Construction method of efficient hybrid simulation system for control design and evaluation of unmanned aerial vehicle

Abstract

The technical scheme of the invention discloses an efficient hybrid simulation system construction method for unmanned aerial vehicle control design and evaluation, and the method can achieve efficient and real turbulence simulation in a boundary-free space, and supports the dynamic convergence and dispersion of multiple unmanned aerial vehicles. Through self-adaptive block and Laplacian initialization, false waves caused by dynamic increase and deletion of grid blocks are remarkably suppressed, and the numerical stability is improved; aLM / ASM is adopted in a near field, so that boundary layer grid refinement is avoided, and a tight support core is coupled to consider both precision and stability; the stability of the convection open boundary under the inflow / outflow coexistence working condition is enhanced, which is superior to the traditional Neumann / original convection condition; simulation has parameters and is efficient, the parameters can be directly calibrated through a gradient method, and the result credibility is improved; compared with a traditional CFD method, the efficiency is extremely high, and design iteration of a real-time or interaction-level controller can be met under the condition that complex turbulent motion can be captured.

Description

Technical Field

[0001] This invention relates to a method for constructing an efficient hybrid simulation system for the control design and evaluation of unmanned aerial vehicles under strong flow field disturbances, so as to achieve efficient hybrid modeling, simulation and optimization, and belongs to the field of fluid simulation and aircraft dynamics modeling technology. Background Technology

[0002] Unmanned Aerial Vehicles (UAVs) have been widely used in recent years, playing a vital role in various fields. Typical applications include aerial photography and videography, agricultural plant protection, logistics delivery, surveying and mapping, environmental monitoring, infrastructure inspection, and military applications. With the continuous development of UAV technology, their structures and configurations are also becoming increasingly diversified. Currently, fixed-wing UAVs, multi-rotor UAVs, and hybrid UAVs combining the characteristics of both fixed-wing and multi-rotor aircraft are widely used. This diversified development trend reflects the continuous improvement in the functionality and applicability of UAVs.

[0003] However, achieving stable and high-performance flight control for specific UAV designs remains a significant challenge. UAVs are affected by complex airflow environments during flight, and their aerodynamic characteristics often exhibit nonlinear and time-varying properties. Especially when designing novel or atypical UAV structures, obtaining a suitable controller to achieve stable flight is a complex task. Directly validating control algorithms on physical UAVs is not only costly and time-consuming, but also carries potential safety risks.

[0004] To address these issues, simulation platforms have become crucial tools for UAV controller design and optimization. Common UAV aerodynamic simulation models primarily employ algebraic empirical models, calculating the aerodynamic forces acting on the UAV by modeling its state. The parameters of these models are typically calibrated through physical experiments or high-precision computational fluid dynamics (CFD) simulations. While these models perform well in relatively stable aerodynamic environments such as stable UAV cruise, their accuracy and applicability are limited in complex scenarios such as turbulent environments, near-ground or near-wall flight, and multi-UAV cooperative operations.

[0005] In theory, a more comprehensive Fluid-Structure Interaction (FSI) simulation method can be used to calculate the aerodynamic distribution of UAVs under different flight conditions. This method achieves synchronous simulation of aerodynamics and airframe motion by coupling the CFD solver with the rigid body dynamics solver. However, existing CFD algorithms have low computational efficiency. To ensure the accuracy of aerodynamic calculations at the boundaries, it is usually necessary to refine the mesh highly in the region near the UAV surface. Even with the help of GPU acceleration and algorithms such as the Lattice Boltzmann Method (LBM), the computational load is still enormous, making it difficult to meet the real-time requirements of flight controller design.

[0006] Currently, although some studies have attempted to improve the computational efficiency of UAV flow fields by refining simulation methods, achieving high-efficiency, low-latency simulation while maintaining physical consistency remains a key challenge in the field of UAV control design. Especially in the design and verification of flight control for multi-UAV cooperative flight and complex airflow environments, existing simulation methods are still insufficient to meet the requirements of real-time or interactive simulation. Summary of the Invention

[0007] The technical problem to be solved by this invention is that existing fluid-structure interaction (FSI) UAV simulations have low computational efficiency, difficulty in handling free and open spaces and multi-UAV scenarios, poor near-wall accuracy, and insufficient controller evaluation portability.

[0008] To address the aforementioned technical problems, the present invention discloses a method for constructing an efficient hybrid simulation system for the control design and evaluation of unmanned aerial vehicles (UAVs). This method achieves high efficiency (interactive / real-time level) while maintaining realistic turbulence effects, and supports boundaryless domains, multiple UAVs, high-speed rotor components, and complex environmental obstacles. Its structural features are as follows: The simulation domain is decomposed into a far-field domain and a near-object domain. The far-field domain uses a GPU-optimized adaptive block lattice Boltzmann solver, while the near-object domain uses a parameterized algebraic empirical model. The far-field and near-object domains rely on the exchange coupling of force and fluid state to form an efficient and stable integrated FSI simulation system.

[0009] Preferably, for the far-field domain, the infinite domain is discretized into non-overlapping uniform cubes, each cube having a size of L, and arrangements are made within each cube. If the grid is uniform, then the size of each cubic unit within the cube is... , where is the actual spatial resolution of the adaptive block-type lattice Boltzmann solver; the adaptive block-type lattice Boltzmann solver solves within cubes, where cubes allocated memory only for simulation are called active cubes, and other cubes are called inactive cubes. In this invention, the adaptive block-type lattice Boltzmann solver uses the D3Q27 discrete lattice velocity set.

[0010] Preferably, collisions are performed locally at each grid node, and the collision model uses a cumulant relaxation model, where higher-order parameters are dynamically calculated based on flow field adaptation to reduce numerical dissipation and dispersion errors. Based on this, a WALE subgrid-scale model is employed to introduce turbulent viscosity, enabling turbulence modeling at high Reynolds numbers.

[0011] Preferably, the active block is dynamically determined based on the flow conditions, and the block boundary is treated in two ways: Adjacent active blocks execute cross-block flow using a conventional distribution function (no interpolation required, maintaining conservation); Adjacent inactive blocks serve as open domain boundaries, employing enhanced convection boundary conditions to satisfy inflow / outflow stability (inflow uses dissipative regression terms, outflow uses normal first-order convection), where: The selection of active blocks adopts a hierarchical strategy based on turbulence characteristics such as local vorticity: a buffer zone with a width of d is set in each block, and the nodes inside and outside the buffer zone are classified according to the "strong / medium / weak" turbulence level, and active blocks are created, maintained or deleted accordingly; the newly created block is initialized with the macro field by solving the Laplace equations of density and velocity and applying the Dirichlet and Neumann conditions of the domain boundary of the adjacent active blocks, and then the distribution function is reconstructed based on the macro field and gradient, thereby avoiding false compression waves caused by constant initialization.

