Unsteady Vortex Lattice Method Research Articles

Accelerating aeroelastic UVLM simulations by inexact Newton algorithms

AbstractWe consider the aeroelastic simulation of flexible mechanical structures submerged in subsonic fluid flows at low Mach numbers. The nonlinear kinematics of flexible bodies are described in the total Lagrangian formulation and discretized by finite elements. The aerodynamic loads are computed using the unsteady vortex-lattice method wherein a free wake is tracked over time. Each implicit time step in the dynamic simulation then requires solving a nonlinear equation system in the structural variables with additional aerodynamic load terms. Our focus here is on the efficient numerical solution of this system by accelerating the Newton algorithm. The particular structure of the aeroelastic nonlinear system suggests the structural derivative as an approximation to the full derivative in the linear Newton system. We investigate and compare two promising algorithms based on this approximation, a quasi-Newton type algorithm and a novel inexact Newton algorithm. Numerical experiments are performed on a flexible plate and on a wind turbine. Our computational results show that the approximation can indeed accelerate the Newton algorithm substantially. Surprisingly, the theoretically preferable inexact Newton algorithm is much slower than the quasi-Newton algorithm, which motivates further research to speed up derivative evaluations.

Rapid dynamic aeroelastic response analysis of the highly flexible wing with distributed propellers influence

Abstract A rapid nonlinear aeroelastic framework for the analysis of the highly flexible wing with distributed propellers is presented, validated and applied to investigate the propeller effects on the wing dynamic response and aeroelastic stability. In the framework, nonlinear beam elements based on the co-rotational method are applied for the large-deformation wing structure, and an efficient cylinder coordinate generation method is proposed for attached propellers at different position. By taking advantage of the relatively slow dynamics of the high-aspect-ratio wing, propeller wake is modeled as a quasi-steady skewed vortex cylinder with no updating process to reduce the high computational cost. Axial and tangential induced velocities are derived and included in the unsteady vortex lattice method. For the numerical cases explored, results indicate that large deformation causes thrust to produce wing negative torsion which limits the displacement oscillation, and slipstream mainly increases the response values. In addition, an improvement of flutter boundary is found with the increase of propeller thrust while slipstream brings a destabilising effect as a result of the increment of dynamic pressure and local lift. The great potential of distributed propellers in gust alleviation and flutter suppression of such aircraft is pointed out and the method as well as conclusions in this paper can provide further guidance.

Development and application of a mesh generator intended for unsteady vortex-lattice method simulations of wind turbines and wind farms

Abstract. In the last decades, the unsteady vortex-lattice method (UVLM) has gained a lot of acceptance to study large onshore–offshore wind turbines (WTs). Furthermore, and due to the development of more powerful computers, parallelization strategies, and algorithms like the fast multipole method, it is possible to use vortex-based methods to analyze and simulate wind farms (WFs). However, UVLM-based solvers require structured meshes, which are generally very tedious to build using classical mesh generators, such as those utilized in the context of finite element methods (FEMs). Wind farm meshing is further complicated by the large number of design parameters associated with the wind turbine (coning angle, tilt angle, blade shape, etc.), farm layout, modeling of the terrain topography (for onshore WFs), and modeling of the sea level surface (for offshore WFs), which makes the use of FEM-oriented meshing tools almost inapplicable. In the literature there is a total absence of meshing tools when it comes to building aerodynamic grids of WTs and WFs to be used along with UVLM-based solvers. Therefore, in this work, we present a detailed description of the geometric modeling and computational implementation of an interactive UVLM-oriented mesh generator, named UVLMeshGen, developed entirely in MATLAB® and easily adaptable to GNU OCTAVE, for wind turbines and onshore–offshore wind farms. The meshing tool developed here consists of (i) a geometric processor in charge of designing and discretizing an entire wind farm and (ii) an independent module in charge of computing the kinematics for the entire WF. The output data provided by the UVLMeshGen consist of nodal coordinates and connectivity arrays, making it especially attractive and useful to be used by other flow potential solvers using vortices, sources and sinks, or dipoles/doublets, among others. The work is completed by providing a series of aerodynamic results related to WTs and WFs to show the capabilities of the mesh generator, without going into detailed discussions of wind turbine aerodynamics, which are not the focus of this paper. The meshing tool developed here is freely available under a Creative Commons Attribution 4.0 International License (Roccia, 2023).

