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NASA NTRS1997

Discretization and Preconditioning Algorithms for the Euler and Navier-Stokes Equations on Unstructured Meshes

Several stabilized discretization procedures for conservation law equations on triangulated domains will be considered. Specifically, numerical schemes based on upwind finite volume, fluctuation splitting, Galerkin least-squares, and space discontinuous Galerkin discretization will be considered in detail. A standard energy analysis for several of these methods will be given via entropy symmetrization. Next, we will present some relatively new theoretical results concerning congruence relationships for left or right symmetrized equations. These results suggest new variants of existing FV, DG, GLS and FS methods which are computationally more efficient while retaining the pleasant theoretical properties achieved by entropy symmetrization. In addition, the task of Jacobian linearization of these schemes for use in Newton's method is greatly simplified owing to exploitation of exact symmetries which exist in the system. These variants have been implemented in the "ELF" library for which example calculations will be shown. The FV, FS and DG schemes also permit discrete maximum principle analysis and enforcement which greatly adds to the robustness of the methods. Some prevalent limiting strategies will be reviewed. Next, we consider embedding these nonlinear space discretizations into exact and inexact Newton solvers which are preconditioned using a nonoverlapping (Schur complement) domain decomposition technique. Elements of nonoverlapping domain decomposition for elliptic problems will be reviewed followed by the present extension to hyperbolic and elliptic-hyperbolic problems. Other issues of practical relevance such the meshing of geometries, code implementation, turbulence modeling, global convergence, etc. will be addressed as needed.

Barth, Timothy↗

NASA NTRS1998

Congruence Approximations for Entrophy Endowed Hyperbolic Systems

Building upon the standard symmetrization theory for hyperbolic systems of conservation laws, congruence properties of the symmetrized system are explored. These congruence properties suggest variants of several stabilized numerical discretization procedures for hyperbolic equations (upwind finite-volume, Galerkin least-squares, discontinuous Galerkin) that benefit computationally from congruence approximation. Specifically, it becomes straightforward to construct the spatial discretization and Jacobian linearization for these schemes (given a small amount of derivative information) for possible use in Newton's method, discrete optimization, homotopy algorithms, etc. Some examples will be given for the compressible Euler equations and the nonrelativistic MHD equations using linear and quadratic spatial approximation.

Barth, Timothy J.↗

NASA NTRS2002

The Delta x B = 0 Constraint Versus Minimization of Numerical Errors in MHD Simulations

The MHD equations are a system of non-strictly hyperbolic conservation laws. The non-convexity of the inviscid flux vector resulted in corresponding Jacobian matrices with undesirable properties. It has previously been shown by Powell et al. (1995) that an 'almost' equivalent MHD system in non-conservative form can be derived. This non-conservative system has a better conditioned eigensystem. Aside from Powell et al., the MHD equations can be derived from basic principles in either conservative or non-conservative form. The Delta x B = 0 constraint of the MHD equations is only an initial condition constraint, it is very different from the incompressible Navier-Stokes equations in which the divergence condition is needed to close the system (i.e., to have the same number of equations and the same number of unknown). In the MHD formulations, if Delta x B = 0 initially, all one needs is to construct appropriate numerical schemes that preserve this constraint at later time evolutions. In other words, one does not need the Delta x B condition to close the MHD system. We formulate our new scheme together with the Cargo & Gallice (1997) form of the MHD approximate Riemann solver in curvilinear grids for both versions of the MHD equations. A novel feature of our new method is that the well-conditioned eigen-decomposition of the non-conservative MHD equations is used to solve the conservative equations. This new feature of the method provides well-conditioned eigenvectors for the conservative formulation, so that correct wave speeds for discontinuities are assured. The justification for using the non-conservative eigen-decomposition to solve the conservative equations is that our scheme has a better control of the numerical error associated with the divergence of the magnetic condition. Consequently, computing both forms of the equations with the same eigen-decomposition is almost equivalent. It will be shown that this approach, using the non-conservative eigensystem when solving the conservative equations, also works well in the context of standard shock-capturing schemes.

