Adjoint Solver Research Articles

Efficient, multimodal, and derivative-free bayesian inference with Fisher–Rao gradient flows

Abstract In this paper, we study efficient approximate sampling for probability distributions known up to normalization constants. We specifically focus on a problem class arising in Bayesian inference for large-scale inverse problems in science and engineering applications. The computational challenges we address with the proposed methodology are: (i) the need for repeated evaluations of expensive forward models; (ii) the potential existence of multiple modes; and (iii) the fact that gradient of, or adjoint solver for, the forward model might not be feasible.

While existing Bayesian inference methods meet some of these challenges individually, we propose a framework that tackles all three systematically. Our approach builds upon the Fisher-Rao gradient flow in probability space, yielding a dynamical system for probability densities that converges towards the target distribution at a uniform exponential rate. This rapid convergence is advantageous for the computational burden outlined in (i). We apply Gaussian mixture approximations with operator splitting techniques to simulate the flow numerically; the resulting approximation can capture multiple modes thus addressing (ii). Furthermore, we employ the Kalman methodology to facilitate a derivative-free update of these Gaussian components and their respective weights, addressing the issue in (iii).

The proposed methodology results in an efficient derivative-free posterior approximation method, flexible enough to handle multi-modal distributions: Gaussian Mixture Kalman Inversion (GMKI). The effectiveness of GMKI is demonstrated both theoretically and numerically in several experiments with multimodal target distributions, including proof-of-concept and two-dimensional examples, as well as a large-scale application: recovering the Navier-Stokes initial condition from solution data at positive times.



Hull shape optimization of autonomous underwater vehicles using a full turbulent continuous adjoint solver

The present study aims at simultaneously optimizing the nose and tail of two distinct Autonomous Underwater Vehicles (AUVs) using a full turbulent continuous adjoint solver. The entire procedure is carried out utilizing two open-source software: Salome (CAD and mesh generator) and OpenFOAM v2206 (CFD solver and optimizer). Reynolds-Averaged Navier–Stokes (RANS) equations closed by the k-ω SST model are employed to simulate the turbulent flow around AUV hulls. Besides, adjointOptimisationFoam solver is used to perform the shape optimization. Two different unconstrained and constrained optimizations of AUV hulls are conducted at an angle of attack (α) equal to 0°. In the unconstrained optimization, the drag force is chosen as an objective function to be minimized while a partial volume is considered as a geometric constraint in the constrained optimization. The predicted drag force is validated against experimental results from the literature, indicating a mere 0.58% discrepancy. In the unconstrained optimizations, the first and second AUV hulls reveal a 4.17% and 3.66% reduction in the drag force, respectively. In the constrained optimizations of the first and second AUV hulls, drag force is reduced by 3.25% and 2.63%, respectively.

Design Optimization of a Novel Flush Boundary Layer Diverter

Abstract A novel boundary layer diverter is designed and optimized for increasing the performance of a typical semi-submerged inlet ingesting large amounts of boundary layer flow. A conceptual diverter configuration based on the prior optimization study is first parametrized with seven design variables. A response surface model is constructed for the pressure recovery at the aerodynamic interface plane (AIP). The flow fields over the diverter and through the inlet duct are computed with SU2 and a baseline diverter is constructed from the response surface model for maximum pressure recovery. The baseline diverter is then placed in a free-form deformation box and shape optimized using the adjoint solver of SU2 with hundreds of design parameters. The optimum configuration provides 2.4% and 7.93% increase in the pressure recovery and the mass flow rate, respectively, together with a significant reduction in flow distortion. The novel diverter is then shape optimized together with the inlet duct. Together with all the performance parameters, the pressure recovery is improved by more than 3%. It is shown that the novel boundary layer diverter by being flush to the inlet surface induces a much smaller drag compared to the conventional diverters.

