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All sourcesNASA NTRSDOE OSTI
NASA NTRS2013

Trajectory Optimization Using Adjoint Method and Chebyshev Polynomial Approximation for Minimizing Fuel Consumption During Climb

This paper describes two methods of trajectory optimization to obtain an optimal trajectory of minimum-fuel- to-climb for an aircraft. The first method is based on the adjoint method, and the second method is based on a direct trajectory optimization method using a Chebyshev polynomial approximation and cubic spine approximation. The approximate optimal trajectory will be compared with the adjoint-based optimal trajectory which is considered as the true optimal solution of the trajectory optimization problem. The adjoint-based optimization problem leads to a singular optimal control solution which results in a bang-singular-bang optimal control.

Chebyshev Polynomial↗
NASA NTRS2017

Robust Acoustic Objective Functions and Sensitivities in Adjoint-Based Design Optimizations

The multidisciplinary design of aircraft typically includes considerations of performance, weight, fuel burn, and noise, among other factors. An objective function is applied to each of these considerations in order to weight the influence of trade-offs between different designs. Higher-order optimization exercises have utilized an adjoint approach to reach an optimal set of objective functions, such as maximum lift and reduced drag. Taking advantage of the adjoint approach significantly reduces the computational time required to find an optimal configuration. Including acoustics in the set of objective functions during an adjoint-based design optimization requires the sensitivity of the acoustic objective function. This document will present an approach for defining the sensitivity of several acoustic metrics and operations that can fill the role of the acoustic objective function. This includes time-integrated metrics such as effective perceived noise level (EPNL) and frequency-integrated metrics such as overall sound pressure level (OASPL). A demonstration case, validation, and details on the implementation in the second generation Aircraft NOise Prediction Program (ANOPP2) are also shown.

Lopes, Leonard V.↗
NASA NTRS2009

Discrete Adjoint-Based Design Optimization of Unsteady Turbulent Flows on Dynamic Unstructured Grids

An adjoint-based methodology for design optimization of unsteady turbulent flows on dynamic unstructured grids is described. The implementation relies on an existing unsteady three-dimensional unstructured grid solver capable of dynamic mesh simulations and discrete adjoint capabilities previously developed for steady flows. The discrete equations for the primal and adjoint systems are presented for the backward-difference family of time-integration schemes on both static and dynamic grids. The consistency of sensitivity derivatives is established via comparisons with complex-variable computations. The current work is believed to be the first verified implementation of an adjoint-based optimization methodology for the true time-dependent formulation of the Navier-Stokes equations in a practical computational code. Large-scale shape optimizations are demonstrated for turbulent flows over a tiltrotor geometry and a simulated aeroelastic motion of a fighter jet.

Nielsen, Eric J.↗
NASA NTRS1997

Constrained Multipoint Aerodynamic Shape Optimization Using an Adjoint Formulation and Parallel Computers

An aerodynamic shape optimization method that treats the design of complex aircraft configurations subject to high fidelity computational fluid dynamics (CFD), geometric constraints and multiple design points is described. The design process will be greatly accelerated through the use of both control theory and distributed memory computer architectures. Control theory is employed to derive the adjoint differential equations whose solution allows for the evaluation of design gradient information at a fraction of the computational cost required by previous design methods. The resulting problem is implemented on parallel distributed memory architectures using a domain decomposition approach, an optimized communication schedule, and the MPI (Message Passing Interface) standard for portability and efficiency. The final result achieves very rapid aerodynamic design based on a higher order CFD method. In order to facilitate the integration of these high fidelity CFD approaches into future multi-disciplinary optimization (NW) applications, new methods must be developed which are capable of simultaneously addressing complex geometries, multiple objective functions, and geometric design constraints. In our earlier studies, we coupled the adjoint based design formulations with unconstrained optimization algorithms and showed that the approach was effective for the aerodynamic design of airfoils, wings, wing-bodies, and complex aircraft configurations. In many of the results presented in these earlier works, geometric constraints were satisfied either by a projection into feasible space or by posing the design space parameterization such that it automatically satisfied constraints. Furthermore, with the exception of reference 9 where the second author initially explored the use of multipoint design in conjunction with adjoint formulations, our earlier works have focused on single point design efforts. Here we demonstrate that the same methodology may be extended to treat complete configuration designs subject to multiple design points and geometric constraints. Examples are presented for both transonic and supersonic configurations ranging from wing alone designs to complex configuration designs involving wing, fuselage, nacelles and pylons.

