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Application of the Laminar Navier–Stokes Equations for Solving 2D and 3D Pathfinding Problems with Static and Dynamic Spatial Constraints: Implementation and Validation in Comsol Multiphysics

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Abstract

Pathfinding problems consist in determining the optimal shortest path, or at least one path, between two points in the space. In this paper, we propose a particular approach, based on methods used in computational fluid dynamics, that intends to solve such problems. In particular, we reformulate pathfinding problems as the motion of a viscous fluid via the use of the laminar Navier–Stokes equations completed with suitable boundary conditions corresponding to some characteristics of the considered problem: position of the initial and final points, a-priori information of the terrain, One-way routes and dynamic spatial configuration. Then, we propose and validate a numerical implementation of this methodology by using Comsol Multiphysics (i.e., a finite element methods software) and by considering various experiments. We compare the obtained results with those returned by a classical pathfinding algorithm. Finally, we perform a sensitivity analysis of the proposed algorithms with respect to some key parameters.

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References

  1. Amutha, B., Ponnavaikko, M.: Location update accuracy in human tracking system using zigbee modules. Int. J. Comput. Sci. Inf. Secur. 6(2), 322–331 (2009)

    Google Scholar 

  2. Arvo, J., Kirk, D.: Fast ray tracing by ray classification. SIGGRAPH Comput. Graph. 21(4), 55–64 (1987)

    Article  Google Scholar 

  3. Batchelor G (2000) An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge. doi:10.1017/CBO9780511800955 (Cambridge Books Online)

  4. Bathe, K.: Computational Fluid and Solid Mechanics. Elsevier Science (2001). https://books.google.es/books?id=Id06Z4YMJLMC

  5. Bretti, G., Natalini, R.: Piccoli B (2007) A fluid-dynamic traffic model on road networks. Arch. Comput. Methods Eng. 14(2), 139–172 (2007). doi:10.1007/s11831-007-9004-8

    Article  MathSciNet  MATH  Google Scholar 

  6. Burns, E.A., Hatem, M., Leighton, M.J., Ruml, W.: Implementing fast heuristic search code. In: Borrajo, D., Felner, A., Korf, R.E., Likhachev, M., Lpez, C.L., Ruml, W., Sturtevant, N.R. (eds.) SOCS. AAAI Press, Palo Alto (2012)

    Google Scholar 

  7. Calvo, C., Villacorta-Atienza, J., Mironov, V., Gallego, V., Makarov, V.: Waves in isotropic totalistic cellular automata: application to real-time robot navigation. Adv. Complex Syst. 19(4), 1650012–1650018 (2016). doi:10.1142/S0219525916500120

    Article  MathSciNet  Google Scholar 

  8. Choset, H., Lynch, K., Hutchinson, S., Kantor, G., Lydia, W., Kavraki, E., Thrun, S.: Principles of Robot Motion: Theory, Algorithms, and Implementation. Intelligent Robotics and Autonomous Agents series. MIT Press, Cambridge (2005)

    MATH  Google Scholar 

  9. Chrpa, L., Novak, P.: Dynamic Trajectory Replanning for Unmanned Aircrafts Supporting Tactical Missions in Urban Environments. Holonic and Multi-Agent Systems for Manufacturing. Springer, Berlin (2011)

    Google Scholar 

  10. Ciarlet, P., Lions, J.: Handbook of Numerical Analysis: Numerical methods for fluids (pt. 3). Handbook of Numerical Analysis. North-Holland (1990). https://books.google.es/books?id=S0Hqp3vOVxkC

  11. Connolly, C., Burns, J., Weiss, R.: Path planning using laplace’s equation. In: 1990 IEEE International Conference on Robotics and Automation, 1990. Proceedings, vol. 3, pp. 2102–2106 (1990)

  12. Connor, D.: Integrating Planning and Control for Constrained Dynamical Systems. PhD., University of Pennsylvania (2007)

  13. Daniel, K., Nash, A., Koenig, S., Felner, A.: Theta*: any-angle path planning on grids. J. Artif. Intell. Res. 39, 533–579 (2010)

    MathSciNet  MATH  Google Scholar 

  14. Dean, W.: Lxxii. the stream-line motion of fluid in a curved pipe (second paper). Lond Edinb. Dublin Philos. Mag. J. Sci. 5(30), 673–695 (1928). doi:10.1080/14786440408564513

    Article  Google Scholar 

  15. Dickmann, D.: On the Near Field Mean Flow Structure of Transverse Jets Issuing Into a Supersonic Freestream. University of Texas at Arlington (2007). https://books.google.es/books?id=4ee-g96_F5gC

