自主泳动弹性绳的轨迹模拟
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摘要
研究柔性结构与流体间耦合作用,可以促进软体机器人的发展.通过速度快、精度高的数值模拟方法模拟水下机器人的实时运动轨迹,可以为真实实验提供测试方向与理论牵引,增大实验成功的可能性.本文研究有自主运动趋势的弹性绳在二维流场中的运动轨迹.首先,对弹性绳离散化建模并同时考虑拉压与扭转弹性力,从能量角度建立动力学方程,此模型可以较为真实地反映弹性绳内力对其运动产生的作用.然后基于半拉格朗日法建立流体求解器.最后,提出简化的基于动量方程的浸入边界法作为耦合算法,通过直接修正网格速度代替浸入边界力法中力源项的作用.使用这种算法求解耦合作用兼具简便性与快速性.对弹性绳模型、流体模型与简化耦合模型依次解算,模拟了正弦形式波动弹性绳在水中的运动轨迹.结果显示,弹性绳在弹性内力与流固相互作用力共同影响下,该种新的浸入边界法可以实现对水下弹性绳运动轨迹的模拟.数值实验显示弹性绳的自主运动参考模型的初相位改变时,其前进方向会发生改变.该仿真模拟算法与平台可以为细长形软体水生机器人的研发提供参考.
The research for flexible structures coupling with fluid simulates and promotes the development of soft robotics.A fast and accurate numerical method is significant for a real-time simulation of robots. This research provides theoretical and critical information for experiments to reduce the possibility of failure, by anticipating the path of the underwater soft robots and the possible requiring parameters for materials. This paper researches for bio-swimming elastic rods coupling with two-dimensional incompressible Newton fluid. First, we discretize elastic rods to extensive springs and rotational springs and stablish the kinetic equations based on energy reflecting the influence of internal force on swimming rods, and solve the governing equation by leap-frog algorithm. Second, semi-Lagrangian method is used to establish a fluid solver.Finally, the simplified coupling algorithm based on immersed-boundary method is raised, calling immersed-boundary method for momentum equations. These equations update the velocity of grid near the coupling interface directly to replace the function source force play in immersed-boundary method. Combining the fluid solver, rods solver and thecoupling algorithm, a integral program solving the problem of the dynamics for immersed rods numerically. Simulating the rods swimming with a referenced pose of sine curvature. Comparing the simulating result to existed experiments of filament swimming and to the swimming trajectory of soft-body fish, we find the numerical result conform to experiment result, leading to the conclusion that this algorithm and dynamic model can simulate the trajectory of discrete elastic rods swimming underwater smoothly under the influence of both internal elastic force and coupling soft body-fluid interaction.Using this program, we test several critical parameters relating to swimming performance of rods, including iteration numbers, frequency and initial phases, finding that changing the initial phase of rods will alter onward direction of elastic rods. These results prove the feasibility and possibility of this algorithm and program being used for guiding development of real soft swimmers.
The research for flexible structures coupling with fluid simulates and promotes the development of soft robotics.A fast and accurate numerical method is significant for a real-time simulation of robots. This research provides theoretical and critical information for experiments to reduce the possibility of failure, by anticipating the path of the underwater soft robots and the possible requiring parameters for materials. This paper researches for bio-swimming elastic rods coupling with two-dimensional incompressible Newton fluid. First, we discretize elastic rods to extensive springs and rotational springs and stablish the kinetic equations based on energy reflecting the influence of internal force on swimming rods, and solve the governing equation by leap-frog algorithm. Second, semi-Lagrangian method is used to establish a fluid solver.Finally, the simplified coupling algorithm based on immersed-boundary method is raised, calling immersed-boundary method for momentum equations. These equations update the velocity of grid near the coupling interface directly to replace the function source force play in immersed-boundary method. Combining the fluid solver, rods solver and thecoupling algorithm, a integral program solving the problem of the dynamics for immersed rods numerically. Simulating the rods swimming with a referenced pose of sine curvature. Comparing the simulating result to existed experiments of filament swimming and to the swimming trajectory of soft-body fish, we find the numerical result conform to experiment result, leading to the conclusion that this algorithm and dynamic model can simulate the trajectory of discrete elastic rods swimming underwater smoothly under the influence of both internal elastic force and coupling soft body-fluid interaction.Using this program, we test several critical parameters relating to swimming performance of rods, including iteration numbers, frequency and initial phases, finding that changing the initial phase of rods will alter onward direction of elastic rods. These results prove the feasibility and possibility of this algorithm and program being used for guiding development of real soft swimmers.
