Adjoint Formulation Research Articles

Noise reduction in car aerodynamics using a surrogate objective function and the continuous adjoint method with wall functions

The purpose of this paper is twofold. In the first part, the continuous adjoint formulation for field integral objective functions used in steady-state, incompressible aerodynamic optimization is developed. The formulation includes the full differentiation of the Spalart–Allmaras turbulence model based on wall functions. In the second part, the developed adjoint method is used for optimizing the side mirror shape of a passenger car, using volumetric B-Splines as the parameterization tool. Based on industrial experience, an appropriate, though approximate, objective function to be minimized is expressed by the integral of the squared turbulent viscosity over a volume residing next to the driver’s window. It should be stressed that if the commonly used “frozen turbulence” assumption was made, by skipping the differentiation of the turbulence model, the adjoint method would not have been able to provide any kind of sensitivity derivatives, since this objective function depends exclusively upon the turbulent viscosity.

On the proper treatment of grid sensitivities in continuous adjoint methods for shape optimization

The continuous adjoint method for shape optimization problems, in flows governed by the Navier–Stokes equations, can be formulated in two different ways, each of which leads to a different expression for the sensitivity derivatives of the objective function with respect to the control variables. The first formulation leads to an expression including only boundary integrals; it, thus, has low computational cost but, when used with coarse grids, its accuracy becomes questionable. The second formulation comprises a sum of boundary and field integrals; due to the field integrals, it has noticeably higher computational cost, obtaining though higher accuracy. In this paper, the equivalence of the two formulations is revisited from the mathematical and, particularly, the numerical point of view. Internal and external aerodynamics cases, in which the objective function is either the total pressure losses or the force exerted on a solid body, are examined and differences in the computed gradients are discussed. After identifying the reason behind these discrepancies, the adjoint formulation is enhanced by the adjoint to a (hypothetical) grid displacement model and the new approach is proved to reproduce the accuracy of the second adjoint formulation while maintaining the low cost of the first one.

Optimum Shape Design for Multirow Turbomachinery Configurations Using a Discrete Adjoint Approach and an Efficient Radial Basis Function Deformation Scheme for Complex Multiblock Grids

This paper proposes a framework for fully automatic gradient-based constrained aerodynamic shape optimization in a multirow turbomachinery environment. The concept of adjoint-based gradient calculation is discussed and the development of the discrete adjoint equations for a turbomachinery Reynolds-averaged Navier–Stokes (RANS) solver, particularly the derivation of flow-consistent adjoint boundary conditions as well as the implementation of a discrete adjoint mixing-plane formulation, are described in detail. A parallelized, automatic grid perturbation scheme utilizing radial basis functions (RBFs), which is accurate and robust as well as able to handle highly resolved complex multiblock turbomachinery grid configurations, is developed and employed to calculate the gradient from the adjoint solution. The adjoint solver is validated by comparing its sensitivities with finite-difference gradients obtained from the flow solver. A sequential quadratic programming (SQP) algorithm is then utilized to determine an improved blade shape based on the gradient information from the objective functional and the constraints. The developed optimization method is used to redesign a single-stage transonic flow compressor in both inviscid and viscous flow. The design objective is to maximize the isentropic efficiency while constraining the mass flow rate and the total pressure ratio.

Bayesian uncertainty quantification of turbulence models based on high-order adjoint

The uncertainties in the parameters of turbulence models employed in computational fluid dynamics simulations are quantified using the Bayesian inference framework and analytical approximations. The posterior distribution of the parameters is approximated by a Gaussian distribution with the most probable value obtained by minimizing the objective function defined by the minus of the logarithm of the posterior distribution. The gradient and the Hessian of the objective function with respect to the parameters are computed using the direct differentiation and the adjoint approach to the flow equations including the turbulence model ones. The Hessian matrix is used both to compute the covariance matrix of the posterior distribution and to initialize the quasi-Newton optimization algorithm used to minimize the objective function. The propagation of uncertainties in output quantities of interest is also presented based on Laplace asymptotic approximations and the adjoint formulation. The proposed method is demonstrated using the Spalart−Allmaras turbulence model parameters in the case of the flat plate flow using DNS data for velocities and the flow through a backward facing step using experimental data for velocities and Reynolds stresses.