A Preliminary Approach towards Rotor Icing Modeling Using the Unsteady Vortex Lattice Method

UAV rotors are at a high risk of ice accumulation during their operations in icing conditions. Thermal ice protection systems (IPSs) are being employed as a means of protecting rotor blades from ice, yet designing the appropriate IPS with the required heating density remains a challenge. In this work, a reduced-order modeling technique based on the Unsteady Vortex Lattice Method (UVLM) is proposed as a way to predicting rotor icing and to calculate the required anti-icing heat loads. The UVLM is gaining recent popularity for aircraft and rotor modeling. This method is flexible enough to model difficult aerodynamic problems, computationally efficient compared to higher-order CFD methods and accurate enough for conceptual design problems. A previously developed implementation of the UVLM for 3D rotor aerodynamic modeling is extended to incorporate a simplified steady-state icing thermodynamic model on the stagnation line of the blade. A viscous coupling algorithm based on a modified α-method incorporates viscous data into the originally inviscid calculations of the UVLM. The algorithm also predicts the effective angle of attack at each blade radial station (r/R), which is, in turn, used to calculate the convective heat transfer for each r/R using a CFD-based correlation for airfoils. The droplet collection efficiency at the stagnation line is calculated using a popular correlation from the literature. The icing mass and heat transfer balance includes terms for evaporation, sublimation, radiation, convection, water impingement, kinetic heating, and aerodynamic heating, as well as an anti-icing heat flux. The proposed UVLM-icing coupling technique is tested by replicating the experimental results for ice accretion and anti-icing of the 4-blade rotor of the APT70 drone. Aerodynamic predictions of the UVLM for the Figure of Merit, thrust, and torque coefficients agree within 10% of the experimental measurements. For icing conditions at −5 °C, the proposed approach overestimates the required anti-icing flux by around 50%, although it sufficiently predicts the effect of aerodynamic heating on the lack of ice formation near the blade tips. At −12 °C, visualizations of ice formation at different anti-icing heating powers agree well with UVLM predictions. However, a large discrepancy was found when predicting the required anti-icing heat load. Discrepancies between the numerical and experimental data are largely owed to the unaccounted transient and 3D effects related to the icing process on the rotating blades, which have been planned for in future work.

Enhanced Unsteady Vortex Lattice Aerodynamics for Nonlinear Flexible Aircraft Dynamic Simulation

Several modeling extensions and numerical improvements to the unsteady vortex-lattice method are discussed for the simulation of very flexible aircraft dynamics. In particular, fuselage aerodynamics are included by means of linear source panels, polar corrections are incorporated on the aerodynamics under potentially large deformations, and a new wake discretization scheme is derived to accelerate the computations. The theory behind each approach is detailed and successfully verified on representative test cases. The resulting modeling environment is then exemplified in a study of the flight dynamics of a flexible aircraft demonstrator model. Results indicate that, for large wing deformations, a noticeable effect of fuselage aerodynamic interference appears on the wing aeroelastic behavior. The polar corrections improve the drag estimations and allow capturing lift decrements due to stall onset at high induced angles of attack, which then affects the gust loads. The wake discretization scheme enables a reduction of the computational time for dynamic simulations on a full aircraft model by about 20% while keeping the arising error negligibly small.

Experimental and numerical flutter analysis of a folding fin with multiple asymmetric free-plays

Experimental folding fin models with an adjustable free-play are tested in a wind tunnel. The fin structure is modeled using the free-interface component mode synthesis method, and its free-play is modeled as four independent nonlinear springs with asymmetric stiffness. A nonplanar unsteady vortex-lattice method considering compressibility is employed to address nonlinear deformation and high subsonic flow. Surface spline interpolation is improved through projection and partition. The aeroelastic characteristics of folding fins with different free-play magnitudes, initial conditions and elastic-axis positions are analyzed using an established time-marching method because of its relatively small computation scale and high precision. The results show good consistency among the presented method, the wind tunnel test and the harmonic balance method. There is a negative correlation between the critical speed of divergent motion and the ratio of the initial condition to the free-play magnitude. If either the free-play magnitude or the initial condition is extreme (tiny or vast), the system nonlinearity degenerates to linearity. Generally, the flutter prevention design of a linear model can be applied to a nonlinear model, such as moving the elastic-axis position aftward. The presented fin configuration exhibits an unstable limit cycle oscillation because the orders of coupled flutter modes do not change with variations in equivalent linear stiffness.

