Adjoint Solver Research Articles

Weighted Gradient-Enhanced Kriging for High-Dimensional Surrogate Modeling and Design Optimization

A novel formulation of gradient-enhanced surrogate model, called weighted gradient-enhanced kriging, is proposed and used in combination with the cheap gradients obtained by the adjoint method to ameliorate the “curse of dimensionality”. The core idea is to build a series of submodels with much smaller correlation matrices and then sum them up with appropriate weight coefficients, aiming to avoid the prohibitive cost associated with decomposing the large correlation matrix of a gradient-enhanced kriging. A self-contained derivation of the proposed method is presented, and then it is verified by surrogate modeling test cases. The present method is integrated into a surrogate-based optimizer and tested for design optimizations. It is further demonstrated for inverse design of a transonic wing, parameterized with a number of design variables in the range from 36 to 108, using Reynolds-averaged Navier–Stokes flow and adjoint solvers. It is observed that, for the wing design with 36 and 54 variables, the weighted and conventional gradient-enhanced kriging are comparable, and both are much more efficient than kriging without using any gradient. For the wing design with 72 and 108 variables, the cost of training a gradient-enhanced kriging increases rapidly and becomes prohibitive. In contrast, the cost of training a weighted gradient-enhanced kriging is kept in an acceptable level, which makes it more practical for higher-dimensional problems.

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.

An arbitrary geometry finite element method for the adjoint neutron transport equation

In this paper a variational formulation for the adjoint even parity neutron transport equation (NTE) based on the generalized least squares method is adopted. The so-called PN method or expansion via Spherical Harmonics Polynomials (SHPs) is then summoned to treat the angular dependency of the equation while Finite Element Method (FEM) is invoked for the spatial domain. Based on our matrix form approach, it is shown that the scalar adjoint flux, ϕ†(r), can be determined through an efficient approach solely or in parallel with the scalar forward flux, ϕ(r). Eigenvalue problems as well as detector readings are of most useful applications of the adjoint calculation which are also discussed in this survey. To verify the equations we upgraded our Even parity Neutron TRANSport code, ENTRANS, with an adjoint solver which is capable of handling one-, two- and three-dimensional problems of arbitrary geometry. Ability to cover any order of multigroup scattering anisotropy (with or without up-scattering) as well as higher order elements is also embedded in the ENTRANS. Finally, several test cases are examined against reference results to illustrate the accuracy and efficiency of the proposed approach.

Adjoint-Based Design Optimisation of an Internal Cooling Channel U-Bend for Minimised Pressure Losses

The success of shape optimisation depends significantly on the parametrisation of the shape. Ideally, it defines a very rich variation in shape, allows for rapid grid generation of high quality, and expresses the shape in a standard Computer Aided Design (CAD) representation. While most existing parametrisation methods fail at least one of these criteria, this work introduces a novel parametrisation method, which satisfies all three. A tri-variate B-spline volume is used to define the volume to be optimised. The position of the external control points are used as design parameters, while the internal control points are repositioned to ensure regularity of the transformation. The grid generation process transforms a Cartesian grid (defined in parametric space) to the physical space using the tri-variate net of control points. This process guarantees a high grid quality even for large deformations, and has extremely low computational cost as it only involves a transformation from parameter space to physical space. This allows the computation of the grid sensitivities with respect to the design variables at a fraction of the cost of a Computational Fluid Dynamics (CFD) iteration, therefore allowing the use of one-shot methods. This novel parametrisation is applied to the shape optimisation of a U-bend passage of a turbine-blade serpentine-cooling channel with the objective to minimise pressure losses. A steady state, Reynolds-Averaged, density-based Navier-Stokes solver is used to predict the pressure losses at a Reynolds number of 40,000. The sensitivities of the objective function with respect to the control points are computed using a hand-derived adjoint solver and geometry generation system. A one-shot approach is used to simultaneously converge flow, gradient and design, resulting in a rapid design approach with a design time equivalent to approximately 10 normal CFD runs, while still maintaining a CAD representation of the geometry. A large reduction in pressure loss is obtained, and the flow in the optimal geometry is analysed in detail.

