Testing SALT Approximations with Numerical Radiation Transfer Code. I. Validity and Applicability
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C. Carr↗
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Variable Step Integration Coupled with the Method of Characteristics Solution for Water-Hammer Analysis, A Case Study
One-dimensional water-hammer modeling involves the solution of two coupled non-linear hyperbolic partial differential equations (PDEs). These equations result from applying the principles of conservation of mass and momentum to flow through a pipe, and usually the assumption that the speed at which pressure waves propagate through the pipe is constant. In order to solve these equations for the interested quantities (i.e. pressures and flow rates), they must first be converted to a system of ordinary differential equations (ODEs) by either approximating the spatial derivative terms with numerical techniques or using the Method of Characteristics (MOC). The MOC approach is ideal in that no numerical approximation errors are introduced in converting the original system of PDEs into an equivalent system of ODEs. Unfortunately this resulting system of ODEs is bound by a time step constraint so that when integrating the equations the solution can only be obtained at fixed time intervals. If the fluid system to be modeled also contains dynamic components (i.e. components that are best modeled by a system of ODEs), it may be necessary to take extremely small time steps during certain points of the model simulation in order to achieve stability and/or accuracy in the solution. Coupled together, the fixed time step constraint invoked by the MOC, and the occasional need for extremely small time steps in order to obtain stability and/or accuracy, can greatly increase simulation run times. As one solution to this problem, a method for combining variable step integration (VSI) algorithms with the MOC was developed for modeling water-hammer in systems with highly dynamic components. A case study is presented in which reverse flow through a dual-flapper check valve introduces a water-hammer event. The predicted pressure responses upstream of the check-valve are compared with test data.
Turpin, Jason B.↗
Approximation theory for LQG (Linear-Quadratic-Gaussian) optimal control of flexible structures
An approximation theory is presented for the LQG (Linear-Quadratic-Gaussian) optimal control problem for flexible structures whose distributed models have bounded input and output operators. The main purpose of the theory is to guide the design of finite dimensional compensators that approximate closely the optimal compensator. The optimal LQG problem separates into an optimal linear-quadratic regulator problem and an optimal state estimation problem. The solution of the former problem lies in the solution to an infinite dimensional Riccati operator equation. The approximation scheme approximates the infinite dimensional LQG problem with a sequence of finite dimensional LQG problems defined for a sequence of finite dimensional, usually finite element or modal, approximations of the distributed model of the structure. Two Riccati matrix equations determine the solution to each approximating problem. The finite dimensional equations for numerical approximation are developed, including formulas for converting matrix control and estimator gains to their functional representation to allow comparison of gains based on different orders of approximation. Convergence of the approximating control and estimator gains and of the corresponding finite dimensional compensators is studied. Also, convergence and stability of the closed-loop systems produced with the finite dimensional compensators are discussed. The convergence theory is based on the convergence of the solutions of the finite dimensional Riccati equations to the solutions of the infinite dimensional Riccati equations. A numerical example with a flexible beam, a rotating rigid body, and a lumped mass is given.
Gibson, J. S.↗
Triangle based TVD schemes for hyperbolic conservation laws
A triangle based total variation diminishing (TVD) scheme for the numerical approximation of hyperbolic conservation laws in two space dimensions is constructed. The novelty of the scheme lies in the nature of the preprocessing of the cell averaged data, which is accomplished via a nearest neighbor linear interpolation followed by a slope limiting procedures. Two such limiting procedures are suggested. The resulting method is considerably more simple than other triangle based non-oscillatory approximations which, like this scheme, approximate the flux up to second order accuracy. Numerical results for linear advection and Burgers' equation are presented.
Durlofsky, Louis J.↗
Triangle based adaptive stencils for the solution of hyperbolic conservation laws
A triangle based total variation diminishing (TVD) scheme for the numerical approximation of hyperbolic conservation laws in two space dimensions is constructed. The novelty of the scheme lies in the nature of the preprocessing of the cell averaged data, which is accomplished via a nearest neighbor linear interpolation followed by a slope limiting procedures. Two such limiting procedures are suggested. The resulting method is considerably more simple than other triangle based non-oscillatory approximations which, like this scheme, approximate the flux up to second order accuracy. Numerical results for linear advection and Burgers' equation are presented.
