Flight Dynamics Simulation Research Articles

Design and testing of a crop growth sensor aboard a fixed-wing unmanned aerial vehicle

• A set of crop growth sensors mounted on a fixed-wing UAV were developed. The system can obtain crop canopies and growth indices, such as LNC and LAI, on-line and in a convenient, high-throughput manner while ensuring that theUAV's airflow does not disturb the crop canopies below. • Flight dynamics computer simulation analysis of the fixed-wing UAV was conducted by the ADAMS software to obtain the deflection angle of the UAV during flight. A ball rolling-type sensor support was designed to ensure that the crop growth sensor is always aimed vertically downwards in-flight. • Due to the adjustment of the sensor support, the system has good dynamic stability and measurement accuracy within the working height range of the sensor. The aim of this study is to overcome disturbance of downwash flow field caused by the low-altitude operation of a multirotor unmanned aerial vehicle (UAV) on crop canopies and interference in spectral reflection information of canopies. For this purpose, a crop growth sensor aboard a fixed-wing UAV was developed through flight dynamics simulation analysis of a fixed-wing UAV. This sensor can collect index data on-line and in real-time including: the ratio vegetation index (RVI) of crop leaves, leaf area index (LAI), leaf dry weight (LDW), and leaf nitrogen content (LNC). Flight dynamics simulation analysis of the fixed-wing UAV was conducted by the automatics dynamic analysis of mechanical system (ADAMS) software to obtain the deflection angle of the UAV during flight. According to the flight characteristics and load on the UAV, a ball rolling-type sensor support was designed to ensure that the crop growth sensor is always aimed vertically downwards in-flight. The field test results show that the crop growth sensor aboard the fixed-wing UAV has good dynamic stability and high measurement accuracy. The RVIs measured by the onboard crop growth sensor in the plots and field were fitted with the results measured by a FieldSpec HandHeld 2 spectroradiometer (ASD, Analytical Spectral Device Co., USA). By analysing the fitted results, the coefficients of determination ( R 2 ) are 0.763 and 0.833 and the root mean square errors (RMSEs) are 0.16 and 0.17, respectively. By linearly fitting RVIs measured by the UAV with rice growth indices including LAI, LDW, and LNC, the coefficients of determination ( R 2 ) are 0.633, 0.581, and 0.528 and RMSEs are 0.18, 0.18, and 0.21, respectively.

ПРОЄКТУВАННЯ ТА СТАТИСТИЧНИЙ АНАЛІЗ КАБІНИ ПІЛОТА ДЛЯ АВІАТРЕНАЖЕРУ

The subject of the study is the cockpit of a dynamic flight simulator. The flight simulator has been used for a long time for training pilots of any type of aircraft. This type of training is more economical. A flight simulator improves control skills in extreme situations with minimum risks for the future pilot. Training on the flight simulator makes it possible to reproduce unfavorable weather conditions at any time. The pilot is on the ground in a special cockpit that is mounted on a movable platform. This system is necessary to provide as close to real flight conditions as possible. The subject of the research in the article is an aviation simulator on the Stewart platform. The aim is to create and analyze the cockpit for the flight simulator with the help of software. The paper presents an already simulated platform and its characteristics. Considering its overall dimensions and weight the cabin is modeled in SolidWorks. The developed 3D model is a cockpit of an airplane with the selected overall dimensions for convenient use. The balsa material from which the cabin is projected, as they are not important aerodynamic characteristics. What matters is the ability of the cabin to withstand the planned loads, the weight of the cabin and its appearance. A static analysis of the loaded cabin was conducted. A loaded cab means that inside the cab the loads are applied in two places. These loads are equal to the weight of the pilot's seat with a person and the weight of the entire electronic part by means of which the movement of the cockpit and visualization of the flight takes place. For the analysis, the cockpit was fixed at the attachment points to the platform. The attachment will be done by welding. The analysis includes the following shear and bending moment diagrams: loads to assess the structure withstand a given weight, displacement, deformation, safety margin to understand what the maximum possible loads the model can withstand without deformation. The research was conducted using SolidWorks Simulation software. Based on the data obtained from the epectra, conclusions were made about the operability and safety of the developed structure

Dynamic Simulation and Conceptual Layout Study on a Quad-Plane in VTOL Mode in Wind Disturbance Environment

