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At least 55 records · Page 3

Analysis of airfoil leading-edge separation bubbles

A local inviscid-viscous interaction technique was developed for the analysis of low speed airfoil leading edge transitional separation bubbles. In this analysis an inverse boundary layer finite difference analysis is solved iteratively with a Cauchy integral representation of the inviscid flow which is assumed to be a linear perturbation to a known global viscous airfoil analysis. Favorable comparisons with data indicate the overall validity of the present localized interaction approach. In addition numerical tests were performed to test the sensitivity of the computed results to the mesh size, limits on the Cauchy integral, and the location of the transition region.

Vatsa, V. N.↗

Solidification interface shape for continuous casting in an offset mold - Two analytical methods

A solution method for finding the unknown solidification interface in manufacturing slab ingots as a continuous casting is presented, which involves a product solution in the potential plane and the use of conjugate harmonic functions. It is argued that the method may be more direct for some geometries than the Cauchy boundary value method. Moreover, the usefulness of the Cauchy boundary value method is demonstrated through the example of a nonsymmetric horizontal mold where the walls are offset to support the lower ingot boundary.

Siegel, R.↗

Numerical solution of random singular integral equation appearing in crack problems

The solution of several elasticity problems, and particularly crack problems, can be reduced to the solution of one-dimensional singular integral equations with a Cauchy-type kernel or to a system of uncoupled singular integral equations. Here a method for the numerical solution of random singular integral equations of Cauchy type is presented. The solution technique involves a Chebyshev series approximation, the coefficients of which are the solutions of a system of random linear equations. This method is applied to the problem of periodic array of straight cracks inside an infinite isotropic elastic medium and subjected to a nonuniform pressure distribution along the crack edges. The statistical properties of the random solution are evaluated numerically, and the random solution is used to determine the values of the stress-intensity factors at the crack tips. The error, expressed as the difference between the mean of the random solution and the deterministic solution, is established. Values of stress-intensity factors at the crack tip for different random input functions are presented.

Sambandham, M.↗

A regularization method for extrapolation of solar potential magnetic fields

The mathematical basis of a Tikhonov regularization method for extrapolating the chromospheric-coronal magnetic field using photospheric vector magnetograms is discussed. The basic techniques show that the Cauchy initial value problem can be formulated for potential magnetic fields. The potential field analysis considers a set of linear, elliptic partial differential equations. It is found that, by introducing an appropriate smoothing of the initial data of the Cauchy potential problem, an approximate Fourier integral solution is found, and an upper bound to the error in the solution is derived. This specific regularization technique, which is a function of magnetograph measurement sensitivities, provides a method to extrapolate the potential magnetic field above an active region into the chromosphere and low corona.

Gary, G. A.↗

Eno-Osher schemes for Euler equations

The combination of the Osher approximate Riemann solver for the Euler equations and various ENO schemes is discussed for one-dimensional flow. The three basic approaches, viz. the ENO scheme using primitive variable reconstruction, either with Cauchy-Kowalewski procedure for time integration or the TVD Runge-Kutta scheme, and the flux-ENO method are tested on different shock tube cases. The shock tube cases were chosen to present a serious challenge to the ENO schemes in order to test their ability to capture flow discontinuities, such as shocks. Also the effect of the ordering of the eigen values, viz. natural or reversed ordering, in the Osher scheme is investigated. The ENO schemes are tested up to fifth order accuracy in space and time. The ENO-Osher scheme using the Cauchy-Kowalewski procedure for time integration is found to be the most accurate and robust compared with the other methods and is also computationally efficient. The tests showed that the ENO schemes perform reasonably well, but have problems in cases where two discontinuities are close together. In that case there are not enough points in the smooth part of the flow to create a non-oscillatory interpolation.

Vandervegt, Jacobus J.↗

ENO-Osher schemes for Euler equations

The combination of the Osher approximate Riemann solver for the Euler equations and various ENO schemes is discussed for one-dimensional flow. The three basic approaches, viz., the ENO scheme using primitive variable reconstruction, either with the Cauchy-Kowalewski procedure for time integration or the TVD Runge-Kutta scheme, and the flux-ENO method are tested on different shock tube cases. The ENO-Osher scheme using the Cauchy-Kowalewski procedure for time integration is found to be the most accurate and robust compared with the other methods and is also computationally efficient. The tests showed that the ENO schemes perform reasonably well, but have problems in cases where two discontinuites are close together. In that case there are not enough points in the smooth part of the flow to create a nonoscillatory interpolation.

