Gradient-based optimization offers significant efficiency advantages for wind turbine blade design, but its application has often been limited by the cost and accuracy of finite-difference derivative calculations, especially when fatigue constraints are considered. In this work, we systematically compare and evaluate four differentiation techniques, namely algorithmic differentiation, implicit differentiation, sparsity exploitation, and parallelization, to determine their effectiveness in computing accurate gradients through time-domain aero-structural simulations. By integrating these techniques with unsteady nonlinear aerodynamic and structural models, we develop software designed for accurate gradient computation. We show that combining these techniques addresses memory and runtime challenges associated with long simulations required by design load cases. Specifically, the most effective combination reduces derivative computation wall time by over an order of magnitude compared to finite differencing while maintaining superior accuracy. We demonstrate this approach in a proof-of-concept aero-structural optimization of a wind turbine blade that improves the cost of energy by 12.78 %. This comparative study establishes a viable approach for fatigue-aware blade design that balances computational efficiency with modeling accuracy.

This paper introduces innovative optimization and deep learning techniques to enhance the prediction of complex wake dynamics in the downstream wind velocity of tilted wind turbines. Traditional methods for calibrating the Bastankhah wake model often lead to increased errors in wind velocity distribution due to overfitting of the local wake characteristics. To address this issue, we propose an additional global optimization step to reduce errors in wind velocity predictions with respect to various wake parameters. Despite this improvement, the Bastankhah model's axisymmetric Gaussian wake shape limits its accuracy for complex wake structures. Therefore, we also propose a deep learning approach, which demonstrates promising results by accurately modeling complex wake shapes across a broader range of tilt angles with minimal computational cost. The deep learning approach achieves near-identical predictions to high-fidelity large-eddy simulations, representing a promising advancement in wake modeling.

Numerical simulation of ordinary differential equations (ODEs) can be challenging when the system exhibits high accelerations and rapidly changing dynamics. Under these conditions the ODE solver often needs to take very small time steps in order to resolve the solution accurately, resulting in increased computational cost. In order to accelerate the simulation of these ODEs we present a novel methodology that uses a pseudo-invertible neural network to map system states into a high-dimensional latent-space. The network is then trained so that the dynamics in this learned latent space are slow, and can be simulated with relatively few function calls. Unlike existing neural methods, the latent dynamic equations are not learned from trajectory data, but derived from the original system equations and the chain rule. This allows the method to generalize better than existing approaches because the derived equations are correct by construction. In this work, we derive latent state equations of motion for any general ODE, and describe the loss function used to enforce slow time evolution of the latent states. We then apply this technique to multiple example ODEs and show that these problems can be solved with 3x to 20x fewer function calls for the same accuracy when simulating in the learned latent space. This reduction in cost could decrease computational demands for scientific simulations across engineering and physics applications.

There are many new and exciting technologies employing distributed elec- tric propulsion, including applications for advanced air mobility (AAM). An important consideration for many AAM applications is the trade between aerodynamic performance and noise generation. Electric ducted fans (EDF) have been suggested as potentially able to meet both performance and noise generation constraints. In this work, we utilize gradient-based optimization employing an axisymmetric ducted rotor aerodynamic model, linear cascade surrogate model, geometrically exact beam structural model, and analytically and empirically based acoustics models to explore aero-structural-acoustic trades for EDF preliminary design. We explore rotor count and duct length configurations while optimizing rotor blade and duct geometry for minimum energy expense across a range of tonal sound power level constraints. Our aero-structural-acoustic optimizations of electric ducted fans show that for the low blade counts studied in this work tonal sound power level sources modeled in this work can be reduced by 10 dB for each additional blade with a 0.6-0.7% increase in energy cost for the flight missions explored. Similarly an increase in duct length equal to the rotor tip radius yields a nearly 40 dB reduction in tonal sound power level with an accompanying 4.8% rise in energy expense. For a given configuration, small changes in rotor and duct geometry can provide a 4 dB decrease in tonal sound power level with less than a 1% rise in energy requirements. This work represents a first step in gradient-based, mid-fidelity, aero-structural-acoustic optimization of electric ducted rotors and sets the stage for future development efforts.

