Synthesis of shaped-beam radiation patterns using the iterative sampling method
Shaped beam radiation patterns synthesis by iterative sampling method with line sources or uniformly spaced antenna arrays
Stutzman, W. L.↗
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Results for “ITERATIVE METHODS”
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Comparison of Two Load Prediction Methods for Strain-Gage Balances
Data from a high-capacity semi-span balance is used to perform a detailed comparison of the load prediction accuracy of two strain-gage balance load prediction methods. Both methods independently obtain their load prediction equations from multivariate least squares fits of balance calibration data. The first method is called Non-Iterative Method. This approach directly uses regression models of the individual load components of a balance for the load prediction. The second method is called Iterative Method. This alternate approach uses a load iteration equation for the load prediction that is constructed from the regression models of the gage outputs of the balance. Basic characteristics of the two methods are reviewed. Afterwards, both methods are applied to calibration and check load data of the chosen balance. Finally, selected analysis results are compared. These comparisons confirmed that the load prediction accuracy of the two methods is the same for all practical purposes.
wind tunnel test↗
Solving Coupled Cluster Equations by the Newton Krylov Method
We describe using the Newton Krylov method to solve the coupled cluster equation. The method uses a Krylov iterative method to compute the Newton correction to the approximate coupled cluster amplitude. The multiplication of the Jacobian with a vector, which is required in each step of a Krylov iterative method such as the Generalized Minimum Residual (GMRES) method, is carried out through a finite difference approximation, and requires an additional residual evaluation. The overall cost of the method is determined by the sum of the inner Krylov and outer Newton iterations. We discuss the termination criterion used for the inner iteration and show how to apply pre-conditioners to accelerate convergence. We will also examine the use of regularization technique to improve the stability of convergence and compare the method with the widely used direct inversion of iterative subspace (DIIS) methods through numerical examples.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
The solution of linear systems of equations with a structural analysis code on the NAS CRAY-2
Two methods for solving linear systems of equations on the NAS Cray-2 are described. One is a direct method; the other is an iterative method. Both methods exploit the architecture of the Cray-2, particularly the vectorization, and are aimed at structural analysis applications. To demonstrate and evaluate the methods, they were installed in a finite element structural analysis code denoted the Computational Structural Mechanics (CSM) Testbed. A description of the techniques used to integrate the two solvers into the Testbed is given. Storage schemes, memory requirements, operation counts, and reformatting procedures are discussed. Finally, results from the new methods are compared with results from the initial Testbed sparse Choleski equation solver for three structural analysis problems. The new direct solvers described achieve the highest computational rates of the methods compared. The new iterative methods are not able to achieve as high computation rates as the vectorized direct solvers but are best for well conditioned problems which require fewer iterations to converge to the solution.
Poole, Eugene L.↗
PyAMG: Algebraic Multigrid Solvers in Python
PyAMG is a Python package of algebraic multigrid (AMG) solvers and supporting tools for approximating the solution to large, sparse linear systems of algebraic equations, Ax = b, where A is an n × n sparse matrix. Sparse linear systems arise in a range of problems in science, from fluid flows to solid mechanics to data analysis. While the direct solvers available in SciPy’s sparse linear algebra package (scipy.sparse.linalg) are highly efficient, in many cases iterative methods are preferred due to overall complexity. However, the iterative methods in SciPy, such as CG and GMRES, often require an efficient preconditioner in order to achieve a lower complexity. Preconditioning is a powerful tool whereby the conditioning of the linear system and convergence rate of the iterative method are both dramatically improved. PyAMG constructs multigrid solvers for use as a preconditioner in this setting. A summary of multigrid and algebraic multigrid solvers can be found in Olson (2015a), in Olson (2015b), and in Falgout (2006); a detailed description can be found in Briggs et al. (2000) and Trottenberg et al. (2001).
97 MATHEMATICS AND COMPUTING↗
Solution of a tridiagonal system of equations on the finite element machine
Two parallel algorithms for the solution of tridiagonal systems of equations were implemented on the Finite Element Machine. The Accelerated Parallel Gauss method, an iterative method, and the Buneman algorithm, a direct method, are discussed and execution statistics are presented.
