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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 NTRS1987

Design sensitivity analysis of nonlinear structural response

A unified theory is described of design sensitivity analysis of linear and nonlinear structures for shape, nonshape and material selection problems. The concepts of reference volume and adjoint structure are used to develop the unified viewpoint. A general formula for design sensitivity analysis is derived. Simple analytical linear and nonlinear examples are used to interpret various terms of the formula and demonstrate its use.

Cardoso, J. B.↗

NASA NTRS2019

Assessing the Impact of Observations in a Multi-Year Reanalysis (MERRA-2)

Operational and quasi-operational weather prediction centers have been routinely assessing the contribution from various observing systems to reducing errors in short-range forecasts for a number of years now. The original technique, Forecast Sensitivity-based Observation Impact (FSOI), involves definition of a forecast error measure and evaluation of sensitivities with respect to changes in the observations that require adjoint operators of both the underlying tangent linear model and corresponding analysis technique. The present work applies FSOI to reanalysis and aims at providing an expanded view of the contribution of various observing systems over nearly 40 years of assimilation. Specifically, this study uses MERRA-2 given that its supporting software includes all ingredients necessary to calculate FSOI. Part of this work shows how the quality of forecasts improves over the course of the reanalysis, and examines forecast sensitivities relevant to FSOI. The assessment here finds, for example, that: conventional observations are a major player in reducing forecast error throughout the 40 years of reanalysis, even when their volume reduces from 45\% in the earlier periods to about 5% in the modern era; satellite radiances, especially microwave instruments are major contributors to error reduction from the early single platform TIROS-N days to the current multi-platform scenario, though their fractional contribution reduces slightly from the early 2000's onward after the increased availability of wind observation from aircraft and atmospheric motion vectors, and the introduction of GPSRO; infrared instruments play a secondary role to microwave but are significant still, with the peculiar result of fractional impacts contribution from modern hyperspectral instruments being roughly similar to those from early infrared instruments. The dependence of results on the chosen error measure is emphasized throughout.

Todling, Ricardo↗

NASA NTRS2014

Aerostructural Level Set Topology Optimization for a Common Research Model Wing

The purpose of this work is to use level set topology optimization to improve the design of a representative wing box structure for the NASA common research model. The objective is to minimize the total compliance of the structure under aerodynamic and body force loading, where the aerodynamic loading is coupled to the structural deformation. A taxi bump case was also considered, where only body force loads were applied. The trim condition that aerodynamic lift must balance the total weight of the aircraft is enforced by allowing the root angle of attack to change. The level set optimization method is implemented on an unstructured three-dimensional grid, so that the method can optimize a wing box with arbitrary geometry. Fast matching and upwind schemes are developed for an unstructured grid, which make the level set method robust and efficient. The adjoint method is used to obtain the coupled shape sensitivities required to perform aerostructural optimization of the wing box structure.

Dunning, Peter D.↗

NASA NTRS2019

Assessing the Impact of Observations in a Multi-Year Reanalysis

Operational and quasi-operational weather prediction centers have been routinely assessing the contribution from various observing systems to reducing errors in short-range forecasts for a number of years now. The original technique, Forecast Sensitivity Observation Impact (FSOI), involves definition of a forecast error measure and evaluation of sensitivities with respect to changes in the observing system that require adjoint operators of both the underlying tangent linear model and corresponding analysis technique. The present work applies FSOI to Reanalysis and aims at providing an expanded view of the contribution of various observing systems over nearly 40 years of assimilation. Specifically, this study uses MERRA-2 given that its supporting software includes all ingredients necessary to calculate FSOI. Part of this work shows how the quality of forecasts improves over the course of the reanalysis, and examines forecast sensitivities relevant to FSOI. The assessment here finds, for example, that: conventional observations are a major player in reducing forecast error throughout the 40 years of reanalysis, even when their volume reduces from 45% in the earlier periods to about 5% in the modern era; satellite radiances, especially microwave instruments are major contributors to error reduction from the early single platform TIROS-N days to the current multi-platform scenario; infrared instruments play a secondary role to microwave but are significant still, with the peculiar result of fractional impacts contribution from modern hyperspectral instruments being roughly similar to those from early infrared instruments. The dependence of results on the chosen error measure is emphasized throughout.