Optimizing wind farms is essential for designing efficient energy systems, especially as farms grow larger and span multiple sites. However, this optimization becomes increasingly challenging due to the rising computational cost associated with more turbines. Gradient-based optimization methods scale better than gradient-free approaches for large problems, but the most computationally expensive component remains the calculation of gradients for the objective function and constraint Jacobians. To address this, we propose leveraging sparsity to accelerate gradient evaluations and reduce the size of the constraint Jacobian. Wind farms naturally exhibit sparsity-many turbines do not influence each other under certain wind directions. However, unlike traditional sparse problems with fixed patterns, wind farm sparsity is dynamic, requiring new strategies to handle changing interactions efficiently. This paper presents a study of sparsity in wind farm optimization and introduces several methods to exploit it. These strategies are tested on multiple farms using the analytic Cumulative Curl model, with gradients computed via automatic differentiation (AD). The same sparsity-aware techniques are also applicable to finite difference (FD) methods, where they can yield even greater speedups due to the high cost of directional evaluations. Results show that sparse methods achieve up to a 10 speedup with less than 5% variance in optimized wake losses compared to traditional methods. These findings suggest that sparsity-aware optimization not only maintains solution quality but also scales efficiently with farm size, enabling more comprehensive design exploration at reduced computational cost.

ABSTRACT Physics‐based design optimization workflows thread the needle between computational cost limitations and simulation complexity, often compromising between modeling detail and the range of operating design conditions. Multipoint aerostructural optimization of wind turbine rotors has so far been confined to low‐fidelity analyses or to high‐fidelity studies with simplified structural models, leaving the most complex design trade‐offs unexplored. We close this gap by performing the first tightly coupled gradient‐based multipoint aerostructural rotor optimization using 3D aerodynamic and structural solvers with discrete coupled adjoints. The optimizer simultaneously varies blade planform, airfoil shapes, and structural thickness through more than 270 design variables, minimizing a weighted combination of rotor mass and power across multiple wind speeds. Applied to a modified DTU 10‐MW benchmark under conservative structural and aerodynamic constraints, our multipoint optimization reduces rotor mass by up to 36% and increases power by 12%–15% across the main operating conditions; biasing the objective toward power yields power gains up to 18% and a 17% mass reduction. For a nominal wind distribution, 3‐point rotor designs accounting for low RPM and high thrust conditions capture dominant trade‐offs and outperform single‐point designs. Adding two off‐design points changes individual‐condition power by less than 3% but leaves the weighted average within 0.5%, and the mass‐power bias has a stronger effect on the final design than the operating‐point weighting itself. Our framework extends naturally to richer load cases and site‐specific wind distributions, providing a basis for high‐fidelity multipoint design earlier in industrial workflows.

Tilt-rotor propulsion system design requires a multidisciplinary approach to tackle important challenges and competing tradeoffs between disciplines. This paper models rotor aerodynamics, blade structures, vehicle drag, electric propulsion, and tonal/broadband acoustics for a tilt-rotor, electric vertical takeoff and landing aircraft using low-to-mid fidelity tools. The authors use gradient-based design optimization with automatic differentiation and parameter sensitivity analyses to explore the design space and complex tradeoffs of tilt-rotor distributed electric propulsion systems, exploring effects of variations in payload/empty weight, battery specific energy, and blade tip speed. This framework models multiple operating points with a mission-focused objective to account for the effects of both hover and cruise conditions on the overall system performance. Additionally, we develop a Pareto front between range and noise and observe that, for the same noise output, modeling tonal and broadband noise increases range by 3.1% when compared to using a Mach tip speed surrogate acoustics model.

Airborne wind energy (AWE) technology aspires to provide increased options for wind energy harvesting at a lower cost than conventional turbines. As AWE technology is still in its infancy, there is relatively little published information concerning its aerodynamic details. In this study, we apply the FLOWUnsteady suite of aerodynamic analysis tools-including a vortex particle method for wakes and actuator line methods for rotors and wings-to the analysis of a fixed-wing, on-board generation, wind harvesting aircraft (or windcraft for short). Specifically, we explore variations in the rotation directions and vertical and horizontal (front to back) spacing of the rotors to understand the complex interactional aerodynamics of a multirotor windcraft design. Our results indicate that for the configurations presented herein, the rotor-on-rotor and wing-on-rotor interactions induce maximum total variations on cumulative rotor performance of less than 3% relative to baseline, with most configurations varying from baseline by 2% or less. In addition, rotor-on-wing interactions cause decreases in wing lift coefficient of up to roughly 7% (for the configurations simulated) relative to the isolated wing. Furthermore, through selecting specific rotor rotation directions, it is possible to prescribe a portion of the shape of the wing lift distribution. Further exploration, perhaps with the assistance of optimization techniques, may yield greater insights into this uncharted space of windcraft interactional aerodynamics.

