On certain approximations in the finite- element method
Finite element method approximations outside variational principle in elasticity problems, examining convergence to exact solution
Mclay, R. W.↗
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Experimental and computational flow-field results for an all-body hypersonic aircraft
A comprehensive test program is defined which is being implemented in the NASA/Ames 3.5 foot Hypersonic Wind Tunnel for obtaining data on a generic all-body hypersonic vehicle for computational fluid dynamics (CFD) code validation. Computational methods (approximate inviscid methods and an upwind parabolized Navier-Stokes code) currently being applied to the all-body model are outlined. Experimental and computational results on surface pressure distributions and Pitot-pressure surveys for the basic sharp-nose model (without control surfaces) at a free-stream Mach number of 7 are presented.
Cleary, Joseph W.↗
Approximate Solution Methods for Spectral Radiative Transfer in High Refractive Index Layers
Some ceramic materials for high temperature applications are partially transparent for radiative transfer. The refractive indices of these materials can be substantially greater than one which influences internal radiative emission and reflections. Heat transfer behavior of single and laminated layers has been obtained in the literature by numerical solutions of the radiative transfer equations coupled with heat conduction and heating at the boundaries by convection and radiation. Two-flux and diffusion methods are investigated here to obtain approximate solutions using a simpler formulation than required for exact numerical solutions. Isotropic scattering is included. The two-flux method for a single layer yields excellent results for gray and two band spectral calculations. The diffusion method yields a good approximation for spectral behavior in laminated multiple layers if the overall optical thickness is larger than about ten. A hybrid spectral model is developed using the two-flux method in the optically thin bands, and radiative diffusion in bands that are optically thick.
Siegel, R.↗
Method for Approximating the Vacuum Motions of Spinning Symmetrical Bodies with Nonconstant Spin Rates
A method for approximating the vacuum motions of spinning rigid symmetrical bodies with varying spin rates and inertias has been completed. The analysis includes the effects of time varying thrust misalignments, mass unbalance, and jet damping. Results are given in the form of equations for space referenced Euler angles, flight-path angles, body referenced attitude rates, and earth-referenced vehicle-trajectory coordinates. The method consists of dividing the problem into intervals during which the time-dependent variables are assumed constant at their mean interval value. In order to check this procedure, solutions for various interval sizes are compared with solutions obtained with numerical methods. Although the method is somewhat lengthy for accurate hand computation in most cases, it is readily programed for machine solutions. Probably more important, the general solutions give insight into the separate effects of the variables and, in many cases, can be quickly used to determine the approximate ranges of the variables required for the desired solution to a given problem. In this respect, equations for determining maximum wobble have been derived for certain input conditions. The method has been shown to compare closely with the numerical solutions of two sample problems. The sample problems also illustrated the relatively large effect of pitch and yaw jet damping on body motions.
VACUUM↗
Generalized moving least squares vs. radial basis function finite difference methods for approximating surface derivatives
Approximating differential operators defined on two-dimensional surfaces is an important problem that arises in many areas of science and engineering. Over the past ten years, localized meshfree methods based on generalized moving least squares (GMLS) and radial basis function finite differences (RBF-FD) have been shown to be effective for this task as they can give high orders of accuracy at low computational cost, and they can be applied to surfaces defined only by point clouds. However, there have yet to be any studies that perform a direct comparison of these methods for approximating surface differential operators (SDOs). The first purpose of this work is to fill that gap. For this comparison, we focus on an RBF-FD method based on polyharmonic spline kernels and polynomials (PHS+Poly) since they are most closely related to the GMLS method. Additionally, we use a relatively new technique for approximating SDOs with RBF-FD called the tangent plane method since it is simpler than previous techniques and natural to use with PHS+Poly RBF-FD. Further, the second purpose of this work is to relate the tangent plane formulation of SDOs to the local coordinate formulation used in GMLS and to show that they are equivalent when the tangent space to the surface is known exactly. The final purpose is to use ideas from the GMLS SDO formulation to derive a new RBF-FD method for approximating the tangent space for a point cloud surface when it is unknown. For the numerical comparisons of the methods, we examine their convergence rates for approximating the surface gradient, divergence, and Laplacian as the point clouds are refined for various parameter choices. We also compare their efficiency in terms of accuracy per computational cost, both when including and excluding setup costs.