Fabio L R Diniz↗

NASA NTRS1995

Implementation of a multiblock sensitivity analysis method in numerical aerodynamic shape optimization

A multiblock sensitivity analysis method is applied in a numerical aerodynamic shape optimization technique. The Sensitivity Analysis Domain Decomposition (SADD) scheme which is implemented in this study was developed to reduce the computer memory requirements resulting from the aerodynamic sensitivity analysis equations. Discrete sensitivity analysis offers the ability to compute quasi-analytical derivatives in a more efficient manner than traditional finite-difference methods, which tend to be computationally expensive and prone to inaccuracies. The direct optimization procedure couples CFD analysis based on the two-dimensional thin-layer Navier-Stokes equations with a gradient-based numerical optimization technique. The linking mechanism is the sensitivity equation derived from the CFD discretized flow equations, recast in adjoint form, and solved using direct matrix inversion techniques. This investigation is performed to demonstrate an aerodynamic shape optimization technique on a multiblock domain and its applicability to complex geometries. The objectives are accomplished by shape optimizing two aerodynamic configurations. First, the shape optimization of a transonic airfoil is performed to investigate the behavior of the method in highly nonlinear flows and the effect of different grid blocking strategies on the procedure. Secondly, shape optimization of a two-element configuration in subsonic flow is completed. Cases are presented for this configuration to demonstrate the effect of simultaneously reshaping interfering elements. The aerodynamic shape optimization is shown to produce supercritical type airfoils in the transonic flow from an initially symmetric airfoil. Multiblocking effects the path of optimization while providing similar results at the conclusion. Simultaneous reshaping of elements is shown to be more effective than individual element reshaping due to the inclusion of mutual interference effects.

Lacasse, James M.↗

NASA NTRS1985

Sensitivity derivatives for static test loading boundary conditions

Attention is given to the application of the adjoint variable method under the special combination of displacement and stress boundary conditions which occur when static test conditions are simulated. In the two-step problem-solving method employed, calculation of internal stresses, displacements and total reaction force at the boundary is followed by the calculation of the applied load and the total reaction force due to unit displacements.

Gurdal, Z.↗

NASA NTRS1992

Multidisciplinary analysis and sensitivity derivatives for isolated helicopter rotors in hover

A new method is presented for using Navier-Stokes results in the system analysis and sensitivity derivative computations for a helicopter rotor in hover. The system coupling includes the interactions between rotor performance, rotor structural properties and aerodynamics (i.e., airfoil shape). Emphasis is placed on the use of high fidelity Navier-Stokes algorithms for the aerodynamic problem. The present formulation is chosen to reduce the number of sensitivity derivative equations and their dimensions as much as possible. In addition, an alternative adjoint formulation is presented which dramatically reduces the computational requirement for disciplines in which there may be a large number of design variables (e.g., aerodynamics). The theoretical method makes use of a combination of derivatives from each discipline; the computation of these derivatives has already been shown to be feasible. Sample computations for the aerodynamic-rotor model coupling and an associated sensitivity derivative are also presented.

Jones, Henry E.↗

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 NTRS2002

Mesoscale Assimilation of TRMM Data with 4DVAR: Preliminary Results

Surface rainfall data, derived from the TRMM Microwave Image (TMI), are assimilated into the PSU/NCAR MM5 model using a 4DVAR technique. Preliminary experiments are performed to incorporate TRMM rainfall data into a hurricane initialization. It is found that the rainfall data assimilation is sensitive to the error characteristics of the data and the physics in the adjoint model. In addition, assimilating the rainfall data alone produces a more realistic eye and rain bands in the hurricane but cannot ensure improvements of hurricane intensity forecasts. Numerical results indicate that it is necessary to incorporate TRMM rainfall data together with other types of data such as wind data into the model, in which case the inclusion of the rainfall data will further improve the intensity forecast of the hurricane. This fact might imply that some proper constraints will be needed for the rainfall assimilation. Relevant results and issues will be presented.

Pu, Zhoa-Xia↗

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 NTRS2017

Development and Applications of the FV3 GEOS-5 Adjoint Modeling System

GMAO has developed a highly sophisticated adjoint modeling system based on the most recent version of the finite volume cubed sphere (FV3) dynamical core. This provides a mechanism for investigating sensitivity to initial conditions and examining observation impacts. It also allows for the computation of singular vectors and for the implementation of hybrid 4DVAR. In this work we will present the scientific assessment of the new adjoint system and show results from a number of research application of the adjoint system.