Quadrotor fixed-wing hybrid vertical take-off and landing (VTOL) UAV, the so-called Quad-Plane, is susceptible to wind disturbance during the VTOL mode, which reduces its stability and limits its application. In order to dynamically analyze its response characteristics to wind disturbance and effectively optimize the wind disturbance rejection performance in VTOL mode, a dynamic flight simulation system based on the 6-DoF equations is established, in which the aerodynamic model of the Quad-Plane is established based on the wind tunnel test, the axial and lateral forces and moments of the rotor in wind disturbance are obtained through CFD simulation, and the Brushless DC (BLDC) motor, wind disturbance, and flight controller are modeled based on actual characteristics. The simulation system is verified and validated by comparing the simulation results with test results of a 5 kg real Quad-Plane. Using this system, the response characteristics of the Quad-Plane to the discrete and continuous wind disturbance are studied. In addition, the influences of major conceptual layout parameters that affect the wind disturbance rejection performance, such as take-off weight, weight distribution, Center of Gravity (CG) position, rotor incline angle, arm length, and arm angle, are studied. Finally, the approaches on designing a Quad-Plane with better wind disturbance rejection performance during the conceptual layout design are presented.

Data-enhanced dynamic flight simulations for flight performance analysis

More often than not, airlines use aircraft in operating scenarios beyond their optimal design conditions, which negatively affects their performance characteristics. These effects are statistically reflected in flight operational data, which are indirectly constrained by air transportation management as well as aircraft design. In this research, we develop a data-enhanced methodology for modeling dynamic flight simulations of aircraft to enable accurate estimation of performance parameters. The relevant flight phases and constraints of these simulations are determined by employing supervised machine learning on the flight data. The methodology is demonstrated by simulating flights across representative short-, medium-, and long-haul sectors using data shared by our airline partner. We compare the fuel burn and flight time calculation results with those from the Bréguet range and endurance equations, from an available open-source flight performance model, and from the reference data for validation. The developed dynamic flight simulation model is designed in such a way to enable accurate flight performance analysis even when high-fidelity force analyses and control models are absent, which is common in preliminary design frameworks. This will further enable incorporating flight data into aircraft design processes.

Enhanced Attitude and Altitude Estimation for Indoor Autonomous UAVs

In recent years the use of Unmanned Aerial Vehicles (UAVs) has considerably grown in the civil sectors, due to their high flexibility of use. Currently, two important key points are making them more and more successful in the civil field, namely the decrease of production costs and the increase in navigation accuracy. In this paper, we propose a Kalman filtering-based sensor fusion algorithm, using a low cost navigation platform that contains an inertial measurement unit (IMU), five ultrasonic ranging sensors and an optical flow camera. The aim is to improve navigation in indoor or GPS-denied environments. A multi-rate version of the Extended Kalman Filter is considered to deal with the use of heterogeneous sensors with different sampling rates, and the presence of non-linearities in the model. The effectiveness of the proposed sensor platform is evaluated by means of numerical tests on the dynamic flight simulator of a quadrotor. Results show high precision and robustness of the attitude estimation algorithm, with a reduced computational cost, being ready to be implemented on low-cost platforms.

The Design of an Automatic Flight Control System and Dynamic Simulation for Fixed-Wing Unmanned Aerial Vehicle (UAV) using X-Plane and LabVIEW

This research focuses on developing an automatic flight control system for a fixed-wing unmanned aerial vehicle (UAV) using a software-in-the-loop method in which the PID controller is implemented in National Instruments LabVIEW software and the flight dynamics of the fixed-wing UAV are simulated using the X-Plane flight simulator. The fixed-wing UAV model is created using the Plane Maker software and is based on existing geometry and propulsion data from the literature. Gain tuning for the PID controller is accomplished using the pole placement technique. In this approach, the controller gain can be calculated using the dynamic parameters in the transfer function model and the desired characteristic equation. The proposed controller designs' performance is validated using attitude, altitude, and velocity hold simulations. The results demonstrate that the technique can be an effective tool for researchers to validate their UAV control algorithms by utilising the realistic UAV or manned aircraft models available in the X-Plane flight simulator.

Fatigue, Emotion, and Cognitive Performance in Simulated Long-Duration, Single-Piloted Flight Missions.

