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Results for “differential algebraic methods”

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At least 127 records · Page 7

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NASA NTRS1972

Analytical treatment of gas flows through multilayer insulation, project 1

A theoretical investigation of gas flow inside a multilayer insulation system was made for the case of the broadside pumping process. A set of simultaneous first-order differential equations for the temperature and pressure of the gas molecules through the perforations on the insulation layers. A modified Runge-Kutta method was used for numerical experiment. The numerical stability problem was also investigated. It was shown that when the relaxation time is less than the time period over which the gas properties change appreciably, the set of differential equations can be replaced by a set of algebraic equations for solution. Numerical examples were given and comparisons with experimental data were made.

Lin, J. T.↗

DOE OSTI2025

Scalable multilevel Monte Carlo methods exploiting parallel redistribution on coarse levels

Here, we study an element agglomeration coarsening strategy that requires data redistribution at coarse levels when the number of coarse elements becomes smaller than the number of MPI processes used on the finest level. The overall procedure generates coarse elements (general unstructured unions of fine grid elements) within the framework of element-based algebraic multigrid methods (or AMGe) studied previously. The AMGe-generated coarse spaces have the ability to exhibit approximation properties of the same order as the fine-level spaces since by construction they contain the piecewise polynomials of the same order as on the fine level. These approximation properties are key for the successful use of AMGe in multilevel solvers for nonlinear partial differential equations as well as for multilevel Monte Carlo (MLMC) simulations. The ability to coarsen without being constrained by the number of MPI processes, as described in the present paper, allows to improve the scalability of these solvers as well as the overall MLMC method. The paper illustrates this latter fact with detailed scalability study of MLMC simulations applied to model Darcy equations with a stochastic log-normal permeability field.

AMGe↗

NASA NTRS1973

Analysis of gas flow through a multilayer insulation system.

A theoretical investigation of gas flow inside a multilayer insulation system has been made for the case of the broadside pumping process. A set of simultaneous first-order differential equations for the temperature and pressure of the gas mixture was obtained by considering the diffusion mechanism of the gas molecules through the perforations on the insulation layers. A modified Runge-Kutta method was used for numerical experiment. The numerical stability problem was investigated. It has been shown that when the relaxation time is small compared with the time period over which the gas properties change appreciably, the set of differential equations can be replaced by a set of algebraic equations for solution. Numerical examples were given and comparison with experimental data was made.

Lin, J. T.↗

NASA NTRS1973

Thermal engineering research

Shih, C. C.↗

NASA NTRS1986

Elliptic generation of composite three-dimensional grids about realistic aircraft

An elliptic method for generating composite grids about realistic aircraft is presented. A body-conforming grid is first generated about the entire aircraft by the solution of Poisson's differential equation. This grid has relatively coarse spacing, and it covers the entire physical domain. At boundary surfaces, cell size is controlled and cell skewness is nearly eliminated by inhomogeneous terms, which are found automatically by the program. Certain regions of the grid in which high gradients are expected, and which map into rectangular solids in the computational domain, are then designated for zonal refinement. Spacing in the zonal grids is reduced by adding points with a simple, algebraic scheme. Details of the grid generation method are presented along with results of the present application, a wing-body configuration based on the F-16 fighter aircraft.

Sorenson, R. L.↗

NASA NTRS1986

Elliptic generation of composite three-dimensional grids about realistic aircraft

An elliptic method for generating composite grids about realistic aircraft is presented. A body-conforming grid is first generated about the entire aircraft by the solution of Poisson's differential equation. This grid has relatively coarse spacing, and it covers the entire physical domain. At boundary surfaces, cell size is controlled and cell skewness is nearly eliminated by inhomogeneous terms, which are found automatically by the program. Certain regions of the grid in which high gradients are expected, and which map into rectangular solids in the computational domain, are then designated for zonal refinement. Spacing in the zonal grids is reduced by adding points with a simple, algebraic scheme. Details of the grid-generation method are presented along with results of the present application, a wing/body configuration based on the F-16 fighter aircraft.

Sorenson, Reese L.↗

NASA NTRS1980

Dynamics and control of a continuum model for a solar power system

An approach for modeling dynamic equations of motion of a plate attached with rigid bodies is presented. The equations of motion are developed using the principle of virtual work. Lagrange multipliers are used as interaction forces and/or moments to maintain prescribed constraints which is the basis of the interconnection between the plate and rigid bodies. The overall approach is unique in the sense that a continuous model described by a family of partial differential equations is established. An approximate formulation by using variational method is established yielding a solution compatible with the assumed degree of approximation. The formulation is useful particularly when parametric study of dynamic response for a satellite power system is desired. As an example, an approximate governing equation of algebraic eigenvalue problem is given for a dual microwave power transmission system. Controller design is discussed.

