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Mathematical procedure leading to better and simpler approximation expansion terms in eigenvalues problem of Ritz and Bubnov- Galerkin method
Improved approximations are incorporated into the inviscid fluid method originated by Spreiter et al (1970) in calculations of the altitude of the Venus ionopause. The altitude calculations are then compared with median altitudes measured by the Pioneer Venus retarding potential analyzer. The calculated ionopause shape is found to closely approximate the measured meridian shape in the solar zenith angle (SZA) range of zero-135 deg. The use of improved ionospheric pressure field ionosheath pressure and Prandtl-Meyer expansion approximations lowers the terminator ionopause altitude to approximately half that obtained with the usual Spreiter approximations. It is also determined that the calculated dawn ionopause is about 300 km higher than the dusk ionopause, and that both ionopauses are close to their respective measured meridian values. It is concluded that median ionopause altitude within the SZA angle range may be calculated without inclusion of a viscous interaction in the theory.
The saturated equilibrium expansion approximation for two phase flow often involves ideal-gas and latent-heat assumptions to simplify the solution procedure. This approach is well documented by Wegener and Mack and works best at low pressures where deviations from ideal-gas behavior are small. A thermodynamic expression for liquid mass fraction that is decoupled from the equations of fluid mechanics is used to compare the effects of the various assumptions on nitrogen-gas saturated equilibrium expansion flow starting at 8.81 atm, 2.99 atm, and 0.45 atm, which are conditions representative of transonic cryogenic wind tunnels. For the highest pressure case, the entire set of ideal-gas and latent-heat assumptions are shown to be in error by 62 percent for the values of heat capacity and latent heat. An approximation of the exact, real-gas expression is also developed using a constant, two phase isentropic expansion coefficient which results in an error of only 2 percent for the high pressure case.
Coefficient of thermal expansion approximates that of glass more closely. Alloy composed of about 60 percent iron, 40 percent nickel, and traces of six other elements. Developed as replacement for Kovar Fe/Ni/Co alloy in ferrule-and-tube assembly, new alloy has same strength, solderability, and compatibility with fuel as does Kovar. Used in glass-to-metal seals without excessive residual stresses. Potential for other applications in which low thermal expansion important; mechanical measuring devices and precise sliding parts that must function over wide temperature ranges.
A simplified analytical technique is presented for modeling the interplanetary scintillation of radio sources of finite angular size with a power-law electron-density-fluctuation power spectrum. The simplification results from the representation of the scintillation spectrum in confluent hypergeometric functions. The approximations presented allow fast numerical evaluation of a spectrum for a weakly scattering but extended medium with less than 10% error over the entire spectrum. Parameters describing anisotropic electron irregularities as well as anisotropic source structure are included, and the dependence of the spectrum normalization on the scales of the medium is derived explicitly. The parametric description of the domains of convergence of the approximate expansions also provides a simple conceptualization of the relative contributions of the scattered radiation along the line of sight to the observed spectrum. This is particularly useful for sources of finite angular size. This technique is applied to previously published observations.
The applicability and usefulness of several classical and other methods for solving the two-point boundary-value problem which arises in non-linear singularly perturbed optimal control are assessed. Specific algorithms of the Picard, Newton and averaging types are formally developed for this class of problem. The computational requirements associated with each algorithm are analysed and compared with the computational requirement of the method of matched asymptotic expansions. Approximate solutions to a linear and a non-linear problem are obtained by each method and compared.
Two variable expansion procedure for approximate solution of nonlinear differential equations with applications to satellite orbital motion
Emission-power series expansion solution and Rosseland approximation applied to radiation problems, discussing errors introduced by approximate methods
The generalized winding number (GWN) is a scalar field that supports robust containment queries on curved geometry, including non-watertight, overlapping, and nested boundary representations. While queries can be easily parallelized over samples, direct evaluation on parametric curves and surfaces remains costly for large and complex models. Fast, state-of-the-art GWN approaches leverage a spatial index to approximate the GWN, typically coupled with a Taylor expansion which approximates the GWN contribution for far clusters of geometric primitives. However, such methods operate only on discrete inputs such as triangle meshes and point clouds, and would introduce containment errors near boundaries if applied to curved input. We extend support for fast GWN evaluation over arbitrary collections of NURBS curves in 2D and trimmed NURBS patches in 3D via a Bounding Volume Hierarchy that stores efficiently precomputed moment data in the hierarchy nodes. When querying the hierarchy, approximations for far clusters are used alongside direct evaluation for nearby NURBS primitives, achieving sub-linear complexity while preserving the geometric features in the vicinity of the query point. Central to our performance improvements is an adaptive subdivision strategy for NURBS primitives during a preprocessing phase, creating better spatial partitions while retaining the same accuracy for containment decisions as a direct evaluation. We demonstrate the performance and accuracy of our approach across a large collection of 2D and 3D datasets.
An asymptotic expansion in inverse powers of the wave number is derived for an integral of a type often encountered in the theory of wave propagation in inhomogeneous media. The procedure, which is essentially a forward-scatter approximation, yields all the terms of the expansion. In contrast, the conventional forward-scatter approximation is found to yield correctly only the first term of the expansion.
