A symmetry reduction technique for higher order Painlev\'e systems

The symmetry reduction of higher order Painlev\'e systems is formulated in terms of Dirac procedure. A set of canonical variables that admit Dirac reduction procedure is proposed for Hamiltonian structures governing the ${A^{(1)}_{2M}}$ and ${A^{(1)}_{2M-1}}$ Painlev\'e systems for $M=2,3,...$.Comment: to appear in Phys. Lett.

Krylov subspaces associated with higher-order linear dynamical systems

A standard approach to model reduction of large-scale higher-order linear dynamical systems is to rewrite the system as an equivalent first-order system and then employ Krylov-subspace techniques for model reduction of first-order systems. This paper presents some results about the structure of the block-Krylov subspaces induced by the matrices of such equivalent first-order formulations of higher-order systems. Two general classes of matrices, which exhibit the key structures of the matrices of first-order formulations of higher-order systems, are introduced. It is proved that for both classes, the block-Krylov subspaces induced by the matrices in these classes can be viewed as multiple copies of certain subspaces of the state space of the original higher-order system

Model reduction in integrated controls-structures design

It is the objective of this paper to present a model reduction technique developed for the integrated controls-structures design of flexible structures. Integrated controls-structures design problems are typically posed as nonlinear mathematical programming problems, where the design variables consist of both structural and control parameters. In the solution process, both structural and control design variables are constantly changing; therefore, the dynamic characteristics of the structure are also changing. This presents a problem in obtaining a reduced-order model for active control design and analysis which will be valid for all design points within the design space. In other words, the frequency and number of the significant modes of the structure (modes that should be included) may vary considerably throughout the design process. This is also true as the locations and/or masses of the sensors and actuators change. Moreover, since the number of design evaluations in the integrated design process could easily run into thousands, any feasible order-reduction method should not require model reduction analysis at every design iteration. In this paper a novel and efficient technique for model reduction in the integrated controls-structures design process, which addresses these issues, is presented

Phonon Dispersion Effects and the Thermal Conductivity Reduction in GaAs/AlAs Superlattices

The experimentally observed order-of-magnitude reduction in the thermal conductivity along the growth axis of (GaAs)_n/(AlAs)_n (or n x n) superlattices is investigated theoretically for (2x2), (3x3) and (6x6) structures using an accurate model of the lattice dynamics. The modification of the phonon dispersion relation due to the superlattice geometry leads to flattening of the phonon branches and hence to lower phonon velocities. This effect is shown to account for a factor-of-three reduction in the thermal conductivity with respect to bulk GaAs along the growth direction; the remainder is attributable to a reduction in the phonon lifetime. The dispersion-related reduction is relatively insensitive to temperature (100 < T < 300K) and n. The phonon lifetime reduction is largest for the (2x2) structures and consistent with greater interface scattering. The thermal conductivity reduction is shown to be appreciably more sensitive to GaAs/AlAs force constant differences than to those associated with molecular masses.Comment: 5 figure

Poisson-Jacobi reduction of homogeneous tensors

The notion of homogeneous tensors is discussed. We show that there is a one-to-one correspondence between multivector fields on a manifold $M$, homogeneous with respect to a vector field $\Delta$ on $M$, and first-order polydifferential operators on a closed submanifold $N$ of codimension 1 such that $\Delta$ is transversal to $N$. This correspondence relates the Schouten-Nijenhuis bracket of multivector fields on $M$ to the Schouten-Jacobi bracket of first-order polydifferential operators on $N$ and generalizes the Poissonization of Jacobi manifolds. Actually, it can be viewed as a super-Poissonization. This procedure of passing from a homogeneous multivector field to a first-order polydifferential operator can be also understood as a sort of reduction; in the standard case -- a half of a Poisson reduction. A dual version of the above correspondence yields in particular the correspondence between $\Delta$-homogeneous symplectic structures on $M$ and contact structures on $N$.Comment: 19 pages, minor corrections, final version to appear in J. Phys. A: Math. Ge

Efficient Analysis of Structures with Rotatable Elements Using Model Order Reduction

This paper presents a novel full-wave technique which allows for a fast 3D finite element analysis of waveguide structures containing rotatable tuning elements of arbitrary shapes. Rotation of these elements changes the resonant frequencies of the structure, which can be used in the tuning process to obtain the S-characteristics desired for the device. For fast commutations of the response as the tuning elements are rotated, the 3D finite element method is supported by multilevel model-order reduction, orthogonal projection at the boundaries of macromodels and the operation called macromodels cloning. All the time-consuming steps are performed only once in the preparatory stage. In the tuning stage, only small parts of the domain are updated, by means of a special meshing technique. In effect, the tuning process is performed extremely rapidly. The results of the numerical experiments confirm the efficiency and validity of the proposed method

Spatial Manifestations of Order Reduction in Runge-Kutta Methods for Initial Boundary Value Problems

This paper studies the spatial manifestations of order reduction that occur when time-stepping initial-boundary-value problems (IBVPs) with high-order Runge-Kutta methods. For such IBVPs, geometric structures arise that do not have an analog in ODE IVPs: boundary layers appear, induced by a mismatch between the approximation error in the interior and at the boundaries. To understand those boundary layers, an analysis of the modes of the numerical scheme is conducted, which explains under which circumstances boundary layers persist over many time steps. Based on this, two remedies to order reduction are studied: first, a new condition on the Butcher tableau, called weak stage order, that is compatible with diagonally implicit Runge-Kutta schemes; and second, the impact of modified boundary conditions on the boundary layer theory is analyzed.Comment: 41 pages, 9 figure