Discrete Adjoint Solver Research Articles

Multi-Row Turbomachinery Aerodynamic Design Optimization by an Efficient and Accurate Discrete Adjoint Solver

This paper proposes an approach that combines manual differentiation (MD) and automatic differentiation (AD) to develop an efficient and accurate multi-row discrete adjoint solver. In this approach, the structures of adjoint codes generated using an AD tool are first analyzed. Then, the AD-generated codes are manually adjusted to reduce memory and CPU time consumption. This manual adjustment is performed by replacing the automatically generated low-efficient differentiated codes with manually developed ones. To demonstrate the effectiveness of the proposed approach, the single-stage transonic compressor–NASA Stage 35 and the 1.5-stage Aachen turbine–are used. The solution information exchange at a rotor-stator/stator-rotor interface is achieved by a conservative, non-reflective, and robust discrete adjoint mixing plane method. The results show that the discrete adjoint solver developed by hybrid automatic and manual differentiation is more economical in computational cost than that developed purely by an AD tool and has higher sensitivity accuracy than the adjoint solver with the constant eddy viscosity (CEV) assumption. Moreover, the multi-row turbomachinery design optimizations can be efficiently performed by the discrete adjoint solver developed by the hybrid automatic and manual differentiation.

New Concept for Design in Turbomachinery Applications Using Full RANS Gradient Methodology

Abstract Centrifugal compressors are mainly designed with the best efficiency point in mind and the focus oriented toward the impeller. The vaned diffuser geometry is usually developed using simple shapes, such as prismatic blades or wedges, which are unable to account for the flow distortion at the impeller outlet. Investigations of diffuser modifications using computational fluid dynamics (CFD) during the design phase can help to extend the operating range. In order to assess the effect of such modifications, simulations in off-design conditions need to be performed. Accurate predictions of local flow structures at such conditions require a robust flow solver with mixing-plane interfaces and advanced turbulence modeling capabilities. In recent years diffuser modifications were performed mainly based on manual trial-and-error approaches and show considerable potential for improvements. To accelerate the process and allow automation, which is crucial in industrial applications, it is suggested to use an adjoint-based optimization algorithm for diffuser shape design. In this work, a discrete adjoint solver implemented into a robust pressure-based CFD framework was used. While many optimization studies follow the frozen-turbulence approach, which essentially ignores the dependency of the sensitivity on the turbulence quantities, the present adjoint solver fully includes such effects. Automatic differentiation even allows for flexibility with respect to the choice of turbulence model. Any model available in the primal solver can also be used in the adjoint solver. In this particular case, the k–ɛ enhanced wall treatment model was used, since it shows good convergence behavior and is capable of predicting the machine characteristic even at off-design conditions. Mesh morphing was realized via the free-form deformation method, which provides stable mesh deformation and offers a simple and fast setup procedure. Several optimizations were conducted for the RWTH Aachen “Radiver” centrifugal compressor case with vaned diffuser. It was shown that an optimization strategy using a multi-operating point approach, considering part- and overload conditions, is capable of improving the peak efficiency as well as the overall shape of the machine characteristic.

A discrete-adjoint framework for optimizing high-fidelity simulations of turbulent reacting flows

An adjoint-based framework is presented that measures exact sensitivity from high-fidelity simulations of turbulent reacting flows. The framework leverages and extends state-of-the-art numerical methods in a manner that is compatible with a discrete adjoint solver. To ensure energy stability and accuracy, high-order narrow-stencil finite-difference operators satisfying the summation-by-parts (SBP) property are combined with simultaneous-approximation-term boundary treatment. An adaptive SBP dissipation operator and its corresponding adjoint are formulated in a way to dampen unresolved modes and preserve scalar boundedness. A flamelet/progress variable approach is employed to handle chemical reactions using tabulated chemistry. The utility of using pre-computed lookup tables is that discrete adjoint sensitivity can be computed efficiently for arbitrary chemical mechanisms. Gradient-based optimization utilizing the sensitivity obtained from the adjoint solution is demonstrated on two configurations. First, momentum actuation is applied upstream in a spatial mixing layer to control the evolution of a passive scalar in a target region downstream. Then, the framework is used to optimize an acoustic actuator in a three-dimensional turbulent jet with the aim of anchoring an H2 lifted flame at a target location. Both cases involve manipulating discrete space-time fields with O(108)−O(109) degrees of freedom, which would not be possible via a brute-force trial-and-error approach.