Holdaway, Daniel↗

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 NTRS2003

Mesoscale Assimilation of TMI Rainfall Data with 4DVAR: Sensitivity Studies

Sensitivity studies are performed on the assimilation of TRMM (Tropical Rainfall Measurement Mission) Microwave Imager (TMI) derived rainfall data into a mesoscale model using a four-dimensional variational data assimilation (4DVAR) technique. A series of numerical experiments is conducted to evaluate the impact of TMI rainfall data on the numerical simulation of Hurricane Bonnie (1998). The results indicate that rainfall data assimilation is sensitive to the error characteristics of the data and the inclusion of physics in the adjoint and forward models. In addition, assimilating the rainfall data alone is helpful for producing a more realistic eye and rain bands in the hurricane but does not ensure improvements in hurricane intensity forecasts. Further study indicated that it is necessary to incorporate TMI rainfall data together with other types of data such as wind data into the model, in which case the inclusion of the rainfall data further improves the intensity forecast of the hurricane. This implies that proper constraints may be needed for rainfall assimilation.

Tao, Wei-Kuo↗

NASA NTRS2018

The Tangent Linear and Adjoint of the FV3 Dynamical Core: Development and Applications

GMAO (NASA's Global Modeling and Assimilation Office) has developed a highly sophisticated adjoint modeling system based on the most recent version of the finite volume cubed sphere (FV3) dynamical core. This provides a mechanism for investigating sensitivity to initial conditions and examining observation impacts. It also allows for the computation of singular vectors and for the implementation of hybrid 4DVAR (4-Dimensional Variational Assimilation). In this work we will present the scientific assessment of the new adjoint system and show results from a number of research application of the adjoint system.

FV3↗

NASA NTRSDate not supplied

Adjoint-Based Minimization of X-59 Sonic Boom Noise Via Control Surfaces

A multidisciplinary design optimization methodology to directly minimize the ground-level noise generated by a supersonic aircraft's sonic boom is presented. A gradient-based optimizer is coupled with a Cartesian Euler flow solver and an atmospheric propagation tool to forge a unique design capability. Adjoint formulations for both the flow solver and propagation tool are also coupled to provide sensitivities in an efficient manner. The design method is demonstrated on the X-59 Low-Boom Flight Demonstrator by optimizing control surface deflections to improve ground-level noise while maintaining trimmed cruise flight. Two noise minimization examples and a noise maximization are presented. All optimized designs result in measurable improvement in the ground-level noise objective. Extensive surveys of the design space confirm that the optimization method is effective in finding a local optimum. Moreover, repeated application of the method with varying initial design points also demonstrates the robustness of the method.

ARMD↗

NASA NTRSDate not supplied

Sonic Boom Ground Noise Minimization Via the Adjoint Method

A multidisciplinary design optimization methodology to directly minimize the ground-level noise generated by the sonic boom of a high altitude supersonic body is presented. A Cartesian Euler flow solver is coupled with an atmospheric propagation tool to create a ground-noise analysis capability for supersonic bodies. Adjoint formulations for both the flow solver and propagation tool are also coupled to compute sensitivities of shape variations in the body in a highly efficient manner. A gradient-based optimizer is then introduced to forge a valuable design capability. Output-based mesh adaptation that is driven directly by ground-level noise is employed to increase accuracy and provide error estimation. The design method is demonstrated first on a simple axisymmetric body with few design variables to evaluate the efficacy of the optimization scheme. Guided by the results of this initial case, the problem is then repeated with somewhat different design variables to further demonstrate the capabilities of the design method. Finally, the method is applied to a real-world problem by optimizing control surface deflection settings of a low-boom aircraft to minimize ground noise while maintaining trimmed, level flight.

ARMD↗

NASA NTRS1998

Aerodynamic Design Optimization on Unstructured Meshes Using the Navier-Stokes Equations

A discrete adjoint method is developed and demonstrated for aerodynamic design optimization on unstructured grids. The governing equations are the three-dimensional Reynolds-averaged Navier-Stokes equations coupled with a one-equation turbulence model. A discussion of the numerical implementation of the flow and adjoint equations is presented. Both compressible and incompressible solvers are differentiated and the accuracy of the sensitivity derivatives is verified by comparing with gradients obtained using finite differences. Several simplifying approximations to the complete linearization of the residual are also presented, and the resulting accuracy of the derivatives is examined. Demonstration optimizations for both compressible and incompressible flows are given.

Nielsen, Eric J.↗