Discrete Adjoint Research Articles

The morphing trailing-edge wing optimization design of the civil aircraft

The research on the aerodynamic optimization design of applying the continuous trailing-edge variable camber wing to the civil aircraft is presented, and the necessity of considering pitch moment trimming in the optimization design is explored. The free-form deform (FFD) technique is used to accomplish the parameterization of the wing shape, the trailing-edge camber and the rotation of the horizontal tail. The discrete adjoint technique based on RANS (Reynolds Averaged Navier-Stokes) equations is used to solve the gradients of aerodynamic coefficients with regard to design variables, and the sequential quadratic programming is used to carry the gradient-based optimization design. Based on the common research model (CRM), a single target optimization with constraints is carried to reduce the aerodynamic drag, and the feasibility of the optimization system is confirmed. Then, in different cruise lift coefficients, the trailing-edge variable camber wing optimizations are carried with and without moment trimming respectively. The optimization results show that by trailing-edge morphing, the spanwise lift coefficient distribution is improved and the wave strength is reduced; in order to achieve the optimized design result, it is a necessity to consider the moment trimming constraint.

Adjoint sensitivity analysis of chaotic dynamical systems with non-intrusive least squares shadowing

This paper presents a discrete adjoint version of the recently developed non-intrusive least squares shadowing (NILSS) algorithm, which circumvents the instability that conventional adjoint methods encounter for chaotic systems. The NILSS approach involves solving a smaller minimization problem than other shadowing approaches and can be implemented with only minor modifications to preexisting tangent and adjoint solvers. Adjoint NILSS is demonstrated on a small chaotic ODE, a one-dimensional scalar PDE, and a direct numerical simulation (DNS) of the minimal flow unit, a turbulent channel flow on a small spatial domain. This is the first application of an adjoint shadowing-based algorithm to a three-dimensional turbulent flow.

Accurate Discrete Adjoint Approach for Optimal Active Separation Control

This paper presents the development of an accurate and robust discrete adjoint approach for the optimization of active flow control mechanism. The adjoint solver is developed by applying algorithmic differentiation techniques to an incompressible unsteady Reynolds-averaged Navier–Stokes solver. Coupled with a quasi-Newton gradient-search optimization algorithm, its applicability is demonstrated by finding the optimal synthetic jet actuation on the suction side of the NACA 4412 airfoil at a Reynolds number and angle of incidence of 20 deg. Numerical results have shown that the actuation sensitivities based on the unsteady discrete adjoint solver are in excellent agreement with the values obtained from the tangent-linear solver and finite differences. Compared to the stalled baseline flow, optimal actuated flow significantly increased the time-averaged lift by 54%, and the stall is delayed from 16 to 22 deg. A detailed analysis of sensitivities and the effect of optimal actuation on the separation is presented.

Waterjet and laser etching: the nonlinear inverse problem.

In waterjet and laser milling, material is removed from a solid surface in a succession of layers to create a new shape, in a depth-controlled manner. The inverse problem consists of defining the control parameters, in particular, the two-dimensional beam path, to arrive at a prescribed freeform surface. Waterjet milling (WJM) and pulsed laser ablation (PLA) are studied in this paper, since a generic nonlinear material removal model is appropriate for both of these processes. The inverse problem is usually solved for this kind of process by simply controlling dwell time in proportion to the required depth of milling at a sequence of pixels on the surface. However, this approach is only valid when shallow surfaces are etched, since it does not take into account either the footprint of the beam or its overlapping on successive passes. A discrete adjoint algorithm is proposed in this paper to improve the solution. Nonlinear effects and non-straight passes are included in the optimization, while the calculation of the Jacobian matrix does not require large computation times. Several tests are performed to validate the proposed method and the results show that tracking error is reduced typically by a factor of two in comparison to the pixel-by-pixel approach and the classical raster path strategy with straight passes. The tracking error can be as low as 2–5% and 1–2% for WJM and PLA, respectively, depending on the complexity of the target surface.