BACKGROUND: Fatigue of air force pilots has become an increasing concern due to changes in mission characteristics. In the current study we investigated fatigue, emotions, and cognitive performance in a simulated 11-h mission in the 39 Gripen fighter aircraft. METHODS: A total of 12 subjects were evaluated in a high-fidelity dynamic flight simulator for 12 consecutive hours. Perceived fatigue was measured by the Samn-Perelli Fatigue Index (SPFI). Emotions were assessed with the Circumplex Affect Space. Cognitive performance was assessed by five cognitive tasks. RESULTS: Significant increase in self-reported fatigue, general decrease in two positive emotional states, as well increase of one negative emotional state occurred after approximately 7 h into the mission. Self-reported fatigue negatively correlated with enthusiasm and cheerfulness (r 0.75; 0.49, respectively) and positively correlated with boredom and gloominess (r 0.61; r 0.30, respectively). Response time in the low-order task negatively correlated with enthusiasm, cheerfulness and calmness (r 0.44; r 0.41; r 0.37, respectively) and positively correlated with boredom and anxiousness (r 0.37; r 0.28, respectively). Mission duration had an adverse impact on emotions in these environmental conditions, particularly after 7 h. DISCUSSION: These results contribute to the understanding of fatigue development in general and of emotion-cognition relationships. These findings emphasize that both emotional states and the type of cognitive tasks to be performed should be considered for planning long-duration missions in single-piloted fighter aircrafts as to increase the probability of missions success. Rosa E, Gronkvist M, Kolegard R, Dahlstrom N, Knez I, Ljung R, Willander J. Fatigue, emotion, and cognitive performance in simulated long-duration, single-piloted flight missions. Aerosp Med Hum Perform. 2021; 92(9):710719.

XGBoost Gradient-Boosted Tree Predictions Using Limited Data for Coaxial Rotor Helicopters

The use of finite-state methods is critical to the development of accurate and efficient inflow models used in rotorcraft flight dynamics simulation and control. Recent work in the finite-state field has allowed for the application of these models to multirotor systems using the adjoint theorem, which involves time delays and adjoint variables. However, the addition of time delays and adjoint variables drives the necessity for the addition of further inflow states to achieve model accuracy. Computation with a higher numbers of inflow states requires greater computing power and therefore limits the ability of real-time analysis. To help mitigate these issues, this paper explores the use of a gradient booted trees in XGBoostTM, as well as the use of varied, lower state training data and limited higher state training data, to accurately predict the velocity on the lower rotor of a coaxial rotor helicopter. The investigation involves XGBoost hyperparameter searches to determine the best model, variation in training and testing subset splits, and use of validation subset comparisons for identifying the best performing model.

Hazardous flight region prediction for a small UAV operated in an urban area using a deep neural network

With an increase of UAVs in logistics and transportation, the safety of UAVs operated in the urban wind environment becomes an important issue. Small UAVs are more sensitive to the wind environment because of their small size, slow cruising speed, and limited endurance. In the unmanned aircraft system traffic management (UTM), a safety risk assessment under bad weather conditions is an important component. In this study, a hazardous flight region prediction system for small UAVs operated in urban areas is developed using a deep neural network (DNN) to support a risk assessment and safe trajectory planning. A large eddy simulation (LES) is applied to reflect the terrain-driven wind environment in the urban area. The result of a weather research and forecasting (WRF) model is used as an initial and boundary condition of the LES to generate a realistic complicated wind environment in an urban area. Furthermore, an iterative nesting algorithm is applied to the LES to obtain a sufficient resolution of the wind environment, which is suitable for the small UAV scale. The deviation distance from the original flight path due to the wind environment is considered as a flight hazard criterion in this study. The proposed system is able to predict deviation distance due to the wind environment over the entire flight space over time by using the DNN model. The training data for the DNN is obtained using the multicopter flight dynamics simulator, which can take into account the influence of a specific wind environment. With the indexes considering this deviation distance and the local topography (distribution of buildings) in the urban area, the hazardous flight region is predicted. The information supplied by the proposed hazardous flight region prediction model can be used for the flight risk assessment and safe flight trajectory planning to increase the flight safety of small UAVs.

