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NASA NTRS1995

Dynamical Approach Study of Spurious Steady-State Numerical Solutions of Nonlinear Differential Equations: Global Asymptotic Behavior of Time Discretizations - Part 2

The global asymptotic nonlinear behavior of 11 explicit and implicit time discretizations for four 2 x 2 systems of first-order autonomous nonlinear ordinary differential equations (ODEs) is analyzed. The objectives are to gain a basic understanding of the difference in the dynamics of numerics between the scalars and systems of nonlinear autonomous ODEs and to set a baseline global asymptotic solution behavior of these schemes for practical computations in computational fluid dynamics. We show how 'numerical' basins of attraction can complement the bifurcation diagrams in gaining more detailed global asymptotic behavior of time discretizations for nonlinear differential equations (DEs). We show how in the presence of spurious asymptotes the basins of the true stable steady states can be segmented by the basins of the spurious stable and unstable asymptotes. One major consequence of this phenomenon which is not commonly known is that this spurious behavior can result in a dramatic distortion and, in most cases, a dramatic shrinkage and segmentation of the basin of attraction of the true solution for finite time steps. Such distortion, shrinkage and segmentation of the numerical basins of attraction will occur regardless of the stability of the spurious asymptotes, and will occur for unconditionally stable implicit linear multistep methods. In other words, for the same (common) steady-state solution the associated basin of attraction of the DE might be very different from the discretized counterparts and the numerical basin of attraction can be very different from numerical method to numerical method. The results can be used as an explanation for possible causes of error, and slow convergence and nonconvergence of steady-state numerical solutions when using the time-dependent approach for nonlinear hyperbolic or parabolic PDEs.

Yee, H. C.↗

NASA NTRS1984

The effect of compact implicit differencing in a baroclinic primitive equations model

In an effort to improve forecast accuracy, the horizontal accuracy of gridpoint forecast models at a number of numerical weather prediction (NWP) centers was increased from second order to fourth order. The current GLAS model uses second-order explicit finite-difference formulas in the vertical and fourth-order explicit finite-difference formulas in the horizontal. Experiments are described which indicate the increased forecast accuracy gained by use of compact fourth-order finite differences in a simple baroclinic model. Tables show an increase in forecast accuracy by a factor of 100-150 by use of fourth-order instead of second-order vertical discretization.

Augenbaum, J. M.↗

NASA NTRS1991

Measured and computed performance of a microstrip filter composed of semi-insulating GaAs on a fused quartz substrate

The performance of a microstrip hammerhead filter that has been fabricated on an electrically thin layer of semiinsulating GaAs backed by a fused quartz substrate was measured and compared to results of a three-dimensional finite-difference time-domain (FD-TD) program used to calculate the response of the filter both with and without the GaAs layer. The program, presented by Sheen et al. (1990), discretizes the entire structure and then simulates the propagation of a Gaussian pulse through the filter. The microstrip filter is intended for applications involving ultrathin lifted-off or etched-back GaAs containing both active devices and passive microstrip circuitry backed by a much thicker mechanically rigid low-loss, low-dielectric-constant substrate. The low-pass characteristics of the hammerhead filter with the intermediate GaAs layer are compared with those of the same filter on quartz alone. Both the measured and computed data show a significant shift in cutoff frequency (about 10 percent at the 3 dB points) for a GaAs layer that is 0.007 wavelengths thick at 4 GHz.

Siegel, Peter H.↗

NASA NTRS1991

Aerodynamic sensitivity analysis methods for the compressible Euler equations

This study presents a mathematical formulation developed for aerodynamic sensitivity coefficients based on a discretized form of the compressible 2D Euler equations. A brief motivating introduction to the aerodynamic sensitivity analysis and the reasons behind an integrated flow/sensitivity analysis for design algorithms are presented. The finite difference approach and the quasi-analytical approach are used to determine the aerodynamic sensitivity coefficients. A new flow prediction concept, which is an outcome of the direct method in the quasi-analytical approach, is developed and illustrated with an example. Surface pressure coefficient distributions of a nozzle-afterbody configuration obtained from the predicted flowfield solution are compared successfully with their corresponding values obtained from a flowfield analysis code and the experimental data.