Aerodynamic shape optimization using a time spectral coupled adjoint for nonlinear aeroelastic problems

In this work, prediction of flutter and Limit Cycle Oscillations (LCO) for the aircraft wing has been carried out using a coupled, time spectral computational fluid dynamics (CFD) and computational structural dynamics (CSD) method. Flutter related shape design has also been carried out using a coupled, adjoint based sensitivity analysis framework. A discrete adjoint solver was developed in time-spectral form to calculate coupled sensitivity of the dynamic aeroelastic motion. The total structural energy was constrained at different levels to determine the flutter onset conditions or different LCO solutions. Compared to the time-marching approach, the time-spectral method not only enables faster convergence for unsteady problems, but also leads to significant improvement in implementation of adjoint based sensitivity analysis. Unlike the time-marching approach, there is no high memory requirement for time-spectral approach. The proposed method was applied to a pitch-plunge airfoil section at transonic flight conditions. The flutter and LCO results showed great agreements with time-marching results. The adjoint based sensitivity values were validated by finite difference based results, showing great agreement with errors less than 2%. Finally, aerodynamic shape optimization was carried out with the design objective of maximizing the flutter velocity, with and without constraint on the drag coefficient, respectively. Significant increase in flutter boundary velocity was observed in both cases.

Discrete Adjoint-Based Aerodynamic Shape Optimization Framework for Natural Laminar Flows

An adjoint-based aerodynamic shape optimization framework for natural laminar flows is developed. The laminar to turbulent-transition onset is predicted by the correlation-based Bas–Cakmakcioglu transition model that is coupled with the Spalart–Allmaras turbulence model. A discrete adjoint implementation is subsequently developed. Automatic differentiation is utilized to construct the partial derivatives in the discrete adjoint formulation. The turbulence and transition models are fully coupled into the sensitivity derivative evaluations, as well as into the objective function evaluations. The sensitivity derivatives evaluated by the discrete adjoint solver are validated against those of the finite central difference method. The discrete adjoint-based aerodynamic shape optimization framework developed for natural laminar flows is successfully employed to optimize aerodynamic characteristics of the NLF(1)-0416 airfoil and a low-aspect-ratio wing.

Balancing Laminar Extension and Wave Drag for Transonic Swept Wings

At a transonic condition, the design of a natural laminar flow (NLF) wing is challenging because the extension of the laminar flow needs to be finely balanced with the potential wave drag increase. To achieve this balance, it is proposed to unlock the wing sweep and introduce a three-dimensional (3-D) contour shock-control bump (SCB) in the optimization of the NLF infinite swept wing aiming at total drag reduction. A 3-D Reynolds-averaged Navier–Stokes flow solver is extended to incorporate laminar–turbulent transition prediction due to streamwise and crossflow instabilities. The flow solver is integrated in a gradient-based optimization framework. The transition criteria including streamwise and crossflow instabilities are coupled in the sensitivity calculation using a discrete-adjoint solver. Transonic design optimization of the sectional profile and wing sweep angle is first conducted at a Mach number of 0.78. The optimization managed to alleviate the shock wave, reducing the pressure drag while it failed to extend the laminar flow. Furthermore, a combined optimization of the wing with a parameterized SCB at the same condition is carried out. The optimized wing with the SCB features a long favorable pressure gradient region and a low-sweep angle, which allows for a large proportion of laminar flow without significant pressure drag. The shock wave is controlled by the 3-D bump, and the SCB has little effect on the upstream flow.

