Discrete Adjoint Research Articles

A discrete adjoint gradient approach for equality and inequality constraints in dynamics

The optimization of multibody systems requires accurate and efficient methods for sensitivity analysis. The adjoint method is probably the most efficient way to analyze sensitivities, especially for optimization problems with numerous optimization variables. This paper discusses sensitivity analysis for dynamic systems in gradient-based optimization problems. A discrete adjoint gradient approach is presented to compute sensitivities of equality and inequality constraints in dynamic simulations. The constraints are combined with the dynamic system equations, and the sensitivities are computed straightforwardly by solving discrete adjoint algebraic equations. The computation of these discrete adjoint gradients can be easily adapted to deal with different time integrators. This paper demonstrates discrete adjoint gradients for two different time-integration schemes and highlights efficiency and easy applicability. The proposed approach is particularly suitable for problems involving large-scale models or high-dimensional optimization spaces, where the computational effort of computing gradients by finite differences can be enormous. Three examples are investigated to validate the proposed discrete adjoint gradient approach. The sensitivity analysis of an academic example discusses the role of discrete adjoint variables. The energy optimal control problem of a nonlinear spring pendulum is analyzed to discuss the efficiency of the proposed approach. In addition, a flexible multibody system is investigated in a combined optimal control and design optimization problem. The combined optimization provides the best possible mechanical structure regarding an optimal control problem within one optimization.

Numerical and experimental investigations on shape optimization of submerged floating tunnels with a discrete adjoint method

The shape of the tube in submerged floating tunnels (SFTs) plays a critical role in determining their performance and safety in marine environments. In this study, a gradient optimization procedure based on the discrete adjoint method is performed to minimize the drag force under uniform currents action. The free form deformation approach is employed to parameterize the design variables. The physical mechanisms of the optimization process are revealed via unsteady numerical simulations and experimental tests. Moreover, the hydrodynamic performance of the tube with the optimal shapes subjected to a wave–current combination is also evaluated. The results show that the drag coefficient is significantly reduced through optimization by reducing the pressure amplitude in the negative-pressure area. Additionally, the lift oscillation is also suppressed to delay structural fatigue, owing to the strength reduction and increased distance of the wake vortex. The experimental analysis indicates the advantage of the optimal shape in reducing the loads under wave–current actions, accompanied by changes in the frequency distribution of the force and vortex structure. The elliptical shape of the SFT's tube provides significant advantages in drag reduction at high Reynold number. Suggestions on the shape design of the section of SFT are given according to different types of constraints.

Adjoint-based aerodynamic shape optimization with hybridized discontinuous Galerkin methods

We present a discrete adjoint approach to aerodynamic shape optimization (ASO) based on a hybridized discontinuous Galerkin (HDG) discretization. Our implementation is designed to tie in as seamlessly as possible into a solver architecture written for general balance laws, thus adding design capability to a tool with a wide range of applicability. Design variables are introduced on designated surfaces using the knots of a 2D spline-based geometry representation, while gradients are computed from the adjoint solution using a difference approximation of residual perturbations. A suitable optimization algorithm, such as an in-house steepest descent or the Preconditioned Sequential Quadratic Programming (PSQP) approach from the pyOpt framework, is then employed to find an improved geometry. We present verification of the implementation, including drag or heat flux minimization in compressible flows, as well as inverse design.

The discrete canonical commutation relationship

We use non-standard finite differences to propose a quantum momentum operator to be used when the spectrum of the operator is discrete. The defined discrete operator complies with the discrete versions of the properties that the continuous variable operator has. The discrete derivative is exact for its eigenfunction, that is, exponential functions. We obtain the discrete adjoint of the momentum operator. The canonical commutation relationship between conjugate operators for discrete variables is diagonal along a particular direction.

Discrete adjoint method for variational integration of constrained ODEs and its application to optimal control of geometrically exact beam dynamics

Direct methods for the simulation of optimal control problems apply a specific discretization to the dynamics of the problem, and the discrete adjoint method is suitable to calculate corresponding conditions to approximate an optimal solution. While the benefits of structure preserving or geometric methods have been known for decades, their exploration in the context of optimal control problems is a relatively recent field of research. In this work, the discrete adjoint method is derived for variational integrators yielding structure preserving approximations of the dynamics firstly in the ODE case and secondly for the case in which the dynamics is subject to holonomic constraints. The convergence rates are illustrated by numerical examples. Thirdly, the discrete adjoint method is applied to geometrically exact beam dynamics, represented by a holonomically constrained PDE.

