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All sourcesNASA NTRSDOE OSTI
NASA NTRS1987

Numerical studies of the thermal design sensitivity calculation for a reaction-diffusion system with discontinuous derivatives

The aim of this study is to find a reliable numerical algorithm to calculate thermal design sensitivities of a transient problem with discontinuous derivatives. The thermal system of interest is a transient heat conduction problem related to the curing process of a composite laminate. A logical function which can smoothly approximate the discontinuity is introduced to modify the system equation. Two commonly used methods, the adjoint variable method and the direct differentiation method, are then applied to find the design derivatives of the modified system. The comparisons of numerical results obtained by these two methods demonstrate that the direct differentiation method is a better choice to be used in calculating thermal design sensitivity.

Hou, Jean W.↗
NASA NTRS1997

Efficient Approximation for Structural Optimization Under Multiple Constraints

The cooperative agreement covered work between August 1995 and August 1997. The focus of the work was efficient approximations of structural response and sensitivity. The effort proceeded in three directions as follows: (1) Development of an approximation extended to efficient sensitivity approximations and demonstrated for structural models for the High Speed Civil Transport; (2) Preliminary development of the adjoint method for calculating sensitivity derivatives; and (3) A review of method for fast exact reanalysis. Attachments of papers which were submitted during this period are included.

Haftka, Raphael T.↗
NASA NTRS1999

Ames Optimized TCA Configuration

Configuration design at Ames was carried out with the SYN87-SB (single block) Euler code using a 193 x 49 x 65 C-H grid. The Euler solver is coupled to the constrained (NPSOL) and the unconstrained (QNMDIF) optimization packages. Since the single block grid is able to model only wing-body configurations, the nacelle/diverter effects were included in the optimization process by SYN87's option to superimpose the nacelle/diverter interference pressures on the wing. These interference pressures were calculated using the AIRPLANE code. AIRPLANE is an Euler solver that uses a unstructured tetrahedral mesh and is capable of computations about arbitrary complete configurations. In addition, the buoyancy effects of the nacelle/diverters were also included in the design process by imposing the pressure field obtained during the design process onto the triangulated surfaces of the nacelle/diverter mesh generated by AIRPLANE. The interference pressures and nacelle buoyancy effects are added to the final forces after each flow field calculation. Full details of the (recently enhanced) ghost nacelle capability are given in a related talk. The pseudo nacelle corrections were greatly improved during this design cycle. During the Ref H and Cycle 1 design activities, the nacelles were only translated and pitched. In the cycle 2 design effort the nacelles can translate vertically, and pitch to accommodate the changes in the lower surface geometry. The diverter heights (between their leading and trailing edges) were modified during design as the shape of the lower wing changed, with the drag of the diverter changing accordingly. Both adjoint and finite difference gradients were used during optimization. The adjoint-based gradients were found to give good direction in the design space for configurations near the starting point, but as the design approached a minimum, the finite difference gradients were found to be more accurate. Use of finite difference gradients was limited by the CPU time limit available on the Cray machines. A typical optimization run using finite difference gradients can use only 30 to 40 design variables and one optimization iteration within the 8 hour queue limit for the chosen grid size and convergence level. The efficiency afforded by the adjoint method allowed for 50-120 design variables and 5-10 optimization iterations in the 8 hour queue. Geometric perturbations to the wing and fuselage were made using the Hicks/Henne (HH) shape functions. The HH functions were distributed uniformly along the chords of the wing defining sections and lofted linearly. During single-surface design, constraints on thickness and volume at selected wing stations were imposed. Both fuselage camber and cross-sectional area distributions were permitted to change during design. The major disadvantage to the use of these functions is the inherent surface waviness produced by repeated use of such functions. Many smoothing operations were required following optimization runs to produce a configuration with reasonable smoothness. Wagner functions were also used on the wing sections but were never used on the fuselage. The Wagner functions are a family of increasingly oscillatory functions that have also been used extensively in airfoil design. The leading and trailing edge regions of the wing were designed by use of polynomial and monomial functions respectively. Twist was attempted but was abandoned because of little performance improvement available from changing the baseline twist.