Gradient based empirical cumulative distribution function approximation for robust aerodynamic design

A robust optimization approach based on the use of the conditional value at risk function is presented, together with an application to a robust transonic aerodynamic design problem of the central section of a Blended Wing-Body configuration. The conditional value at risk is estimated using an approach based on the empirical cumulative probability distribution function. The quantities of interest of the risk function are the aerodynamic characteristics of the airfoil, namely lift, drag, and pitching moment coefficients, computed solving the Reynolds-averaged Navier-Stokes equations with the open-source fluid-dynamic solver SU2. Conditional value at risk computation is costly, so techniques and methods for the reduction of the computational cost are introduced. In particular, the empirical cumulative distribution function is approximated with a first-order series expansion using efficiently calculated gradients from SU2 discrete adjoint solver.

Regularization for Adjoint-Based Unsteady Aerodynamic Optimization Using Windowing Techniques

Unsteady aerodynamic shape optimization presents new challenges in terms of sensitivity analysis of time-dependent objective functions. In this work, we consider periodic unsteady flows governed by the unsteady Reynolds-averaged Navier–Stokes (URANS) equations. Hence, the resulting output functions acting as objective or constraint functions of the optimization are themselves periodic with unknown period length, which may depend on the design parameter of said optimization. Sensitivity analysis on the time average of a function with these properties turns out to be difficult. Therefore, we explore methods to regularize the time average of such a function with the so-called windowing approach. Furthermore, we embed these regularizers into the discrete adjoint solver for the URANS equations of the multiphysics and optimization software SU2. Finally, we exhibit a comparison study between the classical nonregularized optimization procedure and the ones enhanced with regularizers of different smoothness, and we show that the latter result in a more robust optimization.

Efficient Airframe Noise Reduction Framework via Adjoint-Based Shape Optimization

This paper presents an efficient adjoint-based shape optimization framework for airframe noise reduction in which algorithmic differentiation (AD) is applied to the open-source multiphysics solver SU2 to obtain design sensitivities. An AD-based consistent discrete adjoint solver is developed that directly inherits the convergence properties of the primal flow solver due to the differentiation of the entire nonlinear fixed-point iterator. In addition, a 2D and 3D far-field noise prediction framework coupling the Computational Fluid Dynamics (CFD) and Computational Aeroacoustics (CAA) using a permeable surface Ffowcs Williams-Hawkings (FWH) approach in the time domain is also developed. The resultant AD-based discrete adjoint solver is applied to the reduction of noise radiation from 2D and 3D rod–airfoil configurations. The results suggest that the unsteady adjoint information provided by this AD-based discrete adjoint framework is accurate and robust, due to the AD of the entire design chain including the dynamic mesh movement routine and various turbulence models, as well as the hybrid CFD-CAA model. This framework is shown to be highly effective in reducing the tonal noise component due to the turbulent wake impingement on the airfoil leading edge. This study also compares the optimization results obtained on the basis of 3D unsteady Reynolds-averaged Navier–Stokes equation simulations and analyzes them using scale-resolving simulations of higher fidelity.

Aerodynamic-driven topology optimization of compliant airfoils

A strategy for density-based topology optimization of fluid-structure interaction problems is proposed that deals with some shortcomings associated to non stiffness-based design. The goal is to improve the passive aerodynamic shape adaptation of highly compliant airfoils at multiple operating points. A two-step solution process is proposed that decouples global aeroelastic performance goals from the search of a solid-void topology on the structure. In the first step, a reference fully coupled fluid-structure problem is solved without explicitly penalizing non-discreteness in the resulting topology. A regularization step is then performed that solves an inverse design problem, akin to those in compliant mechanism design, which produces a discrete-topology structure with the same response to the fluid loads. Simulations are carried out with the multi-physics suite SU2, which includes Reynolds-averaged Navier-Stokes modeling of the fluid and hyper-elastic material behavior of the geometrically nonlinear structure. Gradient-based optimization is used with the exterior penalty method and a large-scale quasi-Newton unconstrained optimizer. Coupled aerostructural sensitivities are obtained via an algorithmic differentiation based coupled discrete adjoint solver. Numerical examples on a compliant aerofoil with performance objectives at two Mach numbers are presented.