97 MATHEMATICS AND COMPUTING↗
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.↗
An approximate analytical method for vortex-lift and center of pressure on the slender wing
A simplified approximate analytical method for predicting vortex lift and center of pressure on slender wings is presented. Two empirical corrections are proposed. The method is compared with experimental data and gives accurate estimates.
Xieyuan, Y.↗
Classical field approximation of ultralight dark matter: Quantum break times, corrections, and decoherence
The classical field approximation is widely used to better understand the predictions of ultralight dark matter. Here, in this work, we use the truncated Wigner approximation method to test the classical field approximation of ultralight dark matter. This method approximates a quantum state as an ensemble of independently evolving realizations drawn from its Wigner function. The method is highly parallelizable and allows the direct simulation of quantum corrections and decoherence times in systems many times larger than have been previously studied in reference to ultralight dark matter. Our study involves simulation of systems in 1, 2, and 3 spatial dimensions. We simulate three systems, the condensation of a Gaussian random field in three spatial dimensions, a stable collapsed object in three spatial dimensions, and the merging of two stable objects in two spatial dimensions. We study the quantum corrections to the classical field theory in each case. We find that quantum corrections grow exponentially during nonlinear growth with the timescale being approximately equal to the system dynamical time. In stable systems the corrections grow quadratically. We also find that the primary effect of quantum corrections is to reduce the amplitude of fluctuations on the de Broglie scale in the spatial density. Finally, we find that the timescale associated with decoherence due to gravitational coupling to baryonic matter is at least as fast as the quantum corrections due to gravitational interactions. These results are consistent with the predictions of the classical field theory being accurate.
79 ASTRONOMY AND ASTROPHYSICS↗
Scalable Programming Workflows for Validation of Quantum Computers
Hybrid quantum-classical workflows have become standard methods for executing variational algorithms and other quantum simulation techniques, which are key applications for noisy intermediate scale quantum (NISQ) computers. Validating these simulations is an important task which helps gauge the progress of quantum computer development, and classical simulation can serve as a tool to this end. Both exact and more scalable approximate methods with quantifiable error bounds can be used in validation tasks where the applicable metrics include the distance from a calculable ground truth, the quality of an error model fit to data, etc. Here we present a library extension that includes methods for validation of quantum simulations based on scalable hybrid workflows executable on high performance computers. We provide examples that use approximate methods based on tensor networks and stabilizer simulators to bound the error of quantum simulations on NISQ hardware.
Nguyen, Thien↗
Global collocation methods for approximation and the solution of partial differential equations
Polynomial interpolation methods are applied both to the approximation of functions and to the numerical solutions of hyperbolic and elliptic partial differential equations. The derivative matrix for a general sequence of the collocation points is constructed. The approximate derivative is then found by a matrix times vector multiply. The effects of several factors on the performance of these methods including the effect of different collocation points are then explored. The resolution of the schemes for both smooth functions and functions with steep gradients or discontinuities in some derivative are also studied. The accuracy when the gradients occur both near the center of the region and in the vicinity of the boundary is investigated. The importance of the aliasing limit on the resolution of the approximation is investigated in detail. Also examined is the effect of boundary treatment on the stability and accuracy of the scheme.
Solomonoff, A.↗
Numerical solution of 2D-vector tomography problem using the method of approximate inverse
We propose a numerical solution of reconstruction problem of a two-dimensional vector field in a unit disk from the known values of the longitudinal and transverse ray transforms. The algorithm is based on the method of approximate inverse. Numerical simulations confirm that the proposed method yields good results of reconstruction of vector fields.
97 MATHEMATICS AND COMPUTING↗
Modal propagation angles in ducts with soft walls and their connection with suppressor performance
The angles of propagation of the wave fronts associated with duct modes are derived for a cylindrical duct with soft walls (acoustic suppressors) and a uniform steady flow. The angle of propagation with respect to the radial coordinate (angle of incidence on the wall) is shown to be a better correlating parameter for the optimum wall impedance of spinning modes than the previously used mode cutoff ratio. Both the angle of incidence upon the duct wall and the propagation angle with respect to the duct axis are required to describe the attenuation of a propagating mode. Using the modal propagation angles, a geometric acoustics approach to suppressor acoustic performance was developed. Results from this approximate method were compared to exact modal propagation calculations to check the accuracy of the approximate method. The results are favorable except in the immediate vicinity of the modal optimum impedance where the approximate method yields about one-half of the exact maximum attenuation.