Large wind turbines yield more energy but demand careful aeroelastic blade design. Coupled multiphysics design strategies can reduce wind energy costs by exploiting fluid–structure interactions. This work presents the first high-fidelity aerostructural optimization study of a large wind turbine rotor. We use blade-resolved fluid dynamics and structural solvers in a monolithic gradient-based optimization framework to explore steady-state torque and blade mass tradeoffs. The coupled-adjoint approach computes gradients efficiently, enabling the optimization of over 100 structural and geometric parameters simultaneously. Our optimization study modifies a DTU 10 MW benchmark with a simplified structure and isotropic material properties. The tightly coupled optimizations increase torque by 14% while reducing rotor mass by 9% or reduce blade mass by 27% while maintaining torque. Blade-resolved models provide greater design freedom, enabling 5% higher mass reductions than conventional parameterizations at equal torque. This framework paves the way for more detailed high-fidelity optimization studies to complement conventional design approaches.

Tilt-rotor propulsion system design requires a multidisciplinary approach to tackle important challenges and competing tradeoffs between disciplines. This paper models rotor aerodynamics, blade structures, vehicle drag, electric propulsion, and tonal/broadband acoustics for a tilt-rotor, electric vertical takeoff and landing aircraft using low-to-mid fidelity tools. The authors use gradient-based design optimization with automatic differentiation and parameter sensitivity analyses to explore the design space and complex tradeoffs of tilt-rotor distributed electric propulsion systems, exploring effects of variations in payload/empty weight, battery specific energy, and blade tip speed. This framework models multiple operating points with a mission-focused objective to account for the effects of both hover and cruise conditions on the overall system performance. Additionally, we develop a Pareto front between range and noise and observe that, for the same noise output, modeling tonal and broadband noise increases range by 3.1% when compared to using a Mach tip speed surrogate acoustics model.

Large-scale gradient-based Multidisciplinary Design Optimization (MDO) can aid in the exploration of high-dimensional design spaces for novel air vehicle concepts, thereby leading to more efficient and economic designs. This paper builds on past works where we demonstrated large-scale physics-based MDO capabilities and applied these to NASA's lift-plus-cruise electric air taxi concept. We extend this comprehensive mid-fidelity system-level optimization problem with high(er)-fidelity subsystem-level optimizations of various aircraft systems and mission phases, with the aim of further enhancing the accuracy and scope of the aforementioned system-level optimization problem: 1) Power minimization during the transition mission phase; 2) Electrical powertrain topology optimization; 3) Cell chemistry modeling and thermomechanical battery pack topology optimization; 4) Shell-based coupled aero-elastic wing structure optimization. An Analytical Target Cascading-like distributed MDO architecture allows us to couple the system- and subsystem-level optimizations in order to arrive at a consistent and feasible design. We find that the system-level design is most heavily impacted by the reduced battery pack-level energy densities that stem from power peaks in the mission profile. These power peaks seem to result from the inefficient lift rotor blade designs that are needed to satisfy noise constraints. We draw conclusions based on trends in our results and give recommendations for future MDO studies of electrical air taxi vehicle concepts.

Automatic differentiation (AD) is a powerful tool for evaluating numerical derivatives. In particular, reverse-mode AD provides numerical gradients in a way that is insensitive to the number of input variables. This makes reverse-mode AD well-suited for solving large optimization problems. However, reverse-mode AD has a particularly large associated memory cost because most intermediate values in operations need to be cached. This is problematic for large problems such as aerodynamics simulations, though, since the memory requirements can quickly become impractical. The solution implemented in this work is to provide analytic pullback expressions for functions for which many of the intermediate values are not needed. This solution is then applied to a vortex-particle method simulation to obtain numerical derivatives significantly faster.

The vortex particle method (VPM) is a mesh-free approach to computational fluid dynamics (CFD) solving the Navier-Stokes equations in their velocity-vorticity form. The VPM uses a Lagrangian scheme, which not only avoids the hurdles of mesh generation, but it also conserves vortical structures over long distances with minimal numerical dissipation while being orders of magnitude faster than conventional mesh-based CFD. However, VPM is known to be numerically unstable when vortical structures break down close to the turbulent regime. In this study, we reformulate the VPM as a large eddy simulation (LES) in a scheme that is numerically stable, without increasing its computational cost. A new set of VPM governing equations are derived from the LES-filtered Navier-Stokes equations. The new equations reinforce conservation of mass and angular momentum by reshaping the vortex elements subject to vortex stretching. In addition to the VPM reformulation, a new anisotropic dynamic model of subfilter-scale (SFS) vortex stretching is developed. This SFS model is well suited for turbulent flows with coherent vortical structures where the predominant cascade mechanism is vortex stretching. Advection, viscous diffusion, and vortex stretching are validated through simulation of isolated and leapfrogging vortex rings. Mean and fluctuating components of turbulent flow are validated through simulation of a turbulent round jet, where Reynolds stresses are resolved directly and compared to experimental measurements. Finally, the computational efficiency of the scheme is showcased in the simulation of an aircraft rotor in hover, showing our meshless LES to be 100x faster than a mesh-based LES with similar fidelity, while being 10x faster than a low-fidelity unsteady Reynolds-average Navier-Stokes simulation and 1000x faster than a high-fidelity detached-eddy simulation.