Durlofsky, Louis J.↗
The Osher scheme for real gases
An extension of Osher's approximate Riemann solver to include gases with an arbitrary equation of state is presented. By a judicious choice of thermodynamic variables, the Riemann invariats are reduced to quadratures which are then approximated numerically. The extension is rigorous and does not involve any further assumptions or approximations over the ideal gas case. Numerical results are presented to demonstrate the feasibility and accuracy of the proposed method.
Suresh, Ambady↗
Numerical methods and computers used in elastohydrodynamic lubrication
Some of the methods of obtaining approximate numerical solutions to boundary value problems that arise in elastohydrodynamic lubrication are reviewed. The highlights of four general approaches (direct, inverse, quasi-inverse, and Newton-Raphson) are sketched. Advantages and disadvantages of these approaches are presented along with a flow chart showing some of the details of each. The basic question of numerical stability of the elastohydrodynamic lubrication solutions, especially in the pressure spike region, is considered. Computers used to solve this important class of lubrication problems are briefly described, with emphasis on supercomputers.
Hamrock, B. J.↗
Numerical solution of laminar jet mixing with and without free stream.
Systematic numerical solutions of two-dimensional and axisymmetrical laminar jet of an incompressible fluid with and without free stream have been obtained. For the case without free stream, the exact numerical solutions have been obtained for different initial velocity profiles and compared with experimental results and similarity solutions. For the case with free stream, the numerical solutions have been compared with the linearized analytical solutions. An approximate numerical solution for the axial velocity distribution of the three-dimensional laminar jet is proposed. The accuracy of the method has been determined from the exact solutions of the two limiting cases of three-dimensional jets, i.e., the two-dimensional and axisymmetric cases.
Pai, S. I.↗
A sensitivity equation approach to shape optimization in fluid flows
A sensitivity equation method to shape optimization problems is applied. An algorithm is developed and tested on a problem of designing optimal forebody simulators for a 2D, inviscid supersonic flow. The algorithm uses a BFGS/Trust Region optimization scheme with sensitivities computed by numerically approximating the linear partial differential equations that determine the flow sensitivities. Numerical examples are presented to illustrate the method.
Borggaard, Jeff↗
An Inverse Interpolation Method Utilizing In-Flight Strain Measurements for Determining Loads and Structural Response of Aerospace Vehicles
An important and challenging technology aimed at the next generation of aerospace vehicles is that of structural health monitoring. The key problem is to determine accurately, reliably, and in real time the applied loads, stresses, and displacements experienced in flight, with such data establishing an information database for structural health monitoring. The present effort is aimed at developing a finite element-based methodology involving an inverse formulation that employs measured surface strains to recover the applied loads, stresses, and displacements in an aerospace vehicle in real time. The computational procedure uses a standard finite element model (i.e., "direct analysis") of a given airframe, with the subsequent application of the inverse interpolation approach. The inverse interpolation formulation is based on a parametric approximation of the loading and is further constructed through a least-squares minimization of calculated and measured strains. This procedure results in the governing system of linear algebraic equations, providing the unknown coefficients that accurately define the load approximation. Numerical simulations are carried out for problems involving various levels of structural approximation. These include plate-loading examples and an aircraft wing box. Accuracy and computational efficiency of the proposed method are discussed in detail. The experimental validation of the methodology by way of structural testing of an aircraft wing is also discussed.
Shkarayev, S.↗
A numerical solution of the flow field over a transonic airfoil including strong-shock-induced flow separation
An approximate numerical method has been developed for computing the entire transonic flow field over an airfoil, including the boundary layer, the shock-wave/boundary-layer interaction, and the shock-induced-separation. The inviscid flow, the turbulent boundary layer in the weak interaction zone, and the strong interaction elliptic region are solved separately. The inviscid flow is represented by the solution of the transonic small-disturbance equation, while the boundary layer is solved by the generalized Galerkin method, and the strong interaction region (including the wake) is computed from the time-averaged Navier-Stokes equations. A second-order interaction is computed iteratively using the inviscid pressure distribution as a boundary condition for the boundary layer and Navier-Stokes codes, and the viscous displacement thickness to modify the airfoil geometry for the inviscid computation. The computed pressure distributions and shock-wave profiles are compared with experimental data. Agreement is very good except that the pressure recovery on the trailing edge is greater than the experimental data.