Yee, H. C.↗

NASA NTRS2002

The Delta(dot) B = O Constraint vs. Minimization of Numerical Errors in MHD Simulations

The MHD equations are a system of non-strictly hyperbolic conservation laws. The non-convexity of the inviscid flux vector resulted in corresponding Jacobian matrices with undesirable properties. On the other hand, the MHD equations can be derived from basic principles in either conservative or non-conservative form. The non-conservative system has a better conditioned eigensystem. The Delta(dot)B = 0 constraint of the A4HD equations is only an initial condition constraint. One does not need the Delta(dot)B condition to close the MHD system. We formulate our new low dissipative high order scheme together with the Cargo & Gallice (1997) form of the MHD approximate Riemann solver in curvilinear grids for both versions of the MHD equations. A novel feature of our new method is that the well-conditioned eigen-decomposition of the non-conservative MHD equations is used to solve the conservative equations. This new feature of the method provides well-conditioned eigenvectors for the conservative formulation, so that correct wave speeds for discontinuities are assured. The justification for using the non-conservative eigen-decomposition to solve the conservative equations is that our scheme has a better control of the numerical error associated with the Delta(dot)B condition. Consequently, computing both forms of the equations with the same eigen-decomposition is almost equivalent. It will be shown that this approach, using the non-conservative eigensystem when solving the conservative equations, also works well in the context of standard shock-capturing schemes.

Yee, H. C.↗

NASA NTRS2001

Maximally Informative Statistics for Localization and Mapping

This paper presents an algorithm for localization and mapping for a mobile robot using monocular vision and odometry as its means of sensing. The approach uses the Variable State Dimension filtering (VSDF) framework to combine aspects of Extended Kalman filtering and nonlinear batch optimization. This paper describes two primary improvements to the VSDF. The first is to use an interpolation scheme based on Gaussian quadrature to linearize measurements rather than relying on analytic Jacobians. The second is to replace the inverse covariance matrix in the VSDF with its Cholesky factor to improve the computational complexity. Results of applying the filter to the problem of localization and mapping with omnidirectional vision are presented.

Deans, Matthew C.↗

NASA NTRS1999

An Elasticity-Based Mesh Scheme Applied to the Computation of Unsteady Three-Dimensional Spoiler and Aeroelastic Problems

This paper presents a modification of the spring analogy scheme which uses axial linear spring stiffness with selective spring stiffening/relaxation. An alternate approach to solving the geometric conservation law is taken which eliminates the need for storage of metric Jacobians at previous time steps. Efficiency and verification are illustrated with several unsteady 2-D airfoil Euler computations. The method is next applied to the computation of the turbulent flow about a 2-D airfoil and wing with two and three- dimensional moving spoiler surfaces, and the results compared with Benchmark Active Controls Technology (BACT) experimental data. The aeroelastic response at low dynamic pressure of an airfoil to a single large scale oscillation of a spoiler surface is computed. This study confirms that it is possible to achieve accurate solutions with a very large time step for aeroelastic problems using the fluid solver and aeroelastic integrator as discussed in this paper.

Bartels, Robert E.↗

NASA NTRS2005

Solar Sail Attitude Control Performance Comparison

Performance of two solar sail attitude control implementations is evaluated. One implementation employs four articulated reflective vanes located at the periphery of the sail assembly to generate control torque about all three axes. A second attitude control configuration uses mass on a gimbaled boom to alter the center-of-mass location relative to the center-of-pressure producing roll and pitch torque along with a pair of articulated control vanes for yaw control. Command generation algorithms employ linearized dynamics with a feedback inversion loop to map desired vehicle attitude control torque into vane and/or gimbal articulation angle commands. We investigate the impact on actuator deflection angle behavior due to variations in how the Jacobian matrix is incorporated into the feedback inversion loop. Additionally, we compare how well each implementation tracks a commanded thrust profile, which has been generated to follow an orbit trajectory from the sun-earth L1 point to a sub-L1 station.