Multistage Turbomachinery Design Using the Discrete Adjoint Method Within the Open-Source Software SU2

This paper documents a fully turbulent discrete adjoint method for three-dimensional multistage turbomachinery design. The method is based on a duality preserving algorithm and is implemented in the open-source computational fluid dynamics tool SU2. The SU2 Reynolds-averaged Navier–Stokes solver is first extended to treat three-dimensional steady turbomachinery flow using a conservative formulation of the mixing-plane coupled to nonreflective boundary conditions. The numerical features of the flow solver are automatically inherited by the discrete adjoint solver, ensuring the same convergence rate of the primal solver. The flow solver is then validated against experimental data available for three turbine configurations, namely, a one-and-half axial turbine stage, a transonic radial turbine coupled to a downstream diffuser, and a supersonic mini–organic Rankine cycle radial turbine operating with a fluid made by a heavy molecule. Finally the adjoint-based optimization framework is applied to the concurrent shape optimization of three rows of the axial turbine, demonstrating the advantages deriving from adopting multirow automated design methods in the context of turbomachinery design.

Efficient Design Optimization of Acoustic Liners for Engine Noise Reduction

Engine noise reduction using acoustic liner panels is an established technology in the design of bypass turbofan engines. Using acoustic impedance modeling techniques in aeroacoustic simulations, quick assessment of different liner configurations can be made. To minimize the engine noise in the best possible way, optimal liner configurations are ideally determined by employing numerical optimization methods. In general, to exhaust the full potential of the lining technology, it is desired to use a large number of design parameters in the optimization process. In the present work, a multilevel optimization framework is presented for the design of acoustic liner panels, which can efficiently handle large design vectors. The multilevel method combines a global search strategy with a local gradient-based optimization method. The discrete adjoint solver, which is used to evaluate the liner sensitivities required in the local search, has been developed using algorithmic differentiation techniques. This approach allows consistent treatment of boundary conditions in the adjoint part, thereby increasing the robustness and accuracy of the adjoint solver. In this way, the discrete adjoint solver fully incorporates the features of the underlying aeroacoustic solver, such as the impedance modeling and nonreflecting boundary conditions. The feasibility and the efficiency of the multilevel optimization strategy are demonstrated by finding the optimal liner parameters in a turbofan engine bypass duct configuration.

Algorithmic differentiation of an industrial airfoil design tool coupled with the adjoint CFD method

Computer Aided Design (CAD) systems and tools are considered essential for industrial design. They construct and manipulate the geometry of a certain component with an arbitrary set of design parameters. However, it is a challenging task to incorporate the parametric definition in a gradient-based shape optimization loop, since the CAD libraries usually do not provide shape sensitivities w.r.t. the design parameters of the model to be optimized. Typically, these derivatives are evaluated with inaccurate finite differences. On the contrary, to obtain the exact derivative information, algorithmic differentiation (AD) can be applied if the CAD sources are available. In this study, the Rolls-Royce in-house airfoil design and blade generation tool Parablading is differentiated using the AD software tools ADOL-C and Tapenade. The differentiated CAD tool is coupled with a discrete adjoint CFD solver that is part of the Rolls-Royce in-house HYDRA suite of codes, also produced by algorithmic differentiation. This differentiated design chain is used to perform gradient-based shape optimization of the TU Berlin TurboLab stator test-case w.r.t. minimize the total pressure loss and exit angle deviation objectives.

