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He X, Wang C, Jia P, Zhong Z. The Effect of Hindwing Trajectories on Wake-Wing Interactions in the Configuration of Two Flapping Wings in Tandem. Biomimetics (Basel) 2024; 9:406. [PMID: 39056847 PMCID: PMC11274670 DOI: 10.3390/biomimetics9070406] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/23/2024] [Revised: 06/22/2024] [Accepted: 07/01/2024] [Indexed: 07/28/2024] Open
Abstract
The present investigations on tandem wing configurations primarily revolve around the effects of the spacing L and the phase difference φ between the forewing and the hindwing on aerodynamic performance. However, in nature, organisms employing biplane flight, such as dragonflies, demonstrate the ability to achieve superior aerodynamic performance by flexibly adjusting their flapping trajectories. Therefore, this study focuses on the effects of φ, as well as the trajectory of the hindwing, on aerodynamic performance. By summarizing four patterns of wake-wing interaction processes, it is indicated that φ=-90∘ and 0∘ enhances the thrust of the hindwing, while φ=90∘ and 180∘ result in reductions. Furthermore, the wake-wing interactions and shedding modes are summarized corresponding to three kinds of trajectories, including elliptical trajectories, figure-eight trajectories, and double figure-eight trajectories. The results show that the aerodynamic performance of the elliptical trajectory is similar to that of the straight trajectory, while the figure-eight trajectory with positive surging motion significantly enhances the aerodynamic performance of the hindwing. Conversely, the double-figure-eight trajectory degrades the aerodynamic performance of the hindwing.
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Affiliation(s)
- Xu He
- School of Science, Harbin Institute of Technology (Shenzhen), Shenzhen 518055, China; (X.H.); (Z.Z.)
| | - Chao Wang
- School of Mechanical Engineering, Dongguan University of Technology, Dongguan 523808, China
| | - Pan Jia
- School of Science, Harbin Institute of Technology (Shenzhen), Shenzhen 518055, China; (X.H.); (Z.Z.)
| | - Zheng Zhong
- School of Science, Harbin Institute of Technology (Shenzhen), Shenzhen 518055, China; (X.H.); (Z.Z.)
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2
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Zheng M, Peng L, Su G, Pan T, Li Q. Experimental Investigation on Aerodynamic Performance of Inclined Hovering with Asymmetric Wing Rotation. Biomimetics (Basel) 2024; 9:225. [PMID: 38667236 PMCID: PMC11047828 DOI: 10.3390/biomimetics9040225] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/26/2024] [Revised: 04/05/2024] [Accepted: 04/06/2024] [Indexed: 04/28/2024] Open
Abstract
This study presents a model experiment method that can accurately reproduce the flapping motion of insect wings and measure related unsteady aerodynamic data in real time. This method is applied to investigate the aerodynamic characteristics of inclined hovering, which distinguishes it from normal hovering by having asymmetric wing rotation during the two half strokes. In the study of the aerodynamic influence of the downstroke rotational angle, it is found that the rotational angle affects lift generation by changing the angle between the wing surface and the horizontal plane in the mid-downstroke. When the wing is almost parallel to the horizontal plane in the mid-downstroke, the vortex structure can maintain structural integrity and a large magnitude, which is conducive to the generation of high lift. In the study of the aerodynamic effect of the upstroke rotational angle, the windward conversion mechanism is proposed to explain the influence of the upstroke rotational angle on the direction and magnitude of thrust. Obtaining the rotational angle that is most conducive to maintaining the flight state of hovering in the present study can provide guidance for the structural design and kinematic control of micro aerial vehicles.