Three-dimensional shape optimization of a submerged body under wave diffraction

This study explores the use of shape optimization to reduce wave forces on a submerged floating body subjected to wave diffraction. To this end, gradient-based shape optimization is adopted, in which dimensionless wave excitation forces are the optimization objective. A shape parameterization method based on the Fourier-series expansion is developed that permits the representation of an arbitrary three-dimensional floating body. The discrete adjoint method is utilized to calculate the gradient of the objective function with respect to the shape parameters. Using three-dimensional shape optimization, taking the initial shape to be a hemisphere, a significant reduction in surge, heave, and pitch wave forces is achieved, with a maximum reduction of 48.40%, 68.43%, and 46.22%, respectively. Furthermore, optimization effectively suppresses wave run-up, with a maximum reduction of 15.62%. A comprehensive analysis of parameters is performed to reveal the effects of wave number, incident angle, and shape parameters on the final optimized shape and wave load characteristics. This study provides a solid guide to the optimization of floating offshore platforms and the development of innovative structural systems.

Adjoint and Direct Characteristic Equations for Two-Dimensional Compressible Euler Flows

The method of characteristics is a classical method for gaining understanding in the solution of a partial differential equation. It has recently been applied to the adjoint equations of the 2D steady-state Euler equations and the first goal of this paper is to present a linear algebra analysis that greatly simplifies the discussion of the number of independent characteristic equations satisfied along a family of characteristic curves. This method may be applied for both the direct and the adjoint problem. Our second goal is to directly derive in conservative variables the characteristic equations of 2D compressible inviscid flows. Finally, the theoretical results are assessed for a nozzle flow with a classical scheme and its dual consistent discrete adjoint.

A discrete adjoint framework coupled with adaptive PCE for robust aerodynamic optimization of turbomachinery under flow uncertainty

Flow uncertainty is commonly encountered in turbomachinery. To mitigate the negative effects caused by the flow uncertainty, a framework coupled with adaptive polynomial chaos expansion (PCE) and discrete adjoint method for robust aerodynamic design optimization (RADO) of turbomachinery blades is developed. The optimization framework relies on gradient algorithms, which effectively avoids “dimensional curse” problem. Adaptive PCE method improves the efficiency of uncertainty quantification and only once PCE model need to be constructed in per optimization step, making the framework highly efficient. The PCE matrix and chain rule are employed to determine the gradient of the RADO objective function. This framework is employed for optimizing turbine blade, considering both inviscid and viscous flow. Monte Carlo simulation and finite difference PCE are used to validate the reliability of uncertainty quantification and gradient propagation methods. A comparative analysis of the optimization results obtained by both RADO and deterministic aerodynamic design optimization (DADO) is demonstrated. The higher robustness brought by RADO indicates the practicality of this framework in robust optimization of turbomachinery blades.

Shape optimization of the exhaust hood in machining workshops by a discrete adjoint method

Local exhaust system is an effective but energy-consuming method for capturing oil mist particles in machining workshops. To reduce the flow resistance of an exhaust system for minimal fan energy consumption, the method of applying individually shape-optimized exhaust hoods, namely mass-production design, is feasible. However, the combined effect of multiple exhaust hoods in an exhaust system may not be optimal in reducing the flow resistance. This investigation thus firstly validated the shape optimization of an individual exhaust hood by a discrete adjoint method. The discrete adjoint method could adjust the shape of an exhaust hood automatically in the direction of reducing the flow resistance. The design variables were the coordinates of wall boundaries of the exhaust hood. The validation used measured data from a small-scale experiment. This study then applied the validated discrete adjoint method to conduct customized design through the shape optimization of multiple exhaust hoods simultaneously in the exhaust system. The flow resistance under customized design was compared with the method of mass-production design. The results revealed that the customized design led to different shapes of individual exhaust hoods and they were different from the shape of the individually optimized exhaust hood. The flow resistance of the exhaust system under customized design was reduced by 57%. However, only 36.5% reduction in flow resistance was achieved when the mass-production design method was employed. The customized design method was more effective in reducing flow resistance of the exhaust system.