Bladt, Jeff J.↗

NASA NTRS2005

Issues and Methods Concerning the Evaluation of Hypersingular and Near-Hypersingular Integrals in BEM Formulations

It is known that higher order modeling of the sources and the geometry in Boundary Element Modeling (BEM) formulations is essential to highly efficient computational electromagnetics. However, in order to achieve the benefits of hIgher order basis and geometry modeling, the singular and near-singular terms arising in BEM formulations must be integrated accurately. In particular, the accurate integration of near-singular terms, which occur when observation points are near but not on source regions of the scattering object, has been considered one of the remaining limitations on the computational efficiency of integral equation methods. The method of singularity subtraction has been used extensively for the evaluation of singular and near-singular terms. Piecewise integration of the source terms in this manner, while manageable for bases of constant and linear orders, becomes unwieldy and prone to error for bases of higher order. Furthermore, we find that the singularity subtraction method is not conducive to object-oriented programming practices, particularly in the context of multiple operators. To extend the capabilities, accuracy, and maintainability of general-purpose codes, the subtraction method is being replaced in favor of the purely numerical quadrature schemes. These schemes employ singularity cancellation methods in which a change of variables is chosen such that the Jacobian of the transformation cancels the singularity. An example of the sin,oularity cancellation approach is the Duffy method, which has two major drawbacks: 1) In the resulting integrand, it produces an angular variation about the singular point that becomes nearly-singular for observation points close to an edge of the parent element, and 2) it appears not to work well when applied to nearly-singular integrals. Recently, the authors have introduced the transformation u(x(prime))= sinh (exp -1) x(prime)/Square root of ((y prime (exp 2))+ z(exp 2) for integrating functions of the form I = Integral of (lambda(r(prime))((e(exp -jkR))/(4 pi R) d D where A (r (prime)) is a vector or scalar basis function and R = Square root of( (x(prime)(exp2) + (y(prime)(exp2) + z(exp 2)) is the distance between source and observation points. This scheme has all of the advantages of the Duffy method while avoiding the disadvantages listed above. In this presentation we will survey similar approaches for handling singular and near-singular terms for kernels with 1/R(exp 2) type behavior, addressing potential pitfalls and offering techniques to efficiently handle special cases.

Fink, P. W.↗

NASA NTRS2005

Baseline Computational Fluid Dynamics Methodology for Longitudinal-Mode Liquid-Propellant Rocket Combustion Instability

A computational method for the analysis of longitudinal-mode liquid rocket combustion instability has been developed based on the unsteady, quasi-one-dimensional Euler equations where the combustion process source terms were introduced through the incorporation of a two-zone, linearized representation: (1) A two-parameter collapsed combustion zone at the injector face, and (2) a two-parameter distributed combustion zone based on a Lagrangian treatment of the propellant spray. The unsteady Euler equations in inhomogeneous form retain full hyperbolicity and are integrated implicitly in time using second-order, high-resolution, characteristic-based, flux-differencing spatial discretization with Roe-averaging of the Jacobian matrix. This method was initially validated against an analytical solution for nonreacting, isentropic duct acoustics with specified admittances at the inflow and outflow boundaries. For small amplitude perturbations, numerical predictions for the amplification coefficient and oscillation period were found to compare favorably with predictions from linearized small-disturbance theory as long as the grid exceeded a critical density (100 nodes/wavelength). The numerical methodology was then exercised on a generic combustor configuration using both collapsed and distributed combustion zone models with a short nozzle admittance approximation for the outflow boundary. In these cases, the response parameters were varied to determine stability limits defining resonant coupling onset.