引文
1 Lighthill MJ.Note on the swimming of slender fish.Journal of Fluid Mechanics,2006,9(2):305-317
2 Lindsey CC.Form,function,and locomotory habits in fish.Fish Physiology,1978,7:1-100
3 Lighthill MJ.large-amplitude elongated-body theory of fish locomotion.Proceedings of the Royal Society of London,1971,179(1055):125-138
4 Weihs D,Katz J.Hydrodynamic propulsion by large amplitude oscillation of an airfoil with chordwise flexibility.Journal of Fluid Mechanics,1978,88(3):485-497
5 Wu TY.Hydromechanics of swimming propulsion.Part 2.Some optimum shape problems.Journal of Fluid Mechanics,1971,46(3):521-544
6程健宇,庄礼贤,童秉纲.鱼类鳗鲡模式推进的游动性能分析.水动力学研究与进展,1988(3):91-101(Cheng Jianyu,Zhuang Lixian,Tong Binggang.Analysis of swimming performance of fish eel model propulsion.Journal of Hydrodynamics,1988(3):91-101(in Chinese))
7 Mittal R,Dong H,Bozkurttas M,et al.A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries.Journal of Computational Physics,2008,227(10):4825-4852
8 Yamada T.An arbitrary Lagarangian-Eulerian finite element method for geometric nonlinear problems for elastic bodies.Sˉurikaisekikenky¨usho Kˉokyˉuroku,2002(1288):122-135
9 Videler JJ.Fish Swimming.Springer,1993
10 Jones BA,Walker ID.Kinematics for multi-section continuum robots.IEEE Transactions on Robotics,2006,22(1):43-55
11 Lighthill MJ.Note on the swimming of slender fish.Journal of Fluid Mechanics,1960,9(2):305-317
12 Van WS,Strother JA,Flammang BE,et al.Extremely fast prey capture in pipefish is powered by elastic recoil.Journal of the Royal Society Interface,2011,5(20):285
13 Bandringa H.Immersed boundary methods.Ann.Rev.Fluid Mech,2010,37(37):239-261
14 Jia LB,Xiao Q,Wu HJ,et al.Response of a flexible filament in a flowing soap film subject to a forced vibration.Physics of Fluids,2015,27(1):017101
15 Jia LB,Li F,Yin XZ,et al.Coupling modes between two flapping filaments.Journal of Fluid Mechanics,2007,581:199-220
16 Jia LB,Yin XZ.Passive oscillations of two tandem flexible filaments in a flowing soap film.Physical Review Letters,2008,100:228104
17卢振利,李斌.蛇形机器人蜿蜒游动性能动力学仿真分析.机器人,2015,37(6):748-753(Lu Zhenli,Li Bin.Dynamics simulation analysis on serpentine swimming performance of a snake-like robot.Robot,2015,37(6):748-753(in Chinese))
18 Jian LI,Wang ZL,Guo YL,et al.Application of shape memory alloy used for bionic underwater swimming robot.Small&Special Electrical Machines,2014,42(12):82-86
19 Yamakita M,Kamamichi N,Kozuki T,et al.A snake-like swimming robot using IPMC actuator and verification of doping effect.International Conference on Intelligent Robots and Systems,IEEE,2005:2035-2040
20尹婷婷,邓子辰,胡伟鹏等.空间刚性杆-弹簧组合结构轨道、姿态耦合动力学分析.力学学报,2018,50(1):87-98(Yin Tingting,Deng Zichen,Hu Weipeng.Dynamic modeling and simulation of orbit and attitude coupling problems for structure combined of spatial rigid rods and spring.Chinese Journal of Theoretical and Applied Mechanics,2018,50(1):87-98(in Chinese))