Bostic, S. W.↗
Comparison of Two Load Prediction Methods for Strain-Gage Balances
Data from a five-component semi-span balance is used to perform a systematic comparison of the load prediction accuracy of two load prediction methods. Both methods independently obtain the load prediction equations from multivariate least squares fits of balance calibration data. The first method is called the Non-Iterative Method. This approach directly uses regression models of the individual load components of a balance for the load prediction. The second method is called the Iterative Method. This alternate approach uses a load iteration equation for the load prediction that is constructed from the regression coefficients of the gage outputs of the balance. Basic characteristics of the two methods are reviewed. Afterwards, both methods are applied to calibration, check load, and wind tunnel test data of a five-component semi-span balance. Selected analysis results are compared. These comparisons confirm that the accuracy of the two methods is the same for all practical purposes.
strain-gage balance↗
Calibration and Data Analysis of the MC-130 Air Balance
Design, calibration, calibration analysis, and intended use of the MC-130 air balance are discussed. The MC-130 balance is an 8.0 inch diameter force balance that has two separate internal air flow systems and one external bellows system. The manual calibration of the balance consisted of a total of 1854 data points with both unpressurized and pressurized air flowing through the balance. A subset of 1160 data points was chosen for the calibration data analysis. The regression analysis of the subset was performed using two fundamentally different analysis approaches. First, the data analysis was performed using a recently developed extension of the Iterative Method. This approach fits gage outputs as a function of both applied balance loads and bellows pressures while still allowing the application of the iteration scheme that is used with the Iterative Method. Then, for comparison, the axial force was also analyzed using the Non-Iterative Method. This alternate approach directly fits loads as a function of measured gage outputs and bellows pressures and does not require a load iteration. The regression models used by both the extended Iterative and Non-Iterative Method were constructed such that they met a set of widely accepted statistical quality requirements. These requirements lead to reliable regression models and prevent overfitting of data because they ensure that no hidden near-linear dependencies between regression model terms exist and that only statistically significant terms are included. Finally, a comparison of the axial force residuals was performed. Overall, axial force estimates obtained from both methods show excellent agreement as the differences of the standard deviation of the axial force residuals are on the order of 0.001 % of the axial force capacity.
Booth, Dennis↗
A Method to Solve Interior and Exterior Camera Calibration Parameters for Image Resection
An iterative method is presented to solve the internal and external camera calibration parameters, given model target points and their images from one or more camera locations. The direct linear transform formulation was used to obtain a guess for the iterative method, and herein lies one of the strengths of the present method. In all test cases, the method converged to the correct solution. In general, an overdetermined system of nonlinear equations is solved in the least-squares sense. The iterative method presented is based on Newton-Raphson for solving systems of nonlinear algebraic equations. The Jacobian is analytically derived and the pseudo-inverse of the Jacobian is obtained by singular value decomposition.
Samtaney, Ravi↗
Computation of steady axisymmetric flow using a one-dimensional time-dependent method
An iterative numerical method for computing steady, three dimensional, viscous, compressible flow fields, about aerodynamic bodies was studied. In order to develop the iterative method economically, the primary emphasis was directed towards supersonic, axisymmetric flow. However, the technique readily extends to three spatial dimensions. The viscous flow field about a cone-cylinder-flare body was calculated and compared to existing experimental data. Numerical predictions of the cone boundary layer and the flow field shock wave structure agreed with corresponding measurements. A separation was calculated at the cylinder-flare junction in six iterations; however, the size of the vortex did not correspond to the measured size. It was estimated that fifty iterations would be required to properly define the vortex. It was concluded that the iteration technique is of limited value for plane two dimensional and axisymmetrix flows, but of great value for three-dimensional flows.