Shape optimization of annular transonic thrust nozzles via genetic algorithm and adjoint method

In this research, the idea of combining the genetic algorithm and adjoint method was proposed to leverage the benefits of both approaches to improve the accuracy of thrust nozzle shape optimization. Although the genetic algorithm can search the entire design space to approach the global optimum, its computational cost increases considerably with the number of design variables. Meanwhile, the adjoint method is unaffected by the design variables, but it cannot achieve the global optimum directly and relies on the initial guess geometry. To address these challenges, a genetic algorithm was initially employed to approximate the global optimal geometry of a thrust nozzle with a conical center body parameterized by a Bezier curve. A MATLAB code integrated the geometry parameterization, genetic algorithm, and Fluent R2 2021 flow solver. The resulting geometry was then considered as the initial guess for the adjoint method, which accurately achieved the final optimal geometry with increased control points. The genetic algorithm aimed to maximize the thrust coefficient as the objective function, while the adjoint solver aimed to minimize the axial velocity variance at the nozzle outlet as the cost function. Three distinct constraints were incorporated in the optimization process, and the influence of the center body shape on the results was thoroughly examined to consider all possible geometries throughout the optimization procedure. The findings revealed that the conical center body effectively adjusted the flow angle at the exit, significantly affecting the thrust force. Both optimization steps reduced the average velocity and increased the average pressure at the nozzle exit, enhanced the mass flow rate, and finally increased the thrust force. The genetic algorithm and adjoint method contributed to 3 % and 0.6 % increases, respectively, in the thrust force, resulting in a thrust coefficient of 96.15 % for the final optimized nozzle.

Data-driven optimisation of wind farm layout and wake steering with large-eddy simulations

Abstract. Maximising the power production of large wind farms is key to the transition towards net zero. The overarching goal of this paper is to propose a computational method to maximise the power production of wind farms with two practical design strategies. First, we propose a gradient-free method to optimise the wind farm power production with high-fidelity surrogate models based on large-eddy simulations and a Bayesian framework. Second, we apply the proposed method to maximise wind farm power production by both micro-siting (layout optimisation) and wake steering (yaw angle optimisation). Third, we compare the optimisation results with the optimisation achieved with low-fidelity wake models. Finally, we propose a simple multi-fidelity strategy by combining the inexpensive wake models with the high-fidelity framework. The proposed gradient-free method can effectively maximise wind farm power production. Performance improvements relative to wake-model optimisation strategies can be attained, particularly in scenarios of increased flow complexity, such as in the wake steering problem, in which some of the assumptions in the simplified flow models become less accurate. The optimisation with high-fidelity methods takes into account nonlinear and unsteady fluid mechanical phenomena, which are leveraged by the proposed framework to increase the farm output. This paper opens up opportunities for wind farm optimisation with high-fidelity methods and without adjoint solvers.

The Design and Optimization of Additively Manufactured Radial Compressor of a Miniature Gas Turbine Engine

Abstract This paper presents a multidisciplinary design methodology for a single-stage radial compressor of a small-scale gas turbine jet engine. The entire engine, producing more than 600 N thrust with an engine diameter of less than 30 cm, will be manufactured by additive manufacturing (AM) in a single piece. Therefore, it is crucial to pinpoint the design limitations that may arise due to the AM methods. The preliminary design calculations are carried out in the Von Karman Institute for Fluid Dynamics (VKI's) in-house centrifugal compressor off-design (CCOD) code. The detailed design and optimization studies coupled with computational fluid dynamics (CFD) simulations are performed to improve the aerodynamic performance of the radial compressor by using the sequential quadratic programming (SQP) method with an Adjoint solver in the VKI's in-house computer aided design and optimization (CADO) tool. In the final design, a large increase in thrust is obtained and the manufacturing constraints are satisfied.