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Affiliation(s)
- Mengzong Zheng
- Research Institute of Aero-Engine, Beihang University, Beijing 100191, China; (M.Z.); (T.P.); (Q.L.)
| | - Liansong Peng
- China Academy of Information and Communications Technology, Institute for the Integration of Informatization and Industrialization, Beijing 100190, China;
| | - Guanting Su
- School of Energy and Power Engineering, Beihang University, Beijing 100191, China
| | - Tianyu Pan
- Research Institute of Aero-Engine, Beihang University, Beijing 100191, China; (M.Z.); (T.P.); (Q.L.)
| | - Qiushi Li
- Research Institute of Aero-Engine, Beihang University, Beijing 100191, China; (M.Z.); (T.P.); (Q.L.)
- School of Energy and Power Engineering, Beihang University, Beijing 100191, China
- Key Laboratory of Fluid and Power Machinery, Ministry of Education, Xihua University, Chengdu 610039, China
- Engineering Research Center of Intelligent Air-Ground Integration Vehicle and Control, Ministry of Education, Xihua University, Chengdu 610039, China
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3
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Tiwari S, Chandel S. Effect of stroke plane inclination on the hovering aerodynamic performance of tandem flapping foils. BIOINSPIRATION & BIOMIMETICS 2024; 19:026002. [PMID: 38176109 DOI: 10.1088/1748-3190/ad1b2d] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/08/2023] [Accepted: 01/04/2024] [Indexed: 01/06/2024]
Abstract
The four-winged form of dragonfly and damselfly allows them to fly with great agility and endurance, which are accomplished by independently controlling the kinematics of each wing. In this study, we performed numerical simulations of two tandem airfoils oscillating along an inclined stroke plane atRe=157. We investigated the effects of the stroke plane angle (β)of forefoil and hindfoil on the aerodynamic performance of dragonflies (or damselflies) hovering flight. Simulations were conducted for parallel and non-parallel stroke planes of forefoil and hindfoil oscillating with three phase differences:ϕ=0o,90oand180o. For parallel stroke planes, the results show that the total lift increases withβ, whereas the total thrust decreases in same condition. In addition, the total lift and thrust decrease with an increase inϕ. The forefoil performance is affected by the flow induced by hindfoil's leading-edge vortices, and the hindfoil interactions with forefoil's wake vortices significantly affect the hindfoil's performance. The findings demonstrate that non-parallel stroke planes exert detrimental effects on the total lift forϕ=0oand90o. However, forϕ=180o, the lift augmentation of 46% is obtained in the case where the forefoil stroke plane angle is smaller than the hindfoil. The results obtained during this investigation can help in optimizing the wing kinematics during the micro air vehicles development.
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Affiliation(s)
- Shubham Tiwari
- Department of Mechanical Engineering, Defence Institute of Advanced Technology (DU), Girinagar, Pune 411025, Maharashtra, India
| | - Sunil Chandel
- Department of Mechanical Engineering, Defence Institute of Advanced Technology (DU), Girinagar, Pune 411025, Maharashtra, India
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Salami E, Montazer E, Ward TA, Nik Ghazali NN, Anjum Badruddin I. Aerodynamic Performance of a Dragonfly-Inspired Tandem Wing System for a Biomimetic Micro Air Vehicle. Front Bioeng Biotechnol 2022; 10:787220. [PMID: 35662843 PMCID: PMC9159522 DOI: 10.3389/fbioe.2022.787220] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/30/2021] [Accepted: 03/04/2022] [Indexed: 11/16/2022] Open
Abstract
The flying agility demonstrated by dragonflies is accomplished by means of complex aerodynamic forces produced by flapping their four wings arranged in a tandem configuration. The current study presents a novel tandem flapping wing mechanism for a biomimetic air vehicle that was designed and manufactured to experimentally investigate the aerodynamic forces. By optimizing the configuration and using spatial network analysis, it is shown that the designed structure can flap the wings in a linear up–down stroke motion and is capable of maintaining good consistency and aerodynamic performance. Such a mechanism could be used in a future biomimetic micro air vehicle (BMAV) design. The mechanism uses an electromagnetic actuator to flap the wings with a variable beat frequency (30–210 Hz) at various angles of attack (−10°–20°). The results show that the tandem wings generate approximately 50% higher lift than the forewing or hindwing pairs acting alone. Tandem wings also improve stability, which could potentially allow hovering.
