The round jet with swirl
Round laminar jet with swirl, reducing Navier- Stokes equations to ordinary differential equations by similarity transformations
Millsaps, K.↗
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Results for “differential equations”
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A computer program for the geometrically nonlinear static and dynamic analysis of arbitrarily loaded shells of revolution, theory and users manual
A digital computer program known as SATANS (static and transient analysis, nonlinear, shells) for the geometrically nonlinear static and dynamic response of arbitrarily loaded shells of revolution is presented. Instructions for the preparation of the input data cards and other information necessary for the operation of the program are described in detail and two sample problems are included. The governing partial differential equations are based upon Sanders' nonlinear thin shell theory for the conditions of small strains and moderately small rotations. The governing equations are reduced to uncoupled sets of four linear, second order, partial differential equations in the meridional and time coordinates by expanding the dependent variables in a Fourier sine or cosine series in the circumferential coordinate and treating the nonlinear modal coupling terms as pseudo loads. The derivatives with respect to the meridional coordinate are approximated by central finite differences, and the displacement accelerations are approximated by the implicit Houbolt backward difference scheme with a constant time interval. The boundaries of the shell may be closed, free, fixed, or elastically restrained. The program is coded in the FORTRAN 4 language and is dimensioned to allow a maximum of 10 arbitrary Fourier harmonics and a maximum product of the total number of meridional stations and the total number of Fourier harmonics of 200. The program requires 155,000 bytes of core storage.
Ball, R. E.↗
A study on the forced vibration of a Timoshenko beam
By using Galerkin's variational method an approximate solution is developed for a system of two differential equations with linear partial derivatives of the second order. This system of differential equations corresponds to the physical model, known in the literature as the Timoshenko Beam. The results obtained are applied to two particular cases representing respectively: the case of a beam with a rectangular section, with a constant height and a basis with a linear variation; and the case of a beam with a constant basis and a height with cubic variation.
Zainea, B.↗
An implicit finite-difference code for inviscid and viscous cascade flow
An implicit finite-difference code is developed to solve either inviscid or viscous flow about two-dimensional cascade blade elements. General coordinate transformations are used so that boundaries can coincide with coordinate lines, and an automatic grid generation routine based on elliptic partial differential equations is employed to mesh arbitrary cascade elements. Characteristic combinations of the differential equations are used at inflow and outflow boundaries. Computed results for both inviscid and viscous flow are compared with other existing cascade solutions and experimental data.
Steger, J. L.↗
Surface roughness effect on finite oil journal bearings
A theoretical study of the performance of finite oil journal bearings is made, considering the surface roughness effect. The total load supporting ability under such a condition derives from the hydrodynamic as well as asperity contact pressure. These two components of load are calculated separately. The average Reynolds equation for partially lubricated surfaces is used to evaluate hydrodynamic pressure. An analytical expression for average film thickness is obtained and introduced to modify the average Reynolds equation. The resulting differential equation is then solved numerically by finite difference methods for mean hydrodynamic pressure, which in turn gives the hydrodynamic load. Assuming the surface height distribution as Gaussian, the asperity contact pressure is found. The effect of surface roughness parameter, surface pattern, eccentricity ratio, and length to diameter ratio on hydrodynamic load and on side leakage is investigated. It is shown that hydrodynamic load increases with increasing surface roughness when both journal and bearing surfaces have identical roughness structures or when the journal only has a rough surface. The trend of hydrodynamic load is reversed if the journal surface is smooth and the bearing surface is rough.
Majumdar, B. C.↗
Forward and reverse transfer function model synthesis
A process for synthesizing a mathematical model for a linear mechanical system using the forward and reverse Fourier transform functions is described. The differential equation for a system model is given. The Bode conversion of the differential equation, and the frequency and time-domain optimization matching of the model to the forward and reverse transform functions using the geometric simplex method of Nelder and Mead (1965) are examined. The effect of the window function on the linear mechanical system is analyzed. The model is applied to two examples; in one the signal damps down before the end of the time window and in the second the signal has significant energy at the end of the time window.