A duality-preserving adjoint method for segregated Navier–Stokes solvers

Adjoint methods efficiently compute gradients for systems with many inputs and have been widely used for large-scale gradient-based optimization in fluid mechanics. To ensure optimization's numerical robustness, we need to develop an adjoint solution algorithm that has a similar, if not the same, convergence rate as the primal flow solver at each optimization iteration. This consistent primal-adjoint convergence behavior is also called duality-preserving (DP). Existing DP adjoint methods are mostly for fully coupled Navier–Stokes (NS) solvers with compressible flows. However, few DP adjoints have been proposed for segregated NS solvers, e.g., the semi-implicit method for pressure-linked equations (SIMPLE) algorithm solvers widely used for incompressible flow simulations. This study will fill this gap by deriving a fixed-point segregated adjoint formulation that fully preserves the convergence behavior of primal solvers. We first rewrite the steady-state segregated NS primal solution process into a fully coupled left-preconditioned Richardson format. Then, we transpose the preconditioner matrix to obtain a DP iterative adjoint formulation. In addition to algebraic derivation, we create a new graph representation that significantly streamlines the DP adjoint development for new segregated solvers. We prove that the proposed graph representation is equivalent to the above preconditioner transpose approach and guarantees duality. We evaluate our proposed algorithm using three cases with increasing complexity: a miniature-sized problem, an airfoil, and a wing. To quantify duality, we compare the eigenvalues between the primal and adjoint solvers and observe excellent agreements. We also evaluate to what extent various numerical settings (e.g., inner iteration tolerances and non-dual preconditioner) impact the eigenvalue distributions. Our proposed adjoint achieves machine-precision accurate gradient computation with competitive speed. Finally, we incorporate our adjoint solver into a large-scale gradient-based optimization framework and demonstrate its capability for wing aerodynamic shape optimization. The proposed fixed-point adjoint approach has the potential to make the adjoint solution more robust and efficient for any segregated NS solvers.

Shape optimization using the adjoint solver in computational fluid dynamics for additive manufacturing of a pollen filter

Pollen filters play an essential role in protecting people from airborne allergens and ensuring indoor air quality. Allergic reactions to pollen can lead to discomfort, reduced productivity, and increased healthcare costs. A low pressure drop of these pollen filters not only enhances the comfort of individuals using the filters but also contributes to energy savings in ventilation systems, thereby promoting environmental sustainability. This research focuses on the shape optimization of pollen filters using the adjoint solver in computational fluid dynamics, aiming to enhance both human health and environmental sustainability. In a previous study, an approach using the adjoint solver was developed to optimize both the separation efficiency and the pressure drop. In the current work, a methodology is presented that exploits these findings and allows the design of initial structures, subsequent optimization, and detailed experimental and numerical comparisons with a reference filter using the example of a pollen filter. To validate the effectiveness of the optimized filter, the initial geometry and the optimized geometry were fabricated and tested on a test bench. Compared to a reference filter, our filter disk was able to separate 2.9% more particles of size 6 μm and the pressure drop was lower by 34.2%. This research work demonstrates that the developed method can effectively be used to improve the performance of pollen filters. The results obtained from the validation suggest that the optimized geometry of the filter exhibits higher separation efficiency while keeping the pressure drop low compared to state-of-the-art pollen filters.

Design Optimization of Fins of a Sounding Rocket for Maximum Lift-to-Drag Ratio and Minimum Radar Cross- Section Area Using ANSYS

The optimization of a sounding rocket's fins is a crucial part of improving its performance. The geometric optimization of a sounding rocket's fins is presented in this paper. The main goal is to optimize the geometry for the minimum radar cross-section area and maximum lift-to-drag ratio. In CATIA, a 3D model was created. The L/D ratio was maximized using the ANSYS adjoint solver, and the radar cross-section area was minimized using ANSYS optimetrics. To determine the total RCS, the RCS of each fin was determined individually and then added together. Root, leading edge, tip, and trailing edge were the four parameters that were defined for the RCS optimization. The L/D ratio was increased by 8.3 times, and the RCS was decreased by 12% after optimization. Additionally, the body surface can be optimized further. The missile industry can benefit from the paper's findings.