Wing-body junction optimisation with CAD-based parametrisation including a moving intersection

Over the past decades significant progress has been made with adjoint computational fluid dynamics solvers, which are an essential part of efficient high-fidelity aerodynamic shape optimisation. Shape parametrisation is much less mature, in particular the field is lacking efficient and automatic CAD-based parametrisation methods. The paper proposes a novel CAD-based parametrisation with CAD in the design loop such that the CAD shape can ultimately serve as a datum surface in multi-disciplinary optimisation. Wing and fuselage are modelled with B-spline surfaces. The intersection line is calculated using an in-house implementation of a B-spline surface modeller and its derivative is efficiently calculated via finite differences. The proposed parametrisation method is applied to the redesign of the wing–fuselage junction of the DLR-F6 model using an adjoint solver based on Reynolds-averaged Navier–Stokes equations. The moving intersection line capability enables the fuselage surface to be deformed and the resulting intersection line to move along the fixed wing during optimisation. The flow separation in the wing–body junction is substantially suppressed by an improved fuselage shape, at the cost of O(10) steady-state flow and adjoint solutions. The proposed parametrisation method represents an important step towards automated CAD-based optimisation for fully-featured aircraft characterised by complex intersecting surfaces.

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.

Uncertainty analysis with double heterogeneity treatment based on the generalized perturbation theory

Two generalized adjoint solution modules, the generalized adjoint transport equation solver and the generalized adjoint B1 equation solver, were implemented into DeCART for the general response uncertainty analysis on the doubly heterogeneous system. The generalized adjoint transport equation solver for the doubly heterogeneous system was implemented based on the Sanchez-Pomraning method. Moreover, the generalized adjoint B1 equation solution scheme was newly set up using the adjoint B1 equation for an efficient calculation. From the benchmark analysis on the MHTGR-350 Ex. I-1, it was confirmed that the two generalized adjoint solver can provide reliable results for the doubly heterogeneous region. Then, the uncertainties for the kinf and the microscopic cross sections on the same problem were evaluated using DeCART/MUSAD code system. The results reveal that the double heterogeneity does not have a significant effect on the kinf uncertainty but slightly increases the uncertainty of 238U capture compared with the homogeneous case.

Pressure Loss Reduction in U-Bend Using the Adjoint Method

Gas turbine blades are usually cooled by air, which flows through internal passages serpentine shaped, coming from a high pressure compressor to save it from the high temperature. Using 180 degree turns to connect internal paths led to high pressure loss in the cooling system. The purpose of this study is reducing the pressure loss using the adjoint solver method to get the optimum shape of U-shaped paths with and without guide vane (GV). Using three different cases with three different regions of modifications. The optimized shapes showed a reduction in the pressure loss up to 60% in the case of U-bend without guide vanes.

Robust and efficient adjoint solver for complex flow conditions

Abstract A key step in gradient-based aerodynamic shape optimisation using the Reynolds-averaged Navier–Stokes equations is to compute the adjoint solution. Adjoint equations inherit the linear stability and the stiffness of the nonlinear flow equations. Therefore for industrial cases with complex geometries at off-design flow conditions, solving the resulting stiff adjoint equation can be challenging. In this paper, Krylov subspace solvers enhanced by subspace recycling and preconditioned with incomplete lower-upper factorisation are used to solve the stiff adjoint equations arising from typical design and off-design conditions. Compared to the baseline matrix-forming adjoint solver based on the generalized minimal residual method, the proposed algorithm achieved memory reduction of up to a factor of two and convergence speedup of up to a factor of three, on industry-relevant cases. These test cases include the DLR-F6 and DLR-F11 configurations, a wing-body configuration in pre-shock buffet and a large civil aircraft with mesh sizes ranging from 3 to 30 million. The proposed method seems to be particularly effective for the more difficult flow conditions.

Algorithmic Differentiation of Code with Multiple Context-Specific Activities

Algorithmic differentiation (AD) by source-transformation is an established method for computing derivatives of computational algorithms. Static dataflow analysis is commonly used by AD tools to determine the set of active variables, that is, variables that are influenced by the program input in a differentiable way and have a differentiable influence on the program output. In this work, a context-sensitive static analysis combined with procedure cloning is used to generate specialised versions of differentiated procedures for each call site. This enables better detection and elimination of unused computations and memory storage, resulting in performance improvements of the generated code, in both forward- and reverse-mode AD. The implications of this multi-activity AD approach on the static analysis of an AD tool is shown using dataflow equations. The worst-case cost of multi-activity AD on the differentiation process is analysed and practical remedies to avoid running into this worst case are presented. The method was implemented in the AD tool Tapenade, and we present its application to a 3D unstructured compressible flow solver, for which we generate an adjoint solver that performs significantly faster when multi-activity AD is used.

