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A computer program to generate equations of motion matrices, L217 (EOM). Volume 2: Supplemental system design and maintenance document
The equations of motion program L217 (EOM) is described. The program formulates the matrix coefficients for a second order linear differential equation which describes the motion of an airplane relative to its level equilibrium flight condition. Aerodynamic data from FLEXSTAB or Doublet Lattice (L216) programs are used to derive the equations for quasi-steady or complete unsteady aerodynamics.
Structural Dynamics, Stability, and Control of Helicopters
The dynamic synthesis of gyroscopic structures consisting of point-connected substructures is investigated. The objective is to develop a mathematical model capable of an adequate simulation of the modal characteristics of a helicopter using a minimum number of degrees of freedom. The basic approach is to regard the helicopter structure as an assemblage of flexible substructures. The variational equations for the perturbed motion about certain equilibrium solutions are derived. The discretized variational equations can be conveniently exhibited in matrix form, and a great deal of information about the system modal characteristics can be extracted from the coefficient matrices. The derivation of the variational equations requires a monumental amount of algebraic operations. To automate this task a symbolic manipulation program on a digital computer is developed.
Kraus is king: High-order completely positive and trace preserving (CPTP) low rank method for the Lindblad master equation
We design high order accurate methods that exploit low rank structure in the density matrix while respecting the essential structure of the Lindblad equation. Here, our methods preserves complete positivity and are trace preserving.
Structural Analysis and Matrix Interpetive System /SAMIS/ program Technical report, Feb. - Aug. 1966
Development of characteristic equations and error analysis for computer programs contained in structural analysis and matrix interpretive system
Noniterative solutions of integral equations for scattering. I - Single channels
Scattering matrix noniterative integral solutions applied to singlet and triplet s-wave Hartree-Fock phase shifts for electron-H-atom scattering
Application of Krylov exponential propagation to fluid dynamics equations
This paper presents an application of matrix exponentiation via Krylov subspace projection, to the solution of fluid dynamics problems. The main idea is to approximate the operation exp(A)v by means of a projection-like process onto a Krylov subspace. This results in a computation of an exponential matrix vector product similar to the one above but of a much smaller size. Time integration schemes can then be devised to exploit this basic computational kernel. The motivation of this approach is to provide time-integration schemes that are essentially of an explicit nature but which have good stability properties.
State-Dependent Riccati Equation Regulation of Systems with State and Control Nonlinearities
The state-dependent Riccati equations (SDRE) is the basis of a technique for suboptimal feedback control of a nonlinear quadratic regulator (NQR) problem. It is an extension of the Riccati equation used for feedback control of linear problems, with the addition of nonlinearities in the state dynamics of the system resulting in a state-dependent gain matrix as the solution of the equation. In this paper several variations on the SDRE-based method will be considered for the feedback control problem with control nonlinearities. The control nonlinearities may result in complications in the numerical implementation of the control, which the different versions of the SDRE method must try to overcome. The control methods will be applied to three test problems and their resulting performance analyzed.
Linear Hamiltonian systems - The Riccati group and its invariants
The action of the Riccati group on the Riccati differential equation is associated with the action of a subgroup of the symplectic group on a set of Hamiltonian matrices. Within this framework canonical forms are developed for the matrix coefficients of the Riccati differential equation.
The coupling of radiative transfer to quasi 1-D flows with thermochemical nonequilibrium
Quasi-one-dimensional nonequilibrium nozzle flows with coupled radiative transfer are considered. The strongly coupled formulation of radiation and flowfield leads to a governing set of integro-differential equations. A fully implicit numerical method using the full matrix inversion or block iteration methods is presented to solve these equations. The nonequilibrium gas model consists of two chemical species, molecular and atomic nitrogen. The thermodynamic state of the gas is described by two temperatures, translational-rotational and vibrational, and the thermal radiation is assumed to be governed by the vibrational temperature. In radiative transfer, gases are assumed to be absorbing and emitting, and a detailed spectral dependency of the absorption coefficient is prescribed for a non-gray gas. The numerical solutions of strongly radiating nonequilibrium flows are presented for both gray and non-gray gases.
The matrix product approximation for the dynamic cavity method
Stochastic dynamics of classical degrees of freedom, defined on vertices of locally tree-like graphs, can be studied in the framework of the dynamic cavity method which is exact for tree graphs. Such models correspond for example to spin-glass systems, Boolean networks, neural networks, and other technical, biological, and social networks. The central objects in the cavity method are edge messages—conditional probabilities of two vertex variable trajectories. In this paper, we discuss a rather pedagogical derivation for the dynamic cavity method, give a detailed account of the novel matrix product edge message (MPEM) algorithm for the solution of the dynamic cavity equation as introduced, and present optimizations and extensions. Matrix product approximations of the edge messages are constructed recursively in an iteration over time. Computation costs and precision can be tuned by controlling the matrix dimensions of the MPEM in truncations. Without truncations, the dynamics is exact. Data for Glauber–Ising dynamics shows a linear growth of computation costs in time. In contrast to Monte Carlo simulations, the approach has a much better error scaling. Hence, it gives for example access to low probability events and decaying observables like temporal correlations. Here, we discuss optimized truncation schemes and an extension that allows to capture models which have a continuum time limit.
