Conceptual inconsistencies in finite-dimensional quantum and classical mechanics

Utilizing operational dynamic modeling [Phys. Rev. Lett. 109, 190403 (2012); arXiv:1105.4014], we demonstrate that any finite-dimensional representation of quantum and classical dynamics violates the Ehrenfest theorems. Other peculiarities are also revealed, including the nonexistence of the free particle and ambiguity in defining potential forces. Non-Hermitian mechanics is shown to have the same problems. This work compromises a popular belief that finite-dimensional mechanics is a straightforward discretization of the corresponding infinite-dimensional formulation.Comment: 5 pages, 2 figure

Extension of a discontinuous Galerkin finite element method to viscous rotor flow simulations

Heavy vibratory loading of rotorcraft is relevant for many operational aspects of helicopters, such as the structural life span of (rotating) components, operational availability, the pilot's comfort, and the effectiveness of weapon targeting systems. A precise understanding of the source of these vibrational loads has important consequences in these application areas. Moreover, in order to exploit the full potential offered by new vibration reduction technologies, current analysis tools need to be improved with respect to the level of physical modeling of flow phenomena which contribute to the vibratory loads. In this paper, a computational fluid dynamics tool for rotorcraft simulations based on first-principles flow physics is extended to enable the simulation of viscous flows. Viscous effects play a significant role in the aerodynamics of helicopter rotors in high-speed flight. The new model is applied to three-dimensional vortex flow and laminar dynamic stall. The applications clearly demonstrate the capability of the new model to perform on deforming and adaptive meshes. This capability is essential for rotor simulations to accomodate the blade motions and to enhance vortex resolution

Extension of the discontinuous Galerkin finite element method to viscous rotor flow simulations

Heavy vibratory loading of rotorcraft is relevant for many operational aspects of helicopters, such as the structural life span of (rotating) components, op- erational availability, the pilot’s comfort, and the ef- fectiveness of weapon targeting systems. A precise understanding of the source of these vibrational loads has important consequences in these application ar- eas. Moreover, in order to exploit the full poten- tial offered by new vibration reduction technologies, current analysis tools need to be improved with re- spect to the level of physical modeling of flow phe- nomena which contribute to the vibratory loads. In this paper, a computational fluid dynamics tool for rotorcraft simulations based on first-principles flow physics is extended to enable the simulation of vis- cous flows. Viscous effects play a significant role in the aerodynamics of helicopter rotors in high-speed flight. The new model is applied to three-dimensional vortex flow and laminar dynamic stall. The applica- tions clearly demonstrate the capability of the new model to perform on deforming and adaptive meshes. This capability is essential for rotor simulations to accomodate the blade motions and to enhance vor- tex resolution

Classes of hypercomplex polynomials of discrete variable based on the quasi-monomiality principle

With the aim of derive a quasi-monomiality formulation in the context of discrete hypercomplex variables, one will amalgamate through a Clifford-algebraic structure of signature $(0,n)$ the umbral calculus framework with Lie-algebraic symmetries. The exponential generating function ({\bf EGF}) carrying the {\it continuum} Dirac operator D=\sum_{j=1}^n\e_j\partial_{x_j} together with the Lie-algebraic representation of raising and lowering operators acting on the lattice h\BZ^n is used to derive the corresponding hypercomplex polynomials of discrete variable as Appell sets with membership on the space Clifford-vector-valued polynomials. Some particular examples concerning this construction such as the hypercomplex versions of falling factorials and the Poisson-Charlier polynomials are introduced. Certain applications from the view of interpolation theory and integral transforms are also discussed.Comment: 24 pages. 1 figure. v2: a major revision, including numerous improvements throughout the paper was don

High-Order Numerical Solution of Second-Order One-Dimensional Hyperbolic Telegraph Equation Using a Shifted Gegenbauer Pseudospectral Method

We present a high-order shifted Gegenbauer pseudospectral method (SGPM) to solve numerically the second-order one-dimensional hyperbolic telegraph equation provided with some initial and Dirichlet boundary conditions. The framework of the numerical scheme involves the recast of the problem into its integral formulation followed by its discretization into a system of well-conditioned linear algebraic equations. The integral operators are numerically approximated using some novel shifted Gegenbauer operational matrices of integration. We derive the error formula of the associated numerical quadratures. We also present a method to optimize the constructed operational matrix of integration by minimizing the associated quadrature error in some optimality sense. We study the error bounds and convergence of the optimal shifted Gegenbauer operational matrix of integration. Moreover, we construct the relation between the operational matrices of integration of the shifted Gegenbauer polynomials and standard Gegenbauer polynomials. We derive the global collocation matrix of the SGPM, and construct an efficient computational algorithm for the solution of the collocation equations. We present a study on the computational cost of the developed computational algorithm, and a rigorous convergence and error analysis of the introduced method. Four numerical test examples have been carried out in order to verify the effectiveness, the accuracy, and the exponential convergence of the method. The SGPM is a robust technique, which can be extended to solve a wide range of problems arising in numerous applications.Comment: 36 pages, articl