Coupled Simulation of Electric Flight Dynamics and Physics Based Battery Model for Electric Aircraft Battery Pack Sizing Analysis

Electrification of aircraft and aviation systems is a thriving area of research and technology development in recent years1. According to the Airbus Global Market Forecast, air traffic will double every 15 years, leading to increasing fuel consumption causing severe impact due to greenhouse gases and carbon emission2. Besides the environmental impact, as the international demand for fuel increases, and the fact that petroleum resources are limited, fuel prices are projected to rise. On the other side, the drop in the cost of lithium ion batteries makes electric aircraft feasible to commercialize now. This necessitates studying the battery pack sizing problem to build efficient Energy Management System (EMS) for electric aircraft.There is a significant amount of research on EMS for electric aircraft. Most of the current literature related to electric aircraft modeling considers the battery voltage to be linear3. This may lead to inaccurate estimation of state of charge (SOC), a key state variable for batteries and battery pack. In this work, we perform coupled simulation studies of the longitudinal flight dynamics and physics-based battery model (single particle model)4 ,5. This coupling enables accurate estimation of battery pack state of charge. The simulation studies are performed to understand the interplay between battery pack configuration/design parameters and variables/parameters related to flight dynamics, such as altitude, distance, flight path angle, and velocity. We also compare the SOC prediction of the battery pack using single particle model and dynamic empirical battery model. This work will be extended to develop efficient and robust EMS for electric aircraft. Acknowledgements This research was supported by the Texas Materials Institute (TMI) at The University of Texas at Austin. References J. A. Rosero, J. A. Ortega, E. Aldabas, and L. Romeral, IEEE Aerosp. Electron. Syst. Mag., 22, 3–9 (2007).Airbus global market forecase. [Online]., https://www.airbus.com/aircraft/market/global-market-forecast.html.M. Kaptsov and L. Rodrigues, J. Guid. Control. Dyn., 41, 285–290 (2018).S. Santhanagopalan, Q. Guo, P. Ramadass, and R. E. White, J. Power Sources, 156, 620–628 (2006).V. Ramadesigan, P. W. C. Northrop, S. De, S. Santhanagopalan, R. D. Braatz, and V. R. Subramanian, J. Electrochem. Soc., 159, R31–R45 (2012).

Fluid–structure interaction-based aerodynamic modeling for flight dynamics simulation of parafoil system

Prediction of aerodynamic force is a crucial issue for parafoil canopy as the strong nonlinear fluid–structure interaction (FSI) between the flexible canopy material and flow field. Flight tests and wind tunnel experiments are difficult to analyze the aerodynamics of parafoil because of the limitation and difficulty of data measurement in an unknown environment. The objective of this study was to computationally derive the aerodynamic characteristics of parafoil, as an alternative to expensive and unrepeatable test regimes. Different from previous works that assume canopy structure as a rigid body and serve for the design of parafoil, this study focused on the precise dynamic modeling of parafoil based on FSI simulations. To investigate the aerodynamic performance of the full-scale canopy with stabilizers for better control, the strong coupling FSI simulations were performed using the incompressible computational fluid dynamics techniques. The highlight of this paper is to explore the effects of canopy inflation and trailing edge deflections on aerodynamic performance. Then the aerodynamic coefficients are identified by a linear regression method using the obtained database of high fidelity lift and drag forces. Furthermore, an accurate six-degree-of-freedom dynamic model of the parafoil system is implemented based on the estimated coefficients. Simulations are conducted to prove the dynamic stability of the model and the feasibility of trajectory tracking. At last, simulation results of basic motions are compared with airdrop testing data, which demonstrates that the established model is capable of accurately predicting the flight behaviors of the parafoil system.

Aerodynamic and flight dynamic iterative simulation method of a joined wing aircraft

Aerodynamic and flight dynamic iterative simulation method of a joined wing aircraft

Coupled Flexible and Flight Dynamics Modeling and Simulation of a Full-Wing Solar-Powered Unmanned Aerial Vehicle

The full-wing solar-powered unmanned aerial vehicle (UAV) adopts a large aspect ratio wing, a lightweight structural design, and a differential throttle control scheme to maximize flight endurance. Large structural deformation often occurs when the wing is heavily loaded, which affects its flight stability, trajectory tracking accuracy, and flight performance. The traditional rigid-body flight dynamics cannot accurately describe the actual dynamic behavior when the wing is deformed. To fully consider the coupling effect of the structural deformation and the flight motion, we derive a UAV combo model consisting of a flexible wing and rigid fuselage. In the model, we also include strain formulation (s-beam) for structural modeling, finite-state induced flow theory for aerodynamic modeling, static and dynamic combined experiments for engine modeling, and the rigid-body flight dynamic equation. Besides, a model modification method based on flight data is applied to improve the accuracy of the structural parameters. Simulation results show that the wingtip deformation and motion characteristics of the rigid- and combo-system are quite different: the combo model exhibits a certain lag in comparison with the rigid-body, with the amplitude of the motion parameters reduced by 50%, frequency 15%, system kinetic energy 11.8%, and the elevator control efficiency more than 40%.