[0012] Structured arrays (SoA) are used to pre-allocate block-level memory pools on the GPU, storing velocity, density, distribution function, and adjacency index within each block to accelerate constant-time access to neighboring blocks; free blocks are allocated using linear search or priority queues; and a hash table is used to implement a fast mapping from near-object field sampling points to block indices.

[0013] Preferably, for the near-object field, the algebraic empirical model used for the rotor of the unmanned aerial vehicle is the actuation line model (ALM), and the algebraic empirical model used for other components of the unmanned aerial vehicle besides the rotor is the actuation surface model (ASM).

[0014] Preferably, in the actuation line model (ALM), the blade is discretized into several segments along the spanwise direction, and a force point is defined at the center of each segment, based on the relative incoming flow velocity. Local density chord length span With angle of attack Using lift / drag coefficient as well as Calculate the segment force, decompose and accumulate the segment force along the lift / drag direction to obtain the total rotor thrust and torque.

[0015] Preferably, the lift / drag coefficient of the rotor section as well as Lift / drag coefficients of body surface elements, pre-generated and interpolated using XFoil software, or calculated using CFD. as well as Using a simplified approximation, we have , .

[0016] Preferably, in the actuation surface model (ASM), the body surface is uniformly sampled, with each sampling point considered as a small planar element, based on the relative inflow velocity. Local density equivalent area With angle of attack The lift / resistance is calculated using empirical functions and then superimposed.

[0017] Preferably, probe points are set in the near-object field to interpolate local flow velocity and density from the far-field block: the rotor section probe is moved forward by one grid scale along the rotation direction, and the body sampling probe is moved outward by one grid scale along the normal direction to reduce near-wall interference and improve the stability and reliability of the reference flow.

[0018] Preferably, the discrete boundary forces obtained from the algebraic empirical model of the near-object field are diffused and loaded on the far-field grid using a compactly supported kernel function to achieve momentum conservation and boundary effect feedback. In this process, to avoid excessive smoothing under coarse grid, a linear kernel with a radius of 1 after normalization to the grid scale is used to ensure clear and stable near-wall effects. The far-field grid resolution is 0.5 to 2 times the average chord length of the UAV rotor to balance stability and the discriminability of the flow field structure.

[0019] The hybrid simulation system based on the above methods can efficiently simulate aircraft using a near-field algebraic empirical model. However, to ensure the accuracy and practical reference value of the simulation results, calibration is required by adjusting some parameters of the simulator. By using a series of recorded data from real aircraft flights, optimization is performed on some parameters to make the simulation results as close as possible to real flight. Since the simulation is efficient enough, optimization can be directly performed using a differential gradient method. The real flight data used for calibration should cover as many common flight states as possible to prevent overfitting during optimization.

[0020] The aforementioned simulation system, after parameter calibration, offers irreplaceable value for the design and evaluation of flight controllers. For UAV systems with atypical designs, or in situations with strong surrounding turbulent airflow, or in close-range collaborative operations involving multiple UAVs, this simulator can capture UAV motion more closely than reality. Thanks to its efficiency, controller design can be directly optimized and tested in the simulator for different scenarios, ultimately being directly transferred to real hardware – a process known as simulation-to-real-world migration. Furthermore, new controller architecture designs can be efficiently evaluated within this simulator, avoiding the high risks associated with direct real-world experiments.

[0021] This invention achieves high efficiency (interactive / real-time level) while preserving the real effects of turbulence, and supports boundaryless domains, multiple UAVs, high-speed rotor components, and complex environmental obstacles. Compared with existing technologies, it has the following advantages: 1. Efficiency: Achieve real-time performance in single-UAV flight simulation and interactive speed in multi-UAV scenarios.

[0022] 2. Accuracy: Verification with real flight data proves physical consistency, including ground effect, downwash velocity, acceleration and braking maneuvers, and complex trajectory tracking.

[0023] 3. Versatility: Applicable to various types of UAVs, including fixed-wing, multi-rotor, and hybrid configurations; suitable for different flight conditions, including normal cruise, aerodynamic stall, ground / obstacle proximity effects, and multi-UAV systems.

[0024] 4. Calibration: Through an automatic calibration procedure, model parameters can be optimized based on real flight data to ensure alignment with the physical system.

[0025] 5. Portability: Successfully demonstrated the migration from simulation to reality. The flight controller developed in the simulation can be directly deployed to the real UAV system without additional parameter tuning.

[0026] The hybrid simulation system constructed by this invention provides a powerful, efficient, and physically consistent platform for the design, evaluation, and optimization of UAV flight controllers, especially in scenarios involving complex fluid-structure interaction where traditional empirical models are inadequate. Attached Figure Description

[0027] Figure 1 Examples of drone types supported in this invention are shown; Figure 2 A schematic diagram showing the spatial relationship between the far-field and near-field simulation regions in this invention is provided. Figure 3 This diagram illustrates the far-field block partitioning and fluid solution mesh layout in this invention. Figure 4 This diagram illustrates the principle of fluid boundary treatment and block expansion within the far-field block in this invention. Figure 5 This diagram illustrates the effect of the smooth solution method used in creating blocks in this invention. Figure 6 This diagram illustrates the relationship between the near-field boundary model of the present invention and the probe position and far-field grid. Figure 7 This demonstrates a comparative verification of the ground effect force of a quadcopter using the present invention; Figure 8 This demonstrates the verification of the downwash airflow velocity of a quadcopter using the present invention; Figure 9 This demonstrates a comparative verification of using the present invention for quadcopter acceleration-braking flight; Figure 10 This demonstrates a comparative verification of using the present invention for quadcopter trajectory tracking flight; Figure 11 The invention is shown in the simulation of the stall-recovery process of a fixed-wing aircraft. Figure 12 The invention is shown in comparison between flight simulation and experimental results of a quadcopter passing through a narrow space. Figure 13 The invention is shown in various flight process simulations for tiltrotor aircraft. Figure 14 The simulation of the process of transitioning a compound wing from hovering to cruise using the present invention is shown. Figure 15 The simulation of the interaction between the downwash airflow of two quadcopters using the present invention is shown. Figure 16 This invention demonstrates the simultaneous simulation of five aircraft navigating complex obstacles using this invention. Figure 17 This demonstrates the optimization of the trajectory tracking controller using the present invention and its application in experiments; Figure 18 The invention is shown to optimize the aircraft racing controller and is then transferred to experiments. Detailed Implementation

[0028] The present invention will be further illustrated below with reference to specific embodiments. It should be understood that these embodiments are for illustrative purposes only and are not intended to limit the scope of the invention. Furthermore, it should be understood that after reading the teachings of this invention, those skilled in the art can make various alterations or modifications to the invention, and these equivalent forms also fall within the scope defined by the appended claims.