Effects of piezoelectric energy harvesting from a morphing flapping tail on its performance

Monitoring fish migration, which can extend over distances of thousands of kilometers, via fish tags is important to maintain healthy fish stocks and preserve biodiversity. One constraint of current fish tags is the limited power of their batteries. Attaching a piezoelectric element to an oscillating part of the fish body has been proposed to develop self-powered tags. To determine the functionality and potential of this technology, we present an analysis showing variations of the generated voltage with specific aspects of the tail’s response. We also perform numerical simulations to validate the analysis and determine the effects of attaching a piezoelectric element on performance metrics including thrust generation, propulsive efficiency, and harvested electric power. The tail with the attached piezoelectric element is modeled as a unimorph beam moving at a constant forward speed and excited by sinusoidal pitching at its root. The hydrodynamic loads are calculated using three-dimensional unsteady vortex lattice method. These loads are coupled with the equation of motion, which is solved using the finite element method. The implicit finite different scheme is used to discretize the time-dependent generated voltage equation. The analysis shows that the harvested electric power depends on the slope of the trailing edge, a result that is validated with the numerical simulations. The numerical simulations show that, depending on the excitation frequency, attaching a piezoelectric element can increase or decrease the thrust force. The balance of required hydrodynamic power, generated propulsive power and harvested electrical power shows that, depending on the excitation frequency, relatively high levels of harvested power can be harvested without a high adverse impact on the hydrodynamic or propulsive power. For a specified frequency of oscillations, the approach and results can be used to identify design parameters where harvested electrical power by a piezoelectric element will have a minimal adverse impact on the hydrodynamic or propulsive power of a swimming fish.

Assessment of yaw-control effects on wind turbine-wake interaction: A coupled unsteady vortex lattice method and curled wake model analysis

Downstream wind turbines in a wind farm are affected by the wake of the upstream turbine. Therefore, wake steering has been introduced to alleviate the wake effect by yawing the shaft of the upstream wind turbine. In this scenario, the upstream and downstream wind turbines encounter non-uniform inflow in wake steering. In this study, the unsteady vortex lattice method (UVLM) was coupled with the curled wake model to consider the unsteady aerodynamics resulting from the wake steering. The proposed method was validated with the National Renewable Energy Laboratory (NREL) Phase VI experiment and Large Eddy Simulation (LES) data simulated with two-turbine case. Three NREL 5 MW wind turbines were simulated in wake steering conditions, and the power performance of the individual turbines and their root flapwise bending moments were investigated. The flow field investigation indicated that the counter-rotating vortices significantly influence the effective angle of attack and aerodynamic loads. In addition, a secondary effect was observed on the effective angle of attack of the rotor and wake deflection. The simulation results demonstrated the applicability of the UVLM in wake steering and its potential for analyzing the wake effects experienced by multiple wind turbines in a wind farm.

Code-to-Code Benchmark for Simulation Tools Based on the Unsteady Vortex-Lattice Method

Reliable aerodynamic and aeroelastic simulations of advanced aeronautical/mechanical systems require us to predict flow-induced forces as accurately as possible. Nowadays, computational fluid dynamic techniques are quite popular, but at an overwhelming computational cost. Consequently, methods like the unsteady vortex-lattice method (UVLM) became the workhorses for many simulation environments. Then, numerous UVLM-based codes using diverse numerical schemes, enhanced by several add-ons and implemented following different programming paradigms, were available in the literature. However, there is no set of benchmark cases intended for the systematic verification of those codes relying on the UVLM. Therefore, we provide six fully reproducible benchmark cases that can be used for such an end. We also describe two in-house UVLM-based codes that are well suited for aerodynamic simulations and for being encapsulated as an aerodynamic engine within partitioned aeroelastic simulation schemes. Because both codes follow radically different implementation philosophies, these represent excellent candidates to undergo the series of benchmark cases proposed. The work is completed by providing a valuable dataset and comparison criteria to measure to what extent two or more codes are in agreement. Along this path, for very first time, we use a comparison strategy to contrast free-wake methods based on the Hausdorff distance.