Discrete Adjoint Sensitivity Analysis of Hybrid Dynamical Systems With Switching

Sensitivity analysis is an important tool for describing power system dynamic behavior in response to parameter variations. It is a central component in preventive and corrective control applications. The existing approaches for sensitivity calculations, namely, finite-difference and forward sensitivity analysis, require a computational effort that increases linearly with the number of sensitivity parameters. In this paper, we investigate, implement, and test a discrete adjoint sensitivity approach whose computational effort is effectively independent of the number of sensitivity parameters. The proposed approach is highly efficient for calculating sensitivities of larger systems and is consistent, within machine precision, with the function whose sensitivity we are seeking. This is an essential feature for use in optimization applications. Moreover, our approach includes a consistent treatment of systems with switching, such as dc exciters, by deriving and implementing the adjoint jump conditions that arise from state-dependent and time-dependent switchings. The accuracy and the computational efficiency of the proposed approach are demonstrated in comparison with the forward sensitivity analysis approach. This paper focuses primarily on the power system dynamics, but the approach is general and can be applied to hybrid dynamical systems in a broader range of fields.

Tiltrotor CFD Part II - aerodynamic optimisation of tiltrotor blades

ABSTRACTThis paper presents aerodynamic optimisation of tiltrotor blades with high-fidelity computational fluid dynamics. The employed optimisation framework is based on a quasi-Newton method, and the required high-fidelity flow gradients were computed using a discrete adjoint solver. Single-point optimisations were first performed to highlight the contrasting requirements of the helicopter and aeroplane flight regimes. It is then shown how a trade-off blade design can be obtained using a multi-point optimisation strategy. The parametrisation of the blade shape allowed the twist and chord distributions to be modified and a swept tip to be introduced. The work shows how these main blade shape parameters influence the optimal performance of the tiltrotor in helicopter and aeroplane modes, and how an optimised blade shape can increase the overall tiltrotor performance. Moreover, in all the presented cases, the accuracy of the adjoint gradients resulted in a small number of flow evaluations for finding the optimal solution, thus indicating gradient-based optimisation as a viable tool for modern tiltrotor design.

Adjoint-based sensitivity analysis of low-order thermoacoustic networks using a wave-based approach

Strict pollutant emission regulations are pushing gas turbine manufacturers to develop devices that operate in lean conditions, with the downside that combustion instabilities are more likely to occur. Methods to predict and control unstable modes inside combustion chambers have been developed in the last decades but, in some cases, they are computationally expensive. Sensitivity analysis aided by adjoint methods provides valuable sensitivity information at a low computational cost. This paper introduces adjoint methods and their application in wave-based low order network models, which are used as industrial tools, to predict and control thermoacoustic oscillations. Two thermoacoustic models of interest are analyzed. First, in the zero Mach number limit, a nonlinear eigenvalue problem is derived, and continuous and discrete adjoint methods are used to obtain the sensitivities of the system to small modifications. Sensitivities to base-state modification and feedback devices are presented. Second, a more general case with non-zero Mach number, a moving flame front and choked outlet, is presented. The influence of the entropy waves on the computed sensitivities is shown.

A 4D-Var inversion system based on the icosahedral grid model (NICAM-TM 4D-Var v1.0) – Part 1: Offline forward and adjoint transport models

Abstract. A four-dimensional variational (4D-Var) method is a popular algorithm for inverting atmospheric greenhouse gas (GHG) measurements. In order to meet the computationally intense 4D-Var iterative calculation, offline forward and adjoint transport models are developed based on the Nonhydrostatic ICosahedral Atmospheric Model (NICAM). By introducing flexibility into the temporal resolution of the input meteorological data, the forward model developed in this study is not only computationally efficient, it is also found to nearly match the transport performance of the online model. In a transport simulation of atmospheric carbon dioxide (CO2), the data-thinning error (error resulting from reduction in the time resolution of the meteorological data used to drive the offline transport model) is minimized by employing high temporal resolution data of the vertical diffusion coefficient; with a low 6-hourly temporal resolution, significant concentration biases near the surface are introduced. The new adjoint model can be run in discrete or continuous adjoint mode for the advection process. The discrete adjoint is characterized by perfect adjoint relationship with the forward model that switches off the flux limiter, while the continuous adjoint is characterized by an imperfect but reasonable adjoint relationship with its corresponding forward model. In the latter case, both the forward and adjoint models use the flux limiter to ensure the monotonicity of tracer concentrations and sensitivities. Trajectory analysis for high CO2 concentration events are performed to test adjoint sensitivities. We also demonstrate the potential usefulness of our adjoint model for diagnosing tracer transport. Both the offline forward and adjoint models have computational efficiency about 10 times higher than the online model. A description of our new 4D-Var system that includes an optimization method, along with its application in an atmospheric CO2 inversion and the effects of using either the discrete or continuous adjoint method, is presented in an accompanying paper Niwa et al.(2016).