The strong interdependence of electricity, gas, and heating systems can facilitate fault propagation within integrated energy systems (IESs), posing significant challenges to secure operation. This paper proposes a polynomial chaos expansion (PCE)-based approximation method to accurately characterize the IES security region boundary (IES–SRB). By integrating the Karush-Kuhn-Tucker conditions with PCE theory, the IES-SRB approximation problem is reformulated as a set of nonlinear equations concerning the approximation coefficients. Using the Galerkin projection method, these equations are further transformed into a system of projection equations that govern the polynomial approximation coefficients in the IES-SRB approximation. To reduce computational complexity while maintaining high approximation accuracy, a piecewise polynomial approximation method is proposed. Numerical studies on the E39-G20-H6 and E118-G96-H52 IES test systems demonstrate that the proposed method can accurately and effectively construct IES security regions.
The collision operator that appears in the equation of motion for a particle distribution function that was averaged over an ensemble of random Hamiltonians is non-Markovian. It is non-Markovian in that it involves a propagated integral over the past history of the ensemble averaged distribution function. All formal expansions of this nonlinear collision operator to date preserve this non-Markovian character term by term yielding an integro-differential equation that must be converted to a diffusion equation by an additional approximation. An expansion is derived for the collision operator that is strictly Markovian to any finite order and yields a diffusion equation as the lowest nontrivial order. The validity of this expansion is seen to be the same as that of the standard quasilinear expansion.
The computation of inviscid supersonic flow over a two-dimensional airfoil is considered. There are two main nonlinear approaches which lead to approximate solutions. Small-amplitude theory gives valid solutions provided the airfoil thickness is not too great and the Mach number is not too high. The second type of approximation, shock expansion theory, employs the fact that even for flows with strong shocks the effect of the down-running characteristics remains small. This leads to an analytic solution at the airfoil. It is pointed out that the approximate solution is accurate enough for many cases of interest. The numerical method furnishes a rapid correction to the solution in those cases where it is not. The characteristic-streamline coordinate system is useful both for the computation of the approximate solution and the corrections, and is also convenient for displaying and interpreting the results.
The compact extragalactic radio source 3C 345 was observed by very-long-baseline interferometry (wavelength about 3.8 cm) at 12 epochs distributed over the nearly four-year period from February 1971 to October 1974. For one of these epochs, the multibaseline data were sufficient to allow the brightness distribution to be estimated in a model-independent manner; the resultant distribution was clearly dominated by two components. The remaining sets of data were also represented adequately by two-component models. The angular separation of the two components increased during this period from about 1.00 to 1.30 milliarcsec, corresponding to an apparent average speed of expansion of approximately 2.5 c at a fixed position angle of 105 (plus or minus 5) deg. These results, coupled with the fact that contraction has never been observed, seem difficult to reconcile with the so-called Christmas-tree model of the 'superrelativistic' expansion of extragalactic radio sources.
The objectives of the program were to: generate mechanical, thermal, and physical property test data for as-fabricated advanced materials; design and fabricate an accelerated thermal cycling chamber; and determine the effect of thermal cycling on thermomechanical properties and dimensional stability of composites. In the current program, extensive mechanical and thermophysical property tests of various organic matrix, metal matrix, glass matrix, and carbon-carbon composites were conducted, and a reliable database was constructed for spacecraft material selection. Material property results for the majority of the as-fabricated composites were consistent with the predicted values, providing a measure of consolidation integrity attained during fabrication. To determine the effect of thermal cycling on mechanical properties, microcracking, and thermal expansion behavior, approximately 500 composite specimens were exposed to 10,000 cycles between -150 and +150 F. These specimens were placed in a large (18 cu ft work space) thermal cycling chamber that was specially designed and fabricated to simulate one year low earth orbital (LEO) thermal cycling in 20 days. With this rate of thermal cycling, this is the largest thermal cycling unit in the country. Material property measurements of the thermal cycled organic matrix composite laminate specimens exhibited less than 24 percent decrease in strength, whereas, the remaining materials exhibited less than 8 percent decrease in strength. The thermal expansion response of each of the thermal cycled specimens revealed significant reduction in hysteresis and residual strain, and the average CTE values were close to the predicted values.
An Earth-centered, Earth-fixed (ECEF) inertial navigation system must compute the Jacobian of its employed gravitation model with respect to position while time-propagating the error covariance of the system. One commonly used gravitation model is the ‘J2 model’ which is a second-order truncation of the Earth’s spherical harmonic gravitation model. The J2 model is popular because it can quickly and efficiently be evaluated, and the truncation error is small: The ‘J3 term’ --- the third term in the spherical harmonic expansion --- is approximately 1000 times smaller than the J2 term.
Lighthill method with Lagrange expansion for approximative solutions of partial differential equations
Prediction of the shear flow around bodies impulsively set into motion at a uniform velocity. Information is presented on the local wall shear stress, velocity distribution, steady flow times, and thermal response for wedge flows where local flow acceleration occurs. The essential features of the flow field are found to be describable by the approximate series expansion method of Goldstein and Rosenhead (1936). This method would appear to be useful in rapidly calculating the viscous drag on the forward face of various shaped bodies where local flow acceleration occurs.