유연 특성 항공기의 동적 · 정적 거동 분석을 위한 비행 동역학 시뮬레이션 프로그램

현대의 항공기는 고성능이면서도 경량이다. 그래서 유연한 구조물의 특성이 나타나며 비행 성능에 영향을 미치거나 한계를 짓기도 한다. 이러한 유연 특성은 항공기 설계 초기 단계부터 분석이 필요한데, 이를 위해 유연 항공기의 동적 · 정적 거동을 분석할 수 있는 프로그램을 개발하였으며, 그 결과를 제시한다. 다물체 동역학 시뮬레이션 기법에 기반하여, 강체 비행 역학, 구조 진동 거동, 그리고 비정상 공기력 등을 세부 모듈로 개발하고 통합하였다. 마지막으로, 통합 시뮬레이션 프로그램을 이용하여 유연 특성 항공기의 등속 수평 비행과 선회비행을 분석하였다.

PHYSIOLOGICAL CORRELATES OF ALMOST LOSS OF CONSCIOUSNESS (A-LOC) DURING DYNAMIC FLIGHT SIMULATION (DFS) ON HUMAN CENTRIFUGE: A CASE REPORT

Introduction: G-LOC is still a threat for aircrew of high performance aircrafts. It is evoked by a high level of +Gz acceleration causing a deficit of brain oxygenation. Although near-infrared spectroscopy (NIRS) was used to identify brain oxygenation decreases preceding G-LOC, it is not used regularly in aero-medicine research in Poland. Here, we report the physiological correlates of Almost Loss of Consciousness (A-LOC) that occurred during a research project. Method: During Dynamic Flight Simulation training on a human centrifuge one pilot experienced momentaneous confusion which was recorded on a camera. We post-hoc reviewed his vital signs, stroke volumes (SV), cardiac output (CO) obtained with bioimpedance rheography, and brain oxygenation (OX) measures with near-infrared spectroscopy on the forehead. Finally, we reviewed the recording of his eye movements obtained with video-oculography. Results: SV and CO measures were confounded by artefacts induced by anti-G straining manoeuvres (AGSM). However, OX measures demonstrated decreased brain oxygenation that lasted several seconds after the incident. Following the incident, the pilot’s pulse remained decreased for several seconds. Discussion: NIRS methodology seems to be resistant to artefacts created by muscles during AGSM. NIRS systems with a probing rate of 0.1 second might help detect upcoming A-LOC or G-LOC.

Intermittent control strategy can enhance stabilization robustness in bumblebee hovering

Active flight control plays a crucial role in stabilizing the body posture of insects to stay aloft under a complex natural environment. Insects can achieve a closed-loop flight control by integrating the external mechanical system and the internal working system through manipulating wing kinematics according to feedback information from multiple sensors. While studies of proportional derivative/proportional integral derivative-based algorithms are the main subject to explore the continuous flight control mechanisms associated with insect flights, it is normally observed that insects achieve an intermittent spike firing in steering muscles to manipulate wings in flight control discontinuously. Here we proposed a novel intermittent control strategy for a 3 degree of freedom (DoF) pitch-control and explored its stabilization robustness in bumblebee hovering. An integrated computational model was established and validated, which comprises an insect-inspired dynamic flight simulator and a novel discrete feedback controller as well as a simplified free-flight dynamic model. We found that the intermittent control model can achieve an angular-dominant flight control, whereas the continuous control model corresponds to an angular-velocity-dominant one. Given the biological constraints in sensorimotor neurobiology and musculoskeletal mechanics, the intermittent control strategy was examined capable of enhancing the stabilization robustness in terms of sensory latency, stroke derivation, spike interval, and damping strength. Our results indicate that the intermittent control strategy is likely a sophisticated flight control mechanism in insect flights while providing a bioinspired flight-control design for insect size flapping-wing micro air vehicles.