21 Bergou M,Wardetzky M,Robinson S,et al.Discrete elastic rods.Acm Transactions on Graphics,2008,27(3):1-12
22何涛.基于ALE有限元法的流固耦合强耦合数值模拟.力学学报,2018,39(2):1549-1561(He Tao.A partitioned strong coupling algorithm for fluid-structure interaction using arbitrary LagrangianEulerian finete element formulation.Chinese Journal of Theoretical and Applied Mechanics,2018,39(2):1549-1561(in Chinese))
23段松长,赵西增,叶洲腾等.错列角度对双圆柱涡激振动影响的数值模拟研究.力学学报,2018,50(2):244-253(Duan Songchang,Zhao Xizeng,Ye Zhouteng,et al.Numerical study of staggered angle of the vortex-induced vibration of two cylinders.Chinese Journal of Theoretical and Applied Mechanics,2018,50(2):244-253(in Chinese))
24杜特专,王一伟,黄晨光等.航行体水下发射流固耦合效应分析.力学学报,2017,49(4):782-792(Du Tezhuan,Wang Yiwei,Huang Chenguang,et al.Study on coupling effects of underwater lauched vehicle.Chinese Journal of Theoretical and Applied Mechanics,2017,49(4):782-792(in Chinese))
25 Wang J,Xu JZ,Xu Lei.Discussion mechanism based shuffled frog leaping algorithm.Computer&Digital Engineering,2017,45(8):1465-1468
26 Antoci C,Gallati M,Sibilla S.Numerical simulation of fluidstructure interaction by SPH.Computers&Structures,2007,85(11-14):879-890
27 Sarrate J,Huerta A,Donea J.Arbitrary Lagrangian-Eulerian formulation for fluid-rigid body interaction.Computer Methods in Applied Mechanics&Engineering,2001,190(24):3171-3188
28 Stam J.Stable fluids//Proc.ACM SIGGRAPH 1999 Conference Series,1999:121-128
29胡文蓉,童秉纲,马晖扬.鱼类机动运动机理初探//2003空气动力学前沿研究学术研讨会,2013(Hu Wenrong,Tong Bingang,Ma Huiyang.A preliminary study on the mechanism of fish//2013 Symposium on Aerodynamics Frontier Research,2013(in Chinese))
30 Stam J,Jensen HW.Visual simulation of smoke//Conference on Computer Graphics and Interactive Techniques,ACM,2001:15-22
31郝栋伟.基于浸入边界法的“C”型鱼自主游动的数值模拟.[硕士论文].昆明:昆明理工大学,2013(Hao Dongwei.Numerical simulation of self-propelled of“C”type swimming fish using the immersed boundary method.[Master Thesis].Kunming:Kunming University of Science and Technology,2013(in Chinese))
32胡潇毅.一类流固耦合问题的数值算法及风场中摇曳树木的物理仿真.[博士论文].杭州:浙江大学,2008(Hu Xiaoyi.Numerical methods for a class of fluid.structure interaction problems and simulation of swaying tree in wind field.[PhD Thesis].Hangzhou:Zhejiang University,2008(in Chinese))
33王文全,郝栋伟,闫妍等.基于浸入边界法的鱼体自主游动的数值模拟.计算力学学报,2014(5):646-651(Wang Wenquan,Hao Dongwei,Yan Yan,et al.Numerical simulation of self-propelled swimming fish using the immersed boundary method.Chinese Journal of Computational Mechanics,2014(5):646-65(in Chinese))
34郁树梅,王明辉,马书根等.水陆两栖蛇形机器人的研制及其陆地和水下步态.机械工程学报,2012,48(9):18-25(Yu Shumei,Wang Minghui,Ma Shugen,et al.Development of an amphibious snake-like robot and its gaits on ground and in water.Journal of Mechanical Engineering,2012,48(9):18-25(in Chinese))