  16. Dijkstra, E.: A Short Introduction to the Art of Programming. Techn. Hogeschool, Eindhoven (1971)

    Google Scholar 

  17. Eberly, D.: 3D Game Engine Design: A Practical Approach to Real-Time Computer Graphics. CRC Press, Boca Raton (2006)

    Google Scholar 

  18. Fay, J.: Introduction to Fluid Mechanics. MIT Press (1994). https://books.google.es/books?id=XGVpue4954wC

  19. Fuerstman, M., Deschatelets, P., Kane, R., Schwartz, A., Kenis, P., Deutch, J., Whitesides, G.: Solving mazes using microfluidic networks. Langmuir 19(11), 4714–4722 (2003). doi:10.1021/la030054x

    Article  Google Scholar 

  20. Girod, B., Greiner, G., Niemann, H.: Principles of 3D Image Analysis and Synthesis. The Springer International Series in Engineering and Computer Science. Springer, US (2013). https://books.google.es/books?id=jVHuBwAAQBAJ

  21. Glowinski, R., Neittaanmäki, P.: Partial Differential Equations: Modelling and Numerical Simulation. Computational Methods in Applied Sciences. Springer, Netherlands (2008). https://books.google.es/books?id=xKhfyc0Nf54C

  22. Hertzog, D., Ivorra, B., Mohammadi, B., Bakajin, O., Santiago, J.: Optimization of a microfluidic mixer for studying protein folding kinetics. Anal. Chem. 78(13), 4299–4306 (2006). doi:10.1021/ac051903j

    Article  Google Scholar 

  23. Heywood, J.G., Rannacher, R., Turek, S.: Artificial boundaries and flux and pressure conditions for the incompressible navierstokes equations. Int. J. Numer. Methods Fluids 22(5), 325–352 (1996)

    Article  MATH  Google Scholar 

  24. Hunt, B., Lipsman, R., Rosenberg, J.: A Guide to MATLAB: For Beginners and Experienced Users. Cambridge University Press (2001). https://books.google.es/books?id=XhQBx9LJKIAC

  25. Hysing, J., Turek, S.: Evaluation of commercial and academic cfd codes for a two-phase flow benchmark test case. Int. J. Comput. Sci. Eng. 10(4), 387–394 (2015)

    Article  Google Scholar 

  26. Infante, J.A., Ivorra, B., Ramos, A., Rey, J.: On the modelling and simulation of high pressure processes and inactivation of enzymes in food engineering. Math. Models Methods Appl. Sci. 19(12), 2203–2229 (2009). doi:10.1142/S0218202509004091

    Article  MathSciNet  MATH  Google Scholar 

  27. Isebe, D., Azerad, P., Bouchette, F., Ivorra, B.: Mohammadi B (2008) Shape optimization of geotextile tubes for sandy beach protection. Int. J. Numer. Methods Eng. 74(8), 1262–1277 (2008). doi:10.1002/nme.2209

    Article  MATH  Google Scholar 

  28. Ivorra, B., Hertzog, D., Mohammadi, B., Santiago, J.: Semi-deterministic and genetic algorithms for global optimization of microfluidic protein-folding devices. Int. J. Numer. Methods Eng. 66(2), 19–333 (2006). doi:10.1002/nme.1562

    Article  MathSciNet  MATH  Google Scholar 

  29. Ivorra, B., Redondo, J., Santiago, J., Ortigosa, P., Ramos, A.: Two- and three-dimensional modeling and optimization applied to the design of a fast hydrodynamic focusing microfluidic mixer for protein folding. Phys. Fluids 25(3), 032001 (2013). doi:10.1063/1.4793612

    Article  MATH  Google Scholar 

  30. Johnson, R.: Handbook of Fluid Dynamics. Handbook Series for Mechanical Engineering. Taylor & Francis, Oxfordshire (1998)

    Google Scholar 

  31. Katevas, N.: Mobile Robotics in Healthcare. Assistive technology research series. IOS Press (2001). https://books.google.es/books?id=jT__IKy9wTgC

  32. Khatib, O.: Real-time obstacle avoidance for manipulators and mobile robots. Int. J. Robot. Res. 5(1), 90–98 (1986)

    Article  Google Scholar 

  33. Koenig, S., Likhachev, M.: D*lite. In: Eighteenth National Conference on Artificial Intelligence, pp. 476–483. American Association for Artificial Intelligence (2002)

  34. Kwon, H.J.: Use of comsol simulation for undergraduate fluid dynamics course. In: 2012 ASEE Annual Conference & Exposition, San Antonio, Texas. https://peer.asee.org/22167 (2012)

  35. Lee, V., Law, M., Wee, S.: Theory to practice on finite element method and computational fluid dynamics tools. Aust. J. Eng. Educ. 22(2), 123–133 (2015)

    Google Scholar 

  36. Lolla, S.: Path Planning in Time Dependent Flows using Level Set Methods. PhD., University of Massachusetts Institute Of Technology (2012)

  37. Louste, C., Liegeois, A.: Near optimal robust path planning for mobile robots: the viscous fluid method with friction. J. Intell. Robot. Syst. 27(1), 99–112 (2000)