Cliff, Susan E.↗
NASA NTRS1998

Seasonal Cycles of Meridional Overturning and Heat Transport of the Indian Ocean

A general circulation model of the Indian Ocean is fitted to monthly averaged climatological temperatures, salinities, and surface fluxes using the adjoint method. Interannual variability is minimized by penalizing the temporal drift from one seasonal cycle to another during a two-year integration. The resultant meridional overturning and heat transport display large seasonal variations, with maximum amplitudes of 18 and 22 (x 10(exp 6) cubic m/s) for the overturning and 1.8 and 1.4 (x 10(exp 15) W) for heat transport near 10 S and 10 N, respectively. A dynamical decomposition of the overturning and heat transport shows that the time-varying Ekman How plus its barotropic compensation can explain a large part of the seasonal variations in overturning and heat transport. The maximum variations at 10 deg N and 10 deg S are associated with monsoon reversal over the northern Indian Ocean and changes of the easterlies over the southern Indian Ocean. An external mode with variable topography has a moderate contribution where the Somali Current and the corresponding gyre reverse direction seasonally. Contribution front vertical shear (thermal wind and ageostrophic shear) is dominant near the southern boundary and large near the Somali Current latitudes. The dominant balance in the zonally integrated heat budget is between heat storage change and heat transport convergence except south of 15 S. Optimization with seasonal forcings improves estimates of sea surface temperatures, but the annual average overturning and heat transport are very similar to previous results with annual mean forcings. The annual average heat transport consists of roughly equal contributions from time-mean and time-varying fields of meridional velocities and temperatures in the northern Indian Ocean. indicating a significant rectification to the heat transport due to the time-varying fields. The time-mean and time-varying contributions are primarily due to the overturning and horizontal gyre, respectively. Inclusion of TOPEX data enhances the seasonal cycles of the estimated overturning and heat transport in the central Indian Ocean significantly and improves the estimated equatorial zonal flows but leads 10 unrealistic estimates of the velocity structure near the Indonesian Throughflow region, most likely owing to the deficiencies in the lateral boundary conditions.

Lee, Tong↗
NASA NTRS2004

Subduction in an Eddy-Resolving State Estimate of the Northeast Atlantic Ocean

Are eddies an important contributor to subduction in the eastern subtropical gyre? Here, an adjoint model is used to combine a regional, eddy-resolving numerical model with observations to produce a state estimate of the ocean circulation. The estimate is a synthesis of a variety of in- situ observations from the Subduction Experiment, TOPEX/POSEIDON altimetry, and the MTI General Circulation Model. The adjoint method is successful because the Northeast Atlantic Ocean is only weakly nonlinear. The state estimate provides a physically-interpretable, eddy-resolving information source to diagnose subduction. Estimates of eddy subduction for the eastern subtropical gyre of the North Atlantic are larger than previously calculated from parameterizations in coarse-resolution models. Furthermore, eddy subduction rates have typical magnitudes of 15% of the total subduction rate. Eddies contribute as much as 1 Sverdrup to water-mass transformation, and hence subduction, in the North Equatorial Current and the Azores Current. The findings of this thesis imply that the inability to resolve or accurately parameterize eddy subduction in climate models would lead to an accumulation of error in the structure of the main thermocline, even in the relatively-quiescent eastern subtropical gyre.

WATER MASS TRANSFORMATIONS↗
NASA NTRS2004

Advanced Applications of Adifor 3.0 for Efficient Calculation of First-and Second-Order CFD Sensitivity Derivatives