An Adaptive Multipoint Formulation for Robust Parametric Optimization

The performance of a system designed for given functioning conditions often seriously degrades, when operated at other conditions. Therefore, a system operating over a continuous range of conditions should be designed over this range. The aerodynamic shape optimization of an aircraft at multiple altitudes, angles of attack and Mach numbers is a typical case in aerospace. This paper links parametric and multipoint optimizations by the sampling of the operating condition ranges. It is demonstrated that this discrete set of operating conditions, used to formulate a composite objective function, must adequately be chosen. An algorithm is proposed to select these conditions, which ensures a minimal computational cost to the robust optimization. Wing aerodynamic multipoint optimizations using a lifting line model and Reynolds-averaged Navier---Stokes equations, derived with a discrete adjoint formulation, are given as examples.

Unsteady Continuous Adjoint Approach for Aerodynamic Design on Dynamic Meshes

This paper presents the development and application of a new unsteady continuous adjoint formulation for optimal shape design of aerodynamic surfaces in motion, such as rotating or pitching applications. The arbitrary Lagrangian–Eulerian form of the unsteady, compressible Reynolds-averaged Navier–Stokes equations with a generic source term is considered, and from these governing flow equations, an adjoint formulation centered around finding surface sensitivities using shape calculus is derived. This surface formulation provides the gradient information necessary for performing gradient-based aerodynamic shape optimization at a computational cost equivalent to solving the flow equations and with minimal memory overhead. To verify the methodology, gradients provided by the continuous adjoint and finite-differencing approaches are compared. Optimal shape design is demonstrated in both two and three dimensions for pitching airfoil and wing test cases.

Adjoint-based deviational Monte Carlo methods for phonon transport calculations

In the field of linear transport, adjoint formulations exploit linearity to derive powerful reciprocity relations between a variety of quantities of interest. In this paper, we develop an adjoint formulation of the linearized Boltzmann transport equation for phonon transport. We use this formulation for accelerating deviational Monte Carlo simulations of complex, multiscale problems. Benefits include significant computational savings via direct variance reduction, or by enabling formulations which allow more efficient use of computational resources, such as formulations which provide high resolution in a particular phase-space dimension (e.g., spectral). We show that the proposed adjoint-based methods are particularly well suited to problems involving a wide range of length scales (e.g., nanometers to hundreds of microns) and lead to computational methods that can calculate quantities of interest with a cost that is independent of the system characteristic length scale, thus removing the traditional stiffness of kinetic descriptions. Applications to problems of current interest, such as simulation of transient thermoreflectance experiments or spectrally resolved calculation of the effective thermal conductivity of nanostructured materials, are presented and discussed in detail.

Aerodynamic optimization design for high pressure turbines based on the adjoint approach

A first study on the continuous adjoint formulation for aerodynamic optimization design of high pressure turbines based on S2 surface governed by the Euler equations with source terms is presented. The objective function is defined as an integral function along the boundaries, and the adjoint equations and the boundary conditions are derived by introducing the adjoint variable vectors. The gradient expression of the objective function then includes only the terms related to physical shape variations. The numerical solution of the adjoint equation is conducted by a finite-difference method with the Jameson spatial scheme employing the first and the third order dissipative fluxes. A gradient-based aerodynamic optimization system is established by integrating the blade stagger angles, the stacking lines and the passage perturbation parameterization with the quasi-Newton method of Broyden–Fletcher–Goldfarb–Shanno (BFGS). The application of the continuous adjoint method is validated through a single stage high pressure turbine optimization case. The adiabatic efficiency increases from 0.8875 to 0.8931, whilst the mass flow rate and the pressure ratio remain almost unchanged. The optimization design is shown to reduce the passage vortex loss as well as the mixing loss due to the cooling air injection.

Method to Facilitate High-Dimensional Design Space Exploration Using Computationally Expensive Analyses

The emergence of adjoint design methods has been a breakthrough in the field of aerodynamic shape optimization. The strength of such methods lies in the fact that the adjoint formulation effectively removes the curse of dimensionality by drastically reducing the cost of computing the gradient. However, the curse of dimensionality is still very much alive when it comes to design space exploration, where gradient-free methods cannot be avoided. In attempt to solve this problem, this work presents a method called latent space gradient transformation, which facilitates gradient-free exploration for cases where function evaluations are expensive and dimensionality is high. The method builds upon previous work by applying principal component analysis to gradient information, something that had not been done before. This yields a low-rank transformation that maps a high-dimensional design space onto an equivalent, low-dimensional space in which gradient-free methods become more affordable. Concepts and limitations are discussed through analytical examples, and the method is tested on an overwing nacelle integration problem. Results showed that the method is cost-effective, requiring only 150 gradient samples to successfully reduce the problem from 145 design variables to less than 20, with an error of 10 drag counts on average and 25 drag counts at worst with 95% confidence.