2 Lindsey CC.Form,function,and locomotory habits in fish.Fish Physiology,1978,7:1-100
3 Lighthill MJ.large-amplitude elongated-body theory of fish locomotion.Proceedings of the Royal Society of London,1971,179(1055):125-138
4 Weihs D,Katz J.Hydrodynamic propulsion by large amplitude oscillation of an airfoil with chordwise flexibility.Journal of Fluid Mechanics,1978,88(3):485-497
5 Wu TY.Hydromechanics of swimming propulsion.Part 2.Some optimum shape problems.Journal of Fluid Mechanics,1971,46(3):521-544
6程健宇,庄礼贤,童秉纲.鱼类鳗鲡模式推进的游动性能分析.水动力学研究与进展,1988(3):91-101(Cheng Jianyu,Zhuang Lixian,Tong Binggang.Analysis of swimming performance of fish eel model propulsion.Journal of Hydrodynamics,1988(3):91-101(in Chinese))
7 Mittal R,Dong H,Bozkurttas M,et al.A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries.Journal of Computational Physics,2008,227(10):4825-4852
8 Yamada T.An arbitrary Lagarangian-Eulerian finite element method for geometric nonlinear problems for elastic bodies.Sˉurikaisekikenky¨usho Kˉokyˉuroku,2002(1288):122-135
9 Videler JJ.Fish Swimming.Springer,1993
10 Jones BA,Walker ID.Kinematics for multi-section continuum robots.IEEE Transactions on Robotics,2006,22(1):43-55
11 Lighthill MJ.Note on the swimming of slender fish.Journal of Fluid Mechanics,1960,9(2):305-317
12 Van WS,Strother JA,Flammang BE,et al.Extremely fast prey capture in pipefish is powered by elastic recoil.Journal of the Royal Society Interface,2011,5(20):285
13 Bandringa H.Immersed boundary methods.Ann.Rev.Fluid Mech,2010,37(37):239-261
14 Jia LB,Xiao Q,Wu HJ,et al.Response of a flexible filament in a flowing soap film subject to a forced vibration.Physics of Fluids,2015,27(1):017101
15 Jia LB,Li F,Yin XZ,et al.Coupling modes between two flapping filaments.Journal of Fluid Mechanics,2007,581:199-220
16 Jia LB,Yin XZ.Passive oscillations of two tandem flexible filaments in a flowing soap film.Physical Review Letters,2008,100:228104
17卢振利,李斌.蛇形机器人蜿蜒游动性能动力学仿真分析.机器人,2015,37(6):748-753(Lu Zhenli,Li Bin.Dynamics simulation analysis on serpentine swimming performance of a snake-like robot.Robot,2015,37(6):748-753(in Chinese))
18 Jian LI,Wang ZL,Guo YL,et al.Application of shape memory alloy used for bionic underwater swimming robot.Small&Special Electrical Machines,2014,42(12):82-86
19 Yamakita M,Kamamichi N,Kozuki T,et al.A snake-like swimming robot using IPMC actuator and verification of doping effect.International Conference on Intelligent Robots and Systems,IEEE,2005:2035-2040
20尹婷婷,邓子辰,胡伟鹏等.空间刚性杆-弹簧组合结构轨道、姿态耦合动力学分析.力学学报,2018,50(1):87-98(Yin Tingting,Deng Zichen,Hu Weipeng.Dynamic modeling and simulation of orbit and attitude coupling problems for structure combined of spatial rigid rods and spring.Chinese Journal of Theoretical and Applied Mechanics,2018,50(1):87-98(in Chinese))