Optimization of wind plant layouts using an adjoint approach

Abstract. Using adjoint optimization and three-dimensional steady-state Reynolds-averaged Navier–Stokes (RANS) simulations, we present a new gradient-based approach for optimally siting wind turbines within utility-scale wind plants. By solving the adjoint equations of the flow model, the gradients needed for optimization are found at a cost that is independent of the number of control variables, thereby permitting optimization of large wind plants with many turbine locations. Moreover, compared to the common approach of superimposing prescribed wake deficits onto linearized flow models, the computational efficiency of the adjoint approach allows the use of higher-fidelity RANS flow models which can capture nonlinear turbulent flow physics within a wind plant. The steady-state RANS flow model is implemented in the Python finite-element package FEniCS and the derivation and solution of the discrete adjoint equations are automated within the dolfin-adjoint framework. Gradient-based optimization of wind turbine locations is demonstrated for idealized test cases that reveal new optimization heuristics such as rotational symmetry, local speedups, and nonlinear wake curvature effects. Layout optimization is also demonstrated on more complex wind rose shapes, including a full annual energy production (AEP) layout optimization over 36 inflow directions and 5 wind speed bins.

Automatic differentiation based discrete adjoint method for aerodynamic design optimization on unstructured meshes

A systematic methodology for formulating, implementing, solving and verifying discrete adjoint of the compressible Reynolds-averaged Navier-Stokes (RANS) equations for aerodynamic design optimization on unstructured meshes is proposed. First, a general adjoint formulation is constructed for the entire optimization problem, including parameterization, mesh deformation, flow solution and computation of the objective function, which is followed by detailed formulations of matrix-vector products arising in the adjoint model. According to this formulation, procedural components of implementing the required matrix-vector products are generated by means of automatic differentiation (AD) in a structured and modular manner. Furthermore, a duality-preserving iterative algorithm is employed to solve flow adjoint equations arising in the adjoint model, ensuring identical convergence rates for the tangent and the adjoint models. A three-step strategy is adopted to verify the adjoint computation. The proposed method has several remarkable features: the use of AD techniques avoids tedious and error-prone manual derivation and programming; duality is strictly preserved so that consistent and highly accurate discrete sensitivities can be obtained; and comparable efficiency to hand-coded implementation can be achieved. Upon the current discrete adjoint method, a gradient-based optimization framework has been developed and applied to a drag reduction problem.

Design sensitivity analysis for transient response of non-viscously damped dynamic systems

A design sensitivity analysis for the transient response of the non-viscously damped dynamic systems is presented. The non-viscously (viscoelastically) damped system is widely used in structural vibration control. The damping forces in the system depend on the past history of motion via convolution integrals. The non-viscos damping is modeled by the generalized Maxwell model. The transient response is calculated with the implicit Newmark time integration scheme. The design sensitivity analysis method of the history dependent system is developed using the adjoint variable method. The discretize-then-differentiate approach is adopted for deriving discrete adjoint equations. The accuracy and the consistency of the proposed method are demonstrated through a single dof system. The proposed method is also applied to a multi-dof system. The validity and accuracy of the sensitivities from the proposed method are confirmed by finite difference results.

Uncertainty Analysis and Robust Design of Low-Boom Concepts Using Atmospheric Adjoints

This paper seeks to quantify the uncertainty associated with atmospheric conditions when propagating shaped pressure disturbances from a vehicle flying at supersonic speeds. A discrete adjoint formulation is used to obtain sensitivities of boom metrics to atmospheric inputs such as temperature, wind, and relative humidity profiles in addition to deterministic inputs such as the near-field pressure distribution. This study uses a polynomial chaos theory approach to couple these adjoint-derived gradients with uncertainty quantification to enable robust design by using gradient-based optimization techniques. The effectiveness of this approach is demonstrated over an axisymmetric body of revolution and a low-boom concept. Results show that the mean and standard deviation of sonic-boom loudness are simultaneously reduced using robust optimization. Unlike the conventional optimization approaches, the robust optimization approach has the added benefit of generating probability distributions of the sonic-boom metrics under atmospheric uncertainty.