Litchford, R. J.↗

NASA NTRS1994

Active Members to Establish On-Orbit Modal Data for Modal Correlation

The paper presents an approach to experimentally determine the modal sensitivity derivitives with respect to physical parameters of a structure in space or on the ground. The information experimentally validates the Jacobian used to update the analytical model to correlate with the experimental modal data. The approach is applicable to Adaptive Structures and utilizes active members. For verification of the approach, numerical simulations of a 4.5 meter diameter truss structure and experimental verification plan and progress are presented.

space structures modal analysis dynamic models ada↗

NASA NTRS2002

Application of the generalized vertical coordinate ocean model for better representing satellite data

It is found that two adaptive parametric functions can be introduced into the basic ocean equations for utilizing the optimal or hybrid features of commonly used z-level, terrain- following, isopycnal, and pressure coordinates in numerical ocean models. The two parametric functions are formulated by combining three techniques: the arbitrary vertical coordinate system of Kasahara (1 974), the Jacobian pressure gradient formulation of Song (1 998), and a newly developed metric factor that permits both compressible (non-Boussinesq) and incompressible (Boussinesq) approximations. Based on the new formulation, an adaptive modeling strategy is proposed and a staggered finite volume method is designed to ensure conservation of important physical properties and numerical accuracy. Implementation of the combined techniques to SCRUM (Song and Haidvogel1994) shows that the adaptive modeling strategy can be applied to any existing ocean model without incurring computational expense or altering the original numerical schemes. Such a generalized coordinate model is expected to benefit diverse ocean modelers for easily choosing optimal vertical structures and sharing modeling resources based on a common model platform. Several representing oceanographic problems with different scales and characteristics, such as coastal canyons, basin-scale circulation, and global ocean circulation, are used to demonstrate the model's capability for multiple applications. New results show that the model is capable of simultaneously resolving both Boussinesq and non-Boussinesq, and both small- and large-scale processes well. This talk will focus on its applications of multiple satellite sensing data in eddy-resolving simulations of Asian Marginal Sea and Kurosio. Attention will be given to how Topex/Poseidon SSH, TRMM SST; and GRACE ocean bottom pressure can be correctly represented in a non- Boussinesq model.

numerical ocean model generalized coordinate syste↗

NASA NTRS2006

Rate-Based Model Predictive Control of Turbofan Engine Clearance

An innovative model predictive control strategy is developed for control of nonlinear aircraft propulsion systems and sub-systems. At the heart of the controller is a rate-based linear parameter-varying model that propagates the state derivatives across the prediction horizon, extending prediction fidelity to transient regimes where conventional models begin to lose validity. The new control law is applied to a demanding active clearance control application, where the objectives are to tightly regulate blade tip clearances and also anticipate and avoid detrimental blade-shroud rub occurrences by optimally maintaining a predefined minimum clearance. Simulation results verify that the rate-based controller is capable of satisfying the objectives during realistic flight scenarios where both a conventional Jacobian-based model predictive control law and an unconstrained linear-quadratic optimal controller are incapable of doing so. The controller is evaluated using a variety of different actuators, illustrating the efficacy and versatility of the control approach. It is concluded that the new strategy has promise for this and other nonlinear aerospace applications that place high importance on the attainment of control objectives during transient regimes.

