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NASA NTRS1992

Theory and implementation of high-order adaptive hp methods for analysis of incompressible viscous flows

An account is given of 'smart' algorithms for CFD which change in structure and performance with during flow calculations to accommodate changing properties of the solution. Such algorithms prominently include adaptive FEM methods, which are designed to adjust mesh parameters for the control of numerical error; attention is presently given to those which change the mesh size h and the local spectral order p in order to achieve high accuracies with minimal numbers of degrees of freedom. These 'hp methods' produce exponentially convergent approximations through which flow features are resolved by automatically distributing element sizes and spectral orders. This leads to calculation of the local (elementwise) error in the approximation.

Oden, J. T.↗

NASA NTRS1993

Uniform high-order spectral methods for one- and two-dimensional Euler equations

Uniform high order spectral methods to solve multi-dimensional Euler equations for gas dynamics are discussed. Uniform high order spectral approximations with spectral accuracy in smooth regions of solutions are constructed by introducing the idea of the Essentially Non-Oscillatory (ENO) polynomial interpolations into the spectral methods. The authors present numerical results for the inviscid Burgers' equation, and for the one-dimensional Euler equations including the interactions between a shock wave and density disturbance, Sod's and Lax's shock tube problems, and the blast wave problem. The interaction between a Mach 3 two-dimensional shock wave and a rotating vortex is simulated.

Cai, Wei↗

NASA NTRS1993

Actuator and aerodynamic modeling for high-angle-of-attack aeroservoelasticity

Accurate prediction of airframe/actuation coupling is required by the imposing demands of modern flight control systems. In particular, for agility enhancement at high angle of attack and low dynamic pressure, structural integration characteristics such as hinge moments, effective actuator stiffness, and airframe/control surface damping can have a significant effect on stability predictions. Actuator responses are customarily represented with low-order transfer functions matched to actuator test data, and control surface stiffness is often modeled as a linear spring. The inclusion of the physical properties of actuation and its installation on the airframe is therefore addressed in this paper using detailed actuator models which consider the physical, electrical, and mechanical elements of actuation. The aeroservoelastic analysis procedure is described in which the actuators are modeled as detailed high-order transfer functions and as approximate low-order transfer functions. The impacts of unsteady aerodynamic modeling on aeroservoelastic stability are also investigated in this paper by varying the order of approximation, or number of aerodynamic lag states, in the analysis. Test data from a thrust-vectoring configuration of an F/A-18 aircraft are compared to predictions to determine the effects on accuracy as a function of modeling complexity.

Brenner, Martin J.↗

NASA NTRS1993

A comparison of two formulations for high-order accurate essentially non-oscillatory schemes

The finite-volume and finite-difference implementations of high-order accurate essentially nonoscillatory shock-capturing schemes are discussed and compared. Results obtained with fourth-order accurate algorithms based on both formulations are examined for accuracy, sensitivity to grid irregularities, resolution of waves that are oblique to the mesh, and computational efficiency. Some algorithm modifications that may be required for a given application are suggested. Conclusions that pertain to the relative merits of both formulations are drawn, and some circumstances for which each might be useful are noted.

Casper, J.↗

NASA NTRS1993

Effects of spatial order of accuracy on the computation of vortical flowfields

The effect of the order-of-accuracy, used for the spatial discretization, on the resolution of the leading edge vortices over sharp-edged delta wings is investigated. The flowfield is computed using a viscous/inviscid zonal approach. The viscous flow in the vicinity of the wing is computed using the conservative formulation of the compressible, thin-layer Navier-Stokes equations. The leeward-side vortical flowfield and the other flow regions away from the surface are computed as inviscid. The time integration is performed with both an explicit fourth-order Runge-Kutta scheme and an implicit, factorized, iterative scheme. High-order-accurate inviscid fluxes are computed using both a conservative and a non-conservative (primitive variable) formulation. The nonlinear, inviscid terms of the primitive variable form of the governing equations are evaluated with a finite-difference numerical scheme based on the sign of the eigenvalues. High-order, upwind-biased, finite difference formulas are used to evaluate the derivatives of the nonlinear convective terms. Computed results are compared with available experimental data, and comparisons of the flowfield in the vicinity of the vortex cores are presented.

