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NASA NTRS1996

Schwarz Methods: To Symmetrize or not to Symmetrize

A preconditioning theory for Schwarz methods is presented. The theory establishes sufficient conditions for multiplicative and additive Schwarz algorithms to yield self-adjoint positive definite preconditioners. It allows for the analysis and use of non-variational and non-convergent linear methods as preconditioners for conjugate gradient methods, and it is applied to domain decomposition and multigrid. This paper illustrates why symmetrizing may be a bad idea for linear methods. Numerical examples are presented for a test problem.

Holst, Michael↗

NASA NTRS1992

Aerodynamic shape optimization via sensitivity analysis on decomposed computational domains

Direct and iterative method considered to be most applicable to large systems of linear equations arising in discrete sensitivity analysis are assessed. Based on a single-domain grid, computations are performed using a banded matrix solver and an iterative solver, the generalized minimum residual (GMRES) method. The banded matrix solver is found to be generally the most economical method for those applications where the number of right-hand sides is large (i.e., a large number of design variables or a large number of adjoint vectors). For systems of equations that are too large to be solved by direct methods, an approach is proposed whereby the computational domain is divided into small subdomains, and each subdomain is solved separately.

Eleshaky, Mohamed E.↗

NASA NTRS1996

Aerodynamic Shape Optimization of Complex Aircraft Configurations via an Adjoint Formulation

This work describes the implementation of optimization techniques based on control theory for complex aircraft configurations. Here control theory is employed to derive the adjoint differential equations, the solution of which allows for a drastic reduction in computational costs over previous design methods (13, 12, 43, 38). In our earlier studies (19, 20, 22, 23, 39, 25, 40, 41, 42) it was shown that this method could be used to devise effective optimization procedures for airfoils, wings and wing-bodies subject to either analytic or arbitrary meshes. Design formulations for both potential flows and flows governed by the Euler equations have been demonstrated, showing that such methods can be devised for various governing equations (39, 25). In our most recent works (40, 42) the method was extended to treat wing-body configurations with a large number of mesh points, verifying that significant computational savings can be gained for practical design problems. In this paper the method is extended for the Euler equations to treat complete aircraft configurations via a new multiblock implementation. New elements include a multiblock-multigrid flow solver, a multiblock-multigrid adjoint solver, and a multiblock mesh perturbation scheme. Two design examples are presented in which the new method is used for the wing redesign of a transonic business jet.

Reuther, James↗

NASA NTRS1972

A shielding application of perturbation theory to determine changes in neutron and gamma doses due to changes in shield layers

The perturbation theory for fixed sources was applied to radiation shielding problems to determine changes in neutron and gamma ray doses due to changes in various shield layers. For a given source and detector position the perturbation method enables dose derivatives due to all layer changes to be determined from one forward and one inhomogeneous adjoint calculation. The direct approach requires two forward calculations for the derivative due to a single layer change. Hence, the perturbation method for obtaining dose derivatives permits an appreciable savings in computation for a multilayered shield. For an illustrative problem, a comparison was made of the fractional change in the dose per unit change in the thickness of each shield layer as calculated by perturbation theory and by successive direct calculations; excellent agreement was obtained between the two methods.

Fieno, D.↗

NASA NTRS1972

A shielding application of perturbation theory to determine changes in neutron and gamma doses due to changes in shield layers

Perturbation theory for fixed sources was applied to radiation shielding problems to determine changes in neutron and gamma ray doses due to changes in various shield layers. For a given source and detector position, the perturbation method enables dose derivatives due to all layer changes to be determined from one forward and one inhomogeneous adjoint calculation. The direct approach requires two forward calculations for the derivative due to a single layer change. Hence, the perturbation method for a obtaining dose derivatives permits an appreciable savings in computation for a multilayered shield. A comparison was made of the fractional change in the dose per unit change in shield layer thickness as calculated by perturbation theory and by successive direct calculations; excellent agreement was obtained between the two methods.

Fieno, D.↗

NASA NTRS1996

Variational Methods in Sensitivity Analysis and Optimization for Aerodynamic Applications