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Affiliation(s)
- Erfan Salami
- Department of Mechanical Engineering, Faculty of Engineering, University of Malaya, Kuala Lumpur, Malaysia
- *Correspondence: Erfan Salami, ; Nik Nazri Nik Ghazali,
| | - Elham Montazer
- Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran
| | - Thomas A Ward
- School of Engineering and Computer Science, Cedarville University, Cedarville, OH, United States
| | - Nik Nazri Nik Ghazali
- Department of Mechanical Engineering, Faculty of Engineering, University of Malaya, Kuala Lumpur, Malaysia
- *Correspondence: Erfan Salami, ; Nik Nazri Nik Ghazali,
| | - Irfan Anjum Badruddin
- Research Center for Advanced Materials Science (RCAMS), King Khalid University, Abha, Saudi Arabia
- Department of Mechanical Engineering, College of Engineering, King Khalid University, Abha, Saudi Arabia
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Peng L, Pan T, Zheng M, Song S, Su G, Li Q. Kinematics and Aerodynamics of Dragonflies (Pantala flavescens, Libellulidae) in Climbing Flight. Front Bioeng Biotechnol 2022; 10:795063. [PMID: 35372311 PMCID: PMC8966397 DOI: 10.3389/fbioe.2022.795063] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/14/2021] [Accepted: 02/23/2022] [Indexed: 11/13/2022] Open
Abstract
This study presents a detailed analysis of dragonflies’ climbing flight by integratinghigh-speed photogrammetry, three-dimensional reconstruction, and computational fluid dynamics. In this study, a dragonfly’s climbing flight is captured by two high-speed cameras with orthogonal optical axes. Through feature point matching and three-dimensional reconstruction, the body kinematics and wing kinematics of 22 dragonflies in climbing flight are accurately captured. Experimental results show that the climbing angles (η) are distributed from 10° to 80° and are concentrated within two ranges, 60°–70° (36%) and 20°–30° (32%), which are defined as large angle climb (LAC) and small angle climb (SAC), respectively. In order to study the aerodynamic mechanism of the climbing flight based on the biological observation results, the kinematic parameters of the dragonfly during LAC and SAC are selected for analysis and numerical simulation. The results show that the climbing angle η and wing kinematics are related. There are considerable differences in wing kinematics during climbing with different η, while the wing kinematics are unchanged during climbing with similar η. With the increase in η, the phase difference (λ) between the forewing and the hind wing decreases and the amplitude of the positional angle (θmean) of the hind wing increases, while θmean of the forewing remains almost unchanged. Through numerical simulation of LAC and SAC, it can be found that during the climb with different η, the different wing kinematics have a significant influence on aerodynamic performance. During SAC, the increase in λ and the decrease in θmean of the hind wing weaken the aerodynamic disturbance of the forewing by the vortex wing of the hind wing, thus improving the flight efficiency.