A Numerical Unsteady Analysis of a Plunging Wing

Numerical simulation of unsteady motion of wing through undisturbed fluid is formulated using Unsteady Vortex Lattice Method. A free-wake algorithm is developed using a fourth order Adam-Bashforth technique. Rankine vortex model is used to avoid numerical singularities and wake decay algorithm is used to model the viscous dissipation of the shed wake. The resulting MN x MN equations are solved for unknown Γwing using LU decomposition, where M and N are number of divisions of the wing along chord and span respectively. The aerodynamic coefficient of lift, CL, induced drag, CDi , distribution of circulation, Γ and wake geometry are calculated for unsteady plunging motion. Effect of cross-flow is studied. The results are found to be in good agreement with available literature.

Strongly coupled fluid–structure interaction analysis of aquatic flapping wings based on flexible multibody dynamics and the modified unsteady vortex lattice method

Because of their superior efficiency and low detectability compared to conventional submarine-shaped vehicles, bionic underwater vehicles such as Mantabot are playing increasingly important roles in ocean exploration. However, current research on manta-inspired robots is limited to structural design and pure hydrodynamics analysis of pectoral fin performance, with other aspects such as hydroelasticity and optimal design of flapping wings rarely addressed. This work presents a novel method for analyzing aquatic flapping wings by coupling flexible multibody dynamics with a modified unsteady vortex lattice method. The flexible multibody system is solved by using the absolute nodal coordinate formulation, and the conventional unsteady vortex lattice method is modified to accommodate dynamic stall. Compactly supported radial basis functions are used to transfer the hydroelastic forces and structural displacements across the interface meshes while satisfying global energy conservation, and a predictor–corrector method is used to stabilize the iteration procedure. Three different experiments are used to validate the flexible multibody dynamics solver, the modified unsteady vortex lattice solver, and the proposed fluid–structure interaction framework, respectively. Finally, the hydroelastic framework is demonstrated for a fin-like wing, and how flexibility and movement affect the flapping wing dynamics is examined. Numerical examples show that reducing the structural stiffness of the aquatic flapping wing within a certain range improves the propulsive efficiency but also reduces the thrust coefficient, and how the stiffness distribution affects the thrust and propulsive efficiency remains consistent over a range of the Strouhal number.

Parameter Optimization of Foldable Flapping-Wing Mechanism for Maximum Lift

AbstractA lot of flapping-wing mechanisms have been proposed to mimic the flight characteristics of biological flyers. However, it is difficult to find studies that consider the unsteady aerodynamics in the design of the flapping-wing mechanisms. This paper presents a systematic approach to optimize the design parameters of a foldable flapping-wing mechanism (FFWM) with a proper aerodynamics model. For the kinematic model, the eight design parameters are defined to determine the reference configuration of the FFWM. The geometrical constraints of each design parameter are derived, and the kinematic analysis is conducted using the plane vector analysis method. The aerodynamic simulation using an unsteady vortex lattice method is performed to compute the aerodynamic loads induced by the flapping motion. An optimization problem is formulated to search for the optimal design parameters that maximize the average lift force considering the required power corresponding to the aerodynamic torques. The parameter optimization problem is solved for three different length ratios of the outer wing to the inner wing using a genetic algorithm. The optimization results show that increasing the outer wing length can cause a significant loss in the required power. The optimal design parameters found by the proposed approach allow the FFWM to generate maximum lift force with appropriate consideration of the required power.

Multibody-Dynamics Approach to Study the Deformation and Aerodynamics of an Insect Wing

In this study, a multibody-dynamics simulation approach was developed for a hawkmoth flexible wing. The wing structure is modeled as a chain of rigid bodies connected by elastic springs, and the aerodynamic force is measured by the extended unsteady vortex-lattice method. The multibody-dynamics and aerodynamic solvers are combined by an implicit coupling approach, and the quasi-Newtonian method is adopted to solve the system of nonlinear differential equations of motion. For validation, numerical results were compared with measurement data from a robotic wing and a living insect. A parametric analysis was conducted to study the effects of several kinematic parameters on the deformation and aerodynamic performance of the wing in hover. In most cases, using a flexible wing is far more efficient in terms of force production in comparison with its rigid counterpart. In general, wing deformation may cause considerable differences between the wing-tip and wing-base kinematic parameters. In particular, elevation motion as observed in living insects may be due to the passive oscillations of elastic elements, as opposed to a deliberate motion.

