Effective adjoint approaches for computational fluid dynamics

An automatic differentiation tool is used to develop the adjoint code for a three-dimensional computational fluid dynamics solver. Rather than using automatic differentiation to differentiate the entire source code of the computational fluid dynamics solver, we have applied it selectively to produce code that computes the flux Jacobian matrix and the other partial derivatives that are necessary to compute total derivatives using an adjoint method. The resulting linear discrete adjoint system is then solved using the portable, extensible toolkit for scientific computation. This selective application of automatic differentiation is the central idea behind the automatic differentiation adjoint (ADjoint) approach. This approach has the advantage that it is applicable to arbitrary sets of governing equations and cost functions, and that it is exactly consistent with the gradients that would be computed by exact numerical differentiation of the original solver. Furthermore, the approach is largely automatic, thus avoiding the lengthy development times usually required to develop adjoint solvers for partial differential equations. These significant advantages come at the cost of increased memory requirements for the adjoint solver. Derivatives of drag and lift coefficients are validated, and the low computational cost and ease of implementation of the method are shown.

This paper presents an adjoint method for the optimal control of unsteady flows. The goal is to develop the continuous and discrete unsteady adjoint equations and their corresponding boundary conditions for the time-accurate method. First, this paper presents the complete formulation of the time-dependent optimal design problem. Second, we present the time-accurate unsteady continuous and discrete adjoint equations. Third, we present results that demonstrate the application of the theory to a two-dimensional oscillating airfoil. The results are compared with a multipoint approach to illustrate the added benefit of performing full unsteady optimization.

This paper addresses the problem of the design optimization of turbomachinery components under thermo-mechanical constraints, with focus on a radial turbine impeller for turbocharger applications. Typically, turbine components operate at high temperatures and are exposed to important thermal gradients, leading to thermal stresses. Dealing with such structural requirements necessitates the optimization algorithms to operate a coupling between fluid and structural solvers that is computationally intensive. To reduce the cost during the optimization, a novel multiphysics gradient-based approach is developed in this work, integrating a Conjugate Heat Transfer procedure by means of a partitioned coupling technique. The discrete adjoint framework allows for the efficient computation of the gradients of the thermo-mechanical constraint with respect to a large number of design variables. The contribution of the thermal strains to the sensitivities of the cost function extends the multidisciplinary outlook of the optimization and the accuracy of its predictions, with the aim of reducing the empirical safety factors applied to the design process. Finally, a turbine impeller is analyzed in a demanding operative condition and the gradient information results in a perturbation of the grid coordinates, reducing the stresses at the rotor back-plate, as a demonstration of the suitability of the presented method.

Space–time adaptive solution of inverse problems with the discrete adjoint method

The paper is concerned with the application of the space-time discontinuous Galerkin method (STDGM) to the numerical solution of the interaction of a compressible flow and an elastic structure. The flow is described by the system of compressible Navier-Stokes equations written in the conservative form. They are coupled with the dynamic elasticity system of equations describing the deformation of the elastic body, induced by the aerodynamical force on the interface between the gas and the elastic structure. The domain occupied by the fluid depends on time. It is taken into account in the Navier-Stokes equations rewritten with the aid of the arbitrary Lagrangian-Eulerian (ALE) method. The resulting coupled system is discretized by the STDGM using piecewise polynomial approximations of the sought solution both in space and time. The developed method can be applied to the solution of the compressible flow for a wide range of Mach numbers and Reynolds numbers. For the simulation of elastic deformations two models are used: the linear elasticity model and the nonlinear neo-Hookean model. The main goal is to show the robustness and applicability of the method to the simulation of the air flow in a simplified model of human vocal tract and the flow induced vocal folds vibrations. It will also be shown that in this case the linear elasticity model is not adequate and it is necessary to apply the nonlinear model.

