Adjoint-based Optimization Research Articles

Adjoint-based optimisation of detonation initiation by a focusing shock wave

An optimisation study of a shock-wave-focusing geometry is presented in this work. The configuration serves as a reliable and deterministic detonation initiator in a pulsed detonation engine. The combustion chamber consists of a circular pipe with one convergent–divergent axisymmetric nozzle, acting as a focusing device for an incoming shock wave. Geometrical changes are proposed to reduce the minimum shock wave strength necessary for a successful detonation initiation. For that purpose, the adjoint approach is applied. The sensitivity of the initiation to flow variations delivered by this method is used to reshape the obstacle’s form. The thermodynamics is described by a higher-order temperature-dependent polynomial, avoiding the large errors of the constant adiabatic exponent assumption. The chemical reaction of stoichiometric premixed hydrogen-air is modelled by means of a one-step kinetics with a variable pre-exponential factor. This factor is adapted to reproduce the induction time of a complex kinetics model. The optimisation results in a 5% decrease of the incident shock wave threshold for the successful detonation initiation.

Topology-optimized catenary-like metasurface for wide-angle and high-efficiency deflection: from a discrete to continuous geometric phase

We investigate the topology optimization of geometric phase metasurfaces for wide-angle and high-efficiency deflection, where adjoint-based multi-object optimization approach is adopted to improve the absolute efficiency while maintaining the polarization conversion characteristic of geometric phase metasurfaces. We show that, for the initially discrete geometric phase metasurfaces with different materials and working wavelengths, the topology shapes gradually evolve from discrete structures to quasi-continuous arrangements with the increment of optimization iteration operations. More importantly, the finally optimized metasurfaces manifest as catenary-like structure, providing significant improvements of absolute efficiency. Furthermore, for the initial structure with catenary distribution, the corresponding optimized metasurface also has a catenary-like topology shape. Our results on the topology-optimized geometric phase metasurfaces reveal that, from the perspective of numerical optimization, the continuous catenary metasurfaces is superior to the discrete geometric phase metasurfaces.

Dynamic mode decomposition analysis of the common research model with adjoint-based gradient optimization

Aerodynamic shape refinement optimization for passenger aircraft is difficult and requires a significant workload. The adjoint-based gradient optimization method can quickly find local optimal solutions based on the initial shape in these types of problems. The optimization model of the common research model for the drag coefficient minimization and wing thickness constraints with a large-scale grid is established, and the drag coefficient is reduced by 10.2 counts while maintaining the lift coefficient. The stress-blended eddy simulation is used for unsteady simulations, and the optimized configuration can effectively eliminate oscillations in the middle of the upper wing surface. The spanwise flow is reduced and the pressure response on the wing surface is due primarily to shock chordal motion. For aerodynamic analyses with similar shapes, the dynamic mode decomposition (DMD) analysis shows that the upper wing surface mode amplitudes and spanwise instability modes of the optimized design are weaker, and the fluctuations of the pressure are more stable. Therefore, DMD is suitable for refined shape optimization analyses.

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.

Gradient-Based Aerodynamic Optimization of an Airfoil with Morphing Leading and Trailing Edges

This article presents a gradient-based aerodynamic optimization framework and investigates optimum deformations for a transonic airfoil equipped with morphing leading and trailing edges. Specifically, the proposed optimization framework integrates an innovative morphing shape parameterization with a high fidelity Reynolds-averaged Navier–Stokes computational fluid dynamic solver, a hybrid mesh deformation algorithm, and an efficient gradient evaluation method based on continuous adjoint implementation. To achieve a feasible morphing shape, some structural properties of skin and wing-box constraints were introduced into the morphing shape parameterization, which offers skin length control and enables wing-box shape invariance. In this study, the optimum leading and trailing edge deformations with minimization of drag at this cruise stage were searched for using the adjoint-based optimization with a nested feasible morphing procedure, subject to the wing-box, skin length, and airfoil volume constraints. The numerical studies verified the effectiveness of the optimization strategy, and demonstrated the significant aerodynamic performance improvement achieved by using the morphing devices. A lambda shock pattern was observed for the optimized morphing leading edge. That result further indicates the importance of leading edge radius control.