Aerodynamic Modeling and Simulation of Multi-Lifting Surfaces Based on the Unsteady Vortex Lattice Method

Using the unsteady vortex lattice method based on the potential flow theory, a rapid modeling approach is developed for the aerodynamic computation of multi-lifting surfaces. Multiple lifting surfaces with different geometric parameters and grid divisions can be quickly integrated and meshed with the object-oriented data structure. The physical influence between different lifting surfaces was modeled, and the wake–surface interaction was also considered by using different built-in vortex core models. The trajectory data were used to replace the pre-calculated downwash superposition for boundary condition integration, and the instantaneous boundary condition was generated directly from the kinematic states and mesh messages of the model concerned. Considering the direct coupling effect between aerodynamics and rigid body dynamics, the function for free flight was built for medium-fidelity dynamic simulations and aerodynamic data identifications. The proposed high-efficiency modeling and simulation process can be easily applied to models with any number of different lifting surfaces and arbitrary motion modes.

A cost-effective algorithm for the co-simulation of unsteady and non-linear aeroelastic phenomena

Increasingly flexible structures force engineers and scientists to improve simulation capabilities incorporating an adequate description of non-linear phenomena. Furthermore, many critical operating conditions of aeronautical and civil structures are associated with unsteady phenomena, which makes it necessary to tackle this kind of problems. A critical aspect to be analyzed is the stability of the system: the determination of, not only the critical flutter speed, but also the post-critical behavior, is fundamental to guarantee the safety of designs and determine mitigation strategies. In this work, a numerical technique that allows simulating the non-linear and unsteady behavior of highly flexible structures immersed in a fluid at high Reynolds numbers is presented. This problem is addressed using a co-simulation approach, coupling the Unsteady Vortex Lattice Method (as the aerodynamic model) with the Finite Element Method using non-linear beams and rigid bodies (as the structural dynamics model). The coupling is achieved through a weak interaction algorithm. The resulting method has a very good compromise between versatility and computational cost, and is not restricted to periodic motions or linear equations of motion. Nonlinearity is included through the possibility to represent large displacements and rotations of the structure, a visco-elastic material behavior, and a nonlinear relation between the aerodynamic loads and the incidence of the incoming flow. The domains considered in the submodels and the temporal and spatial discretizations used for each one are essentially different; the interaction process must take into account these aspects. The computational implementation is carried out by combining a general-purpose structural dynamics code with an aerodynamic tool oriented to the analysis of three-bladed horizontal axis wind turbines. Examples to validate the code and show the potential of the proposed method are presented in the Results section.

A near-optimal-trim-search method for six-degrees-of-freedom model insects: An application to flight in the lateral-wind condition

This paper presents an improved trim-search method for six-degrees-of-freedom (6-DOF) flapping-wing insects and its application to flight in the lateral-wind condition. A multibody-dynamics (MBD) code was developed and coupled with an extended unsteady vortex-lattice method (UVLM) to create a flapping-flight simulation framework. Using artificial neural networks (ANNs) and the genetic algorithm (GA), the trim-search method was developed to provide a near-optimal solution. The paper shows that the required power of the trim solution obtained by the present method is less than that from the conventional one. Based on the research results, the paper has revealed how a model insect can respond to sustain their hovering flight state in the lateral-wind condition. In general, the insect has to change its left- and right-wing kinematic parameters asymmetrically. The most noticeable differences between the kinematic parameters of the two wings are found in the sweep and rotation amplitudes. The research also showed an increase in the required power of the windward wing and a decrease in that of the leeward wing. However, the total required power generally grows due to the lateral wind.

Quasi three-dimensional deformable blade element and unsteady vortex lattice reduced-order modeling of fluid–structure interaction in flapping wings

Flapping, flexible insect wings deform under inertial and fluid loading. Deformation influences aerodynamic force generation and sensorimotor control, and is thus important to insect flight mechanics. Conventional flapping wing fluid–structure interaction models provide detailed information about wing deformation and the surrounding flow structure, but are impractical in parameter studies due to their considerable computational demands. Here, we develop two quasi three-dimensional reduced-order models (ROMs) capable of describing the propulsive forces/moments and deformation profiles of flexible wings. The first is based on deformable blade element theory (DBET) and the second is based on the unsteady vortex lattice method (UVLM). Both rely on a modal-truncation based structural solver. We apply each model to estimate the aeromechanics of a thin, flapping flat plate with a rigid leading edge, and compare ROM findings to those produced by a coupled fluid dynamics/finite element computational solver. The ROMs predict wing deformation with good accuracy even for relatively large deformations of 25% of the chord length. Aerodynamic loading normal to the wing's rotation plane is well captured by the ROMs, though model errors are larger for in-plane loading. We then perform a parameter sweep to understand how wing flexibility and mass affect peak deflection, mean lift and average power. All models indicate that flexible wings produce less lift but require lower average power to flap. Importantly, these studies highlight the computational efficiency of the ROMs—compared to the convention modeling approach, the UVLM and DBET ROMs solve 4 and 6 orders of magnitude faster, respectively.