Juang, J. N.↗

NASA NTRS1982

Algebraic grid generation

The numerical solution of partial differential equations about irregular geometries and with varying characteristic scales has created the need for coordinate systems and associated transformations which reflect both geometric and physical requirements. The process of finding coordinate transformations in discrete representations is called 'grid generation'. The present investigation is concerned with three algebraic grid generation methods. The methods include transfinite interpolation, the multisurface method, and the two-boundary technique. Interpolation formulas, in terms of homotopic mappings and constraints in terms of point positions and/or derivatives are the essential elements of the techniques. The methods are relatively simple to understand, they are explicit and do not require extensive computational effort, and they have a high degree of generality.

Smith, R. E.↗

NASA NTRS1980

The Finite Analytic Method for steady and unsteady heat transfer problems

A new numerical method called the Finite Analytical Method for solving partial differential equations is introduced. The basic idea of the finite analytic method is the incorporation of the local analytic solution in obtaining the numerical solution of the problem. The finite analytical method first divides the total region of the problem into small subregions in which local analytic solutions are obtained. Then an algebraic equation is derived from the local analytic solution for each subregion relating an interior nodal value at a point P in the subregion to its neighboring nodal values. The assembly of all the local analytic solutions thus provides the finite-analytic numerical solution of the problem. In this paper the finite analytic method is illustrated in solving steady and unsteady heat transfer problems.

Chen, C.-J.↗

NASA NTRS1983

Modeling of steady, rotational, transonic winds from rotating stars and galaxies

The theory of steady transonic winds from condensed bodies is extended to general, two-dimensional, axisymmetric systems. A stream function is used to reduce the gasdynamics equations to a single, second-order differential equation plus an algebraic equation for the density. The approach extends Parker's (1958) quasi-one-dimensional theory, which uses Bernoulli's theorem, to a complete two-dimensional calculation which includes vorticity and rotation. The conceptual basis of the stream function approach is described in detail, and a numerical method for solving the resulting equations is presented. The applications illustrate solutions for stellar (spherical source surface) and galactic (oblate spheroidal source surface) models. Among other things, it is found that for rapidly rotating stellar models the Coriolis force dominates the centrifugal terms and the streamlines bend toward the rotation axis as a consequence of the axisymmetry. For both the rotating and nonrotating galaxy models, the streamlines bend toward the equator.

Kopriva, D. A.↗

NASA NTRS1988

Optical matrix-vector processing for computational fluid dynamics

An optical processor to solve partial differential equations for computational fluid dynamics applications is considered. This application is new and original for optical processors. The algorithms that are used are optical realizations of the Newton-Raphson method for nonlinear equations and a new optical LU direct decomposition and Gauss-Seidel iterative solution to the resultant linear algebraic equations. These algorithms are used to solve Burger's equation (a specific form of the momentum equation in fluid dynamics). The nonlinear equations provide 1-D velocity data at each time step. Simulation results of optical processing with these algorithms on computational fluid dynamics data is included.

Perlee, Caroline J.↗

NASA NTRS1974

Discretized partial differential equations - Examples of control systems defined on modules

The purpose of this paper is to show how the important problems of linear system theory can be solved concisely for a particular class of linear systems, namely block circulant systems, by exploiting the algebraic structure. This type of system arises in lumped approximations to linear partial differential equations. The computation of the transition matrix, the variation of constants formula, observability, controllability, pole allocation, realization theory, stability and quadratic optimal control are discussed. In principle, all questions which are solved here could also be solved by standard methods; the present paper clearly exposes the structure of the solution, and thus permits various savings in computational effort.

Brockett, R. W.↗

NASA NTRS1995

Sensitivity analysis of frictional contact response of axisymmetric composite structures

A computational procedure is presented for evaluating the sensitivity coefficients of the static frictional contact response of axisymmetric composite structures. The structures are assumed to consist of an arbitrary number of perfectly bonded homogeneous anisotropic layers. The material of each layer is assumed to be hyperelastic, and the effect of geometric nonlinearity is included. The sensitivity coefficients measure the sensitivity of the response variations in different material, lamination and geometric parameters of the structure. A displacement finite element model is used for the discretization. The normal contact conditions are incorporated into the formulation by using a perturbed Lagrangian approach with the fundamental unknowns consisting of nodal displacements, and Lagrange multipliers associated with the contact conditions. The Lagrange multipliers are allowed to be discontinuous at interelement boundaries. Tangential contact conditions are incorporated by using a penalty method in conjunction with the classical Coulomb's friction model. The Newton-Raphson iterative scheme is used for the solution of the resulting nonlinear algebraic equations, and for the determination of the contact region, contact conditions (sliding or sticking), and the contact pressures. The sensitivity coefficients are evaluated by using a direct differentiation approach. Numerical results are presented for the frictional contact of a composite spherical cap pressed against a rigid plate.