[0029] This invention proposes an open-domain real-time fluid-structure interaction simulation system for unmanned aerial vehicles (UAVs), which supports one or more UAVs with high-speed rotors to perform real-time or near-real-time flight simulations in an unobstructed, open-boundary three-dimensional space.

[0030] To design and evaluate flight controllers, the aerodynamics around unmanned aerial vehicles (UAVs) are typically assumed to be subsonic and can be modeled using the following unsteady, isothermal, weakly compressible Navier-Stokes (NS) equations:

[0031]

[0032]

[0033] in, , and These represent the fluid velocity field, density field, and pressure field, respectively. It is an external force field. This indicates the corresponding acceleration; Indicates a constant ambient temperature; It is the gas constant; Indicates time; It is the shear stress tensor.

[0034] To develop a fluid-structure interaction (FSI) simulator for controller design and evaluation, this invention requires the simulator to operate in an open space while maintaining sufficiently high simulation efficiency—ideally achieving real-time performance. Furthermore, the FSI simulator should be versatile enough to be suitable for various operating scenarios.

[0035] To address these challenges, this invention introduces a novel concept for forming an efficient hybrid simulator. Specifically, the simulation domain is decomposed into a near-object field and a far-field. To achieve high efficiency, the fluid solver is configured to operate at a relatively coarse resolution across the entire far field, employing a high-efficiency GPU-optimized lattice Boltzmann (LB) solver instead of the traditional Navier-Stokes (NS) solver. Inspired by sparse solvers and adaptive mesh refinement and coarsening, this invention develops a novel adaptive block LB solver. In this solver, blocks of the simulation region can be dynamically created or deleted to enable more flexible moving local domains, adaptable to multi-UAV systems.

[0036] However, applying the same coarse mesh resolution to the near-object field fails to accurately solve the boundary layer flow, leading to inaccurate boundary force calculations, which significantly affects the flight dynamics of the UAV. In this situation, algebraic empirical models play a crucial role. In this invention, two types of localized algebraic empirical models are employed for the near-field near the UAV boundary: a blade model for rapidly rotating thin blades and a body model for other UAV components. These algebraic empirical models are coupled with a far-field LB solver to form a novel hybrid FSI simulator. Because it does not require calculating very fine meshes near the UAV boundary and is optimized for GPUs, the simulator of this invention exhibits extremely high computational efficiency.

[0037] The hybrid simulator of this invention features adjustable model parameters, enabling alignment of physical behavior with the corresponding real-world unmanned aerial vehicle (UAV) system. For the design and evaluation of flight controllers, especially when considering portability to real-world systems, systematically determining appropriate parameters is crucial. This invention proposes an automatic calibration procedure that aligns two systems starting from default parameters.

[0038] I. Far-field fluid simulator I) Foundations of the Lattice Boltzmann Method First, consider the airflow in the far-field region, which is relatively far from the drone's body. This invention chooses to use the Lattice Boltzmann Method (LBM) to solve these dynamic airflow phenomena, utilizing its conserved local dynamics with low dissipation characteristics. This characteristic enables high efficiency when optimized on a GPU.

[0039] In LBM, each grid node The above uses a set of values ​​corresponding to discrete particle velocities. Discrete distribution function To characterize the airflow, these functions evolve over time according to the following dynamic equations:

[0040] The equations are discretized using the D3Q27 structure. This represents the distribution function and macroscopic fluid density. and speed Deterministic discrete collision operators. These macroscopic quantities are obtained by taking the zeroth and first moments of the distribution function:

[0041]

[0042] This equation can be solved by decomposing it into two steps: the flow step and collision steps .

[0043] For real-world scenarios involving high Reynolds numbers, more advanced collision models are needed. This invention employs a cumulant model and adaptive higher-order relaxation rates to significantly reduce numerical dissipation. Simultaneously, a wall-adaptive local eddy viscosity (WALE) model is used to account for subgrid fluid structures in turbulent flow.

[0044] (ii) Adaptive Simulation Domain Block Solver Since this invention focuses on UAV flight simulation, it employs a local domain LB solver to achieve open-domain flight simulation. Inspired by sparse solvers, this invention dynamically allocates or removes simulation domain blocks to track flow near moving objects. This method prioritizes computational resources for critical fluid regions while eliminating inertial forces used in previous local domain LB solvers. However, directly modifying a sparse solver based on the Navier-Stokes equations for LBM is challenging because it assumes incompressibility, requiring new development for its proper application.

[0045] To maintain sufficient simulation efficiency, the local domain LB solver of this invention first assumes an infinite domain, which is divided into regions of size . Non-overlapping uniform cubic blocks. It's important to note that these blocks do not directly correspond to the specific memory portion used to store fluid data. Within each block, [the following is a list of elements] are arranged... A uniform grid with cubic unit size of This is also known as the actual spatial resolution of the solver. It's worth noting that mesh nodes are located at the center of the elements, and the outer surfaces of the boundary elements form the boundary surfaces of the block.

[0046] It's important to note that object boundaries are not considered at this stage because only far-field fluid is being simulated. Far-field flow relies on LBM for efficient fluid flow solving, thus requiring the execution of normal collision and flow procedures. While collisions are performed locally at each mesh node, flow requires special treatment of boundary elements within each block. These elements are treated as normal elements from the flow distribution function of other blocks, or as domain boundary elements requiring treatment of open domains.

[0047] The blocks that actually allocate memory for the simulation are called active blocks. First, criteria need to be specified to dynamically determine active blocks based on flow conditions, allowing them to be adaptively created or deleted. For efficient execution, all blocks are pre-allocated as a memory pool on the GPU before the simulation, and an algorithm is needed to mark the active blocks used in the simulation.