DeCastro, Jonathan A.↗

NASA NTRS2007

Simple and Efficient Numerical Evaluation of Near-Hypersingular Integrals

Recently, significant progress has been made in the handling of singular and nearly-singular potential integrals that commonly arise in the Boundary Element Method (BEM). To facilitate object-oriented programming and handling of higher order basis functions, cancellation techniques are favored over techniques involving singularity subtraction. However, gradients of the Newton-type potentials, which produce hypersingular kernels, are also frequently required in BEM formulations. As is the case with the potentials, treatment of the near-hypersingular integrals has proven more challenging than treating the limiting case in which the observation point approaches the surface. Historically, numerical evaluation of these near-hypersingularities has often involved a two-step procedure: a singularity subtraction to reduce the order of the singularity, followed by a boundary contour integral evaluation of the extracted part. Since this evaluation necessarily links basis function, Green s function, and the integration domain (element shape), the approach ill fits object-oriented programming concepts. Thus, there is a need for cancellation-type techniques for efficient numerical evaluation of the gradient of the potential. Progress in the development of efficient cancellation-type procedures for the gradient potentials was recently presented. To the extent possible, a change of variables is chosen such that the Jacobian of the transformation cancels the singularity. However, since the gradient kernel involves singularities of different orders, we also require that the transformation leaves remaining terms that are analytic. The terms "normal" and "tangential" are used herein with reference to the source element. Also, since computational formulations often involve the numerical evaluation of both potentials and their gradients, it is highly desirable that a single integration procedure efficiently handles both.

Fink, Patrick W.↗

NASA NTRS2006

A Fast Infrared Radiative Transfer Model for Overlapping Clouds

A fast infrared radiative transfer model (FIRTM2) appropriate for application to both single-layered and overlapping cloud situations is developed for simulating the outgoing infrared spectral radiance at the top of the atmosphere (TOA). In FIRTM2 a pre-computed library of cloud reflectance and transmittance values is employed to account for one or two cloud layers, whereas the background atmospheric optical thickness due to gaseous absorption can be computed from a clear-sky radiative transfer model. FIRTM2 is applicable to three atmospheric conditions: 1) clear-sky, 2) single-layered ice or water cloud, and 3) two simultaneous cloud layers in a column (e.g., ice cloud overlying water cloud). Moreover, FIRTM2 outputs the derivatives (i.e., Jacobians) of the TOA brightness temperature with respect to cloud optical thickness and effective particle size. Sensitivity analyses have been carried out to assess the performance of FIRTM2 for two spectral regions, namely the longwave (LW) band (587.3 - 1179.5/cm) and the short-to-medium wave (SMW) band (1180.1 - 2228.9/cm). The assessment is carried out in terms of brightness temperature differences (BTD) between FIRTM2 and the well-known discrete ordinates radiative transfer model (DISORT), henceforth referred to as BTD (F-D). The BTD (F-D) values for single-layered clouds are generally less than 0.8 K. For the case of two cloud layers (specifically ice cloud over water cloud), the BTD(F-D) values are also generally less than 0.8 K except for the SMW band for the case of a very high altitude (>15 km) cloud comprised of small ice particles. Note that for clear-sky atmospheres, FIRTM2 reduces to the clear-sky radiative transfer model that is incorporated into FIRTM2, and the errors in this case are essentially those of the clear-sky radiative transfer model.

Niu, Jianguo↗

NASA NTRS2007

Comparing Consider-Covariance Analysis with Sigma-Point Consider Filter and Linear-Theory Consider Filter Formulations