    Article  MATH  Google Scholar 

  38. Nau, D., Kumar, V., Kanal, L.: General branch and bound, and its relation to A* and AO*. Artif. Intell. 23(1), 29–58 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  39. Pepper, D., Wang, X.: Benchmarking COMSOL Multiphysics 3.5a CFD problems. In: Proceeding of the Cosmol Conference 2009, Boston. Comsol Inc. (2009)

  40. Pimenta, L., Michael, N., Mesquita, R., Pereira, G., Kumar, V.: Control of swarms based on hydrodynamic models. In: IEEE International Conference on Robotics and Automation, 2008. ICRA 2008, pp. 1948–1953 (2008)

  41. Premakumar, P.: A* (A star) search for path planning tutorial. Matlab Central. http://www.mathworks.com/matlabcentral/mlc-downloads/downloads/submissions/26248/versions/3/download/zip (2010)

  42. Ramos Del Olmo, A.: Introducción al análisis matemático del método de elementos finitos. Editorial Complutense, Madrid (2013). ISBN:978-8499381282

  43. Rimon, E., Koditschek, D.: Exact robot navigation using artificial potential functions. IEEE Trans. Robot. Autom. 8(5), 501–518 (1992)

    Article  Google Scholar 

  44. Roussos, G., Dimarogonas, D.V., Kyriakopoulos, K.J.: 3d navigation and collision avoidance for nonholonomic aircraft-like vehicles. Int. J. Adapt. Control Signal Process. 24(10), 900–920 (2010). doi:10.1002/acs.1199

    Article  MathSciNet  MATH  Google Scholar 

  45. Sun, X., Yeoh, W., Uras, T., Koenig, S.: Incremental ara*: an incremental anytime search algorithm for moving-target search. In: International Conference on Automated Planning and Scheduling (2012)

  46. Suzuno, K., Ueyama, D., Branicki, M., Tth, R., Braun, A., Lagzi, I.: Maze solving using fatty acid chemistry. Langmuir 30(31), 9251–9255 (2014). doi:10.1021/la5018467

    Article  Google Scholar 

  47. Szab, C., Sobota, B.: Path-finding algorithm application for route-searching in different areas of computer graphics. In: Zhang, Y. (ed.) New Frontiers in Graph Theory. InTech (2012). ISBN:978-953-51-0115-4

  48. Tabatabaian, M.: Comsol 5 for Engineers. Multiphysics Modeling Series. Mercury Learning & Information (2015). https://books.google.es/books?id=twhSrgEACAAJ

  49. Twizell, E., Bright, N.: Numerical modelling of fan performance. Appl. Math. Model. 5(4), 246–250 (1981). doi:10.1016/S0307-904X(81)80074-1

    Article  MATH  Google Scholar 

  50. Villacorta-Atienza, J., Calvo, C., Makarov, V.: Prediction-for-compaction: navigation in social environments using generalized cognitive maps. Biol. Cybern. 109(3), 307–320 (2015). doi:10.1007/s00422-015-0644-8

    Article  MATH  Google Scholar 

  51. Wang, J., Deng, W.: Optimizing capacity of signalized road network with reversible lanes. Transport (2015). doi:10.3846/16484142.2014.994227

  52. Wu, X., Zhang, S.: The study and application of artificial intelligence pathfinding algorithm in game domain. In: 2011 International Conference on Computer Science and Service System (CSSS), pp. 3772–3774. IEEE (2011). doi:10.1109/CSSS.2011.5974547

  53. Zeng, W., Church, R.L.: Finding shortest paths on real road networks: the case for A*. Int. J. Geogr. Inf. Sci. 23(4), 531–543 (2009). doi:10.1080/13658810801949850

    Article  Google Scholar 

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  1. MOMAT Research Group and Instituto de Matemática Interdisciplinar, Departamento de Matemática Aplicada, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040, Madrid, Spain

    Benjamin Ivorra

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Correspondence to Benjamin Ivorra.

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This work was carried out thanks to the financial support of the Spanish “Ministry of Economy and Competitiveness” under Projects MTM2011-22658 and MTM2015-64865-P; the “Junta de Andalucía” and the European Regional Development Fund through the Project P12-TIC301; and the research group MOMAT (Ref. 910480) supported by “Banco de Santander” and “Universidad Complutense de Madrid”. The author would like to thank Angel M. Ramos del Olmo and Tatiana Diaz Jimenez for their valuable help during this work.

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Ivorra, B. Application of the Laminar Navier–Stokes Equations for Solving 2D and 3D Pathfinding Problems with Static and Dynamic Spatial Constraints: Implementation and Validation in Comsol Multiphysics. J Sci Comput 74, 1163–1187 (2018). https://doi.org/10.1007/s10915-017-0489-5

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