Time dependent adjoint-based optimization for coupled fluid–structure problems

A formulation for sensitivity analysis of fully coupled time-dependent aeroelastic problems is given in this paper. Both forward sensitivity and adjoint sensitivity formulations are derived that correspond to analogues of the fully coupled non-linear aeroelastic analysis problem. Both sensitivity analysis formulations make use of the same iterative disciplinary solution techniques used for analysis, and make use of an analogous coupling strategy. The information passed between fluid and structural solvers is dimensionally equivalent in all cases, enabling the use of the same data structures for analysis, forward and adjoint problems. The fully coupled adjoint formulation is then used to perform rotor blade design optimization for a four bladed HART2 rotor in hover conditions started impulsively from rest. The effect of time step size and mesh resolution on optimization results is investigated.

Entropy minimization in turbine cascade using continuous adjoint formulation

A complete continuous adjoint formulation is presented here for the optimization of the turbulent flow entropy generation rate through a turbine cascade. The adjoint method allows one to have many design variables, but still afford to compute the objective function gradient. The new adjoint system can be applied to different structured and unstructured grids as well as mixed subsonic and supersonic flows. For turbulent flow simulation, the k–ω shear-stress transport turbulence model and Roe's flux function are used. To ensure all possible shape models, a mesh-point method is used for design parameters, and an implicit smoothing function is implemented to avoid the generation of non-smoothed blades. To analyse the capability of the presented algorithm, the shape of a turbine cascade blade is redesigned and a few physical observations are made on how the scheme improves the blade performance.

A practical discrete-adjoint method for high-fidelity compressible turbulence simulations

Methods and computing hardware advances have enabled accurate predictions of complex compressible turbulence phenomena, such as the generation of jet noise that motivates the present effort. However, limited understanding of underlying physical mechanisms restricts the utility of such predictions since they do not, by themselves, indicate a route to design improvements. Gradient-based optimization using adjoints can circumvent the flow complexity to guide designs, though this is predicated on the availability of a sufficiently accurate solution of the forward and adjoint systems. These are challenging to obtain, since both the chaotic character of the turbulence and the typical use of discretizations near their resolution limits in order to efficiently represent its smaller scales will amplify any approximation errors made in the adjoint formulation. Formulating a practical exact adjoint that avoids such errors is especially challenging if it is to be compatible with state-of-the-art simulation methods used for the turbulent flow itself. Automatic differentiation (AD) can provide code to calculate a nominally exact adjoint, but existing general-purpose AD codes are inefficient to the point of being prohibitive for large-scale turbulence simulations. Here, we analyze the compressible flow equations as discretized using the same high-order workhorse methods used for many high-fidelity compressible turbulence simulations, and formulate a practical space–time discrete-adjoint method without changing the basic discretization. A key step is the definition of a particular discrete analog of the continuous norm that defines our cost functional; our selection leads directly to an efficient Runge–Kutta-like scheme, though it would be just first-order accurate if used outside the adjoint formulation for time integration, with finite-difference spatial operators for the adjoint system. Its computational cost only modestly exceeds that of the flow equations. We confirm that its accuracy is limited by computing precision, and we demonstrate it on the aeroacoustic control of a mixing layer with a challengingly broad range of turbulence scales. For comparison, the error from a corresponding discretization of the continuous-adjoint equations is quantified to potentially explain its limited success in past efforts to control jet noise. The differences are illuminating: the continuous-adjoint is shown to suffer from exponential error growth in (reverse) time even for the best-resolved largest turbulence scales. Implications for jet noise reduction and turbulence control in general are discussed.

Bayesian Estimation of a Highly Parameterized Hydraulic Conductivity Field: A Study Case

