Adaptive Grid Approach Research Articles

A Comparison of Approximate Versus Exact Geometrical Representations of Roughness for CFD Calculations of cf and St

Skin friction (cf) and heat transfer (St) predictions were made for a turbulent boundary layer over randomly rough surfaces at Reynolds number of 1×106. The rough surfaces are scaled models of actual gas turbine blade surfaces that have experienced degradation after service. Two different approximations are used to characterize the roughness in the computational model: the discrete element model and full 3D discretization of the surface. The discrete element method considers the total aerodynamic drag on a rough surface to be the sum of shear drag on the flat part of the surface and the form drag on the individual roughness elements. The total heat transfer from a rough surface is the sum of convection on the flat part of the surface and the convection from each of the roughness elements. Correlations are used to model the roughness element drag and heat transfer, thus avoiding the complexity of gridding the irregular rough surface. The discrete element roughness representation was incorporated into a two-dimensional, finite difference boundary layer code with a mixing length turbulence model. The second prediction method employs a viscous adaptive Cartesian grid approach to fully resolve the three-dimensional roughness geometry. This significantly reduces the grid requirement compared to a structured grid. The flow prediction is made using a finite-volume Navier-Stokes solver capable of handling arbitrary grids with the Spalart-Allmaras (S‐A) turbulence model. Comparisons are made to experimentally measured values of cf and St for two unique roughness characterizations. The two methods predict cf to within ±8% and St within ±17%, the RANS code yielding slightly better agreement. In both cases, agreement with the experimental data is less favorable for the surface with larger roughness features. The RANS simulation requires a two to three order of magnitude increase in computational time compared to the DEM method and is not as readily adapted to a wide variety of roughness characterizations. The RANS simulation is capable of analyzing surfaces composed primarily of roughness valleys (rather than peaks), a feature that DEM does not have in its present formulation. Several basic assumptions employed by the discrete element model are evaluated using the 3D RANS flow predictions, namely: establishment of the midheight for application of the smooth wall boundary condition; cD and Nu relations employed for roughness elements; and flow three dimensionality over and around roughness elements.

An adaptive irregular grid approach for 3D deformable image registration

Deformable registration is an important application in medical image analysis and processing. We propose a physics-based parametric approach for deformable image registration, where non-rigid transformations are computed using an irregular grid of control points distributed within the image domain. The image is modelled as a three-dimensional (3D) homogeneous infinite elastic medium. It is assumed that a Gaussian-shaped force is applied at every control point, where the strengths, directions and influence areas of the forces as well as the positions of the control points are considered as free parameters whose optimization leads to maximization of the similarity measure between the images to be registered. For optimization, a computationally efficient Levenberg–Marquardt method is used. The proposed approach has certain advantages over traditional landmark-based methods or the registration methods based on regular grids, for example B-splines, since comparable results can be achieved by using less control points. Experimental results with 3D clinical images demonstrate that our method is capable of successfully coping with complex registration tasks.

