Skip to main content
SpringerLink
Log in

Numerical modeling of wildland surface fire propagation by evolving surface curves

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alexandrian, D.: Vesta - large scale fire simulator. http://www.fireparadox.org/large_scale_fire_simulator.php. Accessed on 25 Oct 2016

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  6. Benninghoff, H., Garcke, H.: Image segmentation using parametric contours with free endpoints. IEEE Trans. Image Process. 25(4), 1639–1648 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  8. Benninghoff, H., Garcke, H.: Segmentation of three-dimensional images with parametric active surfaces and topology changes. J. Sci. Comput., 1–35 (2017)

  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)

  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)

  11. Dziuk, G.: Discrete anisotropic curve shortening flow. SIAM J. Numer. Anal. 36(6), 1808–1830 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  19. McDermott, R., McGrattan, K., Hostikka, S.: Fire dynamics simulator (version