Houghton, J. R.↗
Development of magnetohydrodynamic turbulence in coronal loops
It is proposed that a stationary spectrum of magnetohydrodynamic turbulence is generated in solar coronal loops by the interweaving of magnetic field lines driven by the turbulent velocity field in the convective region. Starting from the set of incompressible magnetohydrodynamic equations, two ordinary differential equations describing the stationary turbulent energy spectrum and the frequency of direct energy cascade for a given kinetic energy spectrum of the subphotospheric convective motions are derived. An estimate of the typical dissipation length scale, at which magnetic energy efficiently heats the plasma by Joule dissipation is obtained. The impact of recent results of two-dimensional magnetohydrodynamic numerical simulations on the coronal heating problem is also discussed.
Gomez, Daniel O.↗
On the spectral stability of time integration algorithms for a class of constrained dynamics problems
Incomplete field formulations have recently been the subject of intense research because of their potential in coupled analysis of independently modeled substructures, adaptive refinement, domain decomposition, and parallel processing. This paper discusses the design and analysis of time-integration algorithms for these formulations and emphasizes the treatment of their inter-subdomain constraint equations. These constraints are shown to introduce a destabilizing effect in the dynamic system that can be analyzed by investigating the behavior of the time-integration algorithm at infinite and zero frequencies. Three different approaches for constructing penalty-free unconditionally stable second-order accurate solution procedures for this class of hybrid formulations are presented, discussed and illustrated with numerical examples. The theoretical results presented in this paper also apply to a large family of nonlinear multibody dynamics formulations. Some of the algorithms outlined herein are important alternatives to the popular technique consisting of transforming differential/algebraic equations into ordinary differential equations via the introduction of a stabilization term that depends on arbitrary constants and that influences the computed so1ution.
Farhat, Charbel↗
Method and apparatus for shape-based energy analysis of solids
A computer-readable medium stores instructions including storing a model of a physical structure and defining a mesh for the model. The mesh includes finite elements, each defined by a respective set of edges. The instructions include, for each finite element, identifying a governing differential equation and a set of complementary functions that exactly satisfy the governing differential equation. The instructions include determining an applied physical stimulus for the physical structure. The instructions include generating an energy optimization model that minimizes a difference between internal energy and external energy of the finite elements in response to the applied physical stimulus. The model includes a matrix of respective scalar multipliers for the complementary functions for each finite element. The instructions include transforming the matrix and calculating a physical parameter of interest. The instructions include, in response to the physical parameter not satisfying a design parameter, updating the model of the physical structure.
Spears, Robert E.↗
An implicit factored scheme for the compressible Navier-Stokes equations. II - The numerical ODE connection
An attempt is made to establish a connection between linear multistep methods for applications to ordinary differential equations and their extension (by approximate factorization) to alternating direction implicit methods for partial differential equations. An earlier implicit factored scheme for the compressible Navier-Stokes equations is generalized by innovations that (1) increase the class of temporal difference schemes to include all linear multistep methods, (2) optimize the class of unconditionally stable factored schemes by a new choice of unknown variable, and (3) improve the computational efficiency by the introduction of quasi-one-leg methods.
Beam, R. M.↗
Preserving nonlinear constraints in variational flow filtering data assimilation
Data assimilation aims to estimate the states of a dynamical system by optimally combining sparse and noisy observations of the physical system with uncertain forecasts produced by a computational model. The states of many dynamical systems of interest obey nonlinear physical constraints, and the corresponding dynamics is confined to a certain sub-manifold of the state space. Standard data assimilation techniques applied to such systems yield posterior states lying outside the manifold, violating the physical constraints. This work focuses on particle flow filters which use stochastic differential equations to evolve state samples from a prior distribution to samples from an observation-informed posterior distribution. The variational Fokker-Planck (VFP)—a generic particle flow filtering framework—is extended to incorporate non-linear, equality state constraints in the analysis. To this end, two algorithmic approaches that modify the VFP stochastic differential equation are discussed: (i) VFPSTAB, to inexactly preserve constraints with the addition of a stabilizing drift term, and (ii) VFPDAE, to exactly preserve constraints by treating the VFP dynamics as a stochastic differential-algebraic equation (SDAE). Additionally, an implicit-explicit time integrator is developed to evolve the VFPDAE dynamics. The strength of the proposed approach for constraint preservation in data assimilation is demonstrated on three test problems: the double pendulum, Korteweg-de-Vries, and the incompressible Navier-Stokes equations.