Van Der Vegt, Jacobus J.↗

Convergence properties of finite-difference hydrodynamics schemes in the presence of shocks

We investigate the asymptotic convergence of finite-difference schemes for the Euler equations when the limiting solution contains shocks. The Lax-Wendroff theorem guarantees that certain conservative schemes converge to correct, physically valid solutions. We focus on two one-dimensional operator-split schemes with explicit artificial-viscosity terms. One, an internal-energy scheme, does not satisfy the assumptions of Lax-Wendroff; the other, a conservative total-energy scheme, does. With viscous lengths chosen proportional to the grid size, we find that both schemes converge to their zero-grid-size limits at the theoretically expected rate, but only the conversative scheme converges toward correct solutions of the inviscid fluid equations. We show that the difference in their behaviors results directly from the presence of shocks in the limiting solution. Empirically, we find that when the viscous lenghts tend toward zero more slowly than the grid size, however the nonconservative scheme also converges toward correct solutions. We characterize the asymptotic behavior of the total-energy scheme in a particular problem in which a shock forms. As the grid is refined, a Cauchy error approaches the expected rate of change slowly. We show that the changes in the artificial viscosity alter the diffusion of small-amplitude waves. The differences associated with such waves make the dominant contribution to the Cauchy error. We formulate an analytic model to relate the rate of approach to the effect of varying diffusion in waves and find quantitative agreement with our numerical results.

Kimoto, Paul A.↗

Advances in Application of Fast Semidirect Computational Methods in Transonic Flow

This paper is intended as a review and summary of the advances made in a recently developed approach for rapid numerical solution of the equations of inviscid transonic aerodynamics. The investigation has been limited to two-dimensional, steady, inviscid flow over airfoils in a subsonic free stream, with emphasis on development of a rapid computational technique, rather than on generality of application. The approach uses finite-difference algorithms called "fast direct elliptic solvers" within an iteration scheme. "Direct" means that the entire computation field is solved at once, rather than in successive traverses over the field as in a point- or line-relaxation method. Such an iterative method is referred to as "semidirect." The iterative convergence can be faster than in other relaxation methods because changes are felt simultaneously at all points in each succeeding iteration. Direct elliptic solvers and semidirect methods have restrictions, but these are gradually being removed. Direct solvers were first developed for solving Poisson's equation on a rectangle without interior boundaries. A method to treat first-order systems, a direct Cauchy-Riemann solver has also been developed. Numerical treatment of part of a system of nonlinear equations by a Poisson solver has been reported. Also Poisson solvers in semidirect methods were used for nonseparable elliptic equations. The semidirect method was extended to the solution of a problem of mixed type, where the improved Murman-Cole transonic small-disturbance difference equations were solved. A slightly supercritical flow over a biconvex airfoil was treated successfully, but the iterations did not converge for more strongly supercritical conditions In another work the addition of terms ot both sides of the difference equations stabilized the iteration for supercritical conditions with large supersonic zones. For this, the Cauchy-Riemann solver was revised to incl,ude the needed terms. Most recently, the evaluation of parameters for rapid convergence and comparisons, with Murman's line-relaxation method was described. The method was extended to full second order accuracy in a fully conservative formulation in another work.

Martin, E. Dale↗

Stress Formulation in Three-Dimensional Elasticity

The theory of elasticity evolved over centuries through the contributions of eminent scientists like Cauchy, Navier, Hooke Saint Venant, and others. It was deemed complete when Saint Venant provided the strain formulation in 1860. However, unlike Cauchy, who addressed equilibrium in the field and on the boundary, the strain formulation was confined only to the field. Saint Venant overlooked the compatibility on the boundary. Because of this deficiency, a direct stress formulation could not be developed. Stress with traditional methods must be recovered by backcalculation: differentiating either the displacement or the stress function. We have addressed the compatibility on the boundary. Augmentation of these conditions has completed the stress formulation in elasticity, opening up a way for a direct determination of stress without the intermediate step of calculating the displacement or the stress function. This Completed Beltrami-Michell Formulation (CBMF) can be specialized to derive the traditional methods, but the reverse is not possible. Elasticity solutions must be verified for the compliance of the new equation because the boundary compatibility conditions expressed in terms of displacement are not trivially satisfied. This paper presents the variational derivation of the stress formulation, illustrates the method, examines attributes and benefits, and outlines the future course of research.

Patnaik, Surya N.↗

Tuning Monotonic Basin Hopping: Improving the Efficiency of Stochastic Search as Applied to Low-Thrust Trajectory Optimization

Trajectory optimization methods using monotonic basin hopping (MBH) have become well developed during the past decade [1, 2, 3, 4, 5, 6]. An essential component of MBH is a controlled random search through the multi-dimensional space of possible solutions. Historically, the randomness has been generated by drawing random variable (RV)s from a uniform probability distribution. Here, we investigate the generating the randomness by drawing the RVs from Cauchy and Pareto distributions, chosen because of their characteristic long tails. We demonstrate that using Cauchy distributions (as first suggested by J. Englander [3, 6]) significantly improves monotonic basin hopping (MBH) performance, and that Pareto distributions provide even greater improvements. Improved performance is defined in terms of efficiency and robustness. Efficiency is finding better solutions in less time. Robustness is efficiency that is undiminished by (a) the boundary conditions and internal constraints of the optimization problem being solved, and (b) by variations in the parameters of the probability distribution. Robustness is important for achieving performance improvements that are not problem specific. In this work we show that the performance improvements are the result of how these long-tailed distributions enable MBH to search the solution space faster and more thoroughly. In developing this explanation, we use the concepts of sub-diffusive, normally-diffusive, and super-diffusive random walks (RWs) originally developed in the field of statistical physics.