Enhanced data efficiency using deep neural networks and Gaussian processes for aerodynamic design optimization

Adjoint-based optimization methods are attractive for aerodynamic shape design primarily due to their computational costs being independent of the dimensionality of the input space and their ability to generate high-fidelity gradients that can then be used in a gradient-based optimizer. This makes them very well suited for high-fidelity simulation based aerodynamic shape optimization of highly parametrized geometries such as aircraft wings. However, the development of adjoint-based solvers involve careful mathematical treatment and their implementation require detailed software development. Furthermore, they can become prohibitively expensive when multiple optimization problems are being solved, each requiring multiple restarts to circumvent local optima. In this work, we propose a machine learning enabled, surrogate-based framework that replaces the expensive adjoint solver, without compromising on predicting predictive accuracy. Specifically, we first train a deep neural network (DNN) from training data generated from evaluating the high-fidelity simulation model on a model-agnostic design of experiments on the geometry shape parameters. The optimum shape may then be computed by using a gradient-based optimizer coupled with the trained DNN. Subsequently, we also perform a gradient-free Bayesian optimization, where the trained DNN is used as the prior mean. We observe that the latter framework (DNN-BO) improves upon the DNN-only based optimization strategy for the same computational cost. Overall, this framework predicts the true optimum with very high accuracy, while requiring far fewer high-fidelity function calls compared to the adjoint-based method. Furthermore, we show that multiple optimization problems can be solved with the same machine learning model with high accuracy, to amortize the offline costs associated with constructing our models. Our methodology finds applications in the early stages of aerospace design.

Adjoint-Free Aerodynamic Shape Optimization of the Common Research Model Wing

Adjoint-free methods are required in aerodynamic shape design optimization if an adjoint solver is unavailable. However, their performance is highly criticized in high-dimensional problems like wing shape design optimization. This work proposes an adjoint-free optimization method using deep-learning techniques to address the issue. A deep-learning-based optimal sampling method is developed to generate various wing shapes subject to both geometric validity and feasibility constraints. To address the curse of dimensionality in adjoint-free optimization, a compact wing shape parameterization method is presented by deriving global wing mode shapes from the sample wings. The proposed method is compared with the adjoint-based optimization method in both single-point and multipoint design of the Common Research Model wing. The proposed adjoint-free optimization method converges within 1000 objective function evaluations. The optimized shapes are close to those obtained by the adjoint-based optimization, and the differences in are all within 0.5 counts. The results show that the proposed adjoint-free optimization method has almost the same efficiency and effectiveness as the adjoint-based optimization method in high-dimensional wing shape design.

Adjoint-based shape optimization for the minimization of flow-induced hemolysis in biomedical applications

This paper reports on the derivation and implementation of a shape optimization procedure for the minimization of hemolysis induction in blood flows through biomedical devices.Despite the significant progress in relevant experimental studies, the ever-growing advances in computational science have made computational fluid dynamics an indispensable tool for the design of biomedical devices. However, even the latter can lead to a restrictive cost when the model requires an extensive number of computational elements or when the simulation needs to be overly repeated. This work aims at the formulation of a continuous adjoint complement to a power-law hemolysis prediction model dedicated to efficiently identifying the shape sensitivity to hemolysis. The proposed approach can accompany any gradient-based optimization method at the cost of approximately one additional flow solution per shape update. The approach is verified against analytical solutions of a benchmark problem and computed sensitivity derivatives are validated by a finite differences study on a generic 2D stenosed geometry. The included application addresses a 3D ducted geometry which features typical characteristics of blood-carrying devices. An optimized shape, leading to a potential improvement up to 22%, is identified. It is shown that the improvement persists for different hemolysis-evaluation parameters.

Using adjoint-based optimization to enhance ignition in non-premixed jets.