Variational methods (VM) sensitivity analysis, which is the continuous alternative to the discrete sensitivity analysis, is employed to derive the costate (adjoint) equations, the transversality conditions, and the functional sensitivity derivatives. In the derivation of the sensitivity equations, the variational methods use the generalized calculus of variations, in which the variable boundary is considered as the design function. The converged solution of the state equations together with the converged solution of the costate equations are integrated along the domain boundary to uniquely determine the functional sensitivity derivatives with respect to the design function. The determination of the sensitivity derivatives of the performance index or functional entails the coupled solutions of the state and costate equations. As the stable and converged numerical solution of the costate equations with their boundary conditions are a priori unknown, numerical stability analysis is performed on both the state and costate equations. Thereafter, based on the amplification factors obtained by solving the generalized eigenvalue equations, the stability behavior of the costate equations is discussed and compared with the state (Euler) equations. The stability analysis of the costate equations suggests that the converged and stable solution of the costate equation is possible only if the computational domain of the costate equations is transformed to take into account the reverse flow nature of the costate equations. The application of the variational methods to aerodynamic shape optimization problems is demonstrated for internal flow problems at supersonic Mach number range. The study shows, that while maintaining the accuracy of the functional sensitivity derivatives within the reasonable range for engineering prediction purposes, the variational methods show a substantial gain in computational efficiency, i.e., computer time and memory, when compared with the finite difference sensitivity analysis.

Ibrahim, A. H.↗

NASA NTRS1990

Statistical analysis of static shape control in space structures

The article addresses the problem of efficient analysis of the statistics of initial and corrected shape distortions in space structures. Two approaches for improving efficiency are considered. One is an adjoint technique for calculating distortion shapes: the second is a modal expansion of distortion shapes in terms of pseudo-vibration modes. The two techniques are applied to the problem of optimizing actuator locations on a 55 m radiometer antenna. The adjoint analysis technique is used with a discrete-variable optimization method. The modal approximation technique is coupled with a standard conjugate-gradient continuous optimization method. The agreement between the two sets of results is good, validating both the approximate analysis and optimality of the results.

Burdisso, Ricardo A.↗

NASA NTRS2016

Development of a High-Order Space-Time Matrix-Free Adjoint Solver

The growth in computational power and algorithm development in the past few decades has granted the science and engineering community the ability to simulate flows over complex geometries, thus making Computational Fluid Dynamics (CFD) tools indispensable in analysis and design. Currently, one of the pacing items limiting the utility of CFD for general problems is the prediction of unsteady turbulent ows.1{3 Reynolds-averaged Navier-Stokes (RANS) methods, which predict a time-invariant mean flowfield, struggle to provide consistent predictions when encountering even mild separation, such as the side-of-body separation at a wing-body junction. NASA's Transformative Tools and Technologies project is developing both numerical methods and physical modeling approaches to improve the prediction of separated flows. A major focus of this e ort is efficient methods for resolving the unsteady fluctuations occurring in these flows to provide valuable engineering data of the time-accurate flow field for buffet analysis, vortex shedding, etc. This approach encompasses unsteady RANS (URANS), large-eddy simulations (LES), and hybrid LES-RANS approaches such as Detached Eddy Simulations (DES). These unsteady approaches are inherently more expensive than traditional engineering RANS approaches, hence every e ort to mitigate this cost must be leveraged. Arguably, the most cost-effective approach to improve the efficiency of unsteady methods is the optimal placement of the spatial and temporal degrees of freedom (DOF) using solution-adaptive methods.

Adjoint↗

NASA NTRS2018

Design of a Modular Monolithic Implicit Solver for Multi-Physics Applications

The design of a modular multi-physics high-order space-time finite-element framework is presented together with its extension to allow monolithic coupling of different physics. One of the main objectives of the framework is to perform efficient high- fidelity simulations of capsule/parachute systems. This problem requires simulating multiple physics including, but not limited to, the compressible Navier-Stokes equations, the dynamics of a moving body with mesh deformations and adaptation, the linear shell equations, non-re effective boundary conditions and wall modeling. The solver is based on high-order space-time - finite element methods. Continuous, discontinuous and C1-discontinuous Galerkin methods are implemented, allowing one to discretize various physical models. Tangent and adjoint sensitivity analysis are also targeted in order to conduct gradient-based optimization, error estimation, mesh adaptation, and flow control, adding another layer of complexity to the framework. The decisions made to tackle these challenges are presented. The discussion focuses first on the "single-physics" solver and later on its extension to the monolithic coupling of different physics. The implementation of different physics modules, relevant to the capsule/parachute system, are also presented. Finally, examples of coupled computations are presented, paving the way to the simulation of the full capsule/parachute system.