Enhancing formula student car performance: Nose shape optimization via adjoint method

For a formula student vehicle, the nose is the first point of flow contact and decides the flow direction to help other aero components work efficiently. Therefore, an optimized nose shape is crucial for the optimum running condition of the vehicle. In recent years, Gradient-based optimization in the automotive industry has emerged with an adjoint approach. The adjoint solver is used to modify the geometry such as the nose of a car to the best optimal shape with respect to other design variables. Hence, the present study encompasses an examination conducted under six distinct Reynolds number conditions, specifically, 8.15 × 105, 9.05 × 105, 9.96 × 105, 10.87 × 105, 11.77 × 105, and 12.68 × 105. The simulation setup is validated using the Ahmed Body's 30° slant angle. The setup uses the k-omega SST turbulence model to capture the airflow phenomenon over the racing car. The simulation setup shows only a 0.07 % variation from the existing experimental findings. The results are represented in terms of different 2D plots, contours and streamline plots. The optimized nose has 27 % less drag and consumes 27.09 % less fuel than the base model at a Reynolds number of 10.8 × 105. To the automotive industry and race community, this optimization technique will be expected to enhance the vehicle's overall performance.

Suppressing instability in a Vlasov–Poisson system by an external electric field through constrained optimization

Maintaining the stability and shape of a plasma is a crucial task in many technological applications ranging from beam shaping to fusion energy. This is often challenging as plasma systems tend to be naturally unstable and kinetic effects can play an important role in the behavior of the instabilities. Due to the large energies involved, the primary way to control plasma systems is the application of external electromagnetic fields.In this work, we deploy a PDE-constrained optimization formulation that uses a kinetic description for plasma dynamics as the constraint. This is to optimize, over all possible controllable external electric fields, the stability of the plasma dynamics under the condition that the Vlasov–Poisson (VP) equation is satisfied. Mathematically we formulate it as a PDE-constrained optimization. For computing the functional derivative with respect to the external field in the optimization updates, the adjoint equation is derived. Furthermore, in the discrete setting, where we employ the semi-Lagrangian method as the forward solver, we also explicitly formulate the corresponding adjoint solver and the gradient as the discrete analogy to the adjoint equation and the Fréchet derivative. A distinct feature we observed of this constrained optimization is the complex landscape of the objective function and the existence of numerous local minima, largely due to the hyperbolic nature of the VP system. To overcome this issue, we utilize a gradient-accelerated genetic algorithm, leveraging the advantages of the genetic algorithm's exploration feature to cover a broader search of the solution space and the fast local convergence aided by the gradient information. We show that our algorithm obtains good electric fields that are able to maintain a prescribed profile in a beam shaping problem and uses nonlinear effects to suppress plasma instability in a two-stream configuration.

Adjoint-based estimation of sensitivity of clinical measures to boundary conditions for arteries

The use of adjoint solvers is considered in order to obtain the sensitivity of clinical measures in aneurysms to incomplete (or unknown) boundary conditions and/or geometry. It is shown that these techniques offer interesting theoretical insights and viable computational tools to obtain these sensitivities. A optimization framework with Navier-Stokes equations (in the laminar regime) as constraints is introduced. Sensitivities with respect to inflow and inflow position are derived. In the specific case of inflow normal to the boundary, the sensitivities are shown to be the function of adjoint pressure. For the Poiseuille flow through a channel, explicit expressions for all the underlying quantities such as solution to the state equations, adjoint equations and sensitivites are derived, which are further confirmed by numerical experiments. In the next two examples, flow inside aneurysm geometries is considered.

Adjoint-Based Design Optimization of a Volute for a Radial Compressor

Numerical optimization methods are widely used for designing turbomachinery components due to the cost and time savings they can provide. In the available literature, the shape optimization of radial compressors mainly focuses on improving the impeller alone. However, it is well-established knowledge that the volute plays a key role in the overall performance of the compressor. The aim of the present paper is to perform an adjoint-based optimization of a volute that is designed for the SRV2-O compressor. The CAD model was first created by using the parametrization of 33 design parameters. Then, a butterfly topology was applied to mesh the computational domain with a multi-block structured grid, and an elliptic smoothing procedure was used to improve the quality of the fluid grid. A steady-state RANS CFD solver with a Spalart-Allmaras turbulence model was used to solve the Navier–Stokes equations, and then the flow sensitivities were computed with an adjoint solver. The objective function consists of minimizing the loss coefficient of the volute. The optimization is performed to obtain an improved design with a 14% loss reduction. A detailed flow and design analysis is carried out to highlight the loss reduction mechanisms, followed by the optimizer. Finally, the compressor map of the full stage is compared between the baseline and the optimized volute from the CFD simulations using a mixing plane interface. This research demonstrates the successful use of a gradient-based optimization technique to improve the volute of a radial compressor and opens the door towards simultaneously optimizing the wheel and the volute.