Baysal, Oktay↗

NASA NTRS1997

An Application of the Difference Potentials Method to Solving External Problems in CFD

Numerical solution of infinite-domain boundary-value problems requires some special techniques that would make the problem available for treatment on the computer. Indeed, the problem must be discretized in a way that the computer operates with only finite amount of information. Therefore, the original infinite-domain formulation must be altered and/or augmented so that on one hand the solution is not changed (or changed slightly) and on the other hand the finite discrete formulation becomes available. One widely used approach to constructing such discretizations consists of truncating the unbounded original domain and then setting the artificial boundary conditions (ABC's) at the newly formed external boundary. The role of the ABC's is to close the truncated problem and at the same time to ensure that the solution found inside the finite computational domain would be maximally close to (in the ideal case, exactly the same as) the corresponding fragment of the original infinite-domain solution. Let us emphasize that the proper treatment of artificial boundaries may have a profound impact on the overall quality and performance of numerical algorithms. The latter statement is corroborated by the numerous computational experiments and especially concerns the area of CFD, in which external problems present a wide class of practically important formulations. In this paper, we review some work that has been done over the recent years on constructing highly accurate nonlocal ABC's for calculation of compressible external flows. The approach is based on implementation of the generalized potentials and pseudodifferential boundary projection operators analogous to those proposed first by Calderon. The difference potentials method (DPM) by Ryaben'kii is used for the effective computation of the generalized potentials and projections. The resulting ABC's clearly outperform the existing methods from the standpoints of accuracy and robustness, in many cases noticeably speed up the multigrid convergence, and at the same time are quite comparable to other methods from the standpoints of geometric universality and simplicity of implementation.

Ryaben 'Kii, Victor S.↗

NASA NTRS1981

Unconditional instability of inflow dependent boundary conditions in difference approximations to hyperbolic systems

The stability of finite difference approximations to initial boundary hyperbolic systems is studied. As is well known, a proper specification of boundary conditions for such systems is essential for their solutions to be well defined. A discrete analogue of the above is proved - if the numerical boundary conditions are consistent with an inflow part of the problem, they render the overall computation unstable. An example of the inviscid gasdynamics equations is considered.

Tadmor, E.↗

NASA NTRS1983

The unconditional instability of inflow-dependent boundary conditions in difference approximations to hyperbolic systems

Tadmor, E.↗

NASA NTRS1995

Micromechanical models for textile composites

Numerical and analytical micromechanical models are presented to predict the thermoelastic behavior of a textile composite. In the numerical model, the unit-cell is discretized with finite elements, and periodic boundary conditions are imposed between opposite faces of the unit-cell. For a thin textile composite, stress gradients effects through the thickness are demonstrated. The consequent difference in the stiffness and strength behavior of thick and thin composites are discussed. The numerical model is implemented to predict 3-D thermo-elastic constants for a thick textile composite, and the plate thermo-mechanical properties for a thin textile composite. The numerical model is extended to compute the thermal residual microstresses due to processing to predict the composite failure envelopes. An analytical model - Selective Averaging Method (SAM) - is proposed, which is based on a judicious combination of stiffness and compliance averaging to estimate the 3-D elastic constants. Both the models are tested and verified for several examples by comparing the stiffness properties with elasticity solutions and available results.

Sankar, Bhavani V.↗

NASA NTRS1999

Practical Aerodynamic Design Optimization Based on the Navier-Stokes Equations and a Discrete Adjoint Method

The technical details are summarized below: Compressible and incompressible versions of a three-dimensional unstructured mesh Reynolds-averaged Navier-Stokes flow solver have been differentiated and resulting derivatives have been verified by comparisons with finite differences and a complex-variable approach. In this implementation, the turbulence model is fully coupled with the flow equations in order to achieve this consistency. The accuracy demonstrated in the current work represents the first time that such an approach has been successfully implemented. The accuracy of a number of simplifying approximations to the linearizations of the residual have been examined. A first-order approximation to the dependent variables in both the adjoint and design equations has been investigated. The effects of a "frozen" eddy viscosity and the ramifications of neglecting some mesh sensitivity terms were also examined. It has been found that none of the approximations yielded derivatives of acceptable accuracy and were often of incorrect sign. However, numerical experiments indicate that an incomplete convergence of the adjoint system often yield sufficiently accurate derivatives, thereby significantly lowering the time required for computing sensitivity information. The convergence rate of the adjoint solver relative to the flow solver has been examined. Inviscid adjoint solutions typically require one to four times the cost of a flow solution, while for turbulent adjoint computations, this ratio can reach as high as eight to ten. Numerical experiments have shown that the adjoint solver can stall before converging the solution to machine accuracy, particularly for viscous cases. A possible remedy for this phenomenon would be to include the complete higher-order linearization in the preconditioning step, or to employ a simple form of mesh sequencing to obtain better approximations to the solution through the use of coarser meshes. . An efficient surface parameterization based on a free-form deformation technique has been utilized and the resulting codes have been integrated with an optimization package. Lastly, sample optimizations have been shown for inviscid and turbulent flow over an ONERA M6 wing. Drag reductions have been demonstrated by reducing shock strengths across the span of the wing.