97 MATHEMATICS AND COMPUTING↗
The Riccati equation, imprimitive actions and symplectic forms
This paper looks at the structure of the solution of a matrix Riccati differential equation under a predefined group of transformations. The group of transformations used is an expanded form of the feedback group. It is shown that this group of transformations is a subgroup of the symplectic group. The orbits of the Riccati differential equation under the action of this group are studied and it is seen how these techniques apply to a decentralized optimal control problem.
Garzia, M. R.↗
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.↗
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. Previously announced in STAR as N83-30109
Phillips, T. N.↗
Prolongation structures of nonlinear evolution equations
A technique is developed for systematically deriving a 'prolongation structure' - a set of interrelated potentials and pseudopotentials - for nonlinear partial differential equations in two independent variables. When this is applied to the Korteweg-de Vries equation, a new infinite set of conserved quantities is obtained. Known solution techniques are shown to result from the discovery of such a structure: related partial differential equations for the potential functions, linear 'inverse scattering' equations for auxiliary functions, Backlund transformations. Generalizations of these techniques will result from the use of irreducible matrix representations of the prolongation structure.
Wahlquist, H. D.↗
Evolution of coupled weakly driven waves in a dissipative plasma
The nonlinear collisional dynamics of coupled driven plasma waves in the presence of background dissipation is studied analytically within kinetic theory. Sufficiently near marginal stability, phase space correlations are poorly preserved and time delays become unimportant. The system is then shown to be governed by two first-order coupled autonomous differential equations of cubic order for the wave amplitudes and two complementary first-order equations for the evolution of their phases. That system of equations can be decoupled and further simplified to a single second-order differential equation of Liénard's type for each amplitude. Numerical solutions for this equation are obtained in the general case, while analytic solutions are obtained for special cases in terms of parameters related to the spacing of the resonances of the two waves in frequency space, e.g., wave lengths and oscillation frequencies. These parameters are further analyzed to find classes of quasi-steady saturation and pulsating scenarios. To classify equilibrium points, local stability analysis is applied, and bifurcation conditions are determined. When the two waves saturate at similar amplitude levels, their combined signal is shown to invariably exhibit amplitude beating and phase jumps of nearly π. In conclusion, the obtained analytical results can be used to benchmark simulations and to interpret eigenmode amplitude measurements in fusion experiments.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Dissipative systems
The abstract theory presented shows how the theory of dissipative systems of ordinary differential equations can be extended to include a wide class of functional and partial differential equations. Since the basic hypotheses are all in terms of boundedness, finding sufficient conditions in terms of Liapunov functions was not difficult. Work is being undertaken to solve some nontrivial examples to illustrate how the theory can applied.
Lasalle, J. P.↗
Analysis and testing of numerical formulas for the initial value problem
Three computer programs for evaluating and testing numerical integration formulas used with fixed stepsize programs to solve initial value systems of ordinary differential equations are described. A program written in PASCAL SERIES, takes as input the differential equations and produces a FORTRAN subroutine for the derivatives of the system and for computing the actual solution through recursive power series techniques. Both of these are used by STAN, a FORTRAN program that interactively displays a discrete analog of the Liapunov stability region of any two dimensional subspace of the system. The derivatives may be used by CLMP, a FORTRAN program, to test the fixed stepsize formula against a good numerical result and interactively display the solutions.
Brown, R. L.↗