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Advances in Computational Mathematics

, Volume 45, Issue 2, pp 1067–1103 | Cite as

Numerical modeling of wildland surface fire propagation by evolving surface curves

  • Martin AmbrozEmail author
  • Martin Balažovjech
  • Matej Medl’a
  • Karol Mikula
Article
  • 31 Downloads

Abstract

We introduce a new approach to wildland fire spread modeling. We evolve a 3-D surface curve, which represents the fire perimeter on the topography, as a projection to a horizontal plane. Our mathematical model is based on the empirical laws of the fire spread influenced by the fuel, wind, terrain slope, and shape of the fire perimeter with respect to the topography (geodesic and normal curvatures). To obtain the numerical solution, we discretize the arising intrinsic partial differential equation by a semi-implicit scheme with respect to the curvature term. For the advection term discretization, we use the so-called inflow-implicit/outflow-explicit approach and an implicit upwind technique which guarantee the solvability of the corresponding linear systems by an efficient tridiagonal solver without any time step restriction and also the robustness with respect to singularities. A fast treatment of topological changes (splitting and merging of the curves) is described and shown on examples as well. We show the experimental order of convergence of the numerical scheme, we demonstrate the influence of the fire spread model parameters on a testing and real topography, and we reconstruct a simulated grassland fire as well.

Keywords

Curve evolution Surface curve Topological changes Wildland fire modeling Geodesic curvature Normal curvature 

Mathematics Subject Classification (2010)

35R01 65M08 53Z05 68U20 

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Notes

Funding information

This work was supported by the grants VEGA 1/0608/15 and APVV-15-0522.