autonomous↗

Tuning Monotonic Basin Hopping: Improving the Efficiency of Stochastic Search as Applied to Low-Thrust Trajectory Optimization

Trajectory optimization methods using MBH have become well developed during the past decade. An essential component of MBH is a controlled random search through the multi-dimensional space of possible solutions. Historically, the randomness has been generated by drawing RVs from a uniform probability distribution. Here, we investigate the generating the randomness by drawing the RVs from Cauchy and Pareto distributions, chosen because of their characteristic long tails. We demonstrate that using Cauchy distributions (as first suggested by Englander significantly improves MBH performance, and that Pareto distributions provide even greater improvements. Improved performance is defined in terms of efficiency and robustness, where efficiency is finding better solutions in less time, and robustness is efficiency that is undiminished by (a) the boundary conditions and internal constraints of the optimization problem being solved, and (b) by variations in the parameters of the probability distribution. Robustness is important for achieving performance improvements that are not problem specific. In this work we show that the performance improvements are the result of how these long-tailed distributions enable MBH to search the solution space faster and more thoroughly. In developing this explanation, we use the concepts of sub-diffusive, normally-diffusive, and super-diffusive RWs originally developed in the field of statistical physics.

mission design↗

Comparison of Pre- and Post-Flight Estimations of Pitch Damping Coefficients for Unguided Entry Vehicles

Introduction: Pre-flight predictions of pitch damping coefficient (𝐶!!+𝐶!"̇)curves are used in entry simulations to determine flight readiness and if any dynamic remediation strategies are required. Accurate characterization of these pitch damping coefficient curves is crucial for mission success. Historically, pre-flight estimations of these curves are derived experimentally either via ballistic range testing or with 1-degree-of-freedom (DOF) free-or forced-oscillation wind tunnel testing. Methodology:With post-flight best-estimated trajectories, a fitting procedure as described in Karlgaard, et. al. [1] is used to compare pre-and post-flight pitch damping coefficients. This technique relies on an assumed analytical form of the trajectory, a limited 3-DOF representation described in Schoenenbergerand Queen [2], which results in anEuler-Cauchy solution form.This solution assumes constant density,thus limited amplitude multi-peak windows of the trajectory are considered to populate the pitch damping-amplitudes pace. Fig. 1 shows preliminary estimations of the pitch damping coefficients estimated using the post-flight best estimated trajectoryfor MER B, using a3-amplitude-peak window. Important to note is that the Euler-Cauchy solution also assumes planar motion, thus coning or 𝛽contribution is not included. Further,the estimated post-flight pitch damping approximations and the pre-flight predictions shown are effective representations of pitch damping, or the integrated effect of the pitch damping coefficient over one pitch cycle at the corresponding cycle’s peak amplitude. Effective pitch damping coefficient curves provide more intuitive insights about where the vehicle is stable and permits readily processed comparisons between data-sets. It is important to note that most pitch damping curves, such as those reported in aerodatabases and utilized in flight mechanics trajectory simulations, are represented in instantaneous angle-of-attack space distinct from what is presented in Fig 1. The extracted post-flight pitch damping coefficient points show a wide range of values across a narrow band of amplitudes (Fig.1a), albeit centered on the published pre-flight approximation. This also applies to the prediction as a function of Mach in Fig1b, where the post-flight scatter points are centered around the pre-flight predictions, but the pre-flight predictions notably lack data aboveMach 5.Further tuning of the technique, such as changing the number of peaks considered in each stencil, resolving density in higher resolution, and employing more accurate fitting procedures (beyond scipy.curve_fit()), will be performed. PosterFocus:This study will apply this reconstruction method to the as-flown trajectories for the following unguided entry vehicles: MER A, MER B, IRVE3, Phoenix, Insight, Pathfinder, and SIAD. The pitch damping coefficient will be derived from these trajectories and compared with the pre-flight predictions. Comparing across different missions, entry environments, and vehicle geometries can inform where our pre-flight testing methodologies accurately capture the flight dynamics. Results will provide valuable insight into the accuracy of pre-flight predicted pitch damping curves. Beyond that, this method can shed light on how accurate these curves need to be, which can inform uncertainty ranges applied to pre-flight curves for other missions.

pitch damping coefficient↗