Ekaterinaris, J. A.↗

NASA NTRS1993

Some aspects of high-order numerical solutions of the linear convection equation with forced boundary conditions

A six-stage low-storage Runge-Kutta time-marching method is presented and shown to be an efficient method for use with high-accuracy spatial difference operators for wave propagation problems. The accuracy of the method for inhomogeneous ordinary differential equations is demonstrated through numerical solutions of the linear convection equation with forced boundary conditions. Numerical experiments are presented simulating a sine wave and a Gaussian pulse propagating into and through the domain. For practical levels of mesh refinement corresponding to roughly ten points per wavelength, the six-stage Runge-Kutta method is more accurate than the popular fourth-order Runge-Kutta method. Further numerical experiments are presented which show that the numerical boundary scheme at an inflow boundary can be a significant source of error when high-accuracy spatial discretizations are used.

Zingg, D. W.↗

NASA NTRS1990

A high-order compact numerical algorithm for supersonic flows

A dissipative compact two-four scheme (second-order time, fourth-order space) similar to the original MacCormack scheme has been developed, that exhibits greater accuracy than conventional fourth-order schemes. The dissipative nature of the scheme allows it to resolve weak discontinuities without artificial damping. A derivation of the scheme is presented, as well as the theoretical stability characteristics. The temporal scheme is then generalized into a steady-state formulation which achieves fourth-order spatial accuracy at steady-state. Several test problems are used to show that the scheme is more accurate than the traditional MacCormack scheme, and is nearly as efficient.

Carpenter, M. H.↗

NASA NTRS1988

Maximum likelihood identification for large space structures

This paper examines the use of on-orbit identification based on Maximum Likelihood Estimation (MLE) to provide these high-order, high-accuracy control design models for large space structures (LSS's). First, it outlines a general MLE identification algorithm, together with a covariance-analysis procedure to assess algorithm performance in terms of systematic and stochastic errors. Next, it examines various simplifications appropriate for the LSS identification application. Simplified analytical performance results are presented, as are numerical results to support these analyses. Finally, a graphical interpretation of these results is given.

Barrett, Michael F.↗

NASA NTRS1993

A comparison of two formulations for high-order accurate essentially non-oscillatory schemes

The finite-volume and finite-difference implementations of high-order accurate essentially non-oscillatory shock-capturing schemes are discussed and compared. Results obtained with fourth-order accurate algorithms based on both formulations are examined for accuracy, sensitivity to grid irregularities, resolution of waves that are oblique to the mesh, and computational efficiency. Some algorithm modifications that may be required for a given application are suggested. Conclusions that pertain to the relative merits of both formulations are drawn, and some circumstances for which each might be useful are noted.

Casper, Jay↗

NASA NTRS1993

Actuator and aerodynamic modeling for high-angle-of-attack aeroservoelasticity

Accurate prediction of airframe/actuation coupling is required by the imposing demands of modern flight control systems. In particular, for agility enhancement at high angle of attack and low dynamic pressure, structural integration characteristics such as hinge moments, effective actuator stiffness, and airframe/control surface damping can have a significant effect on stability predictions. Actuator responses are customarily represented with low-order transfer functions matched to actuator test data, and control surface stiffness is often modeled as a linear spring. The inclusion of the physical properties of actuation and its installation on the airframe is therefore addressed using detailed actuator models which consider the physical, electrical, and mechanical elements of actuation. The aeroservoelastic analysis procedure is described in which the actuators are modeled as detailed high-order transfer functions and as approximate low-order transfer functions. The impacts of unsteady aerodynamic modeling on aeroservoelastic stability are also investigated by varying the order of approximation, or number of aerodynamic lag states, in the analysis. Test data from a thrust-vectoring configuration of an F/A-l8 aircraft are compared to predictions to determine the effects on accuracy as a function of modeling complexity.