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Affiliation(s)
- Liansong Peng
- School of Energy and Power Engineering, Beihang University, Beijing, China
| | - Tianyu Pan
- Reserach Institute of Aero-Engine, Beihang University, Beijing, China
| | - Mengzong Zheng
- Reserach Institute of Aero-Engine, Beihang University, Beijing, China
- *Correspondence: Mengzong Zheng,
| | - Shiying Song
- School of Energy and Power Engineering, Beihang University, Beijing, China
| | - Guanting Su
- School of Energy and Power Engineering, Beihang University, Beijing, China
| | - Qiushi Li
- School of Energy and Power Engineering, Beihang University, Beijing, China
- Reserach Institute of Aero-Engine, Beihang University, Beijing, China
- Key Laboratory of Fluid and Power Machinery, Ministry of Education, Xihua University, Chengdu, China
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Peng L, Zheng M, Pan T, Su G, Li Q. Tandem-wing interactions on aerodynamic performance inspired by dragonfly hovering. ROYAL SOCIETY OPEN SCIENCE 2021; 8:202275. [PMID: 34457328 PMCID: PMC8385352 DOI: 10.1098/rsos.202275] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 01/07/2021] [Accepted: 07/30/2021] [Indexed: 06/13/2023]
Abstract
Dragonflies possess two pairs of wings and the interactions between forewing (FW) and hindwing (HW) play an important role in dragonfly flight. The effects of tandem-wing (TW) interactions on the aerodynamic performance of dragonfly hovering have been investigated. Numerical simulations of single-wing hovering without interactions and TW hovering with interactions are conducted and compared. It is found that the TW interactions reduce the lift coefficient of FW and HW by 7.36% and 20.25% and also decrease the aerodynamic power and efficiency. The above effects are mainly caused by the interaction between the vortex structures of the FW and the HW, which makes the pressure of the wing surface and the flow field near the wings change. During the observations of dragonfly flight, it is found that the phase difference (γ) is not fixed. To explore the influence of phase difference on aerodynamic performance, TW hovering with different phase differences is studied. The results show that at γ = 22.5°, dragonflies produce the maximum lift which is more than 20% of the body weight with high efficiency; at γ = 180°, dragonflies generate the same lift as the body weight.
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Affiliation(s)
- Liansong Peng
- School of Energy and Power Engineering, Beihang University, Beijing 100191, People's Republic of China
| | - Mengzong Zheng
- School of Energy and Power Engineering, Beihang University, Beijing 100191, People's Republic of China
| | - Tianyu Pan
- Research Institute of Aero-Engine, Beihang University, Beijing 100083, People's Republic of China
| | - Guanting Su
- School of Energy and Power Engineering, Beihang University, Beijing 100191, People's Republic of China
| | - Qiushi Li
- School of Energy and Power Engineering, Beihang University, Beijing 100191, People's Republic of China
- Research Institute of Aero-Engine, Beihang University, Beijing 100083, People's Republic of China
- Key Laboratory of Fluid and Power Machinery, Ministry of Education, Xihua University, Chengdu 610039, People's Republic of China
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7
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Khelifa R, Mahdjoub H, M’Gonigle LK, Kremen C. Integrating high-speed videos in capture-mark-recapture studies of insects. Ecol Evol 2021; 11:6033-6040. [PMID: 34141200 PMCID: PMC8207387 DOI: 10.1002/ece3.7372] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/23/2020] [Revised: 01/26/2021] [Accepted: 02/05/2021] [Indexed: 12/04/2022] Open
Abstract
Capture-mark-recapture (CMR) studies have been used extensively in ecology and evolution. While it is feasible to apply CMR in some animals, it is considerably more challenging in small fast-moving species such as insects. In these groups, low recapture rates can bias estimates of demographic parameters, thereby handicapping effective analysis and management of wild populations. Here, we use high-speed videos (HSV) to capture two large dragonfly species, Anax junius and Rhionaeschna multicolor, that rarely land and, thus, are particularly challenging for CMR studies. We test whether HSV, compared to conventional "eye" observations, increases the "resighting" rates and, consequently, improves estimates of both survival rates and the effects of demographic covariates on survival. We show that the use of HSV increases the number of resights by 64% in A. junius and 48% in R. multicolor. HSV improved our estimates of resighting and survival probability which were either under- or overestimated with the conventional observations. Including HSV improved credible intervals for resighting rate and survival probability by 190% and 130% in A. junius and R. multicolor, respectively. Hence, it has the potential to open the door to a wide range of research possibilities on species that are traditionally difficult to monitor with distance sampling, including within insects and birds.