Gradient-based optimization is used to reliably and optimally induce ignition in three examples of laminar non-premixed mixture configurations. Using time-integrated heat release as a cost functional, the non-convex optimization problem identified optimal energy source locations that coincide with the stoichiometric local mixture fraction surface for short optimization horizons, while for longer horizons, the hydrodynamics plays an increasingly important role and a balance between flow and chemistry features determines non-trivial optimal ignition locations. Rather than identifying a single optimal ignition location, the results of this study show that there may be several equally good ignition locations in a given flow configuration.

Adjoint-based topology optimization to reduce the energy consumption of a Hall effect thruster

PurposeThe presented study aims to minimize the energy consumed by a Hall effect thruster (HET) under a constraint which makes it possible to generate a specified magnetic field in a target region of the thruster.Design/methodology/approachHerein topology optimization (TO) is used to reduce the energy consumption of an HET while keeping its performance unchanged. The design variables are the current densities in the coils and the distribution of materials in the polar pieces of the thruster. Intermediate values of material distribution are penalized using the solid isotropic material with penalization method to favor binary solutions. By means of the adjoint method, this paper provides the derivatives of the objective and constraint functions with respect to material distribution and current density variables.FindingsThe TO-based design methodology is developed and validated on a design example involving 2,051 variables. The approach shows its interest and its effectiveness of on a large scale two-criteria problem.Research limitations/implicationsIn this paper, TO is presented as a tool that has allowed to explore new and innovative designs. However, although the design presented is original, its fabrication is not feasible. Despite this, the designs found give a good idea of the starting points for shape and parametric optimization tools.Practical implicationsThrough the HET design problem, TO shows the ability to explore more original design possibilities of a complex magnetostatic design problem and to discover designs that make a HET more efficient with respect to several criteria at the same time.Originality/valueA new way to reduce the energy consumption of a HET is presented. To achieve this, an adjoint-based TO method is developed and then implemented in a simple way. This approach shows that, for efficiency purposes, TO is a key tool for extending the state of the art of HET designs.

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.

An adjoint optimization approach for the topological design of large-scale district heating networks based on nonlinear models

This article deals with the problem of finding the best topology, pipe diameter choices, and operation parameters for realistic district heating networks. Present design tools that employ non-linear flow and heat transport models for topological design are limited to small heating networks with up to 20 potential consumers. We introduce an alternative adjoint-based numerical optimization strategy to enable large-scale nonlinear thermal network optimization. In order to avoid a strong computational cost scaling with the network size, we aggregate consumer constraints with a constraint aggregation strategy. Moreover, to align this continuous optimization strategy with the discrete nature of topology optimization and pipe size choices, we present a numerical continuation strategy that gradually forces the design variables towards discrete design choices. As such, optimal network topology and pipe sizes are determined simultaneously. Finally, we demonstrate the scalability of the algorithm by designing a fictitious district heating network with 160 consumers. As a proof-of-concept, the network is optimized for minimal investment cost and pumping power, while keeping the heat supplied to the consumers within a thermal comfort range of 5%. Starting from a uniform distribution of 15 cm wide piping throughout the network, the novel algorithm finds a network lay-out that reduces piping investment by 23% and pump-related costs by a factor of 14 in less than an hour on a standard laptop. Moreover, the importance of embedding the non-linear transport model is clear from a temperature-induced variation in the consumer flow rates of 72%.

Adjoint-based optimization to minimize scattered pressure fields

Acoustic and elastic metamaterials have shown potential for many different applications in wave manipulation and structural isolation. Minimization of a scattered pressure field has been identified as an important use for such materials, and a wide variety of designs have been proposed for this purpose. Several of these designs are difficult to manufacture (e.g., pentamode designs), and many others involve numerous tunable parameters and a large design parameter space (e.g., distributions of resonators). This presents a challenge for global search-based optimization and raises the need for gradient-based methods. In this work, adjoint-based, PDE-constrained optimization is used to improve the feasibility of minimizing a scattered pressure field via an optimal set of channels in an annulus surrounding a rigid scatterer. Two different optimization strategies are investigated: material identification, where the physical properties of each channel are treated as unknown, and force identification, where the magnitude and phase of the reflected field is treated as unknown at various patches on the annulus. Results are presented for both a single plane wave direction and multiple plane wave directions, as well as for both a single frequency and multiple frequencies. [S.N.L. is managed and operated by NTESS under DOE NNSA Contract No. DE-NA0003525.]