Comprehensive characterization of the spatial variability of aquifer hydraulic parameters is fundamental to correctly design a structure that interacts with the groundwater system, e.g. a detention reservoir. A flood control structure has been planned to dampen the Baganza River floods, upstream the city of Parma (Northern Italy), storing a portion of the flood volume then released at a controlled rate. The location and the geometry of the planned reservoir are already set and the aquifer system beneath the structure is under investigation and monitoring. A large numbers of boreholes identified a sequence of gravel-sand and clayey layers that gives rise to two distinct aquifers (one unconfined and one confined). A numerical model has been developed by means of MODFLOW_2005 in order to simulate the unconfined aquifer under current conditions. The numerical model has been calibrated in steady-state conditions by means of 11 observations of water levels (collected during spring 2013). About 18000 hydraulic conductivity values have been defined and estimated by means of a Bayesian Geostatistical Approach. The inverse methodology estimates the hydraulic conductivity field using both field observations (head data) and prior information on the structure of the unknown parameters. The inverse procedure requires the calculation of the sensitivity of each observation to each of the estimated parameters that has been efficiently evaluated making use of an adjoint state formulation of the forward model MODFLOW_2005-Adjoint. The inverse code is the freeware bgaPEST, a highly parameterized inversion software package developed according to the PEST software concept. The Bayesian Geostatistical Approach is shown to be able to constrain the hydraulic conductivity field with a large number of parameters and just few observations. The results of the inversion are consistent with the alluvial nature of the investigated aquifer and the information content of the data. The calibrated model can then be used as a support tool for the design and the management of the new flood control structure.

A Combined Shape and Liner Optimization of a General Aeroengine Intake for Tone Noise Reduction

A novel acoustic design methodology for turbofan engine inlets is presented. A fan intake tonal noise propagation model is enhanced with its continuous adjoint formulation, having been first applied to design the intake. This is followed by the implement of a liner optimization study on the obtained optimal intake in which far-field noise attenuation due to the presence of a liner is maximized by adjusting the liner impedance. The dependence of optimal liner impedance on frequency is examined. The present paper aims at describing the potential of the combined shape optimization and liner optimization for low noise turbofan duct design. Prior to the optimization process, the implementation of the unsteady aeroacoustic adjoint method in shape optimization is validated by comparing the sensitivity derivative with that obtained by finite differences. The NASA flow impedance tube configuration is selected for benchmarking the time domain impedance condition (TDIBC) used in liner optimization. The examples presented demonstrate that significant noise reduction is achieved for most of the sound field by selecting the optimal shape and liner impedance. The result obtained confirms the effectiveness and efficiency of the proposed combined shape and liner optimization framework.

Adjoint-Based Optimization of Three-Dimensional Flapping-Wing Flows

Optimization of the three-dimensional unsteady viscous flow near a flapping wing in hover is performed using a time-dependent adjoint-based methodology. The sensitivities of a penalized lift coefficient to wing shape and kinematic parameters are computed using a time-dependent discrete-adjoint formulation. The unsteady discrete-adjoint equations required for the calculation of the sensitivity derivatives are integrated backward in time over the entire interval of interest. The gradient of the objective functional obtained using the adjoint formulation is then used to update the values of shape and kinematic design variables. The efficiency of this adjoint-based methodology is demonstrated by optimizing the shape and kinematics of a hovering wing. The numerical results show that the highest improvement in the wing performance is obtained by using combined optimization of the wing shape and kinematics.

Aerodynamic Optimization Design of Multi-stage Turbine Using the Continuous Adjoint Method

AbstractThis paper develops a continuous adjoint formulation for the aerodynamic shape design of a turbine in a multi-stage environment based on S

Sensitivity of aerodynamic forces in laminar and turbulent flow past a square cylinder

We use adjoint-based gradients to analyze the sensitivity of the drag force on a square cylinder. At Re = 40, the flow settles down to a steady state. The quantity of interest in the adjoint formulation is the steady asymptotic value of drag reached after the initial transient, whose sensitivity is computed solving a steady adjoint problem from knowledge of the stable base solution. At Re = 100, the flow develops to the time-periodic, vortex-shedding state. The quantity of interest is rather the time-averaged mean drag, whose sensitivity is computed integrating backwards in time an unsteady adjoint problem from knowledge of the entire history of the vortex-shedding solution. Such theoretical frameworks allow us to identify the sensitive regions without computing the actually controlled states, and provide a relevant and systematic guideline on where in the flow to insert a secondary control cylinder in the attempt to reduce drag, as established from comparisons with dedicated numerical simulations of the two-cylinder system. For the unsteady case at Re = 100, we also compute an approximation to the mean drag sensitivity solving a steady adjoint problem from knowledge of only the mean flow solution, and show the approach to carry valuable information in view of guiding relevant control strategy, besides reducing tremendously the related numerical effort. An extension of this simplified framework to turbulent flow regime is examined revisiting the widely benchmarked flow at Reynolds number Re = 22 000, the theoretical predictions obtained in the frame of unsteady Reynolds-averaged Navier–Stokes modeling being consistent with experimental data from the literature. Application of the various sensitivity frameworks to alternative control objectives such as increasing the lift and reducing the fluctuating drag and lift is also discussed and illustrated with a few selected examples.