[0048] The LB solver can be applied independently to each block, except for elements near the block boundaries which require special handling. Block boundary faces fall into two categories—faces connected to active or inactive blocks.

[0049] For faces adjacent to active blocks, normal flow is performed, and the distribution function of the mesh nodes at the center of the boundary elements flows directly from the nodes of the boundary elements of the adjacent blocks. Note that the boundary elements of the two adjacent blocks are well aligned, resulting in the connected elements forming a larger uniform mesh. Therefore, no interpolation is required during flow to maintain conservation.

[0050] However, for surfaces adjacent to inactive blocks, they are effectively domain boundaries. Therefore, appropriate domain boundary treatment should be applied. This invention assumes that airflow can freely enter and exit the domain boundaries and applies open boundary conditions. These conditions should allow for both laminar and turbulent flow. This invention employs enhanced convective boundary conditions, which are superior to traditional outflow conditions (such as Neumann or pristine convective boundary conditions) in handling inflow and outflow:

[0051] in: Represents macroscopic quantities (velocity) and density ); These are the corresponding static state values; superscript This represents the value at the boundary node. It's the boundary normal. Dissipation parameter. Is entering the direction of quantity The decay rate. In the experiment, the dissipation parameter was set to 0.1. This boundary treatment method exhibits significantly better stability in maneuvering situations involving entry and exit flows.

[0052] As mentioned earlier, only active blocks are allocated memory on the GPU. Therefore, an algorithm is needed that can automatically determine active blocks, which can be further divided into two different types.

[0053] For airflow around the drone, active blocks are needed to ensure the drone is fully immersed in the air domain. Furthermore, airflow near the drone's boundaries is crucial because it generates turbulence and affects the surface forces exerted by the air on the drone's body. To achieve this, a uniform sampling method is first used on the drone's body using a Poisson disk, where the area represented by each sample is defined as the total surface area divided by the total number of samples. Blocks containing surface samples are always designated as active. If a sample is located near a block boundary (identified by a distance threshold), all adjacent blocks should be marked as active to ensure a sufficient fluid region is defined near the drone's boundaries.

[0054] For airflows far from the boundary, active blocks must be deployed with greater adaptability. The criteria for selecting active blocks should ensure that the turbulent wake flow is largely preserved, maintaining the fidelity of the analytical dynamic flow structure while optimizing computational resources; in addition, blocks should be selected to make the airflow near the domain boundary relatively smooth to avoid serious impacts on UAV dynamics.

[0055] To achieve this, a width of [missing value] is defined near the boundary of each block. A buffer is provided. Within each block, the turbulence metric for each grid node is computed in parallel, and the nodes are classified into "strong," "normal," or "weak" categories based on predefined thresholds. In this invention, vorticity is used as the primary metric. However, more complex metrics, such as the Q-criterion or turbulent kinetic energy, may be employed.

[0056] First, examine the grid nodes within the buffer. If any node has a "strong" attribute, mark the current block as active, and also mark adjacent blocks as active. If any node has a "normal" attribute, leave the fluid properties of the current block and its adjacent blocks unchanged; no new active block needs to be created. Then, examine the grid nodes outside the buffer but within the block. If any node exhibits a "strong" attribute, mark the current block as active. Conversely, if a node has a "normal" attribute, the block's activation state remains unchanged.

[0057] Note that if the newly marked active block has not yet been created, memory should be allocated for the block on the GPU and initialized. Finally, if all mesh nodes within the block have "weak" attributes, the block should be removed from memory. Buffer width This affects the number of active blocks created. In this invention, the number is selected empirically. It is 20% of the block size.

[0058] Once a block is marked as active and has not yet been allocated memory on the GPU, it should be created first. Creation involves more than simply allocating memory for all mesh nodes within the block; it also involves initializing macroscopic quantities and distribution function values ​​for each mesh node. Crucially, the macroscopic quantities initialized for the newly created active block should maintain continuity at the block boundaries; otherwise, incorrect compression waves will form, potentially disrupting the simulated flow within the active block.

[0059] To ensure a smooth field transition between blocks, the Laplace equation is first solved on the new block, and then the solution is used as the initialization for the reconstructed distribution function. The solved Laplace equation is for both the density and velocity fields:

[0060]

[0061] Boundary conditions are of two different types. When the boundary face of a new block is adjacent to an existing active block, the Dirichlet boundary condition is applied, where the values ​​of the boundary nodes of the new block are set to be exactly equal to the values ​​of the boundary nodes of the adjacent blocks. For other boundary faces corresponding to domain boundaries, the following Neumann boundary condition is applied:

[0062]

[0063] in, The boundary normal is represented. The Laplace equation is discretized using finite difference on a uniform mesh, and macroscopic quantities are numerically computed using a GPU-based Jacobian solver. The distribution function is then reconstructed based on the macroscopic quantities and their gradients by separately computing the equilibrium and non-equilibrium components.

[0064] Compared to block creation, block deletion is straightforward: simply remove the block from memory by marking it as inactive. During the flow steps of subsequent time steps, domain wall boundary conditions will be automatically applied to the new boundary surfaces formed by the block deletion.

[0065] As previously mentioned, the adaptive block LB solver of this invention requires dynamic block creation and deletion. Allocating GPU memory on-demand for these operations leads to a significant performance degradation. To mitigate this issue, a large contiguous array of GPU memory is pre-allocated for a fixed number of blocks, with each block node storing basic information—fluid velocity, density, distribution function, and related properties. Note that an array structure (SoA) layout is used to store all nodes within a block.

[0066] All pre-allocated blocks are linearized on the GPU and initialized as inactive. For efficient access to adjacent blocks, an index table is maintained within each block, enabling constant-time lookups during fluid solver operations such as flow. Initially, active blocks around the UAV boundary are marked sequentially, starting with the first block. During simulation, blocks are dynamically marked as active or inactive based on selection criteria.

[0067] For block creation, if a block lacks allocated memory, the first inactive block in a pre-allocated array is searched and assigned to the newly active block. While a priority queue could optimize this search, a linear search is employed due to the manageable number of blocks. For block deletion, the block is simply marked as inactive in the pre-allocated array. Finally, since boundary samples are used to locate blocks around the drone, a hash table is constructed to achieve efficient block indexing for a given sample point.