An incremental strategy for calculating consistent discrete CFD sensitivity derivatives
In this preliminary study involving advanced computational fluid dynamic (CFD) codes, an incremental formulation, also known as the 'delta' or 'correction' form, is presented for solving the very large sparse systems of linear equations which are associated with aerodynamic sensitivity analysis. For typical problems in 2D, a direct solution method can be applied to these linear equations which are associated with aerodynamic sensitivity analysis. For typical problems in 2D, a direct solution method can be applied to these linear equations in either the standard or the incremental form, in which case the two are equivalent. Iterative methods appear to be needed for future 3D applications; however, because direct solver methods require much more computer memory than is currently available. Iterative methods for solving these equations in the standard form result in certain difficulties, such as ill-conditioning of the coefficient matrix, which can be overcome when these equations are cast in the incremental form; these and other benefits are discussed. The methodology is successfully implemented and tested in 2D using an upwind, cell-centered, finite volume formulation applied to the thin-layer Navier-Stokes equations. Results are presented for two laminar sample problems: (1) transonic flow through a double-throat nozzle; and (2) flow over an isolated airfoil.
Two-dimensional integrating matrices on rectangular grids
The use of integrating matrices in solving differential equations associated with rotating beam configurations is examined. In vibration problems, by expressing the equations of motion of the beam in matrix notation, utilizing the integrating matrix as an operator, and applying the boundary conditions, the spatial dependence is removed from the governing partial differential equations and the resulting ordinary differential equations can be cast into standard eigenvalue form. Integrating matrices are derived based on two dimensional rectangular grids with arbitrary grid spacings allowed in one direction. The derivation of higher dimensional integrating matrices is the initial step in the generalization of the integrating matrix methodology to vibration and stability problems involving plates and shells.
A covariant multiple scattering series for elastic projectile-target scattering
A covariant formulation of the multiple scattering series for the optical potential is presented. The case of a scalar nucleon interacting with a spin zero isospin zero A-body target through meson exchange, is considered. It is shown that a covariant equation for the projectile-target t-matrix can be obtained which sums the ladder and crossed ladder diagrams efficiently. From this equation, a multiple scattering series for the optical potential is derived, and it is shown that in the impulse approximation, the two-body t-matrix associated with the first order optical potential is the one in which one particle is kept on mass-shell. The meaning of various terms in the multiple scattering series is given. The construction of the first-order optical potential for elastic scattering calculations is described.
Heterogeneous multiphase flow properties of volcanic rocks and implications for noble gas transport from underground nuclear explosions
Of interest to the Underground Nuclear Explosion Signatures Experiment are patterns and timing of explosion-generated noble gases that reach the land surface. The impact of potentially simultaneous flow of water and gas on noble gas transport in heterogeneous fractured rock is a current scientific knowledge gap. This article presents field and laboratory data to constrain and justify a triple continua conceptual model with multimodal multiphase fluid flow constitutive equations that represents host rock matrix, natural fractures, and induced fractures from past underground nuclear explosions (UNEs) at Aqueduct and Pahute Mesas, Nevada National Security Site, Nevada, USA. Capillary pressure from mercury intrusion and direct air–water measurements on volcanic tuff core samples exhibit extreme spatial heterogeneity (i.e., variation over multiple orders of magnitude). Petrographic observations indicate that heterogeneity derives from multimodal pore structures in ash-flow tuff components and post-depositional alteration processes. Comparisons of pre- and post-UNE samples reveal different pore size distributions that are due in part to microfractures. Capillary pressure relationships require a multimodal van Genuchten (VG) constitutive model to best fit the data. Relative permeability estimations based on unimodal VG fits to capillary pressure can be different from those based on bimodal VG fits, implying the choice of unimodal vs. bimodal fits may greatly affect flow and transport predictions of noble gas signatures. The range in measured capillary pressure and predicted relative permeability curves for a given lithology and between lithologies highlights the need for future modeling to consider spatially distributed properties.
A simple method for obtaining heat capacity coefficients of minerals
Abstract Heat capacity data are unavailable or incomplete for many minerals at geologically relevant temperatures. Despite the availability of entropy and enthalpy values in numerous thermodynamic tables (even sometimes at elevated temperatures), there remains need for extrapolation beyond, or interpolation between, temperatures. This approach inevitably results in estimates for entropy and enthalpy values because the heat capacity coefficients required for optimal thermodynamic treatment are less frequently available. Here we propose a simple method for obtaining heat capacity coefficients of minerals. This method requires only the empirically measured temperature-specific heat capacity for calculation via a matrix algorithm. The system of equations solver is written in the Python computing language and has been made accessible in an online repository. Thermodynamically, the solution to a system of equations represents the heat capacity coefficients that satisfy the mineral-specific polynomial. Direct coefficient calculation will result in more robust thermodynamic data, which are not subject to fitting uncertainties. Using hematite as an example, this method provides results that are comparable to conventional means and is applicable to any solid material. Coefficients vary within the traditional large 950 K temperature interval, indicating that best results should instead utilize a smaller 400 K temperature interval. Examples of large-scale implications include the refinement of geothermal gradient estimation in rapidly subsiding sedimentary basins or metamorphic and hydrothermal evolution.
A bound for the spectral radius of a matrix
Bound for spectral radius of matrix derived by constructing integral equation with degenerate kernel
Involutional matrices based on the representation theory of GL/2/.
Mathematical properties of involutional matrix solutions to simple quadratic equation, obtaining symmetry properties, eigenvalues and recursion formulas