Recent literature in applied estimation theory reflects growing interest in the sigma-point (also called unscented ) formulation for optimal sequential state estimation, often describing performance comparisons with extended Kalman filters as applied to specific dynamical problems [c.f. 1, 2, 3]. Favorable attributes of sigma-point filters are described as including a lower expected error for nonlinear even non-differentiable dynamical systems, and a straightforward formulation not requiring derivation or implementation of any partial derivative Jacobian matrices. These attributes are particularly attractive, e.g. in terms of enabling simplified code architecture and streamlined testing, in the formulation of estimators for nonlinear spaceflight mechanics systems, such as filter software onboard deep-space robotic spacecraft. As presented in [4], the Sigma-Point Consider Filter (SPCF) algorithm extends the sigma-point filter algorithm to the problem of consider covariance analysis. Considering parameters in a dynamical system, while estimating its state, provides an upper bound on the estimated state covariance, which is viewed as a conservative approach to designing estimators for problems of general guidance, navigation and control. This is because, whether a parameter in the system model is observable or not, error in the knowledge of the value of a non-estimated parameter will increase the actual uncertainty of the estimated state of the system beyond the level formally indicated by the covariance of an estimator that neglects errors or uncertainty in that parameter. The equations for SPCF covariance evolution are obtained in a fashion similar to the derivation approach taken with standard (i.e. linearized or extended) consider parameterized Kalman filters (c.f. [5]). While in [4] the SPCF and linear-theory consider filter (LTCF) were applied to an illustrative linear dynamics/linear measurement problem, in the present work examines the SPCF as applied to nonlinear sequential consider covariance analysis, i.e. in the presence of nonlinear dynamics and nonlinear measurements. A simple SPCF for orbit determination, exemplifying an algorithm hosted in the guidance, navigation and control (GN&C) computer processor of a hypothetical robotic spacecraft, was implemented, and compared with an identically-parameterized (standard) extended, consider-parameterized Kalman filter. The onboard filtering scenario examined is a hypothetical spacecraft orbit about a small natural body with imperfectly-known mass. The formulations, relative complexities, and performances of the filters are compared and discussed.

Lisano, Michael E.↗

NASA NTRS2008

1D-VAR Retrieval Using Superchannels

Since modern ultra-spectral remote sensors have thousands of channels, it is difficult to include all of them in a 1D-var retrieval system. We will describe a physical inversion algorithm, which includes all available channels for the atmospheric temperature, moisture, cloud, and surface parameter retrievals. Both the forward model and the inversion algorithm compress the channel radiances into super channels. These super channels are obtained by projecting the radiance spectra onto a set of pre-calculated eigenvectors. The forward model provides both super channel properties and jacobian in EOF space directly. For ultra-spectral sensors such as Infrared Atmospheric Sounding Interferometer (IASI) and the NPOESS Airborne Sounder Testbed Interferometer (NAST), a compression ratio of more than 80 can be achieved, leading to a significant reduction in computations involved in an inversion process. Results will be shown applying the algorithm to real IASI and NAST data.

Liu, Xu↗

NASA NTRS2009

Implementation of Radiation, Ablation, and Free Energy Minimization Modules for Coupled Simulations of Hypersonic Flow

A description of models and boundary conditions required for coupling radiation and ablation physics to a hypersonic flow simulation is provided. Chemical equilibrium routines for varying elemental mass fraction are required in the flow solver to integrate with the equilibrium chemistry assumption employed in the ablation models. The capability also enables an equilibrium catalytic wall boundary condition in the non-ablating case. The paper focuses on numerical implementation issues using FIRE II, Mars return, and Apollo 4 applications to provide context for discussion. Variable relaxation factors applied to the Jacobian elements of partial equilibrium relations required for convergence are defined. Challenges of strong radiation coupling in a shock capturing algorithm are addressed. Results are presented to show how the current suite of models responds to a wide variety of conditions involving coupled radiation and ablation.

Gnoffo, Peter A.↗

NASA NTRS2011

Inflow/Outflow Boundary Conditions with Application to FUN3D

Several boundary conditions that allow subsonic and supersonic flow into and out of the computational domain are discussed. These boundary conditions are demonstrated in the FUN3D computational fluid dynamics (CFD) code which solves the three-dimensional Navier-Stokes equations on unstructured computational meshes. The boundary conditions are enforced through determination of the flux contribution at the boundary to the solution residual. The boundary conditions are implemented in an implicit form where the Jacobian contribution of the boundary condition is included and is exact. All of the flows are governed by the calorically perfect gas thermodynamic equations. Three problems are used to assess these boundary conditions. Solution residual convergence to machine zero precision occurred for all cases. The converged solution boundary state is compared with the requested boundary state for several levels of mesh densities. The boundary values converged to the requested boundary condition with approximately second-order accuracy for all of the cases.

Carlson, Jan-Renee↗