[0068] II. Near-field boundary model and far-field coupling 1) Actuation line model of rotor blades Rotor blades, as the propulsion system of various unmanned aerial vehicles (UAVs) such as helicopters, multirotors, and some fixed-wing aircraft, rotate at high speeds during flight and have a thin structure. Direct aerodynamic simulation requires extremely high spatial and temporal resolution to accurately solve for the forces acting on them, which is impractical for UAV control applications. To reproduce the physically consistent effects of surrounding aerodynamic flow on rotor blades, this invention adapts the actuation line model (ALM) into the simulation framework to achieve efficient computation.

[0069] To significantly reduce the computational load, the blade is treated as a finite series of segments along the spanwise direction, and the force of each element is calculated using pre-calculated data corresponding to the respective segment. The force of each segment is then summarized as a function based on the local fluid state. ,in: Indicate the center of each segment; This represents the flow velocity relative to the blade segment in the local coordinate system of the rotating blade. This is the fluid density near the blade boundary. After calculating the force on each element, the total torque and force can be obtained by summing the contributions of all elements.

[0070] Starting from the observation that the cross-section of each blade element can be regarded as a 2D airfoil, Decomposed into lift and resistance ( The lift and drag coefficients can be approximated using the following empirical model (refer to blade element theory):

[0071]

[0072] in: and These are adjustable parameters, and are all set to 1 by default. , , Indicates the first The chord length, span, and angle of attack (AoA) of each element (note) and (It can vary depending on the position of the element's center); It is the relative incoming flow velocity; and It is the corresponding lift and drag coefficient (a function of the angle of attack); and It is a unit vector along the directions of lift and drag. The actuation line defining the force is formed along the element center of the blade.

[0073] This model requires a specified reference flow rate. In the solver of this invention, since the far-field airflow has already been simulated, the airflow of the far-field solver can be directly used as the reference velocity for each element. To obtain reliable In the solver, a probe is defined for each element, located one cell-size in front of the element's center along the rotation direction. The reference velocity is obtained by interpolating the airflow velocity calculated by the far-field solver and then taking the magnitude of that velocity. Angle of attack of each element Defined as the angle between the interpolation velocity projected onto the cross-sectional plane of the element and the chord direction.

[0074] Obtain reliable coefficient functions and This is also crucial. This invention uses XFoil software to obtain different discrete angles of attack. and And it provides interpolation function values ​​for arbitrary angles of attack. For airfoil shapes not supported by XFoil, CFD software (such as OpenFOAM) can be used directly for calculation.

[0075] (ii) Actuation Surface Model Besides the rotating blades, other components of the UAV require additional force calculation models and do not require mesh refinement near the boundaries. This invention employs an Actuated Surface Model (ASM) and further modifies it to account for the effects of far-field flow.

[0076] It is worth noting that the drone body has been sampled evenly, with each sample... With surface normal , representing a sufficiently small local element that can be approximated by a plane. Similarly, the force of each sample By decomposing it into lift and resistance Estimate by components: An empirical model similar to ALM is used to calculate lift and drag:

[0077]

[0078] in: and These are adjustable parameters similar to those in ALM, and are all set to 1 by default. , and Representing samples respectively Reference flow velocity, angle of attack, and surface area; and It is a unit vector along the directions of lift and drag. Because the surface of the UAV is uniformly sampled, It can be approximated by dividing the total surface area by the total number of samples.

[0079] and Based on samples Nearby interpolated far-field velocity calculation. Assuming the sample... The surrounding local area is almost flat, and these samples share the same lift. and resistance The coefficient function can be fitted to the following form:

[0080]

[0081] Similar to ALM, probes are used to determine The probe travels along the sample. The normal direction is positioned at a distance of one cell size to avoid interference from adjacent samples. At this probe location, the far-field velocity relative to the sample velocity is interpolated, and its magnitude is used as the reference flow velocity. Angle of attack Calculated based on the angle between the interpolation rate from the far-field block and the local plane of the sample, this angle is always positive. Note Indicates sample The fluid density at the probe location is also interpolated from the far-field block, which is more reliable.

[0082] The total force and torque can be derived from the forces calculated at the sample and used as input to the rigid body solver.

[0083] (iii) Coupling with the far-field LB solver So far, forces acting on solid surface samples have been estimated using near-field boundary models. However, these forces at the sampling location should also affect the far-field fluid solver, resulting in physically consistent feedback. This invention employs the concept of the immersed boundary method, where force diffusion is used to effectively transfer boundary forces calculated from algebraic empirical models to the surrounding fluid:

[0084] in: It is a far-field fluid node; The total number of surface samples around the index fluid nodes is [number missing]. ; Indicates force, It is the diffusion kernel function, where .

[0085] Due to the coarse mesh resolution used in the far-field solver, the diffusion kernel must remain sharp to avoid excessive smoothing near the boundaries. Therefore, this invention employs a kernel function with the following minimum size for diffusion:

[0086] Note that ALM recommends that the far-field mesh resolution should be comparable to the blade chord length. If the mesh is too fine, applying volume forces at the center of each element may lead to instability during force diffusion. Conversely, a mesh that is too coarse fails to capture the overall trend of turbulence caused by rotating blades. Experimental results show that a moderate mesh resolution of 0.5 to 2 times the average blade chord length results in consistent volume forces and stable simulations within this range.

[0087] III. Calibration of Simulation Parameters The aforementioned FSI solver for UAV dynamics constitutes a hybrid simulation system that integrates a direct adaptive LB solver for the far field and a localized algebraic empirical near-field model. These components are coupled at a relatively coarse mesh resolution and executed on a GPU, enabling extremely high performance for real-time or interactive operation. While reasonable flow field results can be obtained efficiently, ensuring that the simulated UAV dynamics align with the corresponding physical world remains challenging.

[0088] To address this issue, a calibration procedure is needed to find the optimal model parameters (primarily the adjustable parameters in ALM and ASM) using a series of recorded data from real drone flights (also known as system identification). This invention employs gradient-based optimization to derive the optimal model parameters.

[0089] The simulator's internal parameters , , and Assemble into vectors Given that drones may fly in environments with non-zero speeds, using a constant ambient flow velocity in space... As an external parameter To enhance simulation parameters. It should be noted that... It may change over time, and can be used... Initialize the flow rate and continuously specify the domain boundary of the far-field LB solver. To be incorporated into the simulator.