Multi-Objective Aerodynamic and Aeroelastic Coupled Design Optimization Using a Full Viscosity Discrete Adjoint Harmonic Balance Method

Abstract With increasing requirements for high-loading and high-efficiency turbomachines, blades become thinner and thinner and thus design optimizations considering both aerodynamic performances and aeroelastic stability become more and more necessary. In this study, a full viscosity discrete adjoint harmonic balance solver has been developed using algorithmic differentiation (AD), verified by a discrete linear solver based upon duality property, and then adopted to perform multi-disciplinary coupled design optimizations. To this end, a framework of multi-objective adjoint design optimizations has been developed to improve both aerodynamic performances and the aeroelastic stability of turbomachinery blades. This framework is divided into two steps: aeroelastic design initialization and aerodynamic Pareto front determination. First, the blade profiles are optimized to improve the aeroelastic stability only and constrain the variations of aerodynamic performances. Second, the optimized blade profiles in the first step are used as the initial ones and then further optimized with the objective function of aerodynamic parameters and the constraints of aeroelastic parameters. The effectiveness of the multi-objective design optimization method is demonstrated by comparing the optimization results with those from a single-objective aerodynamic and aeroelastic coupled design optimization method. The results from the transonic NASA Rotor 67 subjected to a hypothetical vibration mode show that the multi-objective coupled design optimization method is capable of improving performances in both disciplines.

Effect of constant eddy viscosity assumption on optimization using a discrete adjoint method

This paper presents a thorough study of the effect of the Constant Eddy Viscosity (CEV) assumption on the optimization of a discrete adjoint-based design optimization system. First, the algorithms of the adjoint methods with and without the CEV assumption are presented, followed by a discussion of the two methods’ solution stability. Second, the sensitivity accuracy, adjoint solution stability, and Root Mean Square (RMS) residual convergence rates at both design and off-design operating points are compared between the CEV and full viscosity adjoint methods in detail. Finally, a multi-point steady aerodynamic and a multi-objective unsteady aerodynamic and aeroelastic coupled design optimizations are performed to study the impact of the CEV assumption on optimization. Two gradient-based optimizers, the Sequential Least-Square Quadratic Programming (SLSQP) method and Steepest Descent Method (SDM) are respectively used to draw a firm conclusion. The results from the transonic NASA Rotor 67 show that the CEV assumption can deteriorate RMS residual convergence rates and even lead to solution instability, especially at a near stall point. Compared with the steady cases, the effect of the CEV assumption on unsteady sensitivity accuracy is much stronger. Nevertheless, the CEV adjoint solver is still capable of achieving optimization goals to some extent, particularly if the flow under consideration is benign.

Aerodynamic Optimization Design of Supersonic Wing Based on Discrete Adjoint

Reducing fuel consumption and improving the economy by effectively reducing cruising drag is the main objective of the aerodynamic design of supersonic civil aircraft. In this paper, the aerodynamic optimization design system based on the Reynolds-Averaged Navier–Stokes (RANS) equation and discrete adjoint theory is applied to supersonic wing design. Based on this system, a single-point optimization design study of aerodynamic drag reduction in cruise conditions was carried out for two typical supersonic wing layouts, subsonic leading edge and supersonic leading edge, and the drag reduction reached 3.78% and 4.53%, respectively. The aerodynamic design characteristics of different types of supersonic wings were explored from the perspectives of wing load, twist angle distribution, pressure distribution, airfoil shape characteristics, and flow field characteristics. The optimization results show that the drag reduction of the subsonic leading edge configuration is dominated by the induced drag, while the optimizer mainly focuses on reducing the shock wave drag for the supersonic leading edge configuration. By comparing the sensitivity analysis of lift and drag coefficients to airfoil deformation with the optimization results, the optimized dominant directions of two types of supersonic wings are qualitatively analyzed. The derivatives obtained from discrete adjoint equations are useful to elaborate the design tendency and the reason for the trade-off generation of supersonic wings under specific layouts and engineering constraints, which provides a reference for the design of supersonic wings in the future.

Pure-displacement formulation and bulk modulus propagation for topology optimization with pressure loads

This work proposes a pure-displacement formulation for topology optimization with pressure loads and the use of a continuous boundary propagation technique. This technique allows controlling the non-solid (fluid or void) constitutive behavior by disseminating the desired bulk modulus to holes in the solid domain. The novelty of the proposed approach is to enable the distribution of solid, fluid, and void simultaneously with one design variable field. A density-based topology optimization approach is considered, and a gradient-based interior point algorithm is used. The linear elasticity equilibrium equations are discretized by the finite element method and solved with a sparse direct solver. The sensitivities are calculated by the discrete adjoint method with automatic differentiation. The optimization problems encompass compliance minimization (in 2D and 3D) and a compliant mechanism design to maximize the output displacement. Three examples evaluate the proposed formulation: the internally-pressurized lid, the piston head, and the inverter-compliant mechanism. The results show that different designs are obtained by considering different constitutive behavior for the non-solid phase.