The vortex particle method (VPM) has gained popularity in recent years due to a growing need to predict complex aerodynamic interactions during the preliminary design of electric multirotor aircraft. However, VPM is known to be numerically unstable when vortical structures break down close to the turbulent regime. In recent work, the VPM has been reformulated as a large-eddy simulation (LES) in a scheme that is both meshless and numerically stable without increasing its computational cost. In this study, we build upon this meshless LES scheme to create a solver for interactional aerodynamics. Propeller blades are introduced through an actuator line model following well-established practices for LES. A novel, vorticity-based actuator surface model (ASM) is developed for wings, which is suitable for propeller–wing interactions when a wake impinges on the surface of a wing. This ASM imposes the no-flow-through condition at the airfoil centerline by calculating the circulation that meets this condition and by immersing the associated vorticity in the LES following a pressure-like distribution. Extensive validation of propeller–wing interactions is presented by simulating a tailplane with tip-mounted propellers and a blown wing with propellers mounted midspan.

Electric ducted fans have become an intriguing option for the propulsion systems of clean, quiet advanced air mobility technologies due to their potential benefits in both aerodynamic efficiency and reduced noise profiles compared to open rotor systems. Exploration of conceptual electric ducted fan design in the context of novel, and often complex, applications may be greatly aided by the use of optimization techniques. Specifically, gradient-based optimization lends itself to the exploration of large, complex, multi-disciplinary systems due to its inherent scalability. Despite ducted fans/propellers having been relatively well studied for the last century, modern, gradient-based, optimization-ready analysis tools for ducted fans are scarce. Building from existing ducted fan analysis methods, and utilizing modern code language and differentiation tools, we have been able to develop a gradient-based, optimization-ready analysis tool for low-Mach electric ducted fans. In this work, we present our ducted fan analysis code-dubbed Ducted Axisymmetric Propulsor Evaluation (DuctAPE)-and showcase its suitability for application in gradient-based optimization settings. We verify DuctAPE's accuracy through comparison with the Ducted Fan Design Code (DFDC), showing that DuctAPE matches DFDC within 0.5% for each the various solution methods implemented in DuctAPE. We also showcase DuctAPE's functionality in a gradient-based optimization setting, showing three ducted fan optimizations with different combinations of design variables. From our results we conclude that DuctAPE is a suitable tool for conceptual and preliminary design using modern, gradient-based optimization techniques. Future work will include the application of DuctAPE in multi-disciplinary, gradient-based optimization settings.

Design of vertical takeoff and landing aircraft is challenging in part due to significant aerodynamic interactions between rotors and wings. Computational models can aid in their design, but are computationally expensive. 3-D panel methods coupled with vortex particle wakes offer an attractive solution, but solving for the panel strengths scales poorly for large problems. Multigrid methods, such as Krylov subspace methods in conjunction with the fast multipole method (FMM), have been demonstrated to reduce the scaling to O(N). We explore the performance and limitations of the Krylov-FMM method, traditional matrix-powered GMRES, LU decomposition, and a novel O(N) multigrid approach to solving boundary element problems by combining FMM with Gauss-Seidel iterations. We also explore the effect of warm-starting these methods in unsteady aerodynamic simulation, leading to an order-of-magnitude decrease in cost.

Trajectory optimization of aircraft transition maneuvers can significantly influence the design of these systems, particularly when making decisions about aircraft geometry, propulsion sizing, and control system design. This paper presents the trajectory optimization of air vehicles including electric vertical takeoff and landing (eVTOL) aswell as conventional aircraft. The study evaluates the influence of fidelity in trajectory design by comparing the use of two aerodynamic methods in the context of trajectory optimization. The mid-fidelity method utilizes a vortex lattice method (VLM) to model lifting surfaces while employing blade element momentum theory (BEMT) to model rudimentary rotor-wing interactions. The high-fidelity approach applies a vortex particle method (VPM) to model the wing and propeller geometries while capturing more complex wake interactions. This work also presents a trajectory optimization methodology that is well suited for high fidelity VPM simulations and overcomes many of the challenges associated with traditional methods such as robust convergence and computational cost. The new method is validated through a case study which leverages the VLM-BEMT model to compare results with that of a traditional shooting method. The method is then employed to optimize trajectories using the VPM to model both conventional and eVTOL aircraft.