[0090] Now, the overall parameters of the simulator can be defined as follows: These parameters need to be optimized. This can be achieved by specifying state variables that include the drone's position and orientation. The objective function to be minimized can be expressed as:

[0091] in: This represents the state at discrete-time samples from the simulator and can be affected by all parameters. This represents the corresponding measurement state from actual flight. Note that the second term is a regularization term, preventing the optimized internal parameters from deviating excessively from the default parameters. (A vector consisting entirely of 1s). Used as a regularization parameter for each specific calibration.

[0092] Minimization is achieved through gradient descent starting from the default parameters. We initialize it to zero velocity as an initial guess. Due to efficient simulation, the parameter gradients are calculated by simply performing numerical differentiation.

[0093] When calibrating model parameters for a specific UAV, the aforementioned optimization process can be performed using various flight missions. This invention proposes a two-step general criterion for selecting such missions. First, calibration is performed in a wind tunnel setting under various possible flight conditions, without free motion. Key force contributors, including rotor forces and torque, airframe drag and lift, and control surface forces, can be effectively calibrated in this step. Such a setting is easier to implement in real experiments or high-fidelity CFD simulations and produces a set of parameters reliable for normal flight scenarios.

[0094] The second step involves selecting a complete flight mission to further calibrate overall performance and minimize the gap between simulation and reality. The trajectory selection depends on the UAV type, but it is recommended that it cover a wide range of motion along all three axes and for each control degree of freedom.

[0095] IV. Verification Experiment Due to its high efficiency and low computational resource requirements, the hybrid simulation system of this invention can run on ordinary computers. In all the experiments presented below, the simulations were executed on a computer equipped with an AMD Ryzen 7 5700G CPU, 64 GB of RAM, and an NVIDIA GeForce RTX 2080 Ti GPU with 11 GB of video memory. Furthermore, the system can support laptops with even lower hardware configurations. Depending on the resolution and simulation scale, the system of this invention achieves real-time performance or at least interactive response. Note that when using more advanced GPUs (such as the NVIDIA GeForce RTX 5090), the overall efficiency can be significantly improved, making real-time simulations easier to implement.

[0096] Verification of quadrotor thrust under ground effect, such as Figure 7 As shown: The thrust generated by the quadrotor can be calibrated in the simulator to match experimental values ​​by adjusting the parameters of the ALM (Automatic Ground Effect). Notably, when a plane is positioned below and close to the quadrotor, the thrust increases at the same rotor speed—a phenomenon commonly referred to as ground effect force. The ground effect force encountered by the quadrotor was tested in both simulation and experiments at different distances from the flat floor and different floor tilt angles. The results show strong consistency with physical experiments, verifying the simulator's ability to accurately resolve the additional aerodynamic forces induced by nearby objects on the UAV.

[0097] Downwash flow rate verification, such as Figure 8 As shown: In addition to the forces acting on the UAV, the impact of the UAV on the flow field was compared. The downwash airflow from the rotor can exhibit relatively high speeds and cover a large area, significantly affecting nearby aircraft. A standard case of a stationary quadcopter was evaluated, and the flow velocity beneath it was measured at different distances. This comparison was conducted after model calibration to match the total thrust to experimental data at the same rotor speed. Although significant errors still exist near the rotor, the overall accuracy is sufficient to replicate the downwash effect on other aircraft.

[0098] Quadrotor acceleration and braking verification, such as Figure 9 As shown: In addition to comparisons with static physics experiments, evaluations were also conducted on real-world flight missions. For a quadcopter, an autonomous flight controller is crucial for its stability; otherwise, it cannot even hover briefly due to minor external disturbances. In the comparisons, the PX4 flight controller was deployed on both real and simulated aircraft, sharing the same set of control parameters.

[0099] The mission was designed to accelerate to a certain speed along the heading, maintain flight for a period of time, and then brake to a stop at the current position. Simulated ambient wind speed was also matched with average measurements from outdoor experiments. The simulated trajectory showed a high degree of consistency with the experimental data, indicating that the simulator has sufficient accuracy for basic real-world tasks.

[0100] Quadrotor trajectory tracking verification, such as Figure 10 As shown: Furthermore, a more complex infinite-shape trajectory following task was compared between real flight and simulation, conducted in two different settings. In Setting A, a smaller quadcopter (wheelbase length = 250mm) with a simple position controller was used. At a target speed of 5 seconds per revolution, its trajectory deviated significantly from the target shape due to the limited agility of the quadcopter. The simulated trajectory showed a significant deviation relative to the real flight test, primarily attributed to unstable outdoor wind conditions and the positioning error of the real UAV (sensor noise was not applied in this simulation case). Moreover, the use of a purely inertial-based positioning scheme in this setting further exacerbated the accumulated error from the real flight.

[0101] Conversely, Setup B employs a larger quadcopter (wheelbase length = 600mm) with enhanced wind resistance, operating at a slower speed of 10 seconds per revolution. Furthermore, a centimeter-level precise RTK positioning scheme with virtually no cumulative error is used. The trajectory is closely aligned with an infinitely large target, minimizing the difference between real flight and simulation results.

[0102] These two different settings demonstrate that environmental disturbances and sensing errors can significantly lead to a poor match between simulation and experiment. To generate more robust controllers from simulators, the use of randomization to model this uncertainty is encouraged.

[0103] Previous validation through real-world experiments paired with calibration simulations has demonstrated the acceptable physical consistency of the simulator of this invention. These findings demonstrate the suitability of the simulator of this invention for a wide range of flight control tasks.

[0104] V. Flight Control Applications Stall and recovery of fixed-wing UAVs, such as Figure 11 As shown: In this application example, a PX4 autopilot system was implemented in the simulation system, including a cascaded PID controller for the fixed-wing UAV to regulate the aircraft's speed and attitude. The simulator, using default parameters, was used to replicate the complete aerodynamic stall and recovery process of the fixed-wing UAV. The experimental sequence began with engine (rotor) deactivation after a pull-up maneuver, inducing a complete aerodynamic stall condition. After a brief freefall phase characterized by unstable dynamics, engine restart was able to restore the aircraft to its nominal flight attitude with the assistance of flight control.

[0105] This approach evaluates the operational robustness of flight controllers in extreme scenarios involving strong turbulence, as highly nonlinear stall dynamics evade conventional empirical modeling and render traditional predictive methods insufficient for attitude change analysis. The results highlight the necessity of FSI solutions to resolve complex fluid-structure interactions during these critical flight states to achieve more general flight control.