Walitt, L.↗
Comparison of Electrical Output Format Options for the Analysis of Strain-Gage Balance Calibration Data
Fundamental characteristics of three gage output formats are discussed that may be used to both interpret and analyze wind tunnel strain-gage balance data. The first output format uses raw outputs, i.e., absolute voltage measurements, for the analysis. This choice requires an intercept term in the regression model of the outputs if the Iterative Method is chosen for the load prediction. Raw outputs can be used for the analysis of all known output characteristics as long as the Iterative Method is applied. However, raw outputs cannot be used to analyze data of a balance with bi-directional outputs if the Non-Iterative Method is chosen for the analysis. The second output format, i.e., difference type 1, uses the difference between raw outputs and the natural zeros of the balance gages for the analysis. In this case, the use of the intercept term becomes optional. Analysis results for difference type 1 will exactly match analysis results obtained by using raw outputs as long as identical math models are used for the regression analysis. In addition, difference type 1 may be used to analyze data of a balance with bi directional outputs if an analyst prefers to process data using the Non-Iterative Method. The third output format, i.e., difference type 2, uses the difference between raw outputs and the outputs of the zero load point of a load series for the data analysis. The application of this format is only recommended if the maximum magnitude of all tare loads of a given balance calibration data set is less than two percent of capacity. Data from the calibration of a force balance is used to illustrate the application of the three gage output formats.
Wind Tunnel Strain-Gage Balance↗
Improved Convergence and Robustness of USM3D Solutions on Mixed-Element Grids
Several improvements to the mixed-elementUSM3Ddiscretization and defect-correction schemes have been made. A new methodology for nonlinear iterations, called the Hierarchical Adaptive Nonlinear Iteration Method, has been developed and implemented. The Hierarchical Adaptive Nonlinear Iteration Method provides two additional hierarchies around a simple and approximate preconditioner of USM3D. The hierarchies are a matrix-free linear solver for the exact linearization of Reynolds-averaged Navier-Stokes equations and a nonlinear control of the solution update. Two variants of the Hierarchical Adaptive Nonlinear Iteration Method are assessed on four benchmark cases, namely, a zero-pressure-gradient flat plate, a bump-in-channel configuration, the NACA 0012 airfoil, and a NASA Common Research Model configuration. The new methodology provides a convergence acceleration factor of 1.4 to 13 over the preconditioner-alone method representing the baseline solver technology.
Pandya, Mohagna J.↗
Improved Convergence and Robustness of USM3D Solutions on Mixed-Element Grids
Several improvements to the mixed-element USM3D discretization and defect-correction schemes have been made. A new methodology for nonlinear iterations, called the Hierarchical Adaptive Nonlinear Iteration Method, has been developed and implemented. The Hierarchical Adaptive Nonlinear Iteration Method provides two additional hierarchies around a simple and approximate preconditioner of USM3D. The hierarchies are a matrix-free linear solver for the exact linearization of Reynolds-averaged Navier-Stokes equations and a nonlinear control of the solution update. Two variants of the Hierarchical Adaptive Nonlinear Iteration Method are assessed on four benchmark cases, namely, a zero-pressure-gradient flat plate, a bump-in-channel configuration, the NACA 0012 airfoil, and a NASA Common Research Model configuration. The new methodology provides a convergence acceleration factor of 1.4 to 13 over the preconditioner-alone method representing the baseline solver technology.
Pandya, Mohagna J.↗
Analytic Solution to the Problem of Aircraft Electric Field Mill Calibration
It is by no means a simple task to retrieve storm electric fields from an aircraft instrumented with electric field mill sensors. The presence of the aircraft distorts the ambient field in a complicated way. Before retrievals of the storm field can be made, the field mill measurement system must be "calibrated". In other words, a relationship between impressed (i.e., ambient) electric field and mill output must be established. If this relationship can be determined, it is mathematically inverted so that ambient field can be inferred from the mill outputs. Previous studies have primarily focused on linear theories where the relationship between ambient field and mill output is described by a "calibration matrix" M. Each element of the matrix describes how a particular component of the ambient field is enhanced by the aircraft. For example the product M(sub ix), E(sub x), is the contribution of the E(sub x) field to the i(th) mill output. Similarly, net aircraft charge (described by a "charge field component" E(sub q)) contributes an amount M(sub iq)E(sub q) to the output of the i(th) sensor. The central difficulty in obtaining M stems from the fact that the impressed field (E(sub x), E(sub y), E(sub z), E(sub q) is not known but is instead estimated. Typically, the aircraft is flown through a series of roll and pitch maneuvers in fair weather, and the values of the fair weather field and aircraft charge are estimated at each point along the aircraft trajectory. These initial estimates are often highly inadequate, but several investigators have improved the estimates by implementing various (ad hoc) iterative methods. Unfortunately, none of the iterative methods guarantee absolute convergence to correct values (i.e., absolute convergence to correct values has not been rigorously proven). In this work, the mathematical problem is solved directly by analytic means. For m mills installed on an arbitrary aircraft, it is shown that it is possible to solve for a single 2m-vector that provides all other needed variables (i.e., the unknown fair weather field, the unknown aircraft charge, and the unknown matrix M). Numerical tests of the solution, effects of measurement errors, and studies of solution non-uniqueness are ongoing as of this writing.