Unsteady Adjoint of Pressure Loss for a Fundamental Transonic Turbine Vane

High-fidelity simulations, e.g., large eddy simulation (LES), are often needed for accurately predicting pressure losses due to wake mixing and boundary layer development in turbomachinery applications. An unsteady adjoint of high-fidelity simulations is useful for design optimization in such aerodynamic applications. In this paper, we present unsteady adjoint solutions using a large eddy simulation model for an inlet guide vane from von Karman Institute (VKI) using aerothermal objectives. The unsteady adjoint method is effective in capturing the gradient for a short time interval aerothermal objective, whereas the method provides diverging gradients for long time-averaged thermal objectives. As the boundary layer on the suction side near the trailing edge of the vane is turbulent, it poses a challenge for the adjoint solver. The chaotic dynamics cause the adjoint solution to diverge exponentially from the trailing edge region when solved backward in time. This results in the corruption of the sensitivities obtained from the adjoint solutions. An energy analysis of the unsteady compressible Navier–Stokes adjoint equations indicates that adding artificial viscosity to the adjoint equations can dissipate the adjoint energy while potentially maintaining the accuracy of the adjoint sensitivities. Analyzing the growth term of the adjoint energy provides a metric for identifying the regions in the flow where the adjoint term is diverging. Results for the vane obtained from simulations performed on the Titan supercomputer are demonstrated.

The One‐Shot Method in SU2

AbstractFor PDE‐constrained optimization tasks the one‐shot approach is well‐suited. The solution of the state equation is augmented with an adjoint solver to provide approximate reduced derivatives that can be used in an optimization iteration to change the design. In this paper different one‐shot strategies for SU2 are implemented and tested for aerodynamic shape optimization in the open‐source multi‐physics package SU2 to extend the choice of optimization algorithms. These include the application of the one‐shot approach when considering additional constraints apart from the state equation. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Adjoint-based sensitivity analysis by panel methods and CAS

The work presented in this paper deals with the accuracy of the sensitivity derivatives of aerodynamic cost functions obtained by using an adjoint method. The accuracy of these gradients is evaluated by comparison with gradients computed via finite-differences in a straight-forward manner. The advantages of the use of an adjoint method become extremely clear since the computational effort incurred in the calculation of a complete gradient with respect to an arbitrary number of variables is independent of the number of variables and the only cost involved is the calculation of one flow solution and one adjoint solution, where the adjoint equation is a linear equation and, hence, of reduced complexity. In order to focus on the design and optimization procedure, we limit our study to one of the simplest flows: a subsonic potential flow over an airfoil section. First, for a generic objective function, we present a tool to automatically compute the gradient expression through adjoint techniques based on MAPLE, one of the most powerful computer algebra system (CAS). Second, exploiting the linearity of the governing equation, an adjoint solver is implemented based on the classical Douglas-Neumann panel method. The use of the solver is illustrated for a sample design calculation.

Second-Order Adjoint Sensitivity and Uncertainty Analysis of a Heat Transport Benchmark Problem—I: Analytical Results

This work presents an illustrative application of the second-order adjoint sensitivity analysis methodology (2nd-ASAM) to a heat transport benchmark problem that simulates the radial heat conduction in a heated fuel rod together with the axial heat transport along the coolant channel surrounding the rod. This paradigm benchmark is sufficiently simple to admit an exact solution, thereby making transparent the mathematical derivations underlying the 2nd-ASAM, and also for performing solution verification of computational fluid dynamics simulation tools such as the FLUENT Adjoint Solver, a code that has been used for computing thermal-hydraulic processes within the G4M Reactor. The G4M Reactor is an innovative small modular fast reactor cooled by lead-bismuth eutectic.The benchmark problem has six thermal-hydraulic parameters representative of the G4M Reactor, as well as of a test section that is under design for analyzing thermal-hydraulic phenomena expected within the reactor’s core. Thus, the benchmark exhibits 6 first-order and 21 second-order sensitivities for the temperature distribution at any location within the heated rod (and/or on its surface) and just as many sensitivities for the temperature distribution at any location within the coolant. Locations of particular importance are those where the rod temperature attains its maximum along the rod’s axis or on the rod’s surface, as well as the coolant’s exit, where the coolant temperature attains its maximum.The general theory underlying the 2nd-ASAM indicates that for a physical system comprising Nα parameters, the computation of all of the first- and second-order response sensitivities requires Nα large-scale computations involving correspondingly constructed adjoint systems, which we called second-level adjoint sensitivity systems (2nd-LASS). In practice, however, the actual number of large-scale adjoint computations can be significantly smaller; in particular, all 54 first- and second-order rod and coolant temperature sensitivities for the benchmark presented in this work were obtained using only seven independent adjoint computations. Furthermore, the construction and solution of the 2nd-LASS require very little additional effort beyond the construction of the first-level adjoint sensitivity system (1st-LASS) needed for computing the first-order sensitivities. Very significantly, only the sources on the right side of the differential operator needed to be modified; the left side of the respective differential equations (and hence the solver in large-scale practical applications) remained unchanged from the 1st-LASS and from the original equations when these are linear in the unknown state function.This work also shows that the second-order sensitivities have the following impacts on the computed moments of the response distribution: (a) they cause the expected value of the response to differ from the computed nominal value of the response, (b) they contribute to the response variances and covariances, and (c) they contribute decisively to causing asymmetries in the response distribution. Indeed, neglecting the second-order sensitivities would nullify the third-order response correlations stemming from normally distributed parameters and hence would nullify the skewness of the response; consequently, events occurring in a response’s long and/or short tails could be missed.