21 Bergou M,Wardetzky M,Robinson S,et al.Discrete elastic rods.Acm Transactions on Graphics,2008,27(3):1-12
22何涛.基于ALE有限元法的流固耦合强耦合数值模拟.力学学报,2018,39(2):1549-1561(He Tao.A partitioned strong coupling algorithm for fluid-structure interaction using arbitrary LagrangianEulerian finete element formulation.Chinese Journal of Theoretical and Applied Mechanics,2018,39(2):1549-1561(in Chinese))
23段松长,赵西增,叶洲腾等.错列角度对双圆柱涡激振动影响的数值模拟研究.力学学报,2018,50(2):244-253(Duan Songchang,Zhao Xizeng,Ye Zhouteng,et al.Numerical study of staggered angle of the vortex-induced vibration of two cylinders.Chinese Journal of Theoretical and Applied Mechanics,2018,50(2):244-253(in Chinese))
24杜特专,王一伟,黄晨光等.航行体水下发射流固耦合效应分析.力学学报,2017,49(4):782-792(Du Tezhuan,Wang Yiwei,Huang Chenguang,et al.Study on coupling effects of underwater lauched vehicle.Chinese Journal of Theoretical and Applied Mechanics,2017,49(4):782-792(in Chinese))
25 Wang J,Xu JZ,Xu Lei.Discussion mechanism based shuffled frog leaping algorithm.Computer&Digital Engineering,2017,45(8):1465-1468
26 Antoci C,Gallati M,Sibilla S.Numerical simulation of fluidstructure interaction by SPH.Computers&Structures,2007,85(11-14):879-890
27 Sarrate J,Huerta A,Donea J.Arbitrary Lagrangian-Eulerian formulation for fluid-rigid body interaction.Computer Methods in Applied Mechanics&Engineering,2001,190(24):3171-3188
28 Stam J.Stable fluids//Proc.ACM SIGGRAPH 1999 Conference Series,1999:121-128
29胡文蓉,童秉纲,马晖扬.鱼类机动运动机理初探//2003空气动力学前沿研究学术研讨会,2013(Hu Wenrong,Tong Bingang,Ma Huiyang.A preliminary study on the mechanism of fish//2013 Symposium on Aerodynamics Frontier Research,2013(in Chinese))
30 Stam J,Jensen HW.Visual simulation of smoke//Conference on Computer Graphics and Interactive Techniques,ACM,2001:15-22
31郝栋伟.基于浸入边界法的“C”型鱼自主游动的数值模拟.[硕士论文].昆明:昆明理工大学,2013(Hao Dongwei.Numerical simulation of self-propelled of“C”type swimming fish using the immersed boundary method.[Master Thesis].Kunming:Kunming University of Science and Technology,2013(in Chinese))
32胡潇毅.一类流固耦合问题的数值算法及风场中摇曳树木的物理仿真.[博士论文].杭州:浙江大学,2008(Hu Xiaoyi.Numerical methods for a class of fluid.structure interaction problems and simulation of swaying tree in wind field.[PhD Thesis].Hangzhou:Zhejiang University,2008(in Chinese))
33王文全,郝栋伟,闫妍等.基于浸入边界法的鱼体自主游动的数值模拟.计算力学学报,2014(5):646-651(Wang Wenquan,Hao Dongwei,Yan Yan,et al.Numerical simulation of self-propelled swimming fish using the immersed boundary method.Chinese Journal of Computational Mechanics,2014(5):646-65(in Chinese))
34郁树梅,王明辉,马书根等.水陆两栖蛇形机器人的研制及其陆地和水下步态.机械工程学报,2012,48(9):18-25(Yu Shumei,Wang Minghui,Ma Shugen,et al.Development of an amphibious snake-like robot and its gaits on ground and in water.Journal of Mechanical Engineering,2012,48(9):18-25(in Chinese))