Seginer, A.↗
A numerical method for approximating antenna surfaces defined by discrete surface points
A simple numerical method for the quadratic approximation of a discretely defined reflector surface is described. The numerical method was applied to interpolate the surface normal of a parabolic reflector surface from a grid of nine closest surface points to the point of incidence. After computing the surface normals, the geometrical optics and the aperture integration method using the discrete Fast Fourier Transform (FFT) were applied to compute the radiaton patterns for a symmetric and an offset antenna configurations. The computed patterns are compared to that of the analytic case and to the patterns generated from another numerical technique using the spline function approximation. In the paper, examples of computations are given. The accuracy of the numerical method is discussed.
Lee, R. Q.↗
Error analysis for spectral approximation of the Korteweg-De Vries equation
The conservation and convergence properties of spectral Fourier methods for the numerical approximation of the Korteweg-de Vries equation are analyzed. It is proved that the (aliased) collocation pseudospectral method enjoys the same convergence properties as the spectral Galerkin method, which is less effective from the computational point of view. This result provides a precise mathematical answer to a question raised by several authors in recent years.
Maday, Y.↗
Computational methods for the identification of spatially varying stiffness and damping in beams
A numerical approximation scheme for the estimation of functional parameters in Euler-Bernoulli models for the transverse vibration of flexible beams with tip bodies is developed. The method permits the identification of spatially varying flexural stiffness and Voigt-Kelvin viscoelastic damping coefficients which appear in the hybrid system of ordinary and partial differential equations and boundary conditions describing the dynamics of such structures. An inverse problem is formulated as a least squares fit to data subject to constraints in the form of a vector system of abstract first order evolution equations. Spline-based finite element approximations are used to finite dimensionalize the problem. Theoretical convergence results are given and numerical studies carried out on both conventional (serial) and vector computers are discussed.
Banks, H. T.↗
Numerical Simulation of Unsteady Aerodynamic Models
This report documents the results of the numerical simulations of unsteady aerodynamic models. The results focus on numerical accuracy and efficiency, and the robustness of the numerical methods. The aerodynamic models includes the classical Wagner and Kussner functions and the Leishman-Beddoes dynamic stall model. The simulations includes the numerical approximations of the Duhamel's integrals using both indicial (step) and impulse responses, the numerical integrations of the state-space models, and the exact solutions. The report also presents the conversion among different model representations.
Nguyen, Khanh Q.↗
Implementation of Vigneron's streamwise pressure gradient approximation in the PNS equations
For single sweep parabolized Navier-Stokes solvers, the streamwise pressure gradient must be modified in the subsonic region to eliminate numerical instabilities. The accuracy of this modification on the solution of the parabolized Navier-Stokes equations with Vigneron's technique is shown to depend on how the numerical approximation of the pressure gradient is formed. A simple test case of supersonic laminar flow over a flat plate is computed with two different numerical methods for solving the PNS equations. Significant errors in the temperature profile and skin friction coefficient are demonstrated using a fully conservative differencing treatment of Vigneron's splitting for the pressure gradient typically used in parabolized Navier-Stokes solvers. The physical reason for this error is discussed. An alternate formulation is demonstrated which minimizes these errors.
Morrison, Joseph H.↗
Modeling and control of flexible structures
This monograph presents integrated modeling and controller design methods for flexible structures. The controllers, or compensators, developed are optimal in the linear-quadratic-Gaussian sense. The performance objectives, sensor and actuator locations and external disturbances influence both the construction of the model and the design of the finite dimensional compensator. The modeling and controller design procedures are carried out in parallel to ensure compatibility of these two aspects of the design problem. Model reduction techniques are introduced to keep both the model order and the controller order as small as possible. A linear distributed, or infinite dimensional, model is the theoretical basis for most of the text, but finite dimensional models arising from both lumped-mass and finite element approximations also play an important role. A central purpose of the approach here is to approximate an optimal infinite dimensional controller with an implementable finite dimensional compensator. Both convergence theory and numerical approximation methods are given. Simple examples are used to illustrate the theory.
Gibson, J. S.↗
Wave tilt sounding of a linearly inhomogeneous layered half-space
The wave tilt of a transverse electric (TE) electromagnetic wave over a linearly inhomogeneous lossy layer overlying a homogeneous half-space is studied. Two approaches are used: an exact formulation using solutions of Airy's equation and an approximate numerical solution using a large number of homogeneous layers with a linearly increasing dielectric constant. The numerical results of both solutions are practically identical as long as the thickness of the layers in the approximate model are somewhat smaller than a quarter-wave length.
Warne, L.↗