Journal of Scientific Computing

, Volume 74, Issue 2, pp 1163–1187 | Cite as

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

  • Benjamin Ivorra


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.


Pathfinding Computational fluid dynamics Comsol Multiphysics Spatial constraints Artificial intelligence 


  1. 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. 2.
    Arvo, J., Kirk, D.: Fast ray tracing by ray classification. SIGGRAPH Comput. Graph. 21(4), 55–64 (1987)CrossRefGoogle Scholar
  3. 3.
    Batchelor G (2000) An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge. doi: 10.1017/CBO9780511800955 (Cambridge Books Online)
  4. 4.
    Bathe, K.: Computational Fluid and Solid Mechanics. Elsevier Science (2001).
  5. 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 MathSciNetCrossRefMATHGoogle Scholar
  6. 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. 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 MathSciNetCrossRefGoogle Scholar
  8. 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)MATHGoogle Scholar
  9. 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. 10.
    Ciarlet, P., Lions, J.: Handbook of Numerical Analysis: Numerical methods for fluids (pt. 3). Handbook of Numerical Analysis. North-Holland (1990).
  11. 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)Google Scholar
  12. 12.
    Connor, D.: Integrating Planning and Control for Constrained Dynamical Systems. PhD., University of Pennsylvania (2007)Google Scholar
  13. 13.
    Daniel, K., Nash, A., Koenig, S., Felner, A.: Theta*: any-angle path planning on grids. J. Artif. Intell. Res. 39, 533–579 (2010)MathSciNetMATHGoogle Scholar
  14. 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 CrossRefGoogle Scholar
  15. 15.
    Dickmann, D.: On the Near Field Mean Flow Structure of Transverse Jets Issuing Into a Supersonic Freestream. University of Texas at Arlington (2007).
  16. 16.
    Dijkstra, E.: A Short Introduction to the Art of Programming. Techn. Hogeschool, Eindhoven (1971)Google Scholar
  17. 17.
    Eberly, D.: 3D Game Engine Design: A Practical Approach to Real-Time Computer Graphics. CRC Press, Boca Raton (2006)Google Scholar
  18. 18.
    Fay, J.: Introduction to Fluid Mechanics. MIT Press (1994).
  19. 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 CrossRefGoogle Scholar
  20. 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).
  21. 21.
    Glowinski, R., Neittaanmäki, P.: Partial Differential Equations: Modelling and Numerical Simulation. Computational Methods in Applied Sciences. Springer, Netherlands (2008).
  22. 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 CrossRefGoogle Scholar
  23. 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)CrossRefMATHGoogle Scholar
  24. 24.
    Hunt, B., Lipsman, R., Rosenberg, J.: A Guide to MATLAB: For Beginners and Experienced Users. Cambridge University Press (2001).
  25. 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)CrossRefGoogle Scholar
  26. 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 MathSciNetCrossRefMATHGoogle Scholar
  27. 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 CrossRefMATHGoogle Scholar
  28. 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 MathSciNetCrossRefMATHGoogle Scholar
  29. 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 CrossRefMATHGoogle Scholar
  30. 30.
    Johnson, R.: Handbook of Fluid Dynamics. Handbook Series for Mechanical Engineering. Taylor & Francis, Oxfordshire (1998)Google Scholar
  31. 31.
    Katevas, N.: Mobile Robotics in Healthcare. Assistive technology research series. IOS Press (2001).
  32. 32.
    Khatib, O.: Real-time obstacle avoidance for manipulators and mobile robots. Int. J. Robot. Res. 5(1), 90–98 (1986)CrossRefGoogle Scholar
  33. 33.
    Koenig, S., Likhachev, M.: D*lite. In: Eighteenth National Conference on Artificial Intelligence, pp. 476–483. American Association for Artificial Intelligence (2002)Google Scholar
  34. 34.
    Kwon, H.J.: Use of comsol simulation for undergraduate fluid dynamics course. In: 2012 ASEE Annual Conference & Exposition, San Antonio, Texas. (2012)
  35. 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. 36.
    Lolla, S.: Path Planning in Time Dependent Flows using Level Set Methods. PhD., University of Massachusetts Institute Of Technology (2012)Google Scholar
  37. 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)CrossRefMATHGoogle Scholar
  38. 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)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Pepper, D., Wang, X.: Benchmarking COMSOL Multiphysics 3.5a CFD problems. In: Proceeding of the Cosmol Conference 2009, Boston. Comsol Inc. (2009)Google Scholar
  40. 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)Google Scholar
  41. 41.
    Premakumar, P.: A* (A star) search for path planning tutorial. Matlab Central. (2010)
  42. 42.
    Ramos Del Olmo, A.: Introducción al análisis matemático del método de elementos finitos. Editorial Complutense, Madrid (2013). ISBN:978-8499381282Google Scholar
  43. 43.
    Rimon, E., Koditschek, D.: Exact robot navigation using artificial potential functions. IEEE Trans. Robot. Autom. 8(5), 501–518 (1992)CrossRefGoogle Scholar
  44. 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 MathSciNetCrossRefMATHGoogle Scholar
  45. 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)Google Scholar
  46. 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 CrossRefGoogle Scholar
  47. 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:978-953-51-0115-4Google Scholar
  48. 48.
    Tabatabaian, M.: Comsol 5 for Engineers. Multiphysics Modeling Series. Mercury Learning & Information (2015).
  49. 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 CrossRefMATHGoogle Scholar
  50. 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 CrossRefMATHGoogle Scholar
  51. 51.
    Wang, J., Deng, W.: Optimizing capacity of signalized road network with reversible lanes. Transport (2015). doi: 10.3846/16484142.2014.994227
  52. 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. 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 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.MOMAT Research Group and Instituto de Matemática Interdisciplinar, Departamento de Matemática AplicadaUniversidad Complutense de MadridMadridSpain

Personalised recommendations