[0106] A quadcopter traversing a narrow space, such as Figure 12 As shown: Flying in confined spaces presents unique challenges for any type of unmanned aerial vehicle (UAV). Experimental logs, simulation results, and empirical model predictions were systematically compared for the Crazyflie small quadcopter equipped with a cascaded PID controller derived from a PX4 when it entered a narrow space with a low ceiling. This comparative analysis highlights the key advantages of the simulation framework of this invention in capturing the confined-space aerodynamic interactions that fundamentally alter quadcopter dynamics.

[0107] While traditional empirical models fail to account for the backflow effect caused by the ground / ceiling, the method of this invention successfully replicates experimentally observed ceiling adhesion phenomena with ground effects, as shown in captured images and measured height data. The discrepancy between empirical predictions and experimental / simulation results highlights the limitations of the aerodynamic assumptions employed by traditional empirical models in confined environments. Therefore, the simulation system of this invention becomes an important tool for addressing complex fluid-structure interactions that evade traditional empirical models, providing crucial insights for developing robust navigation strategies in confined spaces.

[0108] Tiltrotor hybrid drone flight, such as Figure 13 , 14 As shown: For tilt-rotor hybrid unmanned aerial vehicles (UAVs), the rotors can dynamically adjust their angles to transition from vertical hovering to forward cruising. Designing a flight controller well-suited for this scenario presents a challenge. This invention demonstrates a tilt-rotor hybrid UAV operating in multiple flight states—including takeoff, turning, and braking maneuvers—with controllable rotor tilt and rotation speed.

[0109] The simulation system implements a cascaded PID controller derived from the PX4, originally designed for a quadcopter to stabilize the aircraft's attitude, with the rotor tilt angle manually set for each operational state. This represents a more complex UAV control challenge due to the intricate aerodynamic coupling between the rotor downwash and the fixed-wing surface. While an ideal flight controller would dynamically modulate the tilt angle into the attitude control loop to enhance stability, the current implementation uses a simplified control architecture to demonstrate the simulator's ability to handle such multifaceted scenarios.

[0110] Note that stabilizing hybrid unmanned aerial vehicles (UAVs) remains an open challenge due to the inherent difficulties in empirical dynamics modeling—resulting from the increased degrees of freedom of the actuator system and the strong rotor-rotor / rotor-body coupling observed during transient maneuvers such as turning and braking. These nonlinear interactions effectively render many model-based control design methods effective in conventional UAV configurations inapplicable. Therefore, the simulator of this invention offers new opportunities to address such complex flight control problems through more advanced methods, including end-to-end learning and data-driven control strategies.

[0111] The downwash interference from two quadcopters, such as Figure 15 As shown: In scenarios involving multiple drones, their formations may dynamically transition between clustered and dispersed configurations. Simulating such scenarios using traditional box-based local domain solvers (such as "FishGym") is challenging. Furthermore, multi-drone systems introduce new control challenges due to the need for coordinated flight over close proximity. The simulator of this invention offers unique advantages due to its adaptive block-based design.

[0112] This invention demonstrates the ability of a system to simulate two flying drones under downwash interference conditions. A smaller quadcopter drone (wheelbase length = 250 mm) flies horizontally at a constant speed and altitude using a controller derived from a PX4, and passes through a downwash zone created by another larger hovering quadcopter drone (wheelbase length = 600 mm). The smaller drone's altitude and pitch angle fluctuate after entering the downwash zone. The fluctuation behavior varies at different speeds.

[0113] The downwash effect is difficult to model empirically because the flow structure beneath the UAV is non-uniform and may be affected by interactions with other UAVs; the forces generated by the propellers in the downwash region are also difficult to predict efficiently without the hybrid simulator of this invention.

[0114] Multiple quadcopters flew through the woods, like Figure 16 As shown: To further demonstrate the capabilities of the hybrid simulation system of this invention for handling multiple UAVs in complex environments, a swarm of five quadcopters traversing forest terrain was showcased. Here, a predefined trajectory generated by cubic spline interpolation of waypoints with velocity profiles was tracked using a cascaded PID controller derived from PX4.

[0115] This scenario intentionally includes extended and approach flight phases, requiring seamless transitions between adaptive mesh blocks. This capability is unattainable in many other FSI solvers due to domain decomposition limitations, while simultaneously testing the flight controller's robustness to aerodynamic coupling effects during close formation flight. This experiment not only validates trajectory tracking performance under more realistic aerodynamic interactions but also provides crucial insights for developing coordinated multi-UAV control strategies in obstacle-rich environments.

[0116] VI. Controller Optimization and Simulation to Actual Migration The simulation system of this invention can optimize not only model parameters but also control parameters. This dual optimization ensures reliable simulation-to-real-world transfer. This invention demonstrates optimization for two tasks using a simple PD controller, and zero-time transfer to real deployment. Specifically, it focuses on the flight control tasks of a quadcopter UAV (wheelbase: 250 mm): complex trajectory tracking and checkpoint racing. The UAV's flight control uses a PD controller with a 30Hz update frequency for position adjustment, while attitude stabilization is handled by a PX4 flight system.

[0117] Trajectory tracking tasks, such as Figure 17 As shown: The target trajectory lies on a hyperbolic paraboloid surface, forming a saddle shape. Initial control parameters are manually assigned as a baseline. Using these nominal gains, the simulation executes the full trajectory and records the tracking performance. To enhance accuracy, gradient-based optimization is employed in the simulation environment. The cost function is defined as the root mean square error (RMSE) between the simulated UAV trajectory and the reference path. In each iteration, the system evaluates its performance under the current control gain, calculates the RMSE, and updates the gain accordingly.

[0118] After convergence, the optimized parameters are directly transferred to the real UAV system without further fine-tuning. The same reference trajectory is used in the real experiment. During flight, the UAV uses an onboard optical flow sensor to estimate its motion, providing real-time position feedback for quantitative evaluation. Comparisons show the simulation and real tracking results using the initial and optimized control parameters. In the baseline case, both simulation and physical experiments show significant deviations from the reference trajectory. After optimization, the UAV exhibits significantly improved tracking performance in both domains.

[0119] Checkpoint race missions, such as Figure 18 As shown: Unlike trajectory tracking, which minimizes the tracking error relative to the desired trajectory, the racing scenario minimizes the time required to reach successive checkpoints. In the demonstration, a sequence of checkpoints arranged in a vertical pentagram is used; this layout challenges the quadcopter with sharp turns and rapid altitude changes. The flight controller uses the next checkpoint as its position setpoint until the quadcopter reaches a specified proximity (0.1 m in this case).