Well Rate and Placement for Optimal Groundwater Remediation Design with A Surrogate Model

A new surrogate-assisted optimization formulation for groundwater remediation design was developed. A stationary Eulerian travel time model was used in lieu of a conservative solute transport model. The decision variables of the management model are well locations and their flow rates. The objective function adjusts the residence time distribution between all pairs of injection-production wells in the remediation system. This goal is achieved by using the Lorenz coefficient as an effective metric to rank the relative efficiency of many remediation policies. A discrete adjoint solver was developed to provide the sensitivity of the objective function with respect to changes in decision variables. The quality management model was checked with simple solutions and then applied to hypothetical two- and three-dimensional test problems. The performance of the simulation-optimization approach was evaluated by comparing the initial and optimal remediation designs using an advective-dispersive solute transport simulator. This study shows that optimal designs simultaneously delay solute transport breakthrough at pumping wells and improve the sweep efficiency leading to smaller cleanup times. Well placement optimization in heterogeneous porous media was found to be more important than well rate optimization. Additionally, optimal designs based on two-dimensional models were found to be more optimistic suggesting a direct use of three-dimensional models in a simulation-optimization framework. The computational budget was drastically reduced because the proposed surrogate-based quality management model is generally cheaper than one single solute transport simulation. The introduced model could be used as a fast, but first-order, approximation method to estimate pump-and-treat capital remediation costs. The results show that physically based low-fidelity surrogate models are promising computational approaches to harness the power of quality management models for complex applications with practical relevance.

Reduced-order model-based convergence acceleration of reverse mode discrete adjoint solvers

This work presents a new technique to reduce the computational cost of sensitivities calculated using a discrete adjoint solver developed via reverse mode automatic differentiation. A fixed-point iterative method is built for the discrete adjoint sensitivity equations by employing the primal time-stepping adjoint approach. The fixed-point sensitivity solution is then accelerated by building a reduced-order model (ROM) that maps the relationship between the sensitivity solution and its corresponding residual. This model is then used to approximate the converged solution, corresponding to a zero residual. While the approximation might not produce the fully converged solution, it typically provides an improved solution that the fixed-point solver can be re-initialized with, which is one of the novel aspects of the present work. After re-initializing the solution, the ROM acceleration technique can be reapplied until the desired convergence criterion is reached. A key feature of the proposed ROM is that it is formed on the fly during a single sensitivity solution. Additionally, its implementation requires only minor modifications to an existing fixed-point iterative solver. The ROM projection technique is evaluated by considering design optimization of horizontal wind turbine blade profiles and cost reductions of 57 to 80% were achieved.

Discrete adjoint gradient evaluations for linear stress and vibration analysis

This paper presents methods used to perform discrete adjoint gradient evaluations for linear stress and vibration analysis. The methods are implemented within the framework of a discrete adjoint structural solver being developed for multidisciplinary adjoint optimizations of turbomachinery components. The code is differentiated using the algorithmic differentiation tool CoDiPack in tandem with manual treatment of the iterative solvers. Stress analysis leads to a linear system of equations that is typically solved by an iterative solver (e.g. GMRES). To ensure accuracy, the adjoint problem is formulated as a new linear system of equations to be solved. Vibration analysis results in a generalized eigenvalue problem that is also typically solved by an interative solver. The adjoint problem takes out the generalized eigenvalue solve and replaces it by one outer product per eigenfrequency, leading to significantly cheap eigenfrequency gradients for vibration analysis.