The Discrete Adjoint Gradient Computation for Optimization Problems in Multibody Dynamics

The adjoint method is a very efficient way to compute the gradient of a cost functional associated to a dynamical system depending on a set of input signals. However, the numerical solution of the adjoint differential equations raises several questions with respect to stability and accuracy. An alternative and maybe more natural approach is the discrete adjoint method (DAM), which constructs a finite difference scheme for the adjoint system directly from the numerical solution procedure, which is used for the solution of the equations of motion. The method delivers the exact gradient of the discretized cost functional subjected to the discretized equations of motion. For the application of the discrete adjoint method to the forward solver, several matrices are necessary. In this contribution, the matrices are derived for the simple Euler explicit method and for the classical implicit Hilber–Hughes–Taylor (HHT) solver.

An approach to computing discrete adjoints for MPI-parallelized models applied to Ice Sheet System Model 4.11

Abstract. Within the framework of sea-level rise projections, there is a strong need for hindcast validation of the evolution of polar ice sheets in a way that tightly matches observational records (from radar, gravity, and altimetry observations mainly). However, the computational requirements for making hindcast reconstructions possible are severe and rely mainly on the evaluation of the adjoint state of transient ice-flow models. Here, we look at the computation of adjoints in the context of the NASA/JPL/UCI Ice Sheet System Model (ISSM), written in C++ and designed for parallel execution with MPI. We present the adaptations required in the way the software is designed and written, but also generic adaptations in the tools facilitating the adjoint computations. We concentrate on the use of operator overloading coupled with the AdjoinableMPI library to achieve the adjoint computation of the ISSM. We present a comprehensive approach to (1) carry out type changing through the ISSM, hence facilitating operator overloading, (2) bind to external solvers such as MUMPS and GSL-LU, and (3) handle MPI-based parallelism to scale the capability. We demonstrate the success of the approach by computing sensitivities of hindcast metrics such as the misfit to observed records of surface altimetry on the northeastern Greenland Ice Stream, or the misfit to observed records of surface velocities on Upernavik Glacier, central West Greenland. We also provide metrics for the scalability of the approach, and the expected performance. This approach has the potential to enable a new generation of hindcast-validated projections that make full use of the wealth of datasets currently being collected, or already collected, in Greenland and Antarctica.

Smoothed truncation error in functional error estimation and correction using adjoint methods in an unstructured finite volume solver

Abstract The goal of this paper is to develop a reliable truncation error estimator for finite-volume schemes and to use this truncation error estimate to correct the output functional of interest. The rough modes in the truncation error for unstructured mesh are dominant and if p-truncation error estimation is used, functional correction has a poor performance. So, we are trying to obtain a smooth estimate of the truncation error to improve the performance of output error estimation. The correction term is based on the truncation error and the adjoint solution. Both discrete and continuous adjoint solutions are used for correcting the functional. Our results for a variety of governing equations suggest that the smoothing scheme can improve the correction process significantly.

Efficient goal-oriented global error estimators for BDF methods using discrete adjoints

In this paper we develop new goal-oriented global error estimators for variable multistep backward differentiation formulae (BDF) methods. These estimators use, for the first time in the context of multistep methods, discrete adjoints computed by adjoint differentiation of the nominal integration scheme. The derivation is based on the recently developed Petrov–Galerkin finite element formulation of BDF methods and their discrete adjoint schemes and makes use of the dual weighted residual methodology. Defect integrals or local truncation errors are used as local error quantities. We prove the asymptotic correctness and optimal convergence of the novel estimators for the one-step BDF method. For multistep BDF methods using a selfstarting procedure we show that the estimator with defect integrals converges but suboptimal and that its effectivity index converges to an offset value. On the other hand, the estimator with local truncation errors is again asymptotically correct and of optimal order. We confirm these results numerically. With a real-world example from chemical engineering, we give promising numerical results for the estimation accuracy in variable BDF-type methods with changing orders and stepsizes. Finally, we give a first use of the novel estimators for goal-oriented global error control of a challenging stiff test problem.

