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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 NTRS2001

Improving Ocean Angular Momentum Estimates Using a Model Constrained by Data

Ocean angular momentum (OAM) calculations using forward model runs without any data constraints have, recently revealed the effects of OAM variability on the Earth's rotation. Here we use an ocean model and its adjoint to estimate OAM values by constraining the model to available oceanic data. The optimization procedure yields substantial changes in OAM, related to adjustments in both motion and mass fields, as well as in the wind stress torques acting on the ocean. Constrained and unconstrained OAM values are discussed in the context of closing the planet's angular momentum budget. The estimation procedure, yields noticeable improvements in the agreement with the observed Earth rotation parameters, particularly at the seasonal timescale. The comparison with Earth rotation measurements provides an independent consistency check on the estimated ocean state and underlines the importance of ocean state estimation for quantitative. studies of the variable large-scale oceanic mass and circulation fields, including studies of OAM.

Ponte, Rui M.↗

NASA NTRS2018

Introduction to Data Assimilation

Atmospheric data assimilation is a class of techniques used for producing descriptions of fields of air temperature, pressure, humidity, wind, etc. on a spatial grid or in terms of a finite functional representation. These are then used to initialize numerical weather forecasts or to analyze the atmosphere for other purposes. The techniques combine past, present, and even future observations in an approximate statistically optimal way. Various types of statistical or physically-based models and their corresponding adjoints are employed to relate diverse fields in both time and space and to relate what is observed to what is being analyzed. Computationally, the problem is very demanding and onstraining on the techniques that can be employed on a routine basis.

Atmospheric data assimilation↗

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↗

NASA NTRSDate not supplied

Influence of the Perception Constraint, Observer Position, and Broadband Self-Noise on Low-Fidelity UAM Vehicle Perception-Influenced-Design (PID) Optimization

This paper will present an approach in design optimization of a proprotor in a forward flight condition with several acoustic objective functions that simulate community response. An adjoint-capable blade element momentum theory (BEMT), implemented in CCBlade.jl, will be used to compute the blade forces and inflow properties required to compute tonal and broadband noise using the second-generation Aircraft NOise Prediction Program (ANOPP2). The influence of broadband noise, observer location, and choice of perception constraint on optimized values of rotation rate, chord distribution, and twist angle will be shown. It is determined that broadband self-noise is an influential source noise mechanism in the design optimization when frequency weighting is used to predict the noise at an out-of-plane observer position.

acoustics↗

NASA NTRSDate not supplied

Influence of the Perception Constraint, Observer Position, and Broadband Self-Noise on Low-Fidelity UAM Vehicle Perception-Influenced-Design (PID) Optimization

acoustics↗

NASA NTRS2018

Sonic Boom Prediction and Mitigation using Three-Dimensional Earth Effects

As the industry progresses towards realizing a commercial supersonic aircraft, the push to reduce noise from sonic boom has intensified. This emphasizes the need to better understand and accurately model the underlying physics and seek ways to reduce modeling gaps in the calculation of sonic boom noise metrics. To answer this need, sonic boom prediction capability of the atmospheric propagation tool sBOOM is enhanced to include three-dimensional Earth effects during propagation. The discrete adjoint capability under three-dimensional effects is also enhanced to be helpful in design optimization. The differences in ground signatures, noise metrics, and carpet widths are compared against results obtained using flat Earth approximation and discussed. Finally, the developed capability is used in CFD based shape optimization to demonstrate the differences in aircraft outer mold line when ellipsoidal Earth effects during shape optimization for sonic boom mitigation.

Rallabhandi, Sriram K.↗

NASA NTRS1985

Design of stiffened composite panels with a fracture constraint

An automated procedure for designing minimum-weight composite panels subject to a local damage constraint under tensile loading was developed. A finite element program based on linear elastic fracture mechanics for calculating stress intensity factors (SIF) was incorporated in the design cycle. Panel fracture toughness was obtained by using a strain based criterion. A general purpose mathematical optimization algorithm was used for the weight minimization. Analytical sensitivity derivatives of the SIF employing the adjoint variable technique was used to enhance the computational efficiency of the procedure. Design results for both unstiffened and stiffened plates are presented.