Koshak, William↗
A Universal Algorithm for the Detection of Bi-directional Gage Output Characteristics
A universal algorithm was developed that may be used to assess the bi-directional characteristics of the gage outputs of a wind tunnel strain-gage balance. The algorithm assumes that balance loads and gage outputs are described in the design format of the balance. It can also be applied to balance calibration data that is processed by using either the Iterative Method or the Non-Iterative Method. The algorithm uses an estimate of the bi-directional part of a gage output at load capacity as input. In addition, the statistical significance of the principle absolute value term in the regression model of either the gage output or the related primary load component is determined. A gage output is assumed to be bi-directional if two conditions are fulfilled: the bi-directional part of the output at load capacity exceeds 0.5 percent of the maximum output at load capacity; the p-value of the principle absolute value term of the regression model of the balance data is less than the threshold of 0.001. Data from the calibration of two six-component force balances and one five-component semi-span balance are used to illustrate the application of the universal detection algorithm.
wind tunnel test↗
A parallel iterative solution method for systems of nonlinear hyperbolic equations
An iterative algorithm suitable for the solution of a system of nonlinear hyperbolic partial differentiation equations in multiple dimensions is discussed. Current numerical methods for systems of nonlinear PDEs have limited parallelism due to strong coupling between the equations. This method decouples the PDEs by linearizing the convention coefficient for a space-time domain. This provides large grain parallelism. The linearization also allows the treatment of some terms in the equations as source terms, providing more freedom to choose from a wider variety of numerical methods. Smaller grain parallelism may be exploited within the solves for each equation. Thus, the method has potential for parallelism at several levels.
Scroggs, Jeffrey S.↗
Iterative PNS method for attached flows with upstream influence
A stable global-iteration procedure is developed by utilizing successive sweeps, from inflow to outflow boundaries, of a parabolized Navier-Stokes code for attached, steady supersonic flow. It is shown that the procedure converges in about ten or fewer iterations, and allows for the upstream influence within the subsonic region of a supersonic boundary layer. An implicit forward difference is employed in the subsonic region to evaluate the pressure-gradient term in the streamwise momentum equation. The pressure-gradient term is normally approximated in standard single-sweep parabolized methods, suppressing the upstream influence. The iterative results obtained demonstrate the validity of the single-sweep method for weak interactions. For hypersonic viscous interaction on a flat plate at a Mach number of 5.8, it is found that the single-sweep method has a small error that vanishes as the Reynolds number of the flow is increased. However, the skin friction has an error of about 10 percent for low Reynolds numbers and for hot wall conditions. The application of this method to laminar two-dimensional flow over weak expansion and compression corners shows that the pressure and skin friction results in the vicinity of the expansion corner compare favorably with a time-dependent Navier-Stokes numerical solution.
Rakich, J. V.↗
A Study of Morrison's Iterative Noise Removal Method
Morrison's iterative noise removal method is studied by characterizing its effect upon systems of differing noise level and response function. The nature of data acquired from a linear shift invariant instrument is discussed so as to define the relationship between the input signal, the instrument response function, and the output signal. Fourier analysis is introduced, along with several pertinent theorems, as a tool to more thorough understanding of the nature of and difficulties with deconvolution. In relation to such difficulties the necessity of a noise removal process is discussed. Morrison's iterative noise removal method and the restrictions upon its application are developed. The nature of permissible response functions is discussed, as is the choice of the response functions used.
Ioup, G. E.↗