Rice, E. J.↗
Modal propagation angles in ducts with soft walls and their connection with suppressor performance
The angles of propagation of the wave fronts associated with duct modes are derived for a cylindrical duct with soft walls (acoustic suppressors) and a uniform steady flow. The angle of propagation with respect to the radial coordinate (angle of incidence on the wall) is shown to be a better correlating parameter for the optimum wall impedance of spinning modes than the previously used mode cutoff ratio. Both the angle of incidence upon the duct wall and the propagation angle with respect to the duct axis are required to describe the attenuation of a propagating mode. Using the modal propagation angles, a geometric acoustics approach to suppressor acoustic performance was developed. Results from this approximate method were compared to exact modal propagation calculations to check the accuracy of the approximate method. The results are favorable except in the immediate vicinity of the modal optimum impedance where the approximate method yields about one-half of the exact maximum attenuation.
Rice, E. J.↗
Differential equation based method for accurate approximations in optimization
A method to efficiently and accurately approximate the effect of design changes on structural response is described. The key to this method is to interpret sensitivity equations as differential equations that may be solved explicitly for closed form approximations, hence, the method is denoted the Differential Equation Based (DEB) method. Approximations were developed for vibration frequencies, mode shapes and static displacements. The DEB approximation method was applied to a cantilever beam and results compared with the commonly-used linear Taylor series approximations and exact solutions. The test calculations involved perturbing the height, width, cross-sectional area, tip mass, and bending inertia of the beam. The DEB method proved to be very accurate, and in most cases, was more accurate than the linear Taylor series approximation. The method is applicable to simultaneous perturbation of several design variables. Also, the approximations may be used to calculate other system response quantities. For example, the approximations for displacements are used to approximate bending stresses.
Pritchard, Jocelyn I.↗
Lorentz drag - an engineering approximation.
Method estimating relative magnitude of magnetoaerodynamic drag/MADD/ or Lorentz drag used for decelerating space vehicle reentry
MAGNETOHYDRODYNAMIC FLOW↗
Eigenvalues of singular differential operators by finite difference methods. I.
Approximation of the eigenvalues of certain self-adjoint operators defined by a formal differential operator in a Hilbert space. In general, two problems are studied. The first is the problem of defining a suitable Hilbert space operator that has eigenvalues. The second problem concerns the finite difference operators to be used.
Baxley, J. V.↗
Partially coherent scattering in stellar chromospheres. II - The first-order escape probability method. III - A second-order escape probability method
Approximate analytic expressions are derived for resonance-line wing diagnostics, accounting for frequency redistribution effects, for homogeneous slabs, and slabs with a constant Planck function gradient. Resonance-line emission profiles from a simplified conceptual standpoint are described in order to elucidate the basic physical parameters of the line-forming layers prior to the performance of detailed numerical calculations. An approximate analytic expression is derived for the dependence on stellar surface gravity of the location of the Ca II and Mg II resonance-line profile peaks. An approximate radiative transfer equation using generalized second-order escape probabilities, applicable even in the presence of nearly coherent scattering in the damping wings of resonance lines, is derived. Approximate analytic solutions that can be applied in special regimes and achieve good agreement with accurate numerical results are found.
Gayley, K. G.↗
Method of approximate determination of the characteristics of nonlinear vibrational systems
Methods are discussed for determining the characteristics of elements of nonlinear vibrational systems, from the results of analyzing forced vibrations which are approximate in themselves, since they are based on the use of experimental data. The constant components of the linearized characteristics and linearization coefficients are determined experimentally by finding the values of these quantities at different vibration amplitudes and frequencies and subsequent approximation of their analytical expressions. Simple and convenient, although approximate, formulas for determining the characteristics of vibrational systems from experimentally known functions are derived.
Atstupenene, R. P.↗