A multi region adjoint-based solver for topology optimization in conjugate heat transfer problems

This work presents an exploration of fluid region optimization within coupled fluid–thermal problems of industrial significance, namely the design of a cooling plate for the thermal management of Printed Circuit Boards (PCB) of electrical propulsion systems. The Topology Optimization technique has been employed through a in-house developed multi-region adjoint solver and a set of customized boundary conditions, allowing the sensitivity computation independently on the problem size. The technique involves the integration of solid material into the computational domain to induce modifications in flow dynamics. This alteration aims to minimize a multi-objective function that considers both the skin temperature and the mechanical power dissipation caused by fluid movement across the domain. The obtained sensitivity values were then employed in optimizing material distribution through the Method of Moving Asymptotes. The derived material distribution was further post-processed to extract the newly optimized configuration of the system. This enabled a thorough evaluation of the optimization methodology’s performance and its effectiveness in enhancing the system’s overall efficiency.

Effect of constant eddy viscosity assumption on optimization using a discrete adjoint method

This paper presents a thorough study of the effect of the Constant Eddy Viscosity (CEV) assumption on the optimization of a discrete adjoint-based design optimization system. First, the algorithms of the adjoint methods with and without the CEV assumption are presented, followed by a discussion of the two methods’ solution stability. Second, the sensitivity accuracy, adjoint solution stability, and Root Mean Square (RMS) residual convergence rates at both design and off-design operating points are compared between the CEV and full viscosity adjoint methods in detail. Finally, a multi-point steady aerodynamic and a multi-objective unsteady aerodynamic and aeroelastic coupled design optimizations are performed to study the impact of the CEV assumption on optimization. Two gradient-based optimizers, the Sequential Least-Square Quadratic Programming (SLSQP) method and Steepest Descent Method (SDM) are respectively used to draw a firm conclusion. The results from the transonic NASA Rotor 67 show that the CEV assumption can deteriorate RMS residual convergence rates and even lead to solution instability, especially at a near stall point. Compared with the steady cases, the effect of the CEV assumption on unsteady sensitivity accuracy is much stronger. Nevertheless, the CEV adjoint solver is still capable of achieving optimization goals to some extent, particularly if the flow under consideration is benign.

Multi-objective shape optimization of doubly offset serpentine diffuser using Adjoint method

Doubly offset serpentine diffusers are needed in the compact designs of modern stealth fighters with highly integrated propulsion systems. In addition, these doubly offset serpentine diffusers are also essential to provide reduced radar cross-section (RCS) due to the blockage of a complete line of sight. In this research, shape optimization of the doubly offset serpentine diffuser is achieved using a multi-objective approach to maximize the pressure recovery (PR) and minimize the flow swirl angle (α) and distortion coefficient (DC) at the aerodynamic interface plane (AIP). A computationally efficient Adjoint-based method was implemented for the optimization. For baseline geometry, an additional offset is added to NASA’s M2129 single serpentine duct to make it doubly offset serpentine diffusers. The freeform deformation technique was used to parametrize the baseline shape using 180 control points. A steady-state flow solution through the diffuser was achieved by solving Reynolds-Averaged Navier-Stokes (RANS) equations using a general-purpose computational analysis tool, i.e., ANSYS Fluent. The sensitivity of objective functions from the Adjoint solver was used to compute the required design change in the direction of minima or maxima of the objective functions using the steepest descent optimization technique. Subsequently, the calculated design changes were implemented through mesh Morpher. The Adjoint-based approach and freeform deformation resulted in a powerful and efficient optimization framework. The optimized shape showed a significant reduction in distortion coefficient and flow swirl angle, along with a slight increase in pressure recovery.