This final report will document the accomplishments of the work of this project. 1) The incremental-iterative (II) form of the reverse-mode (adjoint) method for computing first-order (FO) aerodynamic sensitivity derivatives (SDs) has been successfully implemented and tested in a 2D CFD code (called ANSERS) using the reverse-mode capability of ADIFOR 3.0. These preceding results compared very well with similar SDS computed via a black-box (BB) application of the reverse-mode capability of ADIFOR 3.0, and also with similar SDs calculated via the method of finite differences. 2) Second-order (SO) SDs have been implemented in the 2D ASNWERS code using the very efficient strategy that was originally proposed (but not previously tested) of Reference 3, Appendix A. Furthermore, these SO SOs have been validated for accuracy and computational efficiency. 3) Studies were conducted in Quasi-1D and 2D concerning the smoothness (or lack of smoothness) of the FO and SO SD's for flows with shock waves. The phenomenon is documented in the publications of this study (listed subsequently), however, the specific numerical mechanism which is responsible for this unsmoothness phenomenon was not discovered. 4) The FO and SO derivatives for Quasi-1D and 2D flows were applied to predict aerodynamic design uncertainties, and were also applied in robust design optimization studies.

Taylor, Arthur C., III↗
NASA NTRS2013

Using CFD Surface Solutions to Shape Sonic Boom Signatures Propagated from Off-Body Pressure

The conceptual design of a low-boom and low-drag supersonic aircraft remains a challenge despite significant progress in recent years. Inverse design using reversed equivalent area and adjoint methods have been demonstrated to be effective in shaping the ground signature propagated from computational fluid dynamics (CFD) off-body pressure distributions. However, there is still a need to reduce the computational cost in the early stages of design to obtain a baseline that is feasible for low-boom shaping, and in the search for a robust low-boom design over the entire sonic boom footprint. The proposed design method addresses the need to reduce the computational cost for robust low-boom design by using surface pressure distributions from CFD solutions to shape sonic boom ground signatures propagated from CFD off-body pressure.

Ordaz, Irian↗
NASA NTRS2015

Under-Track CFD-Based Shape Optimization for a Low-Boom Demonstrator Concept

The detailed outer mold line shaping of a Mach 1.6, demonstrator-sized low-boom concept is presented. Cruise trim is incorporated a priori as part of the shaping objective, using an equivalent-area-based approach. Design work is performed using a gradient-driven optimization framework that incorporates a three-dimensional, nonlinear flow solver, a parametric geometry modeler, and sensitivities derived using the adjoint method. The shaping effort is focused on reducing the under-track sonic boom level using an inverse design approach, while simultaneously satisfying the trim requirement. Conceptual-level geometric constraints are incorporated in the optimization process, including the internal layout of fuel tanks, landing gear, engine, and crew station. Details of the model parameterization and design process are documented for both flow-through and powered states, and the performance of these optimized vehicles presented in terms of inviscid L/D, trim state, pressures in the near-field and at the ground, and predicted sonic boom loudness.

Wintzer, Mathias↗
NASA NTRS2016

Adjoint-Based Aerodynamic Design of Complex Aerospace Configurations

An overview of twenty years of adjoint-based aerodynamic design research at NASA Langley Research Center is presented. Adjoint-based algorithms provide a powerful tool for efficient sensitivity analysis of complex large-scale computational fluid dynamics (CFD) simulations. Unlike alternative approaches for which computational expense generally scales with the number of design parameters, adjoint techniques yield sensitivity derivatives of a simulation output with respect to all input parameters at the cost of a single additional simulation. With modern large-scale CFD applications often requiring millions of compute hours for a single analysis, the efficiency afforded by adjoint methods is critical in realizing a computationally tractable design optimization capability for such applications.