Simulation Aided Design of Solar-Powered UAV

This paper describes a distributed flight dynamic simulation method. It needs reasonable computing resources, and can be applied in the design process of solar-powered UAV. A flight dynamics model is established based on the Hamilton’s principle, with the simulation model, this paper discussed the flight characteristics of the large aspect ratio flexible solar-powered UAV, and the influence of different design schemes, then proposed a simulation aided design method, which can be used to select design parameters of the solar-powered UAV, it can also provide reference for similar UAV design.

Advanced Supersonic Parachute Inflation Research Experiment Preflight Trajectory Modeling and Postflight Reconstruction

The Advanced Supersonic Parachute Inflation Research Experiment (ASPIRE) was a series of sounding rocket flights aimed at understanding the dynamics of supersonic parachutes that are used for Mars robotic applications. Although the 2012 Mars Science Laboratory (MSL) had a successful deployment of a supersonic parachute, postflight analysis and the results of the Low-Density Supersonic Decelerator project called into question some of the assumptions underlying MSL’s parachute qualification program. The ASPIRE project was a key part of the risk-reduction program undertaken by Mars 2020, the follow-on to MSL. The first sounding rocket (SR01) flight of ASPIRE occurred near Wallops Island, Virginia on 4 October 2017, and demonstrated the successful deployment and inflation of an MSL build-to-print parachute in flight conditions similar to the 2012 MSL mission. ASPIRE SR02 and SR03 were successful follow-on flights on 31 March 2018 and 7 September 2018 that demonstrated the new, strengthened supersonic parachute designed for the Mars 2020 project. The SR02 and SR03 parachutes were targeted to 100 and 140% of the expected flight limit load for Mars 2020 to confirm new margins expected from the strengthened parachute. Prior to all flights, a multibody flight dynamics simulation was developed to predict the parachute dynamics and was used, in conjunction with other tools, to target Mars-relevant flight conditions. After each flight, the onboard data were used to reconstruct the flight trajectory and to validate the preflight dynamics simulation. Postflight analysis showed that all three tests achieved their targeted conditions and preflight modeling bounded the key performance metrics for the parachute. This paper describes the flight mechanics simulation, postflight reconstruction, and the reconciliation process used to validate the flight models.

Application of multibody simulation tool for dynamical analysis of tethered aerostat

This paper presents a dynamical analysis of a tethered aerostat using a multibody simulation tool called SimscapeMultibody™. Tethered aerostats are lighter than air vehicles that float in the air and are connected to the ground through the tether. Atmospheric winds are a source of disturbance to the aerostat. The wind produces drag force that causes the aerostat to displace from its mean position. This phenomenon is called blow-by. Solving the mathematical model of the motion dynamics with time step is the conventional method used for simulating the flight dynamics of the aerostat. This paper discusses the application of Simscape Multibody™ for the three-degrees-of-freedom flight dynamic simulation of spherical tethered aerostat. Simscape Multibody™, an extension of MATLAB®/Simulink, is a multibody modeling tool in which physical bodies and their interactions have to be defined to simulate the dynamics. The mathematical model of the defined system is automatically generated by the SimscapeMultibody™ solver, and the system dynamics is simulated. The simulation of dynamics using multibody modeling tools provides high fidelity simulation for the tethered aerostats.

Acceleration of unsteady vortex lattice method via dipole panel fast multipole method

The Unsteady Vortex Lattice Method (UVLM) is a medium-fidelity aerodynamic tool that has been widely used in aeroelasticity and flight dynamics simulations. The most time-consuming step is the evaluation of the induced velocity. Supposing that the number of bound and wake lattices is N and the computational cost is ON2, we present an ON Dipole Panel Fast Multipole Method (DPFMM) for the rapid evaluation of the induced velocity in UVLM. The multipole expansion coefficients of a quadrilateral dipole panel have been derived in spherical coordinates, whose accuracy is the same as that of the Biot-Savart kernel at the same truncation degree P. Two methods (the loosening method and the shrinking method) are proposed and tested for space partitioning volumetric panels. Compared with FMM for vortex filaments (with three harmonics), DPFMM is approximately two times faster for N∈[103,106]. The simulation time of a multirotor (N∼104) is reduced from 100 min (with unaccelerated direct solver) to 2 min (with DPFMM).