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Three methods for the application of spectral methods to flows in complex geometries are discussed. They are the spectral iteration method, the spectral embedding method, and the spectral element method. Applications are given to the flow over wavy walls, past a large-eddy breakup device (LEBU), and to flow over a step.
After detailing the construction of spectral approximations to time-dependent mixed initial boundary value problems, a study is conducted of differential equations of the form 'partial derivative of u/partial derivative of t = Lu + f', where for each t, u(t) belongs to a Hilbert space such that u satisfies homogeneous boundary conditions. For the sake of simplicity, it is assumed that L is an unbounded, time-independent linear operator. Attention is given to Fourier methods of both Galerkin and pseudospectral method types, the Galerkin method, the pseudospectral Chebyshev and Legendre methods, the error equation, hyperbolic partial differentiation equations, and time discretization and iterative methods.
This paper describes a robust and efficient subdomain solution procedure for two-dimensional recirculating flows. The solution domain is divided into a number of overlapping subdomains, and a direct fully coupled solution is obtained for each subdomain using a sparse matrix form of LU decomposition. An effective parabolic block correction procedure, which calculates global corrections to the tentative solution by a marching technique similar to that used for boundary layer flows, is used to accelerate the convergence of the basic procedure. The use of effective block correction is found to be essential for the success of the subdomain approach on strongly recirculating flows. In a number of laminar two-dimensional flows, the new block-corrected method performed extremely well, rivaling the best direct methods in execution time, while requiring substantially less computer storage. The new method proved to be from two to ten times faster than conventional iterative methods, while requiring only a moderate increase in storage.
We present a comparison between two numerical methods for the extrapolation of nonlinear force-free magnetic fields, the Iterative Method (IM) and the Progressive Extension Method (PEM). The advantages and disadvantages of these two methods are summarized and the accuracy and numerical instability are discussed. On the basis of this investigation, we claim that the two methods do resemble each other qualitatively.
Ultrasonic thickness measurement of mechanical structures is one of the most popular and commonly used nondestructive methods for various kinds of process control and corrosion monitoring. With ultrasonic propagation speed being temperature-dependent, the thickness measurement can be performed reliably only when the thermal profile is completely known. Most conventional techniques assume the temperature of the test structure is uniform and at room temperature across its thickness. Such assumptions may lead to large errors in the thickness measurement, especially when there are significant temperature variations across the thickness. State-of-the-art techniques use external temperature measurements or implement iterative methods to compensate for the unknown thermal profiles. However, such techniques produce unsatisfactory results when the heat distribution is complex or varies rapidly with time. In this work, we propose a two-sensors technique, using both compressive and shear excitations, with a non-iterative rapid data processing method for accurate thickness measurement under arbitrary time-variant thermal profile. The independent behavior of shear and compressive waves is used to formulate a real-time thickness estimation technique. The developed technique is experimentally validated on a steel plate with fixed acoustic sensors. Test results show that the error in thickness estimation can be reduced by up to 98% compared to conventional thickness gauging methods.
Iterative method for optimal control problems for Markov chains, discussing degenerate nonlinear elliptic equations solution in optimal stochastic control theory
Diffraction-limited astronomical images have been obtained utilizing a variant of the shift-and-add method. It is shown that the matched filter approach for extending the weighted shift-and-add method reduces specklegrams from extended objects and from an object dominated by photon noise. The method is aberration-insensitive and yields very high dynamic range results. The iterative method for arriving at the matched filter does not automatically converge in the case of photon-noisy specklegrams for objects with more than one maximum.
The three-dimensional vortical viscous flow past a sideslipping delta wing at a high angle of attack is numerically investigated. A computational method for calculating low-speed viscous flowfields is developed. The time-iterative method uses an implicit upwind-relaxation finite-difference algorithm with a nonsingular eigensystem to solve the preconditioned, three-dimensional, incompressible Navier-Stokes equations in curvilinear coordinates. An algebraic turbulence model is implemented to account for the eddy viscosity. The technique of local time stepping is incorporated to accelerate the rate of convergence to a steady-state solution. Computed results are compared with experimental data.
An accelerated subspace iteration method for calculating eigenvector derivatives has been developed. Factors affecting the effectiveness and the reliability of the subspace iteration are identified, and effective strategies concerning these factors are presented. The method has been implemented, and the results of a demonstration problem are presented.