Adjoint-based aerodynamic shape optimization including transition to turbulence effects

The inclusion of transition to turbulence effects in aerodynamic shape optimization makes it possible to use it as a tool for the design of airframes with laminar flow. Modified Reynolds-Averaged Navier–Stokes (RANS) models that consider transition to turbulence have gained traction in the computational fluid dynamics (CFD) community. These models enable the computation of transitional flows without the need for external modules. In this work, we use a smooth version of the amplification factor transport (AFT) model, called AFT-S, to perform gradient-based aerodynamic shape optimization (ASO) of airfoils in subsonic and transonic flow conditions. We investigate the benefits of including transition effects into the optimization process and assess the impact of losing laminar flow when early transition to turbulence occurs due to surface contamination. Our results indicate that our design optimization approach yields lower drag airfoils when transition effects are considered. For the transonic case, the optimizer trades between shock wave strength and laminar flow extension to minimize drag.

Natural Laminar-Flow Airfoil Optimization Design Using a Discrete Adjoint Approach

Natural laminar-flow wings are one of the most promising technologies for reducing fuel burn and emissions for commercial aviation. However, there is a lack of tools for performing shape optimization of wings based on computational fluid dynamics considering laminar-to-turbulent transition. To address this need, we develop a discrete adjoint-based optimization framework where transition is modeled. The core of this framework is a Reynolds-averaged Navier–Stokes solver that is coupled with a simplified method to predict Tollmien–Schlichting and laminar separation-induced transition that consists of a laminar boundary-layer code and a database method for flow stability analysis. The transition prediction is integrated with a Spalart–Allmaras turbulence model through a smoothed intermittency function, which makes it suitable for gradient-based optimization. A coupled-adjoint approach that uses transpose Jacobian-vector products derived via automatic differentiation computes the transition prediction derivatives. Lift-constrained drag minimizations of airfoils for a single-point design and a multipoint design problem are performed. The results show that the optimizer successfully reduces the drag coefficient by increasing the extent of laminar flow. The multipoint optimization formulation produces an airfoil with a significant amount of laminar flow that is maintained at several flight conditions. The proposed methods make it possible to perform aerodynamic shape optimization considering laminar-to-turbulent transition in airfoil optimization.

Dynamic Flow Modelling for Model-Predictive Wind Farm Control

We aim to improve wind farm control for power output by building on the results from WFSim for the development of a dynamic wind farm model. This model will be part of a closed-loop, economic model-predictive control approach for wind farms. It is constructed from first principles using open-source tools to be suitable for adjoint-based optimisation of turbine yaw angles.In a steady-state inflow configuration with two turbines, the new control model matches power expectations from high fidelity simulations in SOWFA to within 15 %. Under time-varying wind directions, it shows time delays in wake direction as inflow changes propagate through the farm with the wind speed, although the dynamics still differ from the SOWFA reference. The model runs flow simulations for a wind farm with a 3 x 3 array of turbines at a real-time order of magnitude on a regular laptop computer.The new control model shows dynamic flow behaviour as wake changes propagate through the wind farm. Some further adjustments are necessary to accurately model three-dimensional flow in two dimensions. With more validation of the wake dynamics, it will be suitable for application in a new closed-loop wind farm controller.