Nielsen, Eric J.↗
NASA NTRS2017

Reconfigurable Model Execution in the OpenMDAO Framework

NASA's OpenMDAO framework facilitates constructing complex models and computing their derivatives for multidisciplinary design optimization. Decomposing a model into components that follow a prescribed interface enables OpenMDAO to assemble multidisciplinary derivatives from the component derivatives using what amounts to the adjoint method, direct method, chain rule, global sensitivity equations, or any combination thereof, using the MAUD architecture. OpenMDAO also handles the distribution of processors among the disciplines by hierarchically grouping the components, and it automates the data transfer between components that are on different processors. These features have made OpenMDAO useful for applications in aircraft design, satellite design, wind turbine design, and aircraft engine design, among others. This paper presents new algorithms for OpenMDAO that enable reconfigurable model execution. This concept refers to dynamically changing, during execution, one or more of: the variable sizes, solution algorithm, parallel load balancing, or set of variables-i.e., adding and removing components, perhaps to switch to a higher-fidelity sub-model. Any component can reconfigure at any point, even when running in parallel with other components, and the reconfiguration algorithm presented here performs the synchronized updates to all other components that are affected. A reconfigurable software framework for multidisciplinary design optimization enables new adaptive solvers, adaptive parallelization, and new applications such as gradient-based optimization with overset flow solvers and adaptive mesh refinement. Benchmarking results demonstrate the time savings for reconfiguration compared to setting up the model again from scratch, which can be significant in large-scale problems. Additionally, the new reconfigurability feature is applied to a mission profile optimization problem for commercial aircraft where both the parametrization of the mission profile and the time discretization are adaptively refined, resulting in computational savings of roughly 10% and the elimination of oscillations in the optimized altitude profile.

Hwang, John T.↗
NASA NTRS2019

Uncertainty of Observation Impact Estimation in an Adjoint Model Investigated with an Observing System Simulation Experiment

Adjoint models are often used to estimate the impact of different observations on short-term forecast skill. A common difficulty with the evaluation of short term forecast quality is the choice of verification fields. The use of self-analysis fields for verification is typical but incestuous, and introduces uncertainty due to biases and errors in the analysis field. In this study, an observing system simulation experiment (OSSE) is used to explore the uncertainty in adjoint model estimations of observation impact. The availability of the true state for verification in the OSSE framework in the form of the Nature Run allows calculation of the observation impact without the uncertainties present in self-analysis verification. These impact estimates are compared to estimates calculated using self-analysis verification. The Global Earth Observing System version 5 (GEOS-5) forecast model with Gridpoint Statistical Interpolation (GSI) is used with the National Aeronautics and Space Administration Global Modeling and Assimilation Office (NASA/GMAO) OSSE capability. The adjoint model includes moist processes, with total wet energy selected as the norm for evaluation of observation impacts. The results show that there are measurable but small discrepancies in the adjoint model estimation of observation impact. In general, observations of temperature and winds tend to have overestimated impacts with self-analysis verification, while observations of humidity and moisture-affected observations tend to have underestimated impacts. The small magnitude of the differences in impact estimates supports the robustness of the adjoint method of estimating observation impacts.

N C Prive↗
NASA NTRS2019

High-Fidelity Multidisciplinary Sensitivity Analysis Framework for Multipoint Rotorcraft Optimization

A multidisciplinary, gradient-based sensitivity-analysis methodology is evaluated for optimization of rotorcraft configurations. The tightly coupled discipline models include physics-based fluid dynamics and rotorcraft comprehensive analysis. A discretely-consistent adjoint method accounts for sensitivities of the unsteady flow and unstructured, dynamic, over set grids, while sensitivities of structural responses to aerodynamic loads are computed using a complex-variable method. The methodology is applied to optimize the shape of UH-60A Blackhawk helicopter blades for hover and forward flight conditions. The objective of the multipoint design is to simultaneously increase the rotorcraft figure of merit in a hover flight and reduce the rotor power in a forward flight. Trimmed loose-coupling solutions for the baseline configuration are used to initiate the tight-coupling multidisciplinary analysis. Target thrust and rolling and pitching moments are enforced as optimization constraints. The optimized configuration improves the optimization metrics at both design points. The improved performance and all constraints are maintained over many revolutions beyond the optimization interval, satisfying the required flight conditions. Computational cost of the optimization cycle is assessed in a high-performance computing environment and found affordable for design of rotorcraft in general level-flight conditions.