Brenner, Martin J.↗

NASA NTRS1994

An HP Adaptive Discontinuous Galerkin Method for Hyperbolic Conservation Laws

This dissertation addresses various issues for model classes of hyperbolic conservation laws. The basic approach developed in this work employs a new family of adaptive, hp-version, finite element methods based on a special discontinuous Galerkin formulation for hyperbolic problems. The discontinuous Galerkin formulation admits high-order local approximations on domains of quite general geometry, while providing a natural framework for finite element approximations and for theoretical developments. The use of hp-versions of the finite element method makes possible exponentially convergent schemes with very high accuracies in certain cases; the use of adaptive hp-schemes allows h-refinement in regions of low regularity and p-enrichment to deliver high accuracy, while keeping problem sizes manageable and dramatically smaller than many conventional approaches. The use of discontinuous Galerkin methods is uncommon in applications, but the methods rest on a reasonable mathematical basis for low-order cases and has local approximation features that can be exploited to produce very efficient schemes, especially in a parallel, multiprocessor environment. The place of this work is to first and primarily focus on a model class of linear hyperbolic conservation laws for which concrete mathematical results, methodologies, error estimates, convergence criteria, and parallel adaptive strategies can be developed, and to then briefly explore some extensions to more general cases. Next, we provide preliminaries to the study and a review of some aspects of the theory of hyperbolic conservation laws. We also provide a review of relevant literature on this subject and on the numerical analysis of these types of problems.

Bey, Kim S.↗

NASA NTRS1994

High-order centered difference methods with sharp shock resolution

In this paper we consider high-order centered finite difference approximations of hyperbolic conservation laws. We propose different ways of adding artificial viscosity to obtain sharp shock resolution. For the Riemann problem we give simple explicit formulas for obtaining stationary one and two-point shocks. This can be done for any order of accuracy. It is shown that the addition of artificial viscosity is equivalent to ensuring the Lax k-shock condition. We also show numerical experiments that verify the theoretical results.

Gustafsson, Bertil↗

NASA NTRS1997

Accurate Monotonicity - Preserving Schemes With Runge-Kutta Time Stepping

A new class of high-order monotonicity-preserving schemes for the numerical solution of conservation laws is presented. The interface value in these schemes is obtained by limiting a higher-order polynominal reconstruction. The limiting is designed to preserve accuracy near extrema and to work well with Runge-Kutta time stepping. Computational efficiency is enhanced by a simple test that determines whether the limiting procedure is needed. For linear advection in one dimension, these schemes are shown as well as the Euler equations also confirm their high accuracy, good shock resolution, and computational efficiency.

Suresh, A.↗

NASA NTRS1997

Computational Considerations for the Simulation of Discontinuous Flows

The numerical study of aeroacoustic problems places stringent demands on the choice of a computational algorithm, because it requires the ability to propagate disturbances of small amplitude and short wavelength. The demands are particularly high when shock waves are involved, because the chosen algorithm must also resolve discontinuities in the solution. In a previous work the capabilities and deficiencies of shock-capturing methods for aeroacoustic problems were demonstrated using a high-order essentially nonoscillatory (ENO) numerical method. It was shown that first-order results are obtained when simulating time-dependent flows with discontinuities. The present study reaffirms this conclusion by comparing the ENO results with those obtained using a conventional linear scheme. A sixth-order-accurate compact implicit finite difference scheme is used to investigate various discontinuous flows. The design order of accuracy is achieved in the smooth regions of a steady-state, quasi-one-dimensional Euler test case, as well as in the time-dependent Burgers' equation. However, in the unsteady Euler sound-shock interaction, first-order results are obtained downstream of the shock. A comparison is made between the linear and nonlinear results, noting the advantages of each method. A discontinuous linear model problem is then used to identify the cause of the first-order results. Here, the nature of the solution error is quantified as being predominantly a numerical phase shift, and a post-processing procedure is demonstrated which increases the solution accuracy downstream of the discontinuity to second-order.