Transition Flight Trajectory Optimization for a Flapping-Wing Micro Air Vehicle with Unsteady Vortex-Lattice Method

Flapping-wing micro air vehicles (FWMAVs) have the capability of performing various flight modes like birds and insects. Therefore, it is necessary to understand the various flight modes of FWMAVs in order to fully utilize the capability of the vehicle. The unique flight modes of FWMAVs can be studied through the trajectory optimization. This paper proposes a trajectory optimization framework of an FWMAV. A high-fidelity simulation model is included in the framework to sufficiently consider the complicated dynamics of the FWMAV. The unsteady aerodynamics are modeled with the unsteady panel method (UPM) and the unsteady vortex-lattice method (UVLM). The effect of wing inertia is also considered in the simulation model. In this study, transition flight trajectories are searched with the proposed framework. An optimal control problem is formulated for the transition flight from hovering to forward flight and transcribed to the parameter optimization problem with the direct shooting method. The cost function is defined as energy consumption. The same converged solution can be obtained with different initial guesses. The optimization results show that the FWMAV utilizes the pitch-up maneuver to increase altitude, although the forward speed is reduced. This pitch-up maneuver is performed more actively when the target velocity of transition is low, or the wind condition is favorable to acceleration.

Aeroelastic modeling and analysis of wings in proximity

This study investigates the effects of aerodynamic interaction on the aeroelastic characteristics of wings in proximity which is considered to be aerodynamically more efficient than the single wing case. The structural model of the wing is based on the first-order shear deformation theory and analyzed by the finite element method. The flow surrounding the wings is governed by potential theory and the aerodynamic loads are computed using the unsteady vortex lattice method. The aeroelastic simulation framework is established by integrating the structural and aerodynamic models within the flexible multibody dynamic program to solve the aeroelastic equations of motion in the time domain. Cantilever wings in proximity, arranged in biplane configuration, with different geometric parameters such as gap, sweep angle, dihedral angle, and stagger are considered in the analysis. The numerical results show that the smaller gap between the wings magnifies the aerodynamic interaction effect and decreases the onset of flutter speed. Sweeping the wing backward further reduces the flutter boundaries of the biplane model mainly due to changes in the wing's structural modal whereas the effect of dihedral angles is understated. The effect of stagger is more evident for a smaller gap and mostly leads to an increase in the flutter boundary. The aeroelastic responses are independent of initial disturbance and show a natural tendency to oscillate with a phase difference. This investigation warrants the importance of considering biplane wings as a single system for a gap lower than wing chord length as well as the significance of other geometric parameters in the aeroelastic calculation.

Investigation of chordwise functionally graded flexural rigidity in flapping wings using a two-dimensional pitch–plunge model

Insect wings are heterogeneous structures, with flexural rigidity varying one to two orders of magnitude over the wing surface. This heterogeneity influences the deformation the flapping wing experiences during flight. However, it is not well understood how this flexural rigidity gradient affects wing performance. Here, we develop a simplified 2D model of a flapping wing as a pitching, plunging airfoil using the assumed mode method and unsteady vortex lattice method to model the structural and fluid dynamics, respectively. We conduct parameter studies to explore how variable flexural rigidity affects mean lift production, power consumption and the forces required to flap the wing. We find that there is an optimal flexural rigidity distribution that maximizes lift production; this distribution generally corresponds to a 3:1 ratio between the wing’s flapping and natural frequencies, though the ratio is sensitive to flapping kinematics. For hovering flight, the optimized flexible wing produces 20% more lift and requires 15% less power compared to a rigid wing but needs 20% higher forces to flap. Even when flapping kinematics deviate from those observed during hover, the flexible wing outperforms the rigid wing in terms of aerodynamic force generation and power across a wide range of flexural rigidity gradients. Peak force requirements and power consumption are inversely proportional with respect to flexural rigidity gradient, which may present a trade-off between insect muscle size and energy storage requirements. The model developed in this work can be used to efficiently investigate other spatially variant morphological or material wing features moving forward.