Seeding and adjoining zero-halo partitioned parallel scientific codes

ABSTRACTAlgorithmic differentiation tools can automate the adjoint transformation of parallel message-passing codes [J. Utke, L. Hascoët, P. Heimbach, C. Hill, P. Hovland, and U. Naumann, Toward Adjoinable MPI, in 2009 IEEE International Symposium on Parallel & Distributed Processing, May, IEEE, 2009, pp. 1–8] using the AMPI library. Nevertheless, a non-trivial and manual step after the differentiation is the initialization of the seed and retrieval of the output values from the differentiated code. Ambiguities in seeding occur in programs where the user is unable to expose the complete program flow with a single entry and single exit point to the AD tool. We present the ambiguities associated with seed initialization and output retrieval for adjoint transformation of halo and zero-halo partitioned MPI programs. We introduce a general framework to eliminate ambiguities in seeding and retrieval for shared-node reduction over +, and * operators using a conceptual master-worker model. The model shows the need for new MPI calls for retrieval and eliminate MPI calls for seed initialization. Different implementations for seeding manually assembled adjoints were inferred from the model, namely, partial and unique seeding. We successfully applied the seeding techniques to a 3D zero-halo partitioned unstructured compressible discrete adjoint solver and highlight the merits and demerits of each strategy.

Stabilization of a Time‐Dependent Discrete Adjoint Solver for Chaotic Incompressible Flows

AbstractFollowing the growth of computational fluid dynamics in engineering applications, adjoint methods for sensitivity analysis are being applied to an increasing range of industrial problems. In recent years, growth in computational power has led to widespread use of high‐fidelity, time‐resolved techniques, such as large‐eddy simulation, for applications, including bluff‐body aerodynamics, where time‐averaged equations may be less reliable. The solution of time‐dependent adjoint equations, however, becomes unstable in time as the Reynolds number increases and the flow becomes chaotic. The source of this instability has been explained by analysis of the system's Lyapunov exponents and several solutions have been proposed in the literature based on least‐squares shadowing. Currently, such techniques impose a higher computational cost compared to classical adjoint methods and may become infeasible when the number of unstable Lyapunov exponents is large. In this work, we explore a more recently proposed approach, which balances the adjoint energy source term with added artificial dissipation. The stability of this approach is investigated for a discrete adjoint solver based on the time‐dependent incompressible Navier‐Stokes equations. The accuracy of the resulting shape sensitivities is analyzed using a linear regression technique for the flow around a circular cylinder.

Parallelizable adjoint stencil computations using transposed forward-mode algorithmic differentiation

Algorithmic differentiation (AD) is a tool for generating discrete adjoint solvers, which efficiently compute gradients of functions with many inputs, for example for use in gradient-based optimization. AD is often applied to large computations such as stencil operators, which are an important part of most structured-mesh PDE solvers. Stencil computations are often parallelized, for example by using OpenMP, and optimized by using techniques such as cache-blocking and tiling to fully utilize multicore CPUs and many-core accelerators and GPUs. Differentiating these codes with conventional reverse-mode AD results in adjoint codes that cannot be expressed as stencil operations and may not be easily parallelizable. They thus leave most of the compute power of modern architectures unused. We present a method that combines forward-mode AD and loop transformation to generate adjoint solvers that use the same memory access pattern as the original computation that they are derived from and can benefit from the same optimization techniques. The effectiveness of this method is demonstrated by generating a scalable adjoint CFD solver for multicore CPUs and Xeon Phi accelerators.

A non-intrusive parallel-in-time adjoint solver with the XBraid library

In this paper, an adjoint solver for the multigrid-in-time software library XBraid is presented. XBraid provides a non-intrusive approach for simulating unsteady dynamics on multiple processors while parallelizing not only in space but also in the time domain (XBraid: Parallel multigrid in time, http://llnl.gov/casc/xbraid ). It applies an iterative multigrid reduction in time algorithm to existing spatially parallel classical time propagators and computes the unsteady solution parallel in time. Techniques from Automatic Differentiation are used to develop a consistent discrete adjoint solver which provides sensitivity information of output quantities with respect to design parameter changes. The adjoint code runs backwards through the primal XBraid actions and accumulates gradient information parallel in time. It is highly non-intrusive as existing adjoint time propagators can easily be integrated through the adjoint interface. The adjoint code is validated on advection-dominated flow with periodic upstream boundary condition. It provides similar strong scaling results as the primal XBraid solver and offers great potential for speeding up the overall computational costs for sensitivity analysis using multiple processors.