Second-Order Adjoint Sensitivity and Uncertainty Analysis of a Heat Transport Benchmark Problem—II: Computational Results Using G4M Reactor Thermal-Hydraulic Parameters

This work presents numerical results for the second-order sensitivities of the temperature distributions in a paradigm benchmark problem modeling heat transport in a reactor fuel rod and the surrounding coolant channel. The development of this benchmark problem was originally motivated by the need to verify the numerical results for the first-order sensitivities produced by the FLUENT Adjoint Solver for the G4M Reactor preconceptual design and for a test section designed to investigate thermal-hydraulic phenomena of importance to the safety considerations for this reactor. The relative sensitivities computed using the FLUENT Adjoint Solver had significantly large values, of order unity, thereby motivating the need to investigate the impact of nonlinearities, the bulk of which are quantified by the responses’ second-order sensitivities. However, the current FLUENT Adjoint Solver cannot compute second-order sensitivities, which in turn motivated the derivation of these sensitivities for the heat transport benchmark problem by using the recently developed second-order adjoint sensitivity analysis methodology.The numerical results obtained in this work used thermal-hydraulic parameters having mean values and standard deviations typical of the conditions found in the preliminary conceptual design of the G4M Reactor. These results show that the contributions of the second-order sensitivities to the expected values of the temperature distributions within the rod, on the rod’s surface, and in the coolant are <1% of the corresponding computed nominal values. Similarly, the contributions of the second-order sensitivities to the standard deviations of the temperature distributions within the rod, on the rod’s surface, and in the coolant are also 1%, or less, of the corresponding contributions stemming from the first-order sensitivities, to the respective total standard deviations (uncertainties). These results justify the use of first-order sensitivities for computing expected uncertainties in the temperature distributions within the benchmark problem and, hence, mutatis mutandis, for the test section and G4M Reactor design.On the other hand, the most important impact of the second-order sensitivities is the positive skewnesses they induce in the temperature distributions within the rod, on the rod’s surface, and in the coolant. This implies that all three temperature distributions, particularly in the heated rod, are non-Gaussian, asymmetric, and skewed toward temperatures higher than the respective mean temperatures.

A Heat Conduction and Convection Analytical Benchmark for Adjoint Solution Verification of Computational Fluid Dynamics Codes Used in Reactor Design