A survey is given of extensive research work on cascade-flow problems carried out in recent years in Germany. A considerable part of this work was done in the Variable Density High Speed Cascade Wind Tunnel of the Deutsche Forschungsanstalt fu¨r Luftfahrt at Braunschweig, in which the Reynolds number and the Mach number of the cascade can be varied independently. For compressor cascades with blades of different thickness ratio extensive measurements of the aerodynamic coefficients have been carried out in a wide range of Mach numbers and Reynolds numbers. For very low Reynolds numbers, as they occur for jet engines in high-altitude flight, the influence of turbulence level on loss coefficients has been investigated. Furthermore, comprehensive investigations on secondary-flow losses are reported. The most important parameters in this connection are the ratio of blade length to blade chord, the tip clearance, the Reynolds number, and the deflection of the flow in the cascade. The influence of all these parameters on the secondary-flow losses has been clarified to a certain extent.

Two-dimensional open-cavity flows for free stream Mach number of 0.1 to 1.6 are investigated with direct numerical simulations for cavities of aspect ratio 2 and 6 to characterize the flow stability. We determine the neutral stability curve over a wide range of Mach numbers and Reynolds numbers. In particular, we examine the effects of compressibility on the flow in transonic regime at low Reynolds numbers. Rossiter modes are identified by spectral analysis. The corresponding spatial structures associated with the dominant and subdominant Rossiter modes are extracted with dynamic mode decomposition. The nature of these structures provides us insight to their contribution to instability of the flow. The interaction between the shock waves and the shear layer modes for the transonic flows are investigated.

Eigenvalue problem derivatives computation for a complex matrix using the adjoint method

Aerodynamic shape optimization of aircraft configurations often ignores stability considerations. To address this, a method for the computation of static, dynamic, and transient aircraft stability derivatives and their sensitivities for use in gradient-based optimization is introduced and evaluated. Computational fluid dynamics in the form of a three-dimensional structured-grid multiblock flow solver with both Euler and Reynolds-averaged Navier–Stokes equations is used. To compute the stability derivatives, a time-spectral formulation is used to compute an oscillating solution for the configuration of interest. From this oscillating solution, a series of linear regressions is performed to calculate the various stability derivatives. Because the solution is time dependent, it contains the information required to compute the transient, or “dot,” derivatives for the configuration. An adjoint method is used to compute the gradients of the stability derivatives of interest, enabling gradient-based stability-constrained aerodynamic shape optimization with respect to a large number of design variables. The computed stability derivatives are verified for an airfoil and validated for a generic unmanned combat aerial vehicle. The stability-constrained optimization of a wing demonstrates the viability and usefulness of the method for aircraft design optimization.

Wind turbines are complex systems where component‐level changes can have significant system‐level effects. Effective wind turbine optimization generally requires an integrated analysis approach with a large number of design variables. Optimizing across large variable sets is orders of magnitude more efficient with gradient‐based methods as compared with gradient‐free method, particularly when using exact gradients. We have developed a wind turbine analysis set of over 100 components where 90% of the models provide numerically exact gradients through symbolic differentiation, automatic differentiation, and adjoint methods. This framework is applied to a specific design study focused on downwind land‐based wind turbines.Downwind machines are of potential interest for large wind turbines where the blades are often constrained by the stiffness required to prevent a tower strike. The mass of these rotor blades may be reduced by utilizing a downwind configuration where the constraints on tower strike are less restrictive. The large turbines of this study range in power rating from 5–7MW and in diameter from 105m to 175m. The changes in blade mass and power production have important effects on the rest of the system, and thus the nacelle and tower systems are also optimized. For high‐speed wind sites, downwind configurations do not appear advantageous. The decrease in blade mass (10%) is offset by increases in tower mass caused by the bending moment from the rotor‐nacelle‐assembly. For low‐wind speed sites, the decrease in blade mass is more significant (25–30%) and shows potential for modest decreases in overall cost of energy (around 1–2%). Copyright © 2016 John Wiley & Sons, Ltd.