Parametric and Nonparametric Population Methods

This study examined parametric and nonparametric population modelling methods in three different analyses. The first analysis was of a real, although small, clinical dataset from 17 patients receiving intramuscular amikacin. The second analysis was of a Monte Carlo simulation study in which the populations ranged from 25 to 800 subjects, the model parameter distributions were Gaussian and all the simulated parameter values of the subjects were exactly known prior to the analysis. The third analysis was again of a Monte Carlo study in which the exactly known population sample consisted of a unimodal Gaussian distribution for the apparent volume of distribution (V(d)), but a bimodal distribution for the elimination rate constant (k(e)), simulating rapid and slow eliminators of a drug. For the clinical dataset, the parametric iterative two-stage Bayesian (IT2B) approach, with the first-order conditional estimation (FOCE) approximation calculation of the conditional likelihoods, was used together with the nonparametric expectation-maximisation (NPEM) and nonparametric adaptive grid (NPAG) approaches, both of which use exact computations of the likelihood. For the first Monte Carlo simulation study, these programs were also used. A one-compartment model with unimodal Gaussian parameters V(d) and k(e) was employed, with a simulated intravenous bolus dose and two simulated serum concentrations per subject. In addition, a newer parametric expectation-maximisation (PEM) program with a Faure low discrepancy computation of the conditional likelihoods, as well as nonlinear mixed-effects modelling software (NONMEM), both the first-order (FO) and the FOCE versions, were used. For the second Monte Carlo study, a one-compartment model with an intravenous bolus dose was again used, with five simulated serum samples obtained from early to late after dosing. A unimodal distribution for V(d) and a bimodal distribution for k(e) were chosen to simulate two subpopulations of 'fast' and 'slow' metabolisers of a drug. NPEM results were compared with that of a unimodal parametric joint density having the true population parameter means and covariance. For the clinical dataset, the interindividual parameter percent coefficients of variation (CV%) were smallest with IT2B, suggesting less diversity in the population parameter distributions. However, the exact likelihood of the results was also smaller with IT2B, and was 14 logs greater with NPEM and NPAG, both of which found a greater and more likely diversity in the population studied. For the first Monte Carlo dataset, NPAG and PEM, both using accurate likelihood computations, showed statistical consistency. Consistency means that the more subjects studied, the closer the estimated parameter values approach the true values. NONMEM FOCE and NONMEM FO, as well as the IT2B FOCE methods, do not have this guarantee. Results obtained by IT2B FOCE, for example, often strayed visibly away from the true values as more subjects were studied. Furthermore, with respect to statistical efficiency (precision of parameter estimates), NPAG and PEM had good efficiency and precise parameter estimates, while precision suffered with NONMEM FOCE and IT2B FOCE, and severely so with NONMEM FO. For the second Monte Carlo dataset, NPEM closely approximated the true bimodal population joint density, while an exact parametric representation of an assumed joint unimodal density having the true population means, standard deviations and correlation gave a totally different picture. The smaller population interindividual CV% estimates with IT2B on the clinical dataset are probably the result of assuming Gaussian parameter distributions and/or of using the FOCE approximation. NPEM and NPAG, having no constraints on the shape of the population parameter distributions, and which compute the likelihood exactly and estimate parameter values with greater precision, detected the more likely greater diversity in the parameter values in the population studied. In the first Monte Carlo study, NPAG and PEM had more precise parameter estimates than either IT2B FOCE or NONMEM FOCE, as well as much more precise estimates than NONMEM FO. In the second Monte Carlo study, NPEM easily detected the bimodal parameter distribution at this initial step without requiring any further information. Population modelling methods using exact or accurate computations have more precise parameter estimation, better stochastic convergence properties and are, very importantly, statistically consistent. Nonparametric methods are better than parametric methods at analysing populations having unanticipated non-Gaussian or multimodal parameter distributions.

Wavepacket dynamics on arbitrary Lagrangian–Eulerian grids: Application to an Eckart barrier

An adaptive grid approach to a computational study of the scattering of a wavepacket from a repulsive Eckart barrier is described. The grids move in an arbitrary Lagrangian–Eulerian (ALE) framework and a hybrid of the moving path transform of the Schrodinger equation and the hydrodynamic equations are used for the equations of motion. Boundary grid points follow Lagrangian trajectories and interior grid points follow non-Lagrangian paths. For the hydrodynamic equations the interior grid points are equally spaced between the evolving Lagrangian boundaries. For the moving path transform of the Schrodinger equation interior grid distribution is determined by the principle of equidistribution, and by using a grid smoothing technique these grid points trace a path that continuously adapts to reflect the dynamics of the wavepacket. The moving grid technique is robust and allows accurate computations to be obtained with a small number of grid points for wavepacket propagation times exceeding 5 ps.

Nonlinear solution adaptive structured grids via heat-conduction analogies

This paper discusses an adaptive grid approach, developed using Fortran 77, on quadrilateral meshes for the Euler and Navier-Stokes solvers. Solution adaptation is through two nonlinear heat-conduction analogies applied directly on a two-dimensional surface using the finite volume method. Clustering of the grid generated is controlled by the conductivity in the computational domain, which is related arbitrarily to the geometrical curvature and flow gradient. Three levels of “multigrid” approach are implemented to accelerate convergence as a grid refinement process. The grid quality is accessed by a histogram analysis of maximum angle and aspect ratio distributions within the computational domain. This work assumes that interpolation errors due to numerical approximation of fluxes across the surfaces of a control volume should become significant as the skew angle and aspect ratio increases. Detailed computational results and comparisons with measured data are presented for steady transonic flow over a NACA0012 airfoil, supersonic flow through a DFVLR rotor, and a 15° ramp.

Space-Marching Method for Calculating Steady Supersonic Flows on a Grid Adapted to the Solution

A noniterative implicit space-marching method based on an adaptive grid approach is developed for solving the 2D steady state Euler equations describing supersonic gas flows without streamwise separation. A grid adapted to the solution is generated by using a variational method optimizing such important properties of a grid as smoothness, orthogonality, and grid cell volume variation simultaneously. The weight function in the integral of adaptation is constructed, so that the grid lines are accumulated near strong gradients and curvatures of the solution curve. A special treatment of the boundary points is proposed to determine the location of nodes near slope discontinuities of the boundary mesh line. The noniterative implicit space-marching method of Y. C. Vigneronet al. is applied to solve the Euler equations written in an arbitrary curvilinear system of coordinates. The streamwise and transverse derivatives in the governing equations are approximated by the second-order upwind Richardson and second-order symmetric TVD schemes, respectively. Implicit boundary condition procedure based on the theory of characteristics for hyperbolic systems of equations is employed. Numerical calculations show that the resolution of strong gradient flow fields can significantly be improved by using the grid adapted to the solution, while the number of grid nodes is the same.