Grossman, Bernard↗

NASA NTRS1992

Runge-Kutta methods combined with compact difference schemes for the unsteady Euler equations

Recent development using compact difference schemes to solve the Navier-Stokes equations show spectral-like accuracy. A study was made of the numerical characteristics of various combinations of the Runge-Kutta (RK) methods and compact difference schemes to calculate the unsteady Euler equations. The accuracy of finite difference schemes is assessed based on the evaluations of dissipative error. The objectives are reducing the numerical damping and, at the same time, preserving numerical stability. While this approach has tremendous success solving steady flows, numerical characteristics of unsteady calculations remain largely unclear. For unsteady flows, in addition to the dissipative errors, phase velocity and harmonic content of the numerical results are of concern. As a result of the discretization procedure, the simulated unsteady flow motions actually propagate in a dispersive numerical medium. Consequently, the dispersion characteristics of the numerical schemes which relate the phase velocity and wave number may greatly impact the numerical accuracy. The aim is to assess the numerical accuracy of the simulated results. To this end, the Fourier analysis is to provide the dispersive correlations of various numerical schemes. First, a detailed investigation of the existing RK methods is carried out. A generalized form of an N-step RK method is derived. With this generalized form, the criteria are derived for the three and four-step RK methods to be third and fourth-order time accurate for the non-linear equations, e.g., flow equations. These criteria are then applied to commonly used RK methods such as Jameson's 3-step and 4-step schemes and Wray's algorithm to identify the accuracy of the methods. For the spatial discretization, compact difference schemes are presented. The schemes are formulated in the operator-type to render themselves suitable for the Fourier analyses. The performance of the numerical methods is shown by numerical examples. These examples are detailed. described. The third case is a two-dimensional simulation of a Lamb vortex in an uniform flow. This calculation provides a realistic assessment of various finite difference schemes in terms of the conservation of the vortex strength and the harmonic content after travelling a substantial distance. The numerical implementation of Giles' non-refelctive equations coupled with the characteristic equations as the boundary condition is discussed in detail. Finally, the single vortex calculation is extended to simulate vortex pairing. For the distance between two vortices less than a threshold value, numerical results show crisp resolution of the vortex merging.

Yu, Sheng-Tao↗

NASA NTRS1986

Estimates of the bolometric albedos and radiation balance of Uranus and Neptune

Models possessing an upper haze layer of finite optical depth and a lower cloud layer of infinite optical depth at discrete altitudes are used to bound the wavelength-averaged phase integrals and bolometric albedos of Uranus and Neptune. The models differ in the assumed value of the particles' single scattering phase function and the wavelength dependence of the haze optical depth. A range of phase functions, from the isotropic to those characterizing Titan, Jupiter, and Saturn atmosphere particles, are discussed. The results obtained imply that the meteorological regimes in the observable atmospheres of Uranus and Neptune may differ considerably; internal heat flux could play a much more important role for Neptune than for Uranus.

Pollack, J. B.↗

NASA NTRS1986

Vertical structure of aerosols and clouds in the atmospheres of Uranus and Neptune: Implications for their heat budgets

Models possessing an upper haze layer of finite optical depth and a lower cloud layer of infinite optical depth at discrete altitudes are used to bound the wavelength-averaged phase integrals and bolometric albedos of Uranus and Neptune. The models differ in the assumed value of the particles single scattering phase function and the wavelength dependence of the haze optical depth. A range of phase functions, from the isotropic to those characterizing Titan, Jupiter, and Saturn atmosphere particles, are discussed. The results obtained imply that the meteorological regimes in the observable atmospheres of Uranus and Neptune may differ considerably; internal heat flux could play a much more important role for Neptune than for Uranus.

Pollack, James B.↗

NASA NTRS1992

A fast direct solver for a class of two-dimensional separable elliptic equations on the sphere

An efficient, direct, second-order solver for the discrete solution of two-dimensional separable elliptic equations on the sphere is presented. The method involves a Fourier transformation in longitude and a direct solution of the resulting coupled second-order finite difference equations in latitude. The solver is made efficient by vectorizing over longitudinal wavenumber and by using a vectorized fast Fourier transform routine. It is evaluated using a prescribed solution method and compared with a multigrid solver and the standard direct solver from FISHPAK.

