiltration time, compared with the conventional case where alpha and beta are treated as constants.

Chang, H.-C.↗

NASA NTRS1991

From differential to difference equations for first order ODEs

When constructing an algorithm for the numerical integration of a differential equation, one should first convert the known ordinary differential equation (ODE) into an ordinary difference equation. Given this difference equation, one can develop an appropriate numerical algorithm. This technical note describes the derivation of two such ordinary difference equations applicable to a first order ODE. The implicit ordinary difference equation has the same asymptotic expansion as the ODE itself, whereas the explicit ordinary difference equation has an asymptotic that is similar in structure but different in value when compared with that of the ODE.

Freed, Alan D.↗