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Affiliation(s)
- Rassim Khelifa
- Department of Zoology and Biodiversity Research CentreUniversity of British ColumbiaVancouverBCCanada
| | - Hayat Mahdjoub
- Department of Zoology and Biodiversity Research CentreUniversity of British ColumbiaVancouverBCCanada
| | | | - Claire Kremen
- Department of Zoology and Biodiversity Research CentreUniversity of British ColumbiaVancouverBCCanada
- Institute for Resources, Environment and SustainabilityUniversity of British ColumbiaVancouverBCCanada
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Yanagisawa R, Shigaki S, Yasui K, Owaki D, Sugimoto Y, Ishiguro A, Shimizu M. Wearable Vibration Sensor for Measuring the Wing Flapping of Insects. SENSORS (BASEL, SWITZERLAND) 2021; 21:E593. [PMID: 33467684 PMCID: PMC7829746 DOI: 10.3390/s21020593] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 12/04/2020] [Revised: 01/05/2021] [Accepted: 01/12/2021] [Indexed: 11/16/2022]
Abstract
In this study, we fabricated a novel wearable vibration sensor for insects and measured their wing flapping. An analysis of insect wing deformation in relation to changes in the environment plays an important role in understanding the underlying mechanism enabling insects to dynamically interact with their surrounding environment. It is common to use a high-speed camera to measure the wing flapping; however, it is difficult to analyze the feedback mechanism caused by the environmental changes caused by the flapping because this method applies an indirect measurement. Therefore, we propose the fabrication of a novel film sensor that is capable of measuring the changes in the wingbeat frequency of an insect. This novel sensor is composed of flat silver particles admixed with a silicone polymer, which changes the value of the resistor when a bending deformation occurs. As a result of attaching this sensor to the wings of a moth and a dragonfly and measuring the flapping of the wings, we were able to measure the frequency of the flapping with high accuracy. In addition, as a result of simultaneously measuring the relationship between the behavior of a moth during its search for an odor source and its wing flapping, it became clear that the frequency of the flapping changed depending on the frequency of the odor reception. From this result, a wearable film sensor for an insect that can measure the displacement of the body during a particular behavior was fabricated.
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Affiliation(s)
- Ryota Yanagisawa
- Department of Systems Science, Osaka University, 1-2 Machikaneyama-cho, Toyonaka, Osaka 560-0043, Japan;
| | - Shunsuke Shigaki
- Department of System Innovation, Osaka University, 1-2 Machikaneyama-cho, Toyonaka, Osaka 560-0043, Japan;
| | - Kotaro Yasui
- Frontier Research Institute for Interdisciplinary Sciences, Tohoku University, 6-3 Aramaki aza Aoba, Aoba-ku, Sendai 980-8578, Japan;
- Research Institute of Electrical Communication, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan;
| | - Dai Owaki
- Department of Robotics, Tohoku University, 6-6-01 Aoba, Aramaki, Aoba-ku, Sendai 980-8579, Japan;
| | - Yasuhiro Sugimoto
- Department of Mechanical Engineering, Osaka University, 2-1 Yamadaoka, Suita, Osaka 565-0871, Japan;
| | - Akio Ishiguro
- Research Institute of Electrical Communication, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan;
| | - Masahiro Shimizu
- Department of System Innovation, Osaka University, 1-2 Machikaneyama-cho, Toyonaka, Osaka 560-0043, Japan;
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Su G, Dudley R, Pan T, Zheng M, Peng L, Li Q. Maximum aerodynamic force production by the wandering glider dragonfly ( Pantala flavescens, Libellulidae). J Exp Biol 2020; 223:jeb218552. [PMID: 32457065 DOI: 10.1242/jeb.218552] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/18/2019] [Accepted: 05/19/2020] [Indexed: 11/20/2022]
Abstract
Maximum whole-body force production can influence behavioral outcomes for volant taxa, and may also be relevant to aerodynamic optimization in microair vehicles. Here, we describe a new method for measuring maximum force production in free-flying animals, and present associated data for the wandering glider dragonfly. Flight trajectories were repeatedly acquired from pull-up responses by insects dropped in mid-air with submaximal loads attached beneath the center of body mass. Forces were estimated from calculations of the maximum time-averaged acceleration through time, and multiple estimates were obtained per individual so as to statistically facilitate approximation of maximum capacity through use of the Weibull distribution. On a group level, wandering glider dragonflies were here estimated to be capable of producing total aerodynamic force equal to ∼4.3 times their own body weight, a value which significantly exceeds earlier estimates made for load-lifting dragonflies, and also for other volant taxa in sustained vertical load-lifting experiments. Maximum force production varied isometrically with body mass. Falling and recovery flight with submaximal load represents a new context for evaluating limits to force production by flying animals.