Gurdal, Z.↗

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 NTRS1998

A Preconditioning Method for Shape Optimization Governed by the Euler Equations

We consider a classical aerodynamic shape optimization problem subject to the compressible Euler flow equations. The gradient of the cost functional with respect to the shape variables is derived with the adjoint method at the continuous level. The Hessian (second order derivative of the cost functional with respect to the shape variables) is approximated also at the continuous level, as first introduced by Arian and Ta'asan (1996). The approximation of the Hessian is used to approximate the Newton step which is essential to accelerate the numerical solution of the optimization problem. The design space is discretized in the maximum dimension, i.e., the location of each point on the intersection of the computational mesh with the airfoil is taken to be an independent design variable. We give numerical examples for 86 design variables in two different flow speeds and achieve an order of magnitude reduction in the cost functional at a computational effort of a full solution of the analysis partial differential equation (PDE).

Arian, Eyal↗

NASA NTRSDate not supplied

Influence of the Perception Constraint, Installation Effects, and Broadband SelfNoise on Low-Fidelity, Multi-point, UAM Vehicle Perception-Influenced-Design(PID) Optimization

UAM vehicles represent a unique challenge in acoustics because there are a limited number of full-scale datasets that can be used to calibrate and validate prediction models. In addition, acoustics is considered more often in the early design loop as a higher priority concern in UAM vehicle design, whereas with traditional vehicles, acoustics is typically addressed with post-design constraints. This paper will present an approach in design optimization of a proprotor in two flight conditions, hover, and forward flight, with several acoustic objective functions that simulate community response. An adjoint-capable vortex-lattice method (VLM), implemented in VSPAERO, will be used to compute the blade forces and inflow properties required to compute tonal and broadband noise using the second-generation Aircraft NOise Prediction Program (ANOPP2). In addition, a wing is placed near the proprotor to simulate installation effects on a vehicle’s wing. The influence of rotation rate, chord distribution, twist angle, and sweep on installation, broadband noise, and choice of perception constraint from the design optimization will be shown.

acoustics↗

NASA NTRS1988

Mixed finite element formulation applied to shape optimization

The development presented introduces a general form of mixed formulation for the optimal shape design problem. The associated optimality conditions are easily obtained without resorting to highly elaborate mathematical developments. Also, the physical significance of the adjoint problem is clearly defined with this formulation.

Rodrigues, Helder↗

NASA NTRS1998

Preliminary Results from the Application of Automated Adjoint Code Generation to CFL3D

This report describes preliminary results obtained using an automated adjoint code generator for Fortran to augment a widely-used computational fluid dynamics flow solver to compute derivatives. These preliminary results with this augmented code suggest that, even in its infancy, the automated adjoint code generator can accurately and efficiently deliver derivatives for use in transonic Euler-based aerodynamic shape optimization problems with hundreds to thousands of independent design variables.

Carle, Alan↗

NASA NTRS1993

Sensitivity analysis of Lyapunov and Riccati equations with application to controls-structures integrated design

Controls-structure integrated design is a complicated multidisciplinary design optimization problem which involves the state equations pertaining to open-loop eigenvalues and control laws. In order to alleviate the intensity of the computation, this study uses the adjoint variable method to derive sensitivity equations for the eigenvalue, Liapunov, and Riccati equations. These individual sensitivity equations are then combined together to form the multidisciplinary sensitivity equations for the control structure integrated design problems. A set of linear sensitivity equations, proportional in number to the number of performance functions involved in the optimization process, are solved. This proposed approach may provide a great saving in computer resources. The validity of the newly developed sensitivity equations is verified by numerical examples.

Hou, Gene↗

NASA NTRS2015

Constraining Black Carbon Aerosol over Asia using OMI Aerosol Absorption Optical Depth and the Adjoint of GEOS-Chem