Airfoil Shape Optimization of a Horizontal Axis Wind Turbine Blade using a Discrete Adjoint Solver

In this study, airfoil shape optimization of a wind turbine blade is performed using the ANSYS Fluent Adjoint Solver. The aim of this optimization process is to increase the wind turbine output power, and the objective function is to maximize the airfoil lift to drag ratio (Cl/CD ). This study is applied to the NREL phase VI wind turbine, therefore, the S809 airfoil is used as a reference profile. First, for the validation of the applied numerical model, steady-state simulations are carried out for the S809 airfoil at various angles of attack. Then, the optimization is performed with the airfoil set at a fixed angle of attack, , considering three Reynolds numbers, Re =3 105,4.8 105 and 106. Next, computations are performed for the fluid flow around the optimized airfoils at angles of attack AOA= 6.1° ranging from 0° to 20°. The results show that (i) the lift to drag ratios of the optimized airfoils are significantly improved compared to the baseline S809 airfoil, (ii) this improvement is sensitive to the Reynolds number, and (iii) the Cl/CD ratios are also improved for another angle of attack values. Thereafter, the optimized airfoils are used for the design of the NREL Phase VI blade and the aerodynamic performances of this new wind turbine are assessed using the open-source code QBlade. These latter results indicate that when the blades are designed with the optimized airfoils, the wind turbine aerodynamic performances increase significantly. Indeed, at a wind speed of 10 m/s, the power output of the wind turbine is improved by about 38% compared to that of the original turbine.

Numerical Investigation of Pressure Loss in a Rectangular Channel with a Sharp 180-Degree Turn: Influence of Design Variables and Geometric Shapes

Gas turbine blade cooling typically uses a cooling air passage with a sharp 180° turn in the midchord area of the airfoil. Its geometric shape and dimensions are strictly constrained within the airfoil to ensure both aerodynamic and cooling performance. These characteristics imply the importance of understanding the relationships between the geometric dimensions and the cooling channel performance. In this study, we validated a numerical method using the commercial software, Ansys Fluent 2021 R2, by predicting a total pressure loss coefficient with less than 6% deviation from the experimental results of Metzger et al. for four different Reynolds numbers. Through parameter studies, the divider tip-to-wall clearance was found to be the most significant parameter influencing the pressure loss. Parameter correlations and predictive models between the design variables and the pressure loss were derived by regression analysis using the R language; the regression model predicted the pressure loss to within 2.29% of the numerical method. As the geometries changed, the response surface and the adjoint solver improved the pressure loss by approximately 20.87% and 25.96%, respectively, at the representative Reynolds number of 24,230; this showed that the adjoint solver was a relatively simple and effective method with minimal geometric changes.

Adjoint-Based High-Fidelity Concurrent Aerodynamic Design Optimization of Wind Turbine

To evaluate novel turbine designs, the wind energy sector extensively depends on computational fluid dynamics (CFD). To use CFD in the design optimization process, where lower-fidelity approaches such as blade element momentum (BEM) are more popular, new tools to increase the accuracy must be developed as the latest wind turbines are larger and the aerodynamics and structural dynamics become more complex. In the present study, a new concurrent aerodynamic shape optimization approach towards multidisciplinary design optimization (MDO) that uses a Reynolds-averaged Navier–Stokes solver in conjunction with a numerical optimization methodology is introduced. A multidisciplinary design optimization tool called DAFoam is used for the NREL phase VI turbine as a baseline geometry. Aerodynamic design optimizations in terms of five different schemes, namely, cross-sectional shape, pitch angle, twist, chord length, and dihedral optimization are conducted. Pointwise, a commercial mesh generator is used to create the numerical meshes. As the adjoint approach is strongly reliant on the mesh quality, up to 17.8 million mesh cells were employed during the mesh convergence and result validation processes, whereas 2.65 million mesh cells were used throughout the design optimization due to the computational cost. The Sparse Nonlinear OPTimizer (SNOPT) is used for the optimization process in the adjoint solver. The torque in the tangential direction is the optimization’s merit function and excellent results are achieved, which shows the promising prospect of applying this approach for transient MDO. This work represents the first attempt to implement DAFoam for wind turbine aerodynamic design optimization.

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.