1,455 research outputs found
Three Lectures: Nemd, Spam, and Shockwaves
We discuss three related subjects well suited to graduate research. The
first, Nonequilibrium molecular dynamics or "NEMD", makes possible the
simulation of atomistic systems driven by external fields, subject to dynamic
constraints, and thermostated so as to yield stationary nonequilibrium states.
The second subject, Smooth Particle Applied Mechanics or "SPAM", provides a
particle method, resembling molecular dynamics, but designed to solve continuum
problems. The numerical work is simplified because the SPAM particles obey
ordinary, rather than partial, differential equations. The interpolation method
used with SPAM is a powerful interpretive tool converting point particle
variables to twice-differentiable field variables. This interpolation method is
vital to the study and understanding of the third research topic we discuss,
strong shockwaves in dense fluids. Such shockwaves exhibit stationary
far-from-equilibrium states obtained with purely reversible Hamiltonian
mechanics. The SPAM interpolation method, applied to this molecular dynamics
problem, clearly demonstrates both the tensor character of kinetic temperature
and the time-delayed response of stress and heat flux to the strain rate and
temperature gradients. The dynamic Lyapunov instability of the shockwave
problem can be analyzed in a variety of ways, both with and without symmetry in
time. These three subjects suggest many topics suitable for graduate research
in nonlinear nonequilibrium problems.Comment: 40 pages, with 21 figures, as presented at the Granada Seminar on the
Foundations of Nonequilibrium Statistical Physics, 13-17 September, as three
lecture
Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems
The present article presents a summarizing view at differential-algebraic
equations (DAEs) and analyzes how new application fields and corresponding
mathematical models lead to innovations both in theory and in numerical
analysis for this problem class. Recent numerical methods for nonsmooth
dynamical systems subject to unilateral contact and friction illustrate the
topicality of this development.Comment: Preprint of Book Chapte
Numerical stability of coupled differential equation with piecewise constant arguments
This paper deals with the stability of numerical solutions for a coupled differential equation with piecewise constant arguments. A sufficient condition such that the system is asymptotically stable is derived. Furthermore, when the linear θ-method is applied to this system, it is shown that the linear θ-method is asymptotically stable if and only if 1/2<θ≤1. Finally, some numerical experiments are given
Aeroelastic stability of coupled flap-lag motion of hingeless helicopter blades at arbitrary advance ratios
Equations for large amplitude coupled flap-lag motion of a hingeless elastic helicopter blade in forward flight are derived. Only a torsionally rigid blade excited by quasi-steady aerodynamic loads is considered. The effects of reversed flow together with some new terms due to radial flow are included. Using Galerkin's method the spatial dependence is eliminated and the equations are linearized about a suitable equilibrium position. The resulting system of homogeneous periodic equations is solved using multivariable Floquet-Liapunov theory, and the transition matrix at the end of the period is evaluated by two separate methods. Computational efficiency of the two numerical methods is compared. Results illustrating the effects of forward flight and various important blade parameters on the stability boundaries are presented
Dynamical approach study of spurious steady-state numerical solutions of nonlinear differential equations. Part 1: The ODE connection and its implications for algorithm development in computational fluid dynamics
Spurious stable as well as unstable steady state numerical solutions, spurious asymptotic numerical solutions of higher period, and even stable chaotic behavior can occur when finite difference methods are used to solve nonlinear differential equations (DE) numerically. The occurrence of spurious asymptotes is independent of whether the DE possesses a unique steady state or has additional periodic solutions and/or exhibits chaotic phenomena. The form of the nonlinear DEs and the type of numerical schemes are the determining factor. In addition, the occurrence of spurious steady states is not restricted to the time steps that are beyond the linearized stability limit of the scheme. In many instances, it can occur below the linearized stability limit. Therefore, it is essential for practitioners in computational sciences to be knowledgeable about the dynamical behavior of finite difference methods for nonlinear scalar DEs before the actual application of these methods to practical computations. It is also important to change the traditional way of thinking and practices when dealing with genuinely nonlinear problems. In the past, spurious asymptotes were observed in numerical computations but tended to be ignored because they all were assumed to lie beyond the linearized stability limits of the time step parameter delta t. As can be seen from the study, bifurcations to and from spurious asymptotic solutions and transitions to computational instability not only are highly scheme dependent and problem dependent, but also initial data and boundary condition dependent, and not limited to time steps that are beyond the linearized stability limit
Simulation of time-dependent compressible viscous flows using central and upwind-biased finite-difference techniques
Four time-dependent numerical algorithms for the prediction of unsteady, viscous compressible flows are compared. The analyses are based on the time-dependent Navier-Stokes equations expressed in a generalized curvilinear coordinate system. The methods tested include three traditional central-difference algorithms, and a new upwind-biased algorithm utilizing an implicit, time-marching relaxation procedure based on Newton iteration. Aerodynamic predictions are compared for internal duct-type flows and cascaded turbomachinery flows with spatial periodicity. Two-dimensional internal duct-type flow predictions are performed using an H-type grid system. Planar cascade flows are analyzed using a numerically generated, capped, body-centered, O-type grid system. Initial results are presented for critical and supercritical steady inviscid flow about an isolated cylinder. These predictions are verified by comparisons with published computational results from a similar calculation. Results from each method are then further verified by comparison with experimental data for the more demanding case of flow through a two-dimensional turbine cascade. Inviscid predictions are presented for two different transonic turbine cascade flows. All of the codes demonstrate good agreement for steady viscous flow about a high-turning turbine vane with a leading edge separation. The viscous flow results show a marked improvement over the inviscid results in the region near the separation bubble. Viscous flow results are then further verified in finer detail through comparison with the similarity solution for a flat plate boundary-layer flow. The usefulness of the schemes for the prediction of unsteady flows is demonstrated by examining the unsteady viscous flow resulting from a sinusoidally oscillating flat plate in the vicinity of a stagnant fluid. Predicted results are compared with the analytical solution for this flow. Finally, numerical results are compared with flow visualization and experimental data for the unsteady flow resulting from an impulsively started cylinder. Each algorithm demonstrates unique qualities which may be interpreted as either advantageous or disadvantageous, making it difficult to select an optimum scheme. The preferred method is perhaps best chosen based on the experience of the user and the particular application
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