Adjoint-based optimization in the development of low-emission industrial boilers

ABSTRACT A gradient-based method has been developed and programmed to optimize the NH injections of an existing biomass-fired bubbling fluidized bed boiler, the targets being to minimize both the NO and the NH emissions. In this context, the reactive flow inside the boiler is modelled using a custom-built OpenFOAM solver, and then the NO and NH species are calculated using a post-processing technique. The multiobjective optimization problem is solved by optimizing several weight combinations of the objectives using the gradient-projection method. The required sensitivities were calculated by differentiating the post-processing solver according to the discrete adjoint method. The adjoint-based sensitivities are validated against finite differences calculations. Moreover, in order to evaluate the optimization results, the optimization problem is solved using evolutionary algorithms software. Finally, the optimization results are physically interpreted and the strengths and weaknesses of the proposed method are discussed.

Optimal perturbations and transition energy thresholds in boundary layer shear flows

Subcritical transition to turbulence in spatially developing boundary layer flows can be triggered efficiently by finite amplitude perturbations. In this work, we employ adjoint-based optimization to identify optimal initial perturbations in the Blasius boundary layer, culminating in the computation of the subcritical transition critical energy threshold and the associated fully localized critical optimum in a spatially extended configuration, the so called minimal seed. By dynamically rescaling the variables with the local boundary layer thickness, we show that the identified edge trajectory approaches the same attracting phase space region as previously reported edge trajectories, and reaches the region more efficiently.

Regularized ensemble Kalman methods for inverse problems

Inverse problems are common and important in many applications in computational physics but are inherently ill-posed with many possible model parameters resulting in satisfactory results in the observation space. When solving the inverse problem with adjoint-based optimization, the problem can be regularized by adding additional constraints in the cost function. However, similar regularizations have not been used in ensemble-based methods, where the same optimization is done implicitly through the analysis step rather than through explicit minimization of the cost function. Ensemble-based methods, and in particular ensemble Kalman methods, have gained popularity in practice where physics models typically do not have readily available adjoint capabilities. While the model outputs can be improved by incorporating observations using these methods, the lack of regularization means the inference of the model parameters remains ill-posed. Here we propose a regularized ensemble Kalman method capable of enforcing regularization constraints. Specifically, we derive a modified analysis scheme that implicitly minimizes a cost function with generalized constraints. We demonstrate the method's ability to regularize the inverse problem with three cases of increasing complexity, starting with inferring scalar model parameters. As a final case, we utilize the proposed method to infer the closure field in the Reynolds-averaged Navier–Stokes equations, a problem of significant importance in fluid dynamics and many engineering applications.

Blood flow imaging by optimal matching of computational fluid dynamics to 4D-flow data.

Three-dimensional, time-resolved blood flow measurement (4D-flow) is a powerful research and clinical tool, but improved resolution and scan times are needed. Therefore, this study aims to (1) present a postprocessing framework for optimization-driven simulation-based flow imaging, called 4D-flow High-resolution Imaging with a priori Knowledge Incorporating the Navier-Stokes equations and the discontinuous Galerkin method (4D-flow HIKING), (2) investigate the framework in synthetic tests, (3) perform phantom validation using laser particle imaging velocimetry, and (4) demonstrate the use of the framework in vivo. An optimizing computational fluid dynamics solver including adjoint-based optimization was developed to fit computational fluid dynamics solutions to 4D-flow data. Synthetic tests were performed in 2D, and phantom validation was performed with pulsatile flow. Reference velocity data were acquired using particle imaging velocimetry, and 4D-flow data were acquired at 1.5 T. In vivo testing was performed on intracranial arteries in a healthy volunteer at 7 T, with 2D flow as the reference. Synthetic tests showed low error (0.4%-0.7%). Phantom validation showed improved agreement with laser particle imaging velocimetry compared with input 4D-flow in the horizontal (mean -0.05 vs -1.11 cm/s, P < .001; SD 1.86 vs 4.26 cm/s, P < .001) and vertical directions (mean 0.05 vs -0.04 cm/s, P = .29; SD 1.36 vs 3.95 cm/s, P < .001). In vivo data show a reduction in flow rate error from 14% to 3.5%. Phantom and in vivo results from 4D-flow HIKING show promise for future applications with higher resolution, shorter scan times, and accurate quantification of physiological parameters.