By using a numerical iterative method, it is demonstrated that the mixing ratio H2/He on the giant planets can be inferred from spectral measurements of the intensity emitted by these planets in the far infrared range. The method is successfully applied to synthetic spectra of Saturn computed from atmospheric thermal models. The effect of random and systematic measurement errors on the determination of the mixing ratio is also studied.
In order to reduce the computational difficulty associated with a single grid (SG) solution procedure, the multigrid (MG) technique was identified as a useful means for improving the convergence rate of iterative methods. A full MG full approximation storage (FMG/FAS) algorithm is used to solve the incompressible recirculating flow problems in complex geometries. The algorithm is implemented in conjunction with a pressure correction staggered grid type of technique using the curvilinear coordinates. In order to show the performance of the method, two flow configurations, one a square cavity and the other a channel, are used as test problems. Comparisons are made between the iterations, equivalent work units, and CPU time. Besides showing that the MG method can yield substantial speed-up with wide variations in Reynolds number, grid distributions, and geometry, issues such as the convergence characteristics of different grid levels, the choice of convection schemes, and the effectiveness of the basic iteration smoothers are studied. An adaptive grid scheme is also combined with the MG procedure to explore the effects of grid resolution on the MG convergence rate as well as the numerical accuracy.
Grid smoothing and orthogonalization procedures were developed and implemented in the construction of two and three dimensional grids. The procedures are based on the variational methods of grid generation. The two-dimensional examples were computed using the MSU IRIS Graphics Workstation. It was demonstrated that the elliptic grid generation equations, with arbitrary forcing functions, can be solved, in their variational formulation, using a gradient method. Since gradient methods have a global convergence property, the divergence problems often encountered when using SOR iterative methods can be avoided. It is not to be concluded, however, that SOR methods should be abandoned, since gradient methods tend to converge very slowly. In fact, slow convergence was the major problem encountered in the three-dimensional grids. Further progress was made on the continuing effort to develop conservative interpolation formulas for overlapping grids.
The existing methodologies for robot programming originate primarily from robotic applications to manufacturing, where uncertainties of the robots and their task environment may be minimized by repeated off-line modeling and identification. In space application of robots, however, a higher degree of automation is required for robot programming because of the desire of minimizing the human intervention. We discuss a new paradigm of robotic programming which is based on the concept of machine learning. The goal is to let robots practice tasks by themselves and the operational data are used to automatically improve their motion performance. The underlying mathematical problem is to solve the problem of dynamical inverse by iterative methods. One of the key questions is how to ensure the convergence of the iterative process. There have been a few small steps taken into this important approach to robot programming. We give a representative result on the convergence problem.
Introduction: Life support technology in large closed systems like submarines and space stations catalyzes carbon monoxide (CO) to carbon dioxide, which is easily removed. However, in a small system like the Extravehicular Mobility Unit (EMU), spacesuit, CO from exogenous (contaminated oxygen (O (sub 2) supply) and endogenous (human metabolism) sources will accumulate in the free suit volume. The free volume becomes a sink for CO that is rebreathed by the astronaut. The accumulation through time depends on many variables: the amount absorbed by the astronaut, the amount produced by the astronaut (between 0.28 and 0.34 moles per hour per kilogram)[1], the amount that enters the suit from contaminated O (sub 2), the amount removed through suit leak, the free volume of the suit, and the O (sub 2) partial pressure[2], just to list a few. Contamination of the EMU O (sub 2) supply with no greater than 1 part per million CO was the motivation for empirical measurements from CO pulse oximetry (SpCO) as well as mathematical modeling of the EMU as a rebreather for CO. Methods: We developed a first-order differential mixing equation as well as an iterative method to compute CO accumulation in the EMU. Pre-post measurements of SpCO (Rad-57, Masimo Corporation) from EMU ground training and on-orbit extravehicular activities (EVAs) were collected. Results: Initial modeling without consideration of the astronaut as a sink but only the source of CO showed that after 8 hours breathing 100 percent O (sub 2) with a 10 milliliter per minute (760 millimeters Hg at 21 degrees Centigrade standard) suit leak, an endogenous production rate of 0.23 moles per hour per kilogram for a 70 kilogram person with 42 liters (1.5 cubic feet) free suit volume resulted in a peak CO partial