Wang, Li↗
DOE OSTI2023

An adjoint-based optimization method for jointly inverting heterogeneous material properties and fault slip from earthquake surface deformation data

SUMMARY Analysis of tectonic and earthquake-cycle associated deformation of the crust can provide valuable insights into the underlying deformation processes including fault slip. How those processes are expressed at the surface depends on the lateral and depth variations of rock properties. The effect of such variations is often tested by forward models based on a priori geological or geophysical information. Here, we first develop a novel technique based on an open-source finite-element computational framework to invert geodetic constraints directly for heterogeneous media properties. We focus on the elastic, coseismic problem and seek to constrain variations in shear modulus and Poisson’s ratio, proxies for the effects of lithology and/or temperature and porous flow, respectively. The corresponding nonlinear inversion is implemented using adjoint-based optimization that efficiently reduces the cost function that includes the misfit between the calculated and observed displacements and a penalty term. We then extend our theoretical and numerical framework to simultaneously infer both heterogeneous Earth’s structure and fault slip from surface deformation. Based on a range of 2-D synthetic cases, we find that both model parameters can be satisfactorily estimated for the megathrust setting-inspired test problems considered. Within limits, this is the case even in the presence of noise and if the fault geometry is not perfectly known. Our method lays the foundation for a future reassessment of the information contained in increasingly data-rich settings, for example, geodetic GNSS constraints for large earthquakes such as the 2011 Tohoku-oki M9 event, or distributed deformation along plate boundaries as constrained from InSAR.

Geochemistry & Geophysics↗
NASA NTRS2004

Adjoint Formulation for an Embedded-Boundary Cartesian Method

Many problems in aerodynamic design can be characterized by smooth and convex objective functions. This motivates the use of gradient-based algorithms, particularly for problems with a large number of design variables, to efficiently determine optimal shapes and configurations that maximize aerodynamic performance. Accurate and efficient computation of the gradient, however, remains a challenging task. In optimization problems where the number of design variables dominates the number of objectives and flow- dependent constraints, the cost of gradient computations can be significantly reduced by the use of the adjoint method. The problem of aerodynamic optimization using the adjoint method has been analyzed and validated for both structured and unstructured grids. The method has been applied to design problems governed by the potential, Euler, and Navier-Stokes equations and can be subdivided into the continuous and discrete formulations. Giles and Pierce provide a detailed review of both approaches. Most implementations rely on grid-perturbation or mapping procedures during the gradient computation that explicitly couple changes in the surface shape to the volume grid. The solution of the adjoint equation is usually accomplished using the same scheme that solves the governing flow equations. Examples of such code reuse include multistage Runge-Kutta schemes coupled with multigrid, approximate-factorization, line-implicit Gauss-Seidel, and also preconditioned GMRES. The development of the adjoint method for aerodynamic optimization problems on Cartesian grids has been limited. In contrast to implementations on structured and unstructured grids, Cartesian grid methods decouple the surface discretization from the volume grid. This feature makes Cartesian methods well suited for the automated analysis of complex geometry problems, and consequently a promising approach to aerodynamic optimization. Melvin e t 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 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.

Nemec, Marian↗
NASA NTRS2006

Adjoint Sensitivity Computations for an Embedded-Boundary Cartesian Mesh Method and CAD Geometry

Cartesian-mesh methods 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) tools. Our goal is to combine the automation capabilities of Cartesian methods with an eficient computation of design sensitivities. We address this issue using the adjoint method, where the computational cost of the design sensitivities, or objective function gradients, is esseutially indepeudent of the number of design variables. In previous work, 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 included 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 objective of the present work is to extend our adjoint formulation to problems involving general shape changes. Central to this development is the computation of volume-mesh sensitivities to obtain a reliable approximation of the objective finction gradient. Motivated by the success of mesh-perturbation schemes commonly used in body-fitted unstructured formulations, we propose an approach based on a local linearization of a mesh-perturbation scheme similar to the spring analogy. This approach circumvents most of the difficulties that arise due to non-smooth changes in the cut-cell layer as the boundary shape evolves and provides a consistent approximation tot he exact gradient of the discretized abjective function. A detailed gradient accurace study is presented to verify our approach. Thereafter, we focus on a shape optimization problem for an Apollo-like reentry capsule. The optimization seeks to enhance the lift-to-drag ratio of the capsule by modifyjing 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. This abstract presents only a brief outline of the numerical method and results; full details will be given in the final paper.

Nemec, Marian↗