Carton De Wiart, Corentin↗

NASA NTRS1999

Demonstration of Automatically-Generated Adjoint Code for Use in Aerodynamic Shape Optimization

Gradient-based optimization requires accurate derivatives of the objective function and constraints. These gradients may have previously been obtained by manual differentiation of analysis codes, symbolic manipulators, finite-difference approximations, or existing automatic differentiation (AD) tools such as ADIFOR (Automatic Differentiation in FORTRAN). Each of these methods has certain deficiencies, particularly when applied to complex, coupled analyses with many design variables. Recently, a new AD tool called ADJIFOR (Automatic Adjoint Generation in FORTRAN), based upon ADIFOR, was developed and demonstrated. Whereas ADIFOR implements forward-mode (direct) differentiation throughout an analysis program to obtain exact derivatives via the chain rule of calculus, ADJIFOR implements the reverse-mode counterpart of the chain rule to obtain exact adjoint form derivatives from FORTRAN code. Automatically-generated adjoint versions of the widely-used CFL3D computational fluid dynamics (CFD) code and an algebraic wing grid generation code were obtained with just a few hours processing time using the ADJIFOR tool. The codes were verified for accuracy and were shown to compute the exact gradient of the wing lift-to-drag ratio, with respect to any number of shape parameters, in about the time required for 7 to 20 function evaluations. The codes have now been executed on various computers with typical memory and disk space for problems with up to 129 x 65 x 33 grid points, and for hundreds to thousands of independent variables. These adjoint codes are now used in a gradient-based aerodynamic shape optimization problem for a swept, tapered wing. For each design iteration, the optimization package constructs an approximate, linear optimization problem, based upon the current objective function, constraints, and gradient values. The optimizer subroutines are called within a design loop employing the approximate linear problem until an optimum shape is found, the design loop limit is reached, or no further design improvement is possible due to active design variable bounds and/or constraints. The resulting shape parameters are then used by the grid generation code to define a new wing surface and computational grid. The lift-to-drag ratio and its gradient are computed for the new design by the automatically-generated adjoint codes. Several optimization iterations may be required to find an optimum wing shape. Results from two sample cases will be discussed. The reader should note that this work primarily represents a demonstration of use of automatically- generated adjoint code within an aerodynamic shape optimization. As such, little significance is placed upon the actual optimization results, relative to the method for obtaining the results.

Green, Lawrence↗

NASA NTRS2005

Application of Parallel Adjoint-Based Error Estimation and Anisotropic Grid Adaptation for Three-Dimensional Aerospace Configurations

This paper demonstrates the extension of error estimation and adaptation methods to parallel computations enabling larger, more realistic aerospace applications and the quantification of discretization errors for complex 3-D solutions. Results were shown for an inviscid sonic-boom prediction about a double-cone configuration and a wing/body segmented leading edge (SLE) configuration where the output function of the adjoint was pressure integrated over a part of the cylinder in the near field. After multiple cycles of error estimation and surface/field adaptation, a significant improvement in the inviscid solution for the sonic boom signature of the double cone was observed. Although the double-cone adaptation was initiated from a very coarse mesh, the near-field pressure signature from the final adapted mesh compared very well with the wind-tunnel data which illustrates that the adjoint-based error estimation and adaptation process requires no a priori refinement of the mesh. Similarly, the near-field pressure signature for the SLE wing/body sonic boom configuration showed a significant improvement from the initial coarse mesh to the final adapted mesh in comparison with the wind tunnel results. Error estimation and field adaptation results were also presented for the viscous transonic drag prediction of the DLR-F6 wing/body configuration, and results were compared to a series of globally refined meshes. Two of these globally refined meshes were used as a starting point for the error estimation and field-adaptation process where the output function for the adjoint was the total drag. The field-adapted results showed an improvement in the prediction of the drag in comparison with the finest globally refined mesh and a reduction in the estimate of the remaining drag error. The adjoint-based adaptation parameter showed a need for increased resolution in the surface of the wing/body as well as a need for wake resolution downstream of the fuselage and wing trailing edge in order to achieve the requested drag tolerance. Although further adaptation was required to meet the requested tolerance, no further cycles were computed in order to avoid large discrepancies between the surface mesh spacing and the refined field spacing.

Lee-Rausch, E. M.↗

NASA NTRS1999

Martian Aerocapture Terminal Point Guidance: A Reference Path Optimization Study

An effective method of terminal point guidance is to employ influence coefficients, which are solved from a set of differential equations adjoint to the linearized perturbations of the equations of motion about a reference trajectory. Hence, to optimize this type of guidance, one must first optimize the reference trajectory that the guidance is based upon. This study concentrates on various methods to optimize a reference trajectory for a Martian aerocapture maneuver, including a parametric analysis and first order gradient method. Resulting reference trajectories were tested in separate 2000 6-DOF Monte Carlo runs, using the Atmospheric Guidance Algorithm Testbed for the Mars Surveyor Program 2001 (MSP '01) Orbiter. These results were compared to an August 1998 study using the same terminal point control guidance algorithm and simulation testbed. Satisfactory improvements over the 1998 study are amply demonstrated.