An adjoint method for a high-order discretization of deforming domain conservation laws for optimization of flow problems

The fully discrete adjoint equations and the corresponding adjoint method are derived for a globally high-order accurate discretization of conservation laws on parametrized, deforming domains. The conservation law on the deforming domain is transformed into one on a fixed reference domain by the introduction of a time-dependent mapping that encapsulates the domain deformation and parametrization, resulting in an Arbitrary Lagrangian–Eulerian form of the governing equations. A high-order discontinuous Galerkin method is used to discretize the transformed equation in space and a high-order diagonally implicit Runge–Kutta scheme is used for the temporal discretization. Quantities of interest that take the form of space–time integrals are discretized in a solver-consistent manner. The corresponding fully discrete adjoint method is used to compute exact gradients of quantities of interest along the manifold of solutions of the fully discrete conservation law. These quantities of interest and their gradients are used in the context of gradient-based PDE-constrained optimization.The adjoint method is used to solve two optimal shape and control problems governed by the isentropic, compressible Navier–Stokes equations. The first optimization problem seeks the energetically optimal trajectory of a 2D airfoil given a required initial and final spatial position. The optimization solver, driven by gradients computed via the adjoint method, reduced the total energy required to complete the specified mission nearly an order of magnitude. The second optimization problem seeks the energetically optimal flapping motion and time-morphed geometry of a 2D airfoil given an equality constraint on the x-directed impulse generated on the airfoil. The optimization solver satisfied the impulse constraint to greater than 8 digits of accuracy and reduced the required energy between a factor of 2 and 10, depending on the value of the impulse constraint, as compared to the nominal configuration.

An Efficient Unsteady Adjoint Optimization System for Multistage Turbomachinery

Unsteady blade row interactions play an important role in the performance of multistage turbomachinery. However, most aerodynamic optimizations of multistage turbomachinery are based on mixing-plane steady flow simulations. To take into account the unsteady flow features in the optimization cycle, this paper develops an adjoint-based unsteady aerodynamic optimization system for multistage turbomachinery. To the authors' best knowledge, this is the first work in the literature conducting the unsteady adjoint aerodynamic optimization of multistage turbomachinery. The unsteady flow equations and the discrete adjoint equations are solved using a finite volume code, with the harmonic balance method adopted to reduce the cost of unsteady simulations. The system is applied to the unsteady aerodynamic optimization of a 1.5-stage compressor. Results show the efficiency and capability of the proposed framework for the unsteady aerodynamic optimization of multistage turbomachinery.

Optimisation of Ducted Propellers for Hybrid Air Vehicles Using High-Fidelity CFD

ABSTRACTThis paper presents performance analysis and design of ducted propellers for lighter-than-air vehicles. High-fidelity computational fluid dynamics simulations were first performed on a detailed model of the propulsor, and the results were in very good agreement with available experimental data. Additional simulations were performed using a simplified geometry, to quantify the effect of the duct and of the blade twist on the propeller performance. It was shown that the duct is particularly effective at low flight speed and that the blades with relatively high twist have better performance over the flight envelope. Design of the optimal twist distribution and of the duct shape was also attempted by coupling the flow solver with a quasi-Newton optimisation method. Flow gradients were computed by solving the discrete adjoint of the Reynolds-averaged Navier-Stokes equations using a fixed-point iteration scheme or a nested Krylov method with deflated restarting. The results show that the ducted propeller propulsive efficiency can be increased by 2%.

A note on the dual consistency of the discrete adjoint quasi-one-dimensional Euler equations with cell-centered and cell-vertex central discretizations

A flow discretization is dual-consistent if the associated discrete adjoint equations are consistent with the analytic adjoint equations. We examine here the formulation and numerical solution of the discrete adjoint quasi-one-dimensional Euler equations derived from a second-order, central-difference, finite volume scheme, for both cell-centered and cell-vertex discretizations. It is shown that, while the cell-vertex discretization is dual-consistent, the cell-centered discretization is not, showing oscillations near the boundaries.