This work presents a heat transport benchmark problem when modeling the steady-state radial conduction in a fuel rod coupled to the axial heat convection in a coolant surrounding the rod and flowing along it. This benchmark problem admits exact analytical solutions for the spatially dependent temperature distributions within the rod and the surrounding coolant. The adjoint sensitivity analysis methodology (ASAM) is applied to compute the analytical expressions of the adjoint state functions for this benchmark problem. In turn, these adjoint state functions are used to compute exactly the first-order sensitivities of the various temperature distributions to the benchmark’s thermal-hydraulics parameters. Locations of particular importance are those where the rod, the rod surface, and the coolant temperatures attain their maxima. The analytical expressions of the benchmark sensitivities thus obtained are subsequently used to compute numerical values of the sensitivities of the various temperature distributions that would arise in the preliminary design of the G4M Reactor to thermal-hydraulics parameters characteristic of this reactor.The exact benchmark sensitivities are used for verifying the numerical results produced by the FLUENT Adjoint Solver, a code that has been used for computing thermal-hydraulics processes within the G4M Reactor. This solution verification process indicates that the current FLUENT Adjoint Solver cannot compute any sensitivities for the temperature distribution within the solid rod. However, the FLUENT Adjoint Solver is capable of computing the sensitivities of fluid temperatures to boundary parameters (e.g., boundary temperature, boundary velocity, and boundary pressure), but yields accurate results only for the sensitivities of the fluid outlet temperature and the maximum rod surface temperature to the inlet temperature and inlet velocity, respectively. Even for these sensitivities, the FLUENT Adjoint Solver typically needed over 20 000 iterations to converge to the correct solution. In fact, if the exact sensitivity results had not been known a priori, employment of a user-defined iteration-stopping criterion would have likely produced an erroneous result, which would have been noticed by the user only if the user had had the foresight of computing the respective sensitivities independently, via finite-differences using FLUENT recomputations. Several other important sensitivities, including sensitivities to the boundary heat transfer coefficient and sensitivities to material properties (thermal conductivity and specific heat), cannot be obtained from the current FLUENT postprocessing output.Ideally, the solution verification of the adjoint functions produced by the FLUENT Adjoint Solver would be performed by directly comparing these to the exact expressions of the adjoint functions for the benchmark problem. Such a direct comparison and, hence, a direct solution verification of the FLUENT Adjoint Solver, is currently not possible, because the current FLUENT Adjoint Solver does not provide access to the adjoint functions it computes. Therefore, the results produced by the FLUENT Adjoint Solver can only be verified indirectly, by comparing temperature sensitivities computed using the FLUENT Adjoint Solver to the exact results obtained from the analytical expression of the corresponding benchmark sensitivities. This situation further underscores the need for developing additional thermal-hydraulics benchmark problems that admit exact solutions.

On an extension of one-shot methods to incorporate additional constraints

For design optimization tasks, quite often a so-called one-shot approach is used. It augments the solution of the state equation with a suitable adjoint solver yielding approximate reduced derivatives that can be used in an optimization iteration to change the design. The coordination of these three iterative processes is well established when only the state equation is considered as equality constraint. However, numerous applications require also additional equality constraints. Therefore, we propose a modified augmented Lagrangian function, that defines a simultaneous change of all variables for this extended setting. It is shown that the augmented Lagrangian function proposed in this paper can be used in a gradient-based optimization approach to solve the original design task.

Adjoint Lattice Boltzmann for topology optimization on multi-GPU architecture

In this paper we present a topology optimization technique applicable to a broad range of flow design problems. We propose also a discrete adjoint formulation effective for a wide class of Lattice Boltzmann Methods (LBM). This adjoint formulation is used to calculate sensitivity of the LBM solution to several type of parameters, both global and local. The numerical scheme for solving the adjoint problem has many properties of the original system, including locality and explicit time-stepping. Thus it is possible to integrate it with the standard LBM solver, allowing for straightforward and efficient parallelization (overcoming limitations typical for the discrete adjoint solvers). This approach is successfully used for the channel flow to design a free-topology mixer and a heat exchanger. Both resulting geometries being very complex maximize their objective functions, while keeping viscous losses at acceptable level.

Algorithmic Differentiation of Numerical Methods: Second-order Adjoint Solvers for Parameterized Systems of Nonlinear Equations

Adjoint mode algorithmic (also know as automatic) differentiation (AD) transforms implementations of multivariate vector functions as computer programs into first-order adjoint code. Its reapplication or combinations with tangent mode AD yields higher-order adjoint code. Second derivatives play an important role in nonlinear programming. For example, second-order (Newton-type) nonlinear optimization methods promise faster convergence in the neighborhood of the minimum through taking into account second derivative information. The adjoint mode is of particular interest in large-scale gradient-based nonlinear optimization due to the independence of its computational cost on the number of free variables. Part of the objective function may be given implicitly as the solution of a system of n parameterized nonlinear equations. If the system parameters depend on the free variables of the objective, then second derivatives of the nonlinear system's solution with respect to those parameters are required. The local computational overhead as well as the additional memory requirement for the computation of second-order adjoints of the solution vector with respect to the parameters by AD depends on the number of iterations performed by the nonlinear solver. This dependence can be eliminated by taking a symbolic approach to the differentiation of the nonlinear system.

Optimization of hydraulic turbine diffuser

Hydraulic turbine diffuser recovers pressure energy from residual kinetic energy on turbine runner outlet. Efficiency of this process is especially important for high specific speed turbines, where almost 50% of available head is utilized within diffuser. Magnitude of the coefficient of pressure recovery can be significantly influenced by designing its proper shape. Present paper focuses on mathematical shape optimization method coupled with CFD. First method is based on direct search Nelder-Mead algorithm, while the second method employs adjoint solver and morphing. Results obtained with both methods are discussed and their advantages/disadvantages summarized.