pressure (pCO) of 0.047 millimeters Hg at 4.3 pounds per square inch absolute (222 millimeters Hg). Preliminary results based on a 2008 model[3] with consideration of the astronaut as a sink and source of CO suggests that most of the rebreathed CO stays bound to hemoglobin and myoglobin and; therefore, pCO only increased to 0.002 millimeters Hg in the EMU. Hemoglobin saturation after 8 hours was an insignificant 0.4 percent compared to about 4 percent for cigarette smokers in the general population. This preliminary modeling result supplements 11 pre-post index finger SpCO measurements from EMU ground training (mean 0.5 percent versus 1 percent, probability equal to 0.41 from paired t-test) and 10 on-orbit pre-post EVAs (mean 1.5 percent versus 1.1 percent, probability equal to 0.17 from paired t-test) that showed no consistent increase, at least no increase outside the accuracy of the oximeter (1 percent display resolution with plus or minus 3 percent Standard Deviation). Discussion: Simulations continue, but a preliminary conclusion is that rebreathing endogenous CO accumulating in the EMU is not a serious medical issue during EVA. The absence of CO poisoning signs or symptoms following hundreds of EVAs is also good empirical evidence that corroborates the limited SpCO measurements and preliminary modeling results.
An improved approach is discussed that may be used to directly include first and second order temperature effects in the load prediction algorithm of a wind tunnel strain-gage balance. The improved approach was designed for the Iterative Method that fits strain-gage outputs as a function of calibration loads and uses a load iteration scheme during the wind tunnel test to predict loads from measured gage outputs. The improved approach assumes that the strain-gage balance is at a constant uniform temperature when it is calibrated and used. First, the method introduces a new independent variable for the regression analysis of the balance calibration data. The new variable is designed as the difference between the uniform temperature of the balance and a global reference temperature. This reference temperature should be the primary calibration temperature of the balance so that, if needed, a tare load iteration can be performed. Then, two temperature{dependent terms are included in the regression models of the gage outputs. They are the temperature difference itself and the square of the temperature difference. Simulated temperature{dependent data obtained from Triumph Aerospace's 2013 calibration of NASA's ARC-30K five component semi{span balance is used to illustrate the application of the improved approach.
Formulations and algorithms implemented in the MHOST finite element program are discussed. The code uses a novel concept of the mixed iterative solution technique for the efficient 3-D computations of turbine engine hot section components. The general framework of variational formulation and solution algorithms are discussed which were derived from the mixed three field Hu-Washizu principle. This formulation enables the use of nodal interpolation for coordinates, displacements, strains, and stresses. Algorithmic description of the mixed iterative method includes variations for the quasi static, transient dynamic and buckling analyses. The global-local analysis procedure referred to as the subelement refinement is developed in the framework of the mixed iterative solution, of which the detail is presented. The numerically integrated isoparametric elements implemented in the framework is discussed. Methods to filter certain parts of strain and project the element discontinuous quantities to the nodes are developed for a family of linear elements. Integration algorithms are described for linear and nonlinear equations included in MHOST program.
A boundary element alternating method, denoted herein as BEAM, is presented for two dimensional fracture problems. This is an iterative method which alternates between two solutions. An analytical solution for arbitrary polynomial normal and tangential pressure distributions applied to the crack faces of an embedded crack in an infinite plate is used as the fundamental solution in the alternating method. A boundary element method for an uncracked finite plate is the second solution. For problems of edge cracks a technique of utilizing finite elements with BEAM is presented to overcome the inherent singularity in boundary element stress calculation near the boundaries. Several computational aspects that make the algorithm efficient are presented. Finally, the BEAM is applied to a variety of two dimensional crack problems with different configurations and loadings to assess the validity of the method. The method gives accurate stress intensity factors with minimal computing effort.
Two optimization methods for solving a variational problem in grid generation are described and evaluated. The smoothness, cell volumes, and orthogonality of the variational integrals are examined. The Jacobi-Newton iterative method is compared to the Fletcher-Reeves conjugate gradient method. It is observed that a combination of the Jacobi-Newton iteration and the direct solution of the variational problem produces an algorithm which is easy to program and requires less storage and computer time/iteration than the conjugate gradient method.