Ro, Theodore U.↗

NASA NTRS1986

Preconditioned conjugate residual methods for the solution of spectral equations

Conjugate residual methods for the solution of spectral equations are described. An inexact finite-difference operator is introduced as a preconditioner in the iterative procedures. Application of these techniques is limited to problems for which the symmetric part of the coefficient matrix is positive definite. Although the spectral equation is a very ill-conditioned and full matrix problem, the computational effort of the present iterative methods for solving such a system is comparable to that for the sparse matrix equations obtained from the application of either finite-difference or finite-element methods to the same problems. Numerical experiments are shown for a self-adjoint elliptic partial differential equation with Dirichlet boundary conditions, and comparison with other solution procedures for spectral equations is presented.

Wong, Y. S.↗

NASA NTRS1991

Minimum-fuel, power-limited transfers between coplanar elliptical orbits

Two techniques of computing coplanar, minimum-fuel, power-limited transfers are presented, based on approximate solutions obtained by the averaging method. For the first technique, the average solution provides estimates of the initial adjoint variables, and the second provides approximations of the optimal controls in feedback form. The accuracy of the techniques for computing coplanar, minimum-fuel, power-limited transfers is evaluated for a number of initial and final orbits.

Mease, Kenneth D.↗

NASA NTRS1972

Shielding application of perturbation theory to determine changes in neutron and gamma doses due to changes in shield layers

Perturbation theory formulas were derived and applied to determine changes in neutron and gamma-ray doses due to changes in various radiation shield layers for fixed sources. For a given source and detector position, the perturbation method enables dose derivatives with respect to density, or equivalently thickness, for every layer to be determined from one forward and one inhomogeneous adjoint calculation. A direct determination without the perturbation approach would require two forward calculations to evaluate the dose derivative due to a change in a single layer. Hence, the perturbation method for obtaining dose derivatives requires fewer computations for design studies of multilayer shields. For an illustrative problem, a comparison was made of the fractional change in the dose per unit change in the thickness of each shield layer in a two-layer spherical configuration as calculated by perturbation theory and by successive direct calculations; excellent agreement was obtained between the two methods.

Fieno, D.↗

NASA NTRS2006

Adjoint-Based Algorithms for Adaptation and Design Optimizations on Unstructured Grids

Schemes based on discrete adjoint algorithms present several exciting opportunities for significantly advancing the current state of the art in computational fluid dynamics. Such methods provide an extremely efficient means for obtaining discretely consistent sensitivity information for hundreds of design variables, opening the door to rigorous, automated design optimization of complex aerospace configuration using the Navier-Stokes equation. Moreover, the discrete adjoint formulation provides a mathematically rigorous foundation for mesh adaptation and systematic reduction of spatial discretization error. Error estimates are also an inherent by-product of an adjoint-based approach, valuable information that is virtually non-existent in today's large-scale CFD simulations. An overview of the adjoint-based algorithm work at NASA Langley Research Center is presented, with examples demonstrating the potential impact on complex computational problems related to design optimization as well as mesh adaptation.

Nielsen, Eric J.↗

NASA NTRS2008

Output-Adaptive Tetrahedral Cut-Cell Validation for Sonic Boom Prediction

A cut-cell approach to Computational Fluid Dynamics (CFD) that utilizes the median dual of a tetrahedral background grid is described. The discrete adjoint is also calculated, which permits adaptation based on improving the calculation of a specified output (off-body pressure signature) in supersonic inviscid flow. These predicted signatures are compared to wind tunnel measurements on and off the configuration centerline 10 body lengths below the model to validate the method for sonic boom prediction. Accurate mid-field sonic boom pressure signatures are calculated with the Euler equations without the use of hybrid grid or signature propagation methods. Highly-refined, shock-aligned anisotropic grids were produced by this method from coarse isotropic grids created without prior knowledge of shock locations. A heuristic reconstruction limiter provided stable flow and adjoint solution schemes while producing similar signatures to Barth-Jespersen and Venkatakrishnan limiters. The use of cut-cells with an output-based adaptive scheme completely automated this accurate prediction capability after a triangular mesh is generated for the cut surface. This automation drastically reduces the manual intervention required by existing methods.

Park, Michael A.↗

NASA NTRS1982

Iterative spectral methods and spectral solutions to compressible flows

A spectral multigrid scheme is described which can solve pseudospectral discretizations of self-adjoint elliptic problems in O(N log N) operations. An iterative technique for efficiently implementing semi-implicit time-stepping for pseudospectral discretizations of Navier-Stokes equations is discussed. This approach can handle variable coefficient terms in an effective manner. Pseudospectral solutions of compressible flow problems are presented. These include one dimensional problems and two dimensional Euler solutions. Results are given both for shock-capturing approaches and for shock-fitting ones.

Hussaini, M. Y.↗