Carpenter Mark H.↗

NASA NTRS1998

Nonlinear Comparison of High-Order and Optimized Finite-Difference Schemes

The effect of reducing the formal order of accuracy of a finite-difference scheme in order to optimize its high-frequency performance is investigated using the I-D nonlinear unsteady inviscid Burgers'equation. It is found that the benefits of optimization do carry over into nonlinear applications. Both explicit and compact schemes are compared to Tam and Webb's explicit 7-point Dispersion Relation Preserving scheme as well as a Spectral-like compact scheme derived following Lele's work. Results are given for the absolute and L2 errors as a function of time.

Hixon, R.↗

NASA NTRS1995

Towards Distributed Fluids/Controls Simulations

Computational prototyping of aircraft in nonlinear regimes will require the coupling of high-order disciplines on high performance computers. This effort investigates the coupling of the Reynolds-averaged Navier-Stokes equations with six degree-of-freedom nonlinear body dynamics and controls on distributed architectures. Towards this goal, solutions obtained on a wide-node IBM SP-2, an Ethernet-linked cluster of workstations, and a Cray C90 are compared in terms of accuracy and throughput.

Atwood, Christopher A.↗

NASA NTRS2000

Validation of a High-Order Prefactored Compact Scheme on Nonlinear Flows with Complex Geometries

Three benchmark problems are solved using a sixth-order prefactored compact scheme employing an explicit 10th-order filter with optimized fourth-order Runge-Kutta time stepping. The problems solved are the following: (1) propagation of sound waves through a transonic nozzle; (2) shock-sound interaction; and (3) single airfoil gust response. In the first two problems, the spatial accuracy of the scheme is tested on a stretched grid, and the effectiveness of boundary conditions is shown. The solution stability and accuracy near a shock discontinuity is shown as well. Also, 1-D nonlinear characteristic boundary conditions will be evaluated. In the third problem, a nonlinear Euler solver will be used that solves the equations in generalized curvilinear coordinates using the chain rule transformation. This work, continuing earlier work on flat-plate cascades and Joukowski airfoils, will focus mainly on the effect of the grid and boundary conditions on the accuracy of the solution. The grids were generated using a commercially available grid generator, GridPro/az3000.

Hixon, Ray↗

NASA NTRS1995

A 3D High-Order Unstructured Finite-Volume Algorithm for Solving Maxwell's Equations

A three-dimensional finite-volume algorithm based on arbitrary basis functions for time-dependent problems on general unstructured grids is developed. The method is applied to the time-domain Maxwell equations. Discrete unknowns are volume integrals or cell averages of the electric and magnetic field variables. Spatial terms are converted to surface integrals using the Gauss curl theorem. Polynomial basis functions are introduced in constructing local representations of the fields and evaluating the volume and surface integrals. Electric and magnetic fields are approximated by linear combinations of these basis functions. Unlike other unstructured formulations used in Computational Fluid Dynamics, the new formulation actually does not reconstruct the field variables at each time step. Instead, the spatial terms are calculated in terms of unknowns by precomputing weights at the beginning of the computation as functions of cell geometry and basis functions to retain efficiency. Since no assumption is made for cell geometry, this new formulation is suitable for arbitrarily defined grids, either smooth or unsmooth. However, to facilitate the volume and surface integrations, arbitrary polyhedral cells with polygonal faces are used in constructing grids. Both centered and upwind schemes are formulated. It is shown that conventional schemes (second order in Cartesian grids) are equivalent to the new schemes using first degree polynomials as the basis functions and the midpoint quadrature for the integrations. In the new formulation, higher orders of accuracy are achieved by using higher degree polynomial basis functions. Furthermore, all the surface and volume integrations are carried out exactly. Several model electromagnetic scattering problems are calculated and compared with analytical solutions. Examples are given for cases based on 0th to 3rd degree polynomial basis functions. In all calculations, a centered scheme is applied in the interior, while an upwind matching scheme is employed at material interfaces and the Engquist-Majda non-reflecting boundary condition is implemented at the numerical outer boundaries. The staggered leapfrog scheme and the Runge-Kutta methods are utilized for the time integration. Excellent agreements are found between the numerical and analytical solutions.

Liu, Yen↗