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Affiliation(s)
- Guanting Su
- National Key Laboratory of Science and Technology on Aero-Engine Aero-Thermodynamics, School of Energy and Power Engineering, Beihang University, Beijing 100191, China
- Department of Integrative Biology, University of California, Berkeley, Berkeley, CA 94720, USA
| | - Robert Dudley
- Department of Integrative Biology, University of California, Berkeley, Berkeley, CA 94720, USA
- Smithsonian Tropical Research Institute, Balboa, Republic of Panama
| | - Tianyu Pan
- National Key Laboratory of Science and Technology on Aero-Engine Aero-Thermodynamics, School of Energy and Power Engineering, Beihang University, Beijing 100191, China
| | - Mengzong Zheng
- National Key Laboratory of Science and Technology on Aero-Engine Aero-Thermodynamics, School of Energy and Power Engineering, Beihang University, Beijing 100191, China
| | - Liansong Peng
- National Key Laboratory of Science and Technology on Aero-Engine Aero-Thermodynamics, School of Energy and Power Engineering, Beihang University, Beijing 100191, China
| | - Qiushi Li
- National Key Laboratory of Science and Technology on Aero-Engine Aero-Thermodynamics, School of Energy and Power Engineering, Beihang University, Beijing 100191, China
- Key Laboratory of Fluid and Power Machinery, Xihua University, Chengdu 610039, China
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Jeong D, Li Y, Kim S, Choi Y, Lee C, Kim J. Mathematical modeling and computer simulation of the three-dimensional pattern formation of honeycombs. Sci Rep 2019; 9:20364. [PMID: 31889154 PMCID: PMC6937337 DOI: 10.1038/s41598-019-56942-6] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/03/2019] [Accepted: 12/19/2019] [Indexed: 11/09/2022] Open
Abstract
We present a mathematical model, a numerical scheme, and computer simulations of the three-dimensional pattern formation of a honeycomb structure by using the immersed boundary method. In our model, we assume that initially the honeycomb cells have a hollow hemisphere mounted by a hollow circular cylinder shape at their birth and there is force acting upon the entire side of the cell. The net force from the individual cells is a key factor in their transformation from a hollow hemisphere mounted by a hollow circular cylinder shape to a rounded rhombohedral surfaces mounted by a hexagonal cylinder shape. Numerical simulations of the proposed mathematical model equation produce the rounded rhombohedral surfaces mounted by a hexagonal cylinder patterns observed in honeybee colonies.
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Affiliation(s)
- Darae Jeong
- Department of Mathematics, Kangwon National University, Gangwon-do, 24341, Republic of Korea
| | - Yibao Li
- School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049, China
| | - Sangkwon Kim
- Department of Mathematics, Korea University, Seoul, 02841, Republic of Korea
| | - Yongho Choi
- Department of Mathematics and Big Data, Daegu University, Gyeongsan-si, Gyeongsangbuk-do, 38453, Republic of Korea
| | - Chaeyoung Lee
- Department of Mathematics, Korea University, Seoul, 02841, Republic of Korea
| | - Junseok Kim
- Department of Mathematics, Korea University, Seoul, 02841, Republic of Korea.
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