Karaoglan, I.↗

DOE OSTI2022

Electromagnetic Transient (EMT) Simulation Algorithms for Evaluation of Large-Scale Extreme Fast Charging Systems (T&D Models)

Simulation of high-fidelity models of extreme fast charging (XFC) systems and large-area power grids with many XFCs can be time consuming in traditional simulators. Traditional simulators use a single method of discretization for all the components that results in imposing a large computational burden of inverting a large matrix as well as increased computations related to single method of discretization (that is typically a trapezoidal method). To overcome the problem of simulating large-area power grids with many XFCs, in this paper, advanced numerical simulation algorithms are applied for the first time together to reduce the dimension of matrix inversion. Here, the algorithms include numerical stiffness-based segregation, time constant-based segregation, clustering and aggregation on differential algebraic equations (DAEs), and multi-order integration approaches. These algorithms apply multiple discretization algorithms rather than a single discretization algorithm that further reduces the computational burden. The approaches mentioned here have resulted in speed-up of up to 18x in the simulation of a single distribution system with 15 XFCs and of up to 271x in the simulation of a transmission-distribution system with 300 XFCs in multiple distribution feeders with respect to conventional simulators (like power systems computer aided design [PSCAD]).

42 ENGINEERING↗

DOE OSTI2026

Tracking discontinuities in parameter space

We develop a geometric framework in Feynman-parameter space to determine constraints on the sequential discontinuities of Feynman integrals. Our method is based on tracking the deformation of the integration contour as external kinematics are analytically continued. This procedure imposes powerful constraints on the analytic structure of Feynman integrals, providing crucial inputs for their bootstrap. We demonstrate the usefulness of this framework by applying it to integrals in dimensional regularization, with higher propagator powers, and to examples with non-uniform transcendental weight. The method is illustrated with several one- and two-loop calculations.

Differential and Algebraic Geometry↗

NASA NTRS1991

Homotopy approach to optimal, linear quadratic, fixed architecture compensation

Optimal linear quadratic Gaussian compensators with constrained architecture are a sensible way to generate good multivariable feedback systems meeting strict implementation requirements. The optimality conditions obtained from the constrained linear quadratic Gaussian are a set of highly coupled matrix equations that cannot be solved algebraically except when the compensator is centralized and full order. An alternative to the use of general parameter optimization methods for solving the problem is to use homotopy. The benefit of the method is that it uses the solution to a simplified problem as a starting point and the final solution is then obtained by solving a simple differential equation. This paper investigates the convergence properties and the limitation of such an approach and sheds some light on the nature and the number of solutions of the constrained linear quadratic Gaussian problem. It also demonstrates the usefulness of homotopy on an example of an optimal decentralized compensator.

Mercadal, Mathieu↗

NASA NTRS1976

Heat transfer to a full-coverage film-cooled surface with 30 degree slant-hole injection

Heat transfer behavior was studied in a turbulent boundary layer with full coverage film cooling through an array of discrete holes and with injection 30 deg to the wall surface in the downstream direction. Stanton numbers were measured for a staggered hole pattern with pitch-to-diameter ratios of 5 and 10, an injection mass flux ratio range of 0.1 to 1.3, and a range of Reynolds number Re sub x of 150,000 to 5 million. Air was used as the working fluid, and the mainstream velocity varied from 9.8 to 34.2 m/sec (32 to 112 ft/sec). The data were taken for secondary injection temperature equal to the wall temperature and also equal to the mainstream temperature. The data may be used to obtain Stanton number as a continuous function of the injectant temperature by use of linear superposition theory. The heat transfer coefficient is defined on the basis of a mainstream-to-wall temperature difference. This definition permits direct comparison of performance between film cooling and transpiration cooling. A differential prediction method was developed to predict the film cooling data base. The method utilizes a two-dimensional boundary layer program with routines to model the injection process and turbulence augmentation. The program marches in the streamwise direction, and when a row of holes is encountered, it stops and injects fluid into the boundary layer. The turbulence level is modeled by algebraically augmenting the mixing length, with the augmentation keyed to a penetration distance for the injected fluid.

Crawford, M. E.↗

NASA NTRS1989

Algebraic multigrid methods applied to problems in computational structural mechanics

The development of algebraic multigrid (AMG) methods and their application to certain problems in structural mechanics are described with emphasis on two- and three-dimensional linear elasticity equations and the 'jacket problems' (three-dimensional beam structures). Various possible extensions of AMG are also described. The basic idea of AMG is to develop the discretization sequence based on the target matrix and not the differential equation. Therefore, the matrix is analyzed for certain dependencies that permit the proper construction of coarser matrices and attendant transfer operators. In this manner, AMG appears to be adaptable to structural analysis applications.

Mccormick, Steve↗