As a high-fidelity approach to transition prediction, the coupled Reynolds-averaged Navier–Stokes (RANS) and linear stability theory (LST)-based method is widely used in engineering applications and is the preferred method for laminar flow optimization. However, the further development of gradient-based laminar flow wing optimization schemes is hindered by a lack of efficient and accurate derivative computation methods for LST-based eigenvalue problems with a large number of design variables. To address this deficiency and to compute the derivatives in the LST-based solution solver, we apply the adjoint method and analytical reverse algorithm differentiation (RAD), which scale well with the number of inputs. The core of this paper is the computation of the standard eigenvalue and eigenvector derivatives for the LST problem, which involves a complex matrix. We develop an adjoint method to compute these derivatives, and we couple this method with RAD to reduce computational costs. In addition, we incorporate the LST-based partial derivatives into the laminar–turbulent transition prediction framework for the computation of total derivatives. We verify our proposed method with reference to finite difference (FD) results for an infinite swept wing. Both the intermediate derivatives from the transition module and total derivatives agree with the FD reference results to at least three digits, demonstrating the accuracy of our proposed approach. The fully adjoint and the coupled adjoint–RAD methods both have considerable advantages in terms of computational efficiency compared with iterative RAD and FD methods. The LST-based transition method and the proposed method for efficient and accurate derivative computations have prospects for wide application to laminar flow optimization in aerodynamic design.

The boom structure is a key component of giant boom cranes, and the stability-ensured topology optimization is critical to its lightweight design. The finite difference method, direct differentiation or adjoint method needs many time-consuming nonlinear analyses for this problem with a large number of design variables and constraints, and the last two methods are difficult to implement in off-the-shelf softwares. To overcome these challenges, this work first defines a global stability index to measure the global stability of the whole structure, and a compression member stability index to identify the buckling of compression members. Numerical and experimental verifications of these two stability indices are conducted by analyzing a simple three-dimensional frame. Next, the anti-buckling mechanism of boom structures is analyzed to develop the precedence order of freezing relative web members. The stability indices and the freezing measure are then utilized as a part of a novel Stability-Ensured Soft Kill Option (SSKO) algorithm, built upon the existing Soft Kill Option (SKO) method. The objective is to minimize the discrepancy between structural volume and predetermined target volume, while the global stability and stress are regarded as constraints. Lastly, the SSKO algorithm with different scenarios is applied to topology optimization problems of four-section frames and a ring crane boom; in both cases the consistent and stable topologies exhibit applicability of the proposed algorithm.

This work details a computational framework for gradient-based optimization of a nonlinear flapping wing structure with a large number of design variables. System response is computed with a corotational form of the updated Lagrangian formulation, using implicit integration techniques. This is followed by an analytical sensitivity analysis of the discrete finite element system, using the adjoint method. Such an undertaking can be very expensive, and so two techniques are used to ameliorate the cost. The first projects the finite element system onto a reduced basis formed around a series of POD modes. The second uses a monolithic time formulation with spectral elements, and can be used to compute only the desired time-periodic response. Results are given in terms of the accuracy of these methods for both system response and adjoint computations. The work concludes with the structural design of a flapping wing: the elastic deformation at the wingtip is minimized through the flapping stroke by varying the thickness of each finite element. The design variables alter the elastic forces of the flexible multibody system through stiffness effects and the inertial forces through mass effects. Significant improvements in computational cost are obtained at little expense to the accuracy of the results obtained via design optimization.

The problem of constrained optimization via the gradient-based discrete adjoint steepest descent method is studied under the assumption that the constraint equations are solved inexactly. Error propagation from the constraint equations to the gradient is studied analytically, as is the convergence rate of the inexactly constrained algorithm as it relates to the exact algorithm. A method is developed for adapting the residual tolerance to which the constraint equations are solved. The adaptive tolerance method is applied to two simple test cases to demonstrate the potential gains in computational efficiency.

ABSTRACTThe aerodynamic performance of a deployable and low-cost unmanned aerial vehicle (UAV) is investigated and improved in present work. The parameters of configuration, such as airfoil and winglet, are determined via an optimising process based on a discrete adjoint method. The optimised target is locked on an increasing lift-to-drag ratio with a limited variation of pitching moments. The separation that will lead to a stall is delayed after optimisation. Up to 128 design variables are used by the optimised solver to give enough flexibility of the geometrical transformation. As much as 20% enhancement of lift-to-drag ratio is gained at the cruise angle-of-attack, that is, a significant improvement in the lift-to-drag ratio adhering to the preferred configuration is obtained with increasing lift and decreasing drag coefficients, essentially entailing an improved aerodynamic performance.