[0120] The optimization process is similar to a trajectory tracking task, except that the objective is to minimize the task completion time rather than the trajectory tracking error. Compared to the initial control, the optimized control significantly reduces the task completion time, enabling the quadcopter to smoothly and quickly pass through all checkpoints.

[0121] The above results verify the ability of the simulation system of this invention to support closed-loop controller design and optimization, and demonstrate strong consistency in simulation-to-real-world migration. This framework enables efficient iteration in simulation and reliable deployment on hardware, providing a practical pipeline for the design, optimization, and migration of UAV flight controllers.

[0122] This invention enables efficient and realistic turbulence simulation in boundless space, supporting dynamic convergence and divergence of multiple UAVs. Through adaptive blocks and Laplace initialization, it significantly suppresses spurious waves caused by the dynamic addition and deletion of mesh blocks, improving numerical stability. The near-field simulation employs ALM / ASM, eliminating the need for boundary layer mesh refinement, and tightly coupled support kernels balance accuracy and stability. It enhances the stability of convective open boundaries under coexisting inflow / outflow conditions, outperforming traditional Neumann / primitive convection conditions. The simulation is parametric and efficient, allowing direct parameter calibration via gradient methods, thus improving result reliability. Compared to traditional CFD methods, it is extremely efficient, capable of capturing complex turbulent motions and satisfying real-time or interactive controller design iterations.

Claims

1. A method for constructing an efficient hybrid simulation system for the control design and evaluation of unmanned aerial vehicles, characterized by decomposing the simulation domain into a far-field domain and a near-object field domain, wherein, The far-field domain employs a GPU-optimized adaptive block lattice Boltzmann solver, while the near-object domain uses a parameterized algebraic empirical model. The far- and near-object domains rely on the exchange and coupling of force and fluid state to form an efficient and stable integrated FSI simulation system.

2. The method for constructing an efficient hybrid simulation system for unmanned aerial vehicle control design and evaluation as described in claim 1, characterized in that, For the far-field domain, the infinite domain is discretized into non-overlapping uniform cubes, each cube having a size of L, and arrangements are placed within each cube. If the grid is uniform, then the size of each cubic unit within the cube is... , where is the actual spatial resolution of the adaptive block lattice Boltzmann solver; the adaptive block lattice Boltzmann solver solves within cubes, where cubes allocated memory only for simulation are called active cubes, and other cubes are called inactive cubes.

3. The method for constructing an efficient hybrid simulation system for unmanned aerial vehicle control design and evaluation as described in claim 2, characterized in that, Collisions are performed locally at each grid node. The collision model uses a cumulative relaxation model, in which higher-order parameters are dynamically calculated based on flow field adaptation to reduce numerical dissipation and dispersion errors. On this basis, the WALE subgrid scale model is adopted to introduce turbulent viscosity and realize turbulence modeling in the case of high Reynolds number.

4. The method for constructing an efficient hybrid simulation system for unmanned aerial vehicle control design and evaluation as described in claim 3, characterized in that, Active blocks are dynamically determined based on flow conditions, and the block boundaries are treated in two ways: adjacent active block surfaces are subjected to conventional distribution functions for cross-block flow; adjacent inactive block surfaces are treated as open domain boundaries, and enhanced convection boundary conditions are adopted to satisfy inflow / outflow stability. The selection of active blocks adopts a hierarchical strategy based on turbulence characteristics such as local vorticity: a buffer zone of width d is set in each block, and the nodes inside and outside the buffer zone are classified according to the "strong / medium / weak" turbulence level, and active blocks are created, maintained, or deleted accordingly. The newly created block is initialized with macroscopic field by solving the Laplace equations of density and velocity and applying the Dirichlet and Neumann conditions of adjacent active blocks and the domain boundary. Then, the distribution function is reconstructed based on the macroscopic field and gradient, thereby avoiding spurious compression waves caused by constant initialization.

5. The method for constructing an efficient hybrid simulation system for unmanned aerial vehicle control design and evaluation as described in claim 1, characterized in that, For the near-object field, the algebraic empirical model used for the rotor of the unmanned aerial vehicle is the actuation line model (ALM), while the algebraic empirical model used for other components of the unmanned aerial vehicle besides the rotor is the actuation surface model (ASM).

6. The method for constructing an efficient hybrid simulation system for unmanned aerial vehicle control design and evaluation as described in claim 5, characterized in that, In the Actuation Line Model (ALM), the blade is discretized into several segments along the spanwise direction, and a force point is defined at the center of each segment, based on the relative incoming flow velocity. Local density chord length span With angle of attack Using lift / drag coefficient as well as Calculate the segment force, decompose and accumulate the segment force along the lift / drag direction to obtain the total rotor thrust and torque.

7. The method for constructing an efficient hybrid simulation system for unmanned aerial vehicle control design and evaluation as described in claim 6, characterized in that, Lift / drag coefficient of rotor section as well as Lift / drag coefficients of body surface elements, pre-generated and interpolated using XFoil software, or calculated using CFD. as well as Using a simplified approximation, we have , .

8. The method for constructing an efficient hybrid simulation system for unmanned aerial vehicle control design and evaluation as described in claim 5, characterized in that, In the Actuation Surface Model (ASM), the surface of the machine is uniformly sampled, and each sampling point is regarded as a small plane element, based on the relative incoming flow velocity. Local density equivalent area With angle of attack The lift / resistance is calculated using empirical functions and then superimposed.

9. The method for constructing an efficient hybrid simulation system for unmanned aerial vehicle control design and evaluation as described in claim 1, characterized in that, Probe points are set in the near-object field to interpolate local flow velocity and density from the far-field block: the rotor section probe is moved forward by one grid scale along the rotation direction, and the body sampling probe is moved outward by one grid scale along the normal direction to reduce near-wall interference and improve the stability and reliability of the reference flow.

10. The method for constructing an efficient hybrid simulation system for unmanned aerial vehicle control design and evaluation as described in claim 1, characterized in that, Discrete boundary forces obtained from the algebraic empirical model of the near-object field are diffused and loaded on the far-field grid using a compact support kernel function to achieve momentum conservation and boundary effect feedback. In this process, to avoid excessive smoothing under coarse grid, a linear kernel with a radius of 1 after normalization to the grid scale is used to ensure that the near-wall action is clear and stable. The grid resolution in the far field is 0.5 to 2 times the average chord length of the UAV rotor to balance stability and the discriminability of the flow field structure.

Images