A discrete adjoint full potential formulation for fast aerostructural optimization in preliminary aircraft design

Preliminary aircraft design is often carried out with the help of multidisciplinary optimization processes. Because there is a strong coupling between the flow and the structure of the aircraft, and because new composite materials have a unique capability of being designed for anisotropic directional stiffness and strength, these processes must model the aeroelastic behavior of the aircraft in order to be effective. Since many design variables are involved during the preliminary design stage, the optimization problems, formulated using high or medium fidelity models, can be solved using the adjoint method. Moreover, the level of fidelity of the fluid model and the associated simulation technique must also be selected with care, as they tend to be the main contributors to the computational cost. The novel contribution of the present work is twofold. Firstly, the discretized gradients of the full potential flow equation, which is a medium-fidelity model, are derived analytically so that they can be used in adjoint optimization problems. Moreover, the full potential flow solution and the computation of the gradients are implemented in an open-source and readily available finite element code. Secondly, aerodynamic shape and aerostructural optimization calculations are carried out on example wings to demonstrate the effectiveness and the computational efficiency of the proposed method. Overall, the results show that the newly implemented discrete adjoint nonlinear potential flow formulation is able to quickly optimize both the shape and the structural parameters of a typical wing. More specifically, the twist distribution along the wingspan is adapted to reduce the induced drag, the airfoils become more supercritical so that the shock strength and the associated wave drag are reduced, and the thickness of the structural elements is tailored to the loads to reduce the internal stresses. The present methodology is therefore able to deliver results at a low computational cost which are sufficiently accurate for the early design stages. Furthermore, the results obtained using the proposed methodology could be used as a starting point for the optimization calculations performed in later design stages.

Probability of Initiation in Neutron Transport

We discuss the numerical solution of the nonlinear integro-differential equation for the probability of a divergent neutron chain in a stationary system (i.e., the probability of initiation (POI)). We follow the development described in Bell’s classic paper on the stochastic theory of neutron transport. As noted by Bell, the linearized form of this equation resembles the linear adjoint neutron transport equation. A matrix formalism for the discretized steady state (or forward) neutron equation in slab geometry is first developed, and is then used to derive the discrete adjoint equation. A main advantage of this discrete development is that the resulting discrete adjoint equation does not depend upon how the multigroup cross sections for the forward problem are obtained. That is, we derive the discrete adjoint directly from the discrete forward equations rather than discretizing directly the adjoint equation. This also guarantees that the discrete adjoint operator is consistent with the inner product used to define the adjoint operator. We discuss three approaches for the numerical solution of the POI equations, and present numerical results on several test problems. The three solution methods are a simple fixed point iteration, a second approach that is akin to a nonlinear Power iteration, and a third approach which uses a Newton-Krylov nonlinear solver. We also give sufficient conditions to guarantee the existence and uniqueness of nontrivial solutions to our discrete POI equations when the discrete system is supercritical, and that only the trivial solution exists when the discrete system is subcritical. Our approach is modeled after the analysis presented for the continuous POI equations by Mokhtar-Kharroubi and Jarmouni-Idrissi, and by Pazy and Rabinowitz.

Multi-point aero-structural design optimization of wings considering drag-divergence constraints

Considering the drag-divergence performance as a constraint, this paper presents a multi-point aero-structural design optimization of wings on a wing-body-tail-engine configuration of long-range dual-aisle civil aircraft by using a gradient-based method based on the discrete adjoint method. Firstly, the accuracy of the coupled aero-structural analysis method used in this paper was validated with DLR-F6 wing-body configuration. Then a wing aero-structural design optimization based on a dual-aisle civil aircraft is offered. The drag coefficient of the optimized configuration in every state decreased, which was reduced by 13.67 counts at the cruise condition, the difference in drag coefficient from Mach 0.85 to Mach 0.87 was decreased from 28.52 counts to 18.98 counts, indicating its drag-divergence performance has been improved undoubtedly. Finally, we compared the performance among the multi-point aero-structural optimized configuration, the single-point aerodynamic optimized configuration and the single-point structural optimized configuration. The results show that the multi-point aero-structural design optimization considering drag-divergence performance has great potential to gain a design configuration with better comprehensive and practical performance compared with a single-point optimization in a single discipline.