The Spherical Harmonics Discrete Ordinate Method for Three-Dimensional Atmospheric Radiative Transfer

A new algorithm for modeling radiative transfer in inhomogeneous three-dimensional media is described. The spherical harmonics discrete ordinate method uses a spherical harmonic angular representation to reduce memory use and time computing the source function. The radiative transfer equation is integrated along discrete ordinates through a spatial grid to model the streaming of radiation. An adaptive grid approach, which places additional points where they are most needed to improve accuracy, is implemented. The solution method is a type of successive order of scattering approach or Picard iteration. The model computes accurate radiances or fluxes in either the shortwave or longwave regions, even for highly peaked phase functions. Broadband radiative transfer is computed efficiently with a k distribution. The results of validation tests and examples illustrating the efficiency and accuracy of the algorithm are shown for simple geometries and realistic simulated clouds.

Adaptive Grid Approaches and Multilevel Iterative Methods for SOL Transport Codes

Adaptive Grid Approaches and Multilevel Iterative Methods for SOL Transport Codes

Application of computational fluid dynamics to sonic boom near- and mid-field prediction

A technique combining a quasilinear extrapolation theory and a three-dimensional parabolized Navier-Stokes (PNS) code has been used to calculate the supersonic overpressure from three different geometries at near- and mid-fields. Wind-tunnel data is used for code validation. Comparison of the computed results with different grid refinements, and different extrapolation distances, are shown in this article. It is observed that a large number of grid points is needed to resolve the tail shock/expansion fan interaction. Therefore, an adaptive grid approach is employed to calculate the flowfield. The effects of a thin, attached boundary layer and the sting of the wind-tunnel model to the sonic boom have also been studied in this article. The agreement between the results and the wind-tunnel data confirms that this technique can be applied to the problem of sonic-boom prediction.

A grid overlapping scheme for flowfield computations about multicomponent configurations

An efficient grid interfacing zonal algorithm has been developed for computing the transonic flowfield about three-dimensional multicomponent configurations. The algorithm uses the full-potential formulation and the AF2 fully implicit approximate factorization scheme. The flowfield solution is computed using a component- adaptive grid approach in which separate grids are employed for the individual components in the multicom- ponent configuration, where each component grid is optimized for a particular geometry. The component grids are allowed to overlap, and flowfield information is transmitted from one grid to another through the overlap region. An overlapping grid scheme has been implemented for an isolated wing and a noninteracting wing/pylon/nacelle configuration . Numerical results show that the present algorithm is stable, accurate, and promises to be effective in computing the flowfield about complex configurations. ELIABLE and efficient three-dimensional transonic analysis methods are needed to make realistic and cost- effective predictions of aircraft aerodynamics. Early efforts to predict the transonic flowfield about aircraft multiple- component configurations are based on the transonic small- disturbance formulation.1-3 This allows the geometry of the configuration to be greatly simplified and the surface boundary condition to be applied on a mean approximate surface. Accurate prediction of such flowfields, however, requires the use of the full-potential formulation and the generation of a suitable surf ace-fitted grid. Because each aircraft component (wing, nacelle, fuselage) requires, in general, a grid system that is usually incompatible with the grid systems of the other components, the generation of a single surface-fitted grid for the entire configuration is a difficult task. In such a global grid, control of grid point distribution, skewness, and clustering will be difficult to achieve. Efforts to predict the flowfield about a complete aircraft configuration using a single-grid approach have been made recently by Yu.4 In the present paper the alternate approach of using a component adaptive grid scheme is investigated. The basic idea of this approach is to develop a separate grird for each component of a complex configuration where each individual grid is optimized for a particular component. The flow solver for each component employs the AF2 fully implicit ap- proximate factorization scheme.5 Earlier studies in two dimensions showed that by allowing the component grids to overlap, fast convergence can be achieved.6 This approach has been generalized herein to three dimensions, where an overlapped grid scheme has been im- plemented for an isolated wing and a wing/pylon/nac elle configuration. The application of an embedded grid scheme to an isolated wing geometry was performed primarily as a proof-of- concept study to verify the grid interfacing logic in three dimensions and to determine if single-grid results could be replicated. The application to the wing/pylon/nacelle con- figuration represents a more realistic use of the grid em- bedding technique. The full interacting algorithm has been