References

  1. 1.
    Alexandrian, D.: Vesta - large scale fire simulator. http://www.fireparadox.org/large_scale_fire_simulator.php. Accessed on 25 Oct 2016
  2. 2.
    Balažovjech, M., Mikula, K.: A higher order scheme for a tangentially stabilized plane curve shortening flow with a driving force. SIAM J. Sci. Comput. 33 (5), 2277–2294 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Balažovjech, M., Mikula, K., Petrášová, M., Urbán, J.: Lagrangean method with topological changes for numerical modelling of forest fire propagation. In: Proceedings of ALGORITMY, pp. 42–52 (2012)Google Scholar
  4. 4.
    Barrett, J.W., Garcke, H., Nürnberg, R.: Numerical approximation of gradient flows for closed curves in \(\mathbb {{R}}^d\). IMA J. Numer. Anal. 30(1), 4–60 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Benninghoff, H., Garcke, H.: Efficient image segmentation and restoration using parametric curve evolution with junctions and topology changes. SIAM J. Imag. Sci. 7(3), 1451–1483 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Benninghoff, H., Garcke, H.: Image segmentation using parametric contours with free endpoints. IEEE Trans. Image Process. 25(4), 1639–1648 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Benninghoff, H., Garcke, H.: Segmentation and restoration of images on surfaces by parametric active contours with topology changes. J. Math. Imag. Vis. 55(1), 105–124 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Benninghoff, H., Garcke, H.: Segmentation of three-dimensional images with parametric active surfaces and topology changes. J. Sci. Comput., 1–35 (2017)Google Scholar
  9. 9.
    Bose, C., Bryce, R., Dueck, G.: Untangling the Prometheus nightmare. In: Proc. 18th IMACS World Congress MODSIM09 and International Congress on Modelling and Simulation, pp. 13–17. Cairns (2009)Google Scholar
  10. 10.
    Butler, B., Anderson, W., Catchpole, E.: Influence of slope on fire spread rate. In: The Fire Environment–Innovations, Management, and Policy; Conference Proceedings, pp. 75–82 (2007)Google Scholar
  11. 11.
    Dziuk, G.: Discrete anisotropic curve shortening flow. SIAM J. Numer. Anal. 36(6), 1808–1830 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Finney, M.A., et al.: FARSITE Fire Area Simulator–model development and evaluation, vol. 3. US Department of Agriculture, Forest Service, Rocky Mountain Research Station Ogden, UT (1998)Google Scholar
  13. 13.
    Hou, T.Y., Klapper, I., Si, H.: Removing the stiffness of curvature in computing 3-d filaments. J. Comput. Phys. 143(2), 628–664 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hou, T.Y., Lowengrub, J.S., Shelley, M.J.: Removing the stiffness from interfacial flows with surface tension. J. Comput. Phys. 114(2), 312–338 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Subirana, J.S., Zornoza, J.M.J., Hernández-Pajares, M.: Ellipsoidal and cartesian coordinates conversion. http://www.navipedia.net/index.php/Ellipsoidal_and_Cartesian_Coordinates_Conversion (s). Accessed 10 Jun 2016
  16. 16.
    Krasnow, K., Schoennagel, T., Veblen, T.T.: Forest fuel mapping and evaluation of LANDFIRE fuel maps in Boulder County, Colorado, USA. For. Ecol. Manage. 257(7), 1603–1612 (2009)CrossRefGoogle Scholar
  17. 17.
    Krivá, Z., Mikula, K., Peyriéras, N., Rizzi, B., Sarti, A., Stašová, O.: 3d early embryogenesis image filtering by nonlinear partial differential equations. Med. Image Anal. 14(4), 510–526 (2010)CrossRefGoogle Scholar
  18. 18.
    Lopes, A., Cruz, M., Viegas, D.: Firestation: An integrated software system for the numerical simulation of fire spread on complex topography. Environ. Modell. Software 17(3), 269–285 (2002)CrossRefGoogle Scholar
  19. 19.
    McDermott, R., McGrattan, K., Hostikka, S.: Fire dynamics simulator (version 5) technical reference guide NIST. Spec. Publ. 1018(5) (2008)Google Scholar
  20. 20.
    Mell, W.E., et al.: The wildland–urban interface fire problem–current approaches and research needs. Int. J. Wildland Fire 19(2), 238–251 (2010)CrossRefGoogle Scholar
  21. 21.
    Mikula, K., Ohlberger, M.: A new level set method for motion in normal direction based on a semi-implicit forward-backward diffusion approach. SIAM J. Sci. Comput. 32(3), 1527–1544 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Mikula, K., Ohlberger, M., Urbán, J.: Inflow-implicit/outflow-explicit finite volume methods for solving advection equations. Appl. Numer. Math. 85, 16–37 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Mikula, K., Ševčovič, D.: Evolution of plane curves driven by a nonlinear function of curvature and anisotropy. SIAM J. Appl. Math. 61(5), 1473–1501 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Mikula, K., Ševčovič, D.: Computational and qualitative aspects of evolution of curves driven by curvature and external force. Comput. Vis. Sci. 6(4), 211–225 (2004)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Mikula, K., Ševčovič, D.: A direct method for solving an anisotropic mean curvature flow of plane curves with an external force. Math. Methods Appl. Sci. 27(13), 1545–1565 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Mikula, K., Ševčovič, D.: Evolution of curves on a surface driven by the geodesic curvature and external force. Appl. Anal. 85(4), 345–362 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Mikula, K., Urbán, J.: New fast and stable lagrangean method for image segmentation. In: 2012 5th International Congress on Image and Signal Processing (CISP), pp. 688–696. IEEE (2012)Google Scholar
  28. 28.
    Mikula, K., Urbán, J.: A new tangentially stabilized 3D curve evolution algorithm and its application in virtual colonoscopy. Adv. Comput. Math. 40(4), 819–837 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Monoši, M., Majlingová, A., Kapusniak, J.: Lesné požiare žilinská univerzita v žiline (2015)Google Scholar
  30. 30.
    Nakamura, G., Potthast, R.: Inverse Modeling, pp. 2053–2563. IOP Publishing (2015).  https://doi.org/10.1088/978-0-7503-1218-9
  31. 31.
    Osher, S., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces. Applied Mathematical Sciences. Springer, New York (2002)zbMATHGoogle Scholar
  32. 32.
    Pauš, P., Beneš, M.: Algorithm for topological changes of parametrically described curves. In: Proceedings of ALGORITMY, pp. 176–184 (2009)Google Scholar
  33. 33.
    Prichard, S.J., et al.: Fuel characteristic classification system version 3.0: Technical documentation. Tech. rep., U.S. Department of Agriculture, Forest Service, Pacific Northwest Research Station (2013)Google Scholar
  34. 34.
    Scott, J.H., Burgan, R.E.: Standard fire behavior fuel models: a comprehensive set for use with rothermel’s surface fire spread model. The Bark Beetles, Fuels, and Fire Bibliography, p. 66 (2005)Google Scholar
  35. 35.
    Sethian, J.A.: Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science. Cambridge University Press (1999)Google Scholar
  36. 36.
    Sullivan, A.: A review of wildland fire spread modelling, 1990-present 3: Mathematical analogues and simulation models. arXiv:0706.4130(2007)
  37. 37.
    Vakalis, D., et al.: A GIS based operational system for wildland fire crisis management i. Mathematical modelling and simulation. Appl. Math. Model. 28(4), 389–410 (2004)CrossRefzbMATHGoogle Scholar
  38. 38.
    Viegas, D., et al.: Slope and wind effects on fire spread. In: IVth International Forest Fire Conference. Coimbra (Portugal). FFR & Wildland Fire Safety. Millpress, Rotterdam (2002)Google Scholar
  39. 39.
    Zhang, J.W., Han, G.Q., Wo, Y.: Image registration based on generalized and mean hausdorff distances. In: Proceedings of 2005 International Conference on Machine Learning and Cybernetics, 2005, vol. 8, pp. 5117–5121. IEEE (2005)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Civil EngineeringSlovak University of TechnologyBratislavaSlovak Republic
  2. 2.Algoritmy:SK s.r.o.BratislavaSlovak Republic

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