Moorthi, Shrinivas↗

NASA NTRS2009

Ablative Thermal Response Analysis Using the Finite Element Method

A review of the classic techniques used to solve ablative thermal response problems is presented. The advantages and disadvantages of both the finite element and finite difference methods are described. As a first step in developing a three dimensional finite element based ablative thermal response capability, a one dimensional computer tool has been developed. The finite element method is used to discretize the governing differential equations and Galerkin's method of weighted residuals is used to derive the element equations. A code to code comparison between the current 1-D tool and the 1-D Fully Implicit Ablation and Thermal Response Program (FIAT) has been performed.

Dec John A.↗

NASA NTRS1975

Stress analysis and stress intensity factors for finite geometry solids containing rectangular surface cracks

The line method of analysis is applied to the Navier-Cauchy equations of elastic equilibrium to calculate the displacement field in a finite geometry bar containing a variable depth rectangular surface crack under extensionally applied uniform loading. The application of this method to these equations leads to coupled sets of simultaneous ordinary differential equations whose solutions are obtained along sets of lines in a discretized region. Using the obtained displacement field, normal stresses and the stress intensity factor variation along the crack periphery are calculated for different crack depth to bar thickness ratios. Crack opening displacements and stress intensity factors are also obtained for a through-thickness, center cracked bar with variable thickness. The reported results show a considerable potential for using this method in calculating stress intensity factors for commonly encountered surface crack geometries in finite solids.

Gyekenyesi, J. P.↗

NASA NTRS1995

Nonlinear wave propagation using three different finite difference schemes (category 2 application)

Three common finite difference schemes are used to examine the computation of one-dimensional nonlinear wave propagation. The schemes are studied for their responses to numerical parameters such as time step selection, boundary condition implementation, and discretization of governing equations. The performance of the schemes is compared and various numerical phenomena peculiar to each is discussed.

Pope, D. Stuart↗

NASA NTRS1997

Mechanical Characterization and Micromechanical Modeling of Woven Carbon/Copper Composites

The present investigation examines the in-plane mechanical behavior of a particular woven metal matrix composite (MMC); 8-harness (8H) satin carbon/copper (C/Cu). This is accomplished via mechanical testing as well as micromechanical modeling. While the literature is replete with experimental and modeling efforts for woven and braided polymer matrix composites, little work has been done on woven and braided MMC's. Thus, the development and understanding of woven MMC's is at an early stage. 8H satin C/Cu owes its existence to the high thermal conductivity of copper and low density and thermal expansion of carbon fibers. It is a candidate material for high heat flux applications, such as space power radiator panels. The experimental portion of this investigation consists of monotonic and cyclic tension, compression, and Iosipescu shear tests, as well as combined tension-compression tests. Tests were performed on composite specimens with three copper matrix alloy types: pure Cu, Cu-0.5 weight percent Ti (Cu-Ti), and Cu-0.7 weight percent Cr (Cu-Cr). The small alloying additions are present to promote fiber/matrix interfacial bonding. The analytical modeling effort utilizes an approach in which a local micromechanical model is embedded in a global micromechanical model. This approach differs from previously developed analytical models for woven composites in that a true repeating unit cell is analyzed. However, unlike finite element modeling of woven composites, the geometry is sufficiently idealized to allow efficient geometric discretization and efficient execution.

Bednarcyk, Brett A.↗

NASA NTRS1977

Stress analysis and stress-intensity factors for finite geometry solids containing rectangular surface cracks

The line method of analysis is applied to the Navier-Cauchy equations of elastic equilibrium to calculate the displacement field in a finite geometry bar containing a variable depth rectangular surface crack under extensionally applied uniform loading. The application of this method to these equations leads to coupled sets of simultaneous ordinary differential equations whose solutions are obtained along sets of lines in a discretized region. Using the obtained displacement field, normal stresses, and the stress-intensity factor variation along the crack periphery are calculated for different crack depth to bar thickness ratios. Crack opening displacements and stress-intensity factors are also obtained for a through-thickness, center-cracked bar with variable thickness. The reported results show a considerable potential for using this method in calculating stress-intensity factors for commonly encountered surface crack geometries in finite solids

Gyekenyesi, J. P.↗