Accurate estimates of the emissions and distribution of black carbon (BC) in the region referred to here as Southeastern Asia (70degE-l50degE, 11degS-55degN) are critical to studies of the atmospheric environment and climate change. Analysis of modeled BC concentrations compared to in situ observations indicates levels are underestimated over most of Southeast Asia when using any of four different emission inventories. We thus attempt to reduce uncertainties in BC emissions and improve BC model simulations by developing top-down, spatially resolved, estimates of BC emissions through assimilation of OMI observations of aerosol absorption optical depth (AAOD) with the GEOS-Chem model and its adjoint for April and October of 2006. Overwhelming enhancements, up to 500%, in anthropogenic BC emissions are shown after optimization over broad areas of Southeast Asia in April. In October, the optimization of anthropogenic emissions yields a slight reduction (1-5%) over India and parts of southern China, while emissions increase by 10-50% over eastern China. Observational data from in situ measurements and AERONET observations are used to evaluate the BC inversions and assess the bias between OMI and AERONET AAOD. Low biases in BC concentrations are improved or corrected in most eastern and central sites over China after optimization, while the constrained model still underestimates concentrations in Indian sites in both April and October, possibly as a. consequence of low prior emissions. Model resolution errors may contribute up to a factor of 2.5 to the underestimate of surface BC concentrations over northern India. We also compare the optimized results using different anthropogenic emission inventories and discuss the sensitivity of top-down constraints on anthropogenic emissions with respect to biomass burning emissions. In addition, the impacts of brown carbon, the formulation of the observation operator, and different a priori constraints on the optimization are investigated. Overall, despite these limitations and uncertainties, using OMI AAOD to constrain BC sources improves model representation of BC distributions, particularly over China.

Aerosols↗

NASA NTRS1996

Quasi-Optimal Schwarz Methods for the Conforming Spectral Element Discretization

Fast methods are proposed for solving the system K(sub N)x = b resulting from the discretization of self-adjoint elliptic equations in three dimensional domains by the spectral element method. The domain is decomposed into hexahedral elements, and in each of these elements the discretization space is formed by polynomials of degree N in each variable. Gauss-Lobatto-Legendre (GLL) quadrature rules replace the integrals in the Galerkin formulation. This system is solved by the preconditioned conjugate gradients method. The conforming finite element space on the GLL mesh consisting of piecewise Q(sub 1) elements produces a stiffness matrix K(sub h) that is spectrally equivalent to the spectral element stiffness matrix K(sub N). The action of the inverse of K(sub h) is expensive for large problems, and is therefore replaced by a Schwarz preconditioner B(sub h) of this finite element stiffness matrix. The preconditioned operator then becomes B(sub h)(exp -l)K(sub N). The technical difficulties stem from the nonregularity of the mesh. Tools to estimate the convergence of a large class of new iterative substructuring and overlapping Schwarz preconditioners are developed. This technique also provides a new analysis for an iterative substructuring method proposed by Pavarino and Widlund for the spectral element discretization.

Casarin, Mario↗

NASA NTRS2018

High-Fidelity Multidisciplinary Design Optimization Methodology with Application to Rotor Blades

A multidisciplinary design optimization procedure has been developed and applied to rotorcraft simulations involving tightly-coupled, high-fidelity computational fluid dynamics and comprehensive analysis. A discretely-consistent, adjoint-based sensitivity analysis available in the fluid dynamics solver provides sensitivities arising from unsteady turbulent flows on unstructured, dynamic, overset meshes, while a complex-variable approach is used to compute structural sensitivities with respect to aerodynamic loads. The multidisciplinary sensitivity analysis is conducted through integrating the sensitivity components from each discipline of the coupled system. Accuracy of the coupled system for high-fidelity rotorcraft analysis is verified; simulation results exhibit good agreement with established solutions. A constrained gradient-based design optimization for a HART-II rotorcraft configuration is demonstrated. The computational cost for individual components of the multidisciplinary sensitivity analysis is assessed and improved.

Wang, Li↗

NASA NTRS1991

One shot methods for optimal control of distributed parameter systems 1: Finite dimensional control

The efficient numerical treatment of optimal control problems governed by elliptic partial differential equations (PDEs) and systems of elliptic PDEs, where the control is finite dimensional is discussed. Distributed control as well as boundary control cases are discussed. The main characteristic of the new methods is that they are designed to solve the full optimization problem directly, rather than accelerating a descent method by an efficient multigrid solver for the equations involved. The methods use the adjoint state in order to achieve efficient smoother and a robust coarsening strategy. The main idea is the treatment of the control variables on appropriate scales, i.e., control variables that correspond to smooth functions are solved for on coarse grids depending on the smoothness of these functions. Solution of the control problems is achieved with the cost of solving the constraint equations about two to three times (by a multigrid solver). Numerical examples demonstrate the effectiveness of the method proposed in distributed control case, pointwise control and boundary control problems.

Taasan, Shlomo↗