Adjoint-based robust optimization design of laminar flow wing under flight condition uncertainties

It is an inherent uncertainty problem that the application of laminar flow technology to the wing of large passenger aircraft is affected by flight conditions. In order to seek a more robust natural laminar flow control effect, it is necessary to develop an effective optimization design method. Meanwhile, attention must be given to the impact of crossflow (CF) instability brought on by the sweep angle. This paper constructs a robust optimization design framework based on discrete adjoint methods and non-intrusive polynomial chaos. Transition prediction is implemented by coupled Reynolds-Averaged Navier-Stokes (RANS) and simplified eN method, which can consider both Tollmien-Schlichting (TS) wave and crossflow vortex instability. We have performed gradient enhancement processing on the general Polynomial Chaos Expansion (PCE), which is advantageous to reduce the computational cost of single uncertainty propagation. This processing takes advantage of the gradient information obtained by solving the coupled adjoint equations considering transition. The statistical moment gradient solution used for the robust optimization design also uses the derivatives of coupled adjoint equations. The framework is applied to the robust design of a 25° swept wing with infinite span in transonic flow. The uncertainty quantification and sensitivity analysis on the baseline wing shows that the uncertainty quantification method in this paper has high accuracy, and qualitatively reveals the factors that dominate in different flow field regions. By the robust optimization design, the mean and standard deviation of the drag coefficient can be reduced by 29% and 45%, respectively, and compared with the deterministic optimization design results, there is less possibility of forming shock waves under flight condition uncertainties. Robust optimization results illustrate the trade-off between the transition delay and the wave drag reduction.

Adjoint Based Aerodynamic Shape Optimisation Using Kinetic Meshfree Method

The gradient based optimisation algorithms combined with the finite volume or element based adjoint approaches have been very successful in aerodynamic shape optimization (ASO). The meshfree least squares kinetic upwind method (LSKUM), which works on a cloud of points and its connectivity, has the advantage in terms of flexibility of generating a cloud of points and having an arbitrary number of points in the connectivity. The LSKUM based meshfree solvers have been successfully applied to problems involving multi-body configurations and moving boundaries. In the present work, the LSKUM based primal and discrete adjoint meshfree solvers for steady Euler flows have been used to perform aerodynamic shape optimisation. It is well-known that the raw shape sensitivities of a defined objective function are highly oscillatory. In this research, we have used the Sobolev gradient smoothing. An approach is developed to find a suitable choice for the smoothing parameter in the Sobolev gradient algorithm. Numerical results are presented for the shape optimisation of the NACA0012 airfoil at transonic and supersonic flows with aerodynamic and geometric constraints. The performance of the discrete adjoint LSKUM solver in accurate computation of shape sensitivities is demonstrated. The advantages of using LSKUM in aerodynamic shape optimisation have been shown.

Free-Form Deformation Parameterization on the Aerodynamic Optimization of Morphing Trailing Edge

Every aerodynamic optimization is proceeded by a parameterization of the studied aerial object, and due to its influence on the final optimization process, careful attention should be made in choosing the appropriate parameterization method. An aerodynamic optimization of a morphing trailing edge is performed using a free-form deformation parameterization technique with the purpose of examining the influence of the initial conditions of the parameterization on the optimization results, namely on the number of control points. High-fidelity gradient-based optimization using the discrete adjoint method is established by the coupling of OpenFOAM and Python within the DAFoam optimization framework. The results indicate that the number of control points has a considerable effect on the optimization process, in particular on the convergence, objective function value, and on the deformation feasibility.

Machine learning for RANS turbulence modeling of variable property flows

This paper presents a machine learning methodology to improve the predictions of traditional RANS turbulence models in channel flows subject to strong variations in their thermophysical properties. The developed formulation contains several improvements over the existing Field Inversion Machine Learning (FIML) frameworks described in the literature. We first showcase the use of efficient optimization routines to automatize the process of field inversion in the context of CFD, combined with the use of symbolic algebra solvers to generate sparse-efficient algebraic formulas to comply with the discrete adjoint method. The proposed neural network architecture is characterized by the use of an initial layer of logarithmic neurons followed by hyperbolic tangent neurons, which proves numerically stable. The machine learning predictions are then corrected using a novel weighted relaxation factor methodology, that recovers valuable information from otherwise spurious predictions. Additionally, we introduce L2 regularization to mitigate over-fitting and to reduce the importance of non-essential features. In order to analyze the results of our deep learning system, we utilize the K-fold cross-validation technique, which is beneficial for small datasets. The results show that the machine learning model acts as an excellent non-linear interpolator for DNS cases well-represented in the training set. In the most successful case, the L-infinity modeling error on the velocity profile was reduced from 23.4% to 4.0%. It is concluded that the developed machine learning methodology corresponds to a valid alternative to improve RANS turbulence models in flows with strong variations in their thermophysical properties without introducing prior modeling assumptions into the system.