Constrained multi-scale turnover Lagrangian map for anisotropic synthetic turbulence: A priori tests

Synthetic turbulence has been useful in the modelling and simulation of turbulence, and as a surrogate to understand the dynamics of real hydrodynamic turbulence. In a recently proposed Multiscale Turnover Lagrangian Map (MTLM) method, an initial random field is transformed into a synthetic field after a series of simple mappings, with moderate computational cost. It has been shown that the resulted fields reproduce highly realistic statistics on many aspects of isotropic hydrodynamic turbulence, including small-scale intermittency, geometric statistics, and pressure statistics. Thus, it is of great interests to generalize the method to model inhomogeneous turbulence. In this paper, we formulate the problem as an optimization problem, where the initial random field is taken as the control variable, and the additional features presented in inhomogeneous turbulence are taken as a target function to be matched by the synthetic fields. The goal is to find the optimal control variable which minimizes the difference between the target function and the synthetic field. Using the adjoint formulation, we derive the optimality system of the problem, which formulates a procedure to generate inhomogeneous synthetic turbulence. The procedure, named the Constrained MTLM, is applied to synthesize two Kolmogorov flows where persistent large scale structures produce nontrivial mean flow statistics and local anisotropy in small scales. We compare the synthetic fields with direct numerical simulation data, and show that the former reproduces closely the mean flow statistics such as Reynolds stress distribution and mean turbulent kinetic energy balance. They also reproduce the effects of inhomogeneity on small scale structures, which is manifested in the distributions of mean subgrid-scale energy dissipation, and the alignment between the subgrid-scale stress tensor and the filtered strain rate tensor, among others. We conclude that the method is useful to further extend the applicability of synthetic turbulence.

Full waveform inversion and the truncated Newton method: quantitative imaging of complex subsurface structures

ABSTRACTFull waveform inversion is a powerful tool for quantitative seismic imaging from wide‐azimuth seismic data. The method is based on the minimization of the misfit between observed and simulated data. This amounts to the solution of a large‐scale nonlinear minimization problem. The inverse Hessian operator plays a crucial role in this reconstruction process. Accounting accurately for the effect of this operator within the minimization scheme should correct for illumination deficits, restore the amplitude of the subsurface parameters, and help to remove artefacts generated by energetic multiple reflections. Conventional minimization methods (nonlinear conjugate gradient, quasi‐Newton methods) only roughly approximate the effect of this operator. In this study, we are interested in the truncated Newton minimization method. These methods are based on the computation of the model update through a matrix‐free conjugate gradient solution of the Newton linear system. We present a feasible implementation of this method for the full waveform inversion problem, based on a second‐order adjoint state formulation for the computation of Hessian‐vector products. We compare this method with conventional methods within the context of 2D acoustic frequency full waveform inversion for the reconstruction of P‐wave velocity models. Two test cases are investigated. The first is the synthetic BP 2004 model, representative of the Gulf of Mexico geology with high velocity contrasts associated with the presence of salt structures. The second is a 2D real data‐set from the Valhall oil field in North sea. Although, from a computational cost point of view, the truncated Newton method appears to be more expensive than conventional optimization algorithms, the results emphasize its increased robustness. A better reconstruction of the P‐wave velocity model is provided when energetic multiple reflections make it difficult to interpret the seismic data. A better trade‐off between regularization and resolution is obtained when noise contamination of the data requires one to regularize the solution of the inverse problem.

Volumetric b-splines shape parametrization for aerodynamic shape design

In this paper, volumetric b-splines are employed to deform the aerodynamic surface. The geometry is enclosed in a volumetric parallelepiped, referred to as control box, which in, essence, is similar to Free Form Deformation (FFD). The use of generic volumetric b-splines requires to calculate the parametric coordinates, but offers some important advantages over other parameterization techniques, such as deformation locality. This parameterization can be applied to the design of complex three-dimensional configurations and also offers the possibility to incorporate the geometric description extracted from the Computational Aid Design (CAD) files, which most frequently are in the form of Non-Uniform Rational B-Splines (NURBS), in what we call multilevel deformation; that is, the control box deforms the NURBS, while the NURBS deforms the computational grid, following the chain rule. In this work, a volumetric b-spline parameterization has been applied in an automatic gradient-based optimization strategy in order to optimize the aerodynamic surface of aeronautical configurations. The design variables are the control points, while the gradients are calculated using the continuous adjoint formulation.