Reuther, James↗
NASA NTRS1995

Supersonic wing and wing-body shape optimization using an adjoint formulation

This paper describes the implementation of optimization techniques based on control theory for wing and wing-body design of supersonic configurations. The work represents an extension of our earlier research in which control theory is used to devise a design procedure that significantly reduces the computational cost by employing an adjoint equation. In previous studies it was shown that control theory could be used to~eviseransonic design methods for airfoils and wings in which the shape and the surrounding body-fitted mesh are both generated analytically, and the control is the mapping function. The method has also been implemented for both transonic potential flows and transonic flows governed by the Euler equations using an alternative formulation which employs numerically generated grids, so that it can treat more general configurations. Here results are presented for three-dimensional design cases subject to supersonic flows governed by the Euler equation.

Reuther, James↗
NASA NTRS2001

Isolating Curvature Effects in Computing Wall-Bounded Turbulent Flows

The flow over the zero-pressure-gradient So-Mellor convex curved wall is simulated using the Navier-Stokes equations. An inviscid effective outer wall shape, undocumented in the experiment, is obtained by using an adjoint optimization method with the desired pressure distribution on the inner wall as the cost function. Using this wall shape with a Navier-Stokes method, the abilities of various turbulence models to simulate the effects of curvature without the complicating factor of streamwise pressure gradient can be evaluated. The one-equation Spalart-Allmaras turbulence model overpredicts eddy viscosity, and its boundary layer profiles are too full. A curvature-corrected version of this model improves results, which are sensitive to the choice of a particular constant. An explicit algebraic stress model does a reasonable job predicting this flow field. However, results can be slightly improved by modifying the assumption on anisotropy equilibrium in the model's derivation. The resulting curvature-corrected explicit algebraic stress model possesses no heuristic functions or additional constants. It lowers slightly the computed skin friction coefficient and the turbulent stress levels for this case (in better agreement with experiment), but the effect on computed velocity profiles is very small.

Rumsey, Christopher L.↗
NASA NTRS2007

Anisotropic Solution Adaptive Unstructured Grid Generation Using AFLR

An existing volume grid generation procedure, AFLR3, was successfully modified to generate anisotropic tetrahedral elements using a directional metric transformation defined at source nodes. The procedure can be coupled with a solver and an error estimator as part of an overall anisotropic solution adaptation methodology. It is suitable for use with an error estimator based on an adjoint, optimization, sensitivity derivative, or related approach. This offers many advantages, including more efficient point placement along with robust and efficient error estimation. It also serves as a framework for true grid optimization wherein error estimation and computational resources can be used as cost functions to determine the optimal point distribution. Within AFLR3 the metric transformation is implemented using a set of transformation vectors and associated aspect ratios. The modified overall procedure is presented along with details of the anisotropic transformation implementation. Multiple two-and three-dimensional examples are also presented that demonstrate the capability of the modified AFLR procedure to generate anisotropic elements using a set of source nodes with anisotropic transformation metrics. The example cases presented use moderate levels of anisotropy and result in usable element quality. Future testing with various flow solvers and methods for obtaining transformation metric information is needed to determine practical limits and evaluate the efficacy of the overall approach.

Marcum, David L.↗
NASA NTRS2017

Aerodynamic Design of the Hybrid Wing Body Propulsion-Airframe Integration

A hybrid wingbody (HWB) concept is being considered by NASA as a potential subsonic transport aircraft that meets aerodynamic, fuel, emission, and noise goals in the time frame of the 2030s. While the concept promises advantages over conventional wing-and-tube aircraft, it poses unknowns and risks, thus requiring in-depth and broad assessments. Specifically, the configuration entails a tight integration of the airframe and propulsion geometries; the aerodynamic impact has to be carefully evaluated. With the propulsion nacelle installed on the (upper) body, the lift and drag are affected by the mutual interference effects between the airframe and nacelle. The static margin for longitudinal stability is also adversely changed. We develop a design approach in which the integrated geometry of airframe (HWB) and propulsion is accounted for simultaneously in a simple algebraic manner, via parameterization of the planform and airfoils at control sections of the wingbody. In this paper, we present the design of a 300-passenger transport that employs distributed electric fans for propulsion. The trim for stability is achieved through the use of the wingtip twist angle. The geometric shape variables are determined through the adjoint optimization method by minimizing the drag while subject to lift, pitch moment, and geometry constraints. The design results clearly show the influence on the aerodynamic characteristics of the installed nacelle and trimming for stability. A drag minimization with the trim constraint yields a reduction of 10 counts in the drag coefficient.

hybrid wing↗
NASA NTRS2018

Comparison of MERRA-2 and ECCO-V4 Ocean Surface Heat Fluxes: Consequences of Different Forcing Feedbacks on Ocean Circulation and Implications for Climate Data Assimilation

Forcing ocean models with reanalysis data is a common practice in ocean modeling. As part of this practice, prescribed atmospheric state variables and interactive ocean SST (Sea Surface Temperature) are used to calculate fluxes between the ocean and the atmosphere. When forcing an ocean model with reanalysis fields, errors in the reanalysis data, errors in the ocean model and errors in the forcing formulation will generate a different solution compared to other ocean reanalysis solutions (which also have their own errors). As a first step towards a consistent coupled ocean-atmosphere reanalysis, we compare surface heat fluxes from a state-of-the-art atmospheric reanalysis, the Modern-Era Retrospective analysis for Research and Applications, Version 2 (MERRA-2), to heat fluxes from a state-of-the-art oceanic reanalysis, the Estimating the Circulation and Climate of the Ocean Version 4, Release 2 (ECCO-v4). Then, we investigate the errors associated with the MITgcm (Massachusetts Institute of Technology general circulation model) ocean model in its ECCO-v4 ocean reanalysis configuration (1992-2011) when it is forced with MERRA- 2 atmospheric reanalysis fields instead of with the ECCO-v4 adjoint optimized ERA-interim state variables. This is done by forcing ECCO-v4 ocean with and without feedbacks from MERRA-2 related to turbulent fluxes of heat and moisture and the outgoing long wave radiation. In addition, we introduce an intermediate forcing method that includes only the feedback from the interactive outgoing long wave radiation. The resulting ocean circulation is compared with ECCO-v4 reanalysis and in-situ observations. We show that, without feedbacks, imbalances in the energy and the hydrological cycles of MERRA-2 (which are directly related to the fact it was created without interactive ocean) result in considerable SST drifts and a large reduction in sea level. The bulk formulae and interactive outgoing long wave radiation, although providing air-sea feedbacks and reducing model-data misfit, strongly relax the ocean to observed SST and may result in unwanted features such as large change in the water budget. These features have implications in a desired forcing recipe to be used. The results strongly and unambiguously argue for next generation data assimilation climate studies to involve fully coupled systems.

MERRA-2↗
NASA NTRSDate not supplied

Investigation of a Smooth Local Correlation-based Transition Model in a Discrete-Adjoint Aerodynamic Shape Optimization Algorithm

A smooth local correlation-based transition model is fully coupled to a RANS-based Newton-Krylov flow solver and discrete-adjoint gradient-based optimization algorithm. The free-transition optimization framework is evaluated using lift-constrained drag minimizations of airfoils at design conditions ranging from light to single-aisle aircraft and an infinite swept wing at design conditions representative of a transonic strut-braced wing aircraft. The impact of the streamwise grid resolution on the ability of the optimization algorithm to delay boundary-layer transition is investigated, with the results demonstrating that streamwise grid resolution requirements increase as the transition length decreases with increasing Reynolds number. The optimization problem at the light aircraft design conditions is demonstrated to be multi-modal, with the optimization algorithm producing two distinct designs: one with a thin, reflexed trailing edge and steep pressure recovery regions, the other with increased aft loading, with the latter design outperforming the former. A drag minimization of an airfoil at transonic design conditions demonstrates that the optimization algorithm successfully trades a decrease in viscous drag by delaying boundary-layer transition with an increase in wave drag, while the drag minimization of an infinite swept wing demonstrates the capability of the optimizational gorithm to delay both Tollmien-Schlichting and stationary crossflow instabilities.

AATT↗
NASA NTRS2009

Local-in-Time Adjoint-Based Method for Optimal Control/Design Optimization of Unsteady Compressible Flows

.We study local-in-time adjoint-based methods for minimization of ow matching functionals subject to the 2-D unsteady compressible Euler equations. The key idea of the local-in-time method is to construct a very accurate approximation of the global-in-time adjoint equations and the corresponding sensitivity derivative by using only local information available on each time subinterval. In contrast to conventional time-dependent adjoint-based optimization methods which require backward-in-time integration of the adjoint equations over the entire time interval, the local-in-time method solves local adjoint equations sequentially over each time subinterval. Since each subinterval contains relatively few time steps, the storage cost of the local-in-time method is much lower than that of the global adjoint formulation, thus making the time-dependent optimization feasible for practical applications. The paper presents a detailed comparison of the local- and global-in-time adjoint-based methods for minimization of a tracking functional governed by the Euler equations describing the ow around a circular bump. Our numerical results show that the local-in-time method converges to the same optimal solution obtained with the global counterpart, while drastically reducing the memory cost as compared to the global-in-time adjoint formulation.

Yamaleev, N. K.↗
NASA NTRSDate not supplied

Toward Adjoint-Based Aeroacoustic Optimization for Propeller and Rotorcraft Applications

The goal of the present project is to build a multidisciplinary, rapid, robust, and accurate computational tool to optimize wing-mounted propeller designs. The full Farassat’s formulation F1Afor aeroacoustic analysis is implemented in the open-source software SU2.This extension enables the prediction of far-field noise generated by moving sources. The formulation is verified, for a stationary and rotating sphere in a wind tunnel and for a tiltrotor in forward flight, by comparing the acoustic predictions ofSU2 with the predictions computed by NASA’s aeroacoustics code ANOPP2. The algorithmic differentiation capability of SU2 provides discretely consistent, adjoint-based sensitivity analysis for this formulation. The adjoint-based sensitivities are verified through comparison with complex-step sensitivities

R Omur Icke↗
NASA NTRS2004

Adjoint Algorithm for CAD-Based Shape Optimization Using a Cartesian Method

Adjoint solutions of the governing flow equations are becoming increasingly important for the development of efficient analysis and optimization algorithms. A well-known use of the adjoint method is gradient-based shape optimization. Given an objective function that defines some measure of performance, such as the lift and drag functionals, its gradient is computed at a cost that is essentially independent of the number of design variables (geometric parameters that control the shape). More recently, emerging adjoint applications focus on the analysis problem, where the adjoint solution is used to drive mesh adaptation, as well as to provide estimates of functional error bounds and corrections. The attractive feature of this approach is that the mesh-adaptation procedure targets a specific functional, thereby localizing the mesh refinement and reducing computational cost. Our focus is on the development of adjoint-based optimization techniques for a Cartesian method with embedded boundaries.12 In contrast t o implementations on structured and unstructured grids, Cartesian methods decouple the surface discretization from the volume mesh. This feature makes Cartesian methods well suited for the automated analysis of complex geometry problems, and consequently a promising approach to aerodynamic optimization. Melvin et developed an adjoint formulation for the TRANAIR code, which is based on the full-potential equation with viscous corrections. More recently, Dadone and Grossman presented an adjoint formulation for the Euler equations. In both approaches, a boundary condition is introduced to approximate the effects of the evolving surface shape that results in accurate gradient computation. Central to automated shape optimization algorithms is the issue of geometry modeling and control. The need to optimize complex, "real-life" geometry provides a strong incentive for the use of parametric-CAD systems within the optimization procedure. In previous work, we presented an effective optimization framework that incorporates a direct-CAD interface. In this work, we enhance the capabilities of this framework with efficient gradient computations using the discrete adjoint method. We present details of the adjoint numerical implementation, which reuses the domain decomposition, multigrid, and time-marching schemes of the flow solver. Furthermore, we explain and demonstrate the use of CAD in conjunction with the Cartesian adjoint approach. The final paper will contain a number of complex geometry, industrially relevant examples with many design variables to demonstrate the effectiveness of the adjoint method on Cartesian meshes.

Nemec, Marian↗
NASA NTRS2000

Adjoint Techniques for Topology Optimization of Structures Under Damage Conditions

The objective of this cooperative agreement was to seek computationally efficient ways to optimize aerospace structures subject to damage tolerance criteria. Optimization was to involve sizing as well as topology optimization. The work was done in collaboration with Steve Scotti, Chauncey Wu and Joanne Walsh at the NASA Langley Research Center. Computation of constraint sensitivity is normally the most time-consuming step of an optimization procedure. The cooperative work first focused on this issue and implemented the adjoint method of sensitivity computation (Haftka and Gurdal, 1992) in an optimization code (runstream) written in Engineering Analysis Language (EAL). The method was implemented both for bar and plate elements including buckling sensitivity for the latter. Lumping of constraints was investigated as a means to reduce the computational cost. Adjoint sensitivity computation was developed and implemented for lumped stress and buckling constraints. Cost of the direct method and the adjoint method was compared for various structures with and without lumping. The results were reported in two papers (Akgun et al., 1998a and 1999). It is desirable to optimize topology of an aerospace structure subject to a large number of damage scenarios so that a damage tolerant structure is obtained. Including damage scenarios in the design procedure is critical in order to avoid large mass penalties at later stages (Haftka et al., 1983). A common method for topology optimization is that of compliance minimization (Bendsoe, 1995) which has not been used for damage tolerant design. In the present work, topology optimization is treated as a conventional problem aiming to minimize the weight subject to stress constraints. Multiple damage configurations (scenarios) are considered. Each configuration has its own structural stiffness matrix and, normally, requires factoring of the matrix and solution of the system of equations. Damage that is expected to be tolerated is local and represents a small change in the stiffness matrix compared to the baseline (undamaged) structure. The exact solution to a slightly modified set of equations can be obtained from the baseline solution economically without actually solving the modified system.. Shennan-Morrison-Woodbury (SMW) formulas are matrix update formulas that allow this (Akgun et al., 1998b). SMW formulas were therefore used here to compute adjoint displacements for sensitivity computation and structural displacements in damaged configurations.

Akgun, Mehmet A.↗
NASA NTRS2006

Reentry-Vehicle Shape Optimization Using a Cartesian Adjoint Method and CAD Geometry

A DJOINT solutions of the governing flow equations are becoming increasingly important for the development of efficient analysis and optimization algorithms. A well-known use of the adjoint method is gradient-based shape. Given an objective function that defines some measure of performance, such as the lift and drag functionals, its gradient is computed at a cost that is essentially independent of the number of design variables (e.g., geometric parameters that control the shape). Classic aerodynamic applications of gradient-based optimization include the design of cruise configurations for transonic and supersonic flow, as well as the design of high-lift systems. are perhaps the most promising approach for addressing the issues of flow solution automation for aerodynamic design problems. In these methods, the discretization of the wetted surface is decoupled from that of the volume mesh. This not only enables fast and robust mesh generation for geometry of arbitrary complexity, but also facilitates access to geometry modeling and manipulation using parametric computer-aided design (CAD). In previous work on Cartesian adjoint solvers, Melvin et al. developed an adjoint formulation for the TRANAIR code, which is based on the full-potential equation with viscous corrections. More recently, Dadone and Grossman presented an adjoint formulation for the two-dimensional Euler equations using a ghost-cell method to enforce the wall boundary conditions. In Refs. 18 and 19, we presented an accurate and efficient algorithm for the solution of the adjoint Euler equations discretized on Cartesian meshes with embedded, cut-cell boundaries. Novel aspects of the algorithm were the computation of surface shape sensitivities for triangulations based on parametric-CAD models and the linearization of the coupling between the surface triangulation and the cut-cells. The accuracy of the gradient computation was verified using several three-dimensional test cases, which included design variables such as the free stream parameters and the planform shape of an isolated wing. The objective of the present work is to extend our adjoint formulation to problems involving general shape changes. Factors under consideration include the computation of mesh sensitivities that provide a reliable approximation of the objective function gradient, as well as the computation of surface shape sensitivities based on a direct-CAD interface. We present detailed gradient verification studies and then focus on a shape optimization problem for an Apollo-like reentry vehicle. The goal of the optimization is to enhance the lift-to-drag ratio of the capsule by modifying the shape of its heat-shield in conjunction with a center-of-gravity (c.g.) offset. This multipoint and multi-objective optimization problem is used to demonstrate the overall effectiveness of the Cartesian adjoint method for addressing the issues of complex aerodynamic design.

Nemec, Marian↗
NASA NTRS1997

Admitting the Inadmissible: Adjoint Formulation for Incomplete Cost Functionals in Aerodynamic Optimization

We derive the adjoint equations for problems in aerodynamic optimization which are improperly considered as "inadmissible." For example, a cost functional which depends on the density, rather than on the pressure, is considered "inadmissible" for an optimization problem governed by the Euler equations. We show that for such problems additional terms should be included in the Lagrangian functional when deriving the adjoint equations. These terms are obtained from the restriction of the interior PDE to the control surface. Demonstrations of the explicit derivation of the adjoint equations for "inadmissible" cost functionals are given for the potential, Euler, and Navier-Stokes equations.

Arian, Eyal↗
NASA NTRS2012

Unsteady Adjoint Approach for Design Optimization of Flapping Airfoils

This paper describes the work for optimizing the propulsive efficiency of flapping airfoils, i.e., improving the thrust under constraining aerodynamic work during the flapping flights by changing their shape and trajectory of motion with the unsteady discrete adjoint approach. For unsteady problems, it is essential to properly resolving time scales of motion under consideration and it must be compatible with the objective sought after. We include both the instantaneous and time-averaged (periodic) formulations in this study. For the design optimization with shape parameters or motion parameters, the time-averaged objective function is found to be more useful, while the instantaneous one is more suitable for flow control. The instantaneous objective function is operationally straightforward. On the other hand, the time-averaged objective function requires additional steps in the adjoint approach; the unsteady discrete adjoint equations for a periodic flow must be reformulated and the corresponding system of equations solved iteratively. We compare the design results from shape and trajectory optimizations and investigate the physical relevance of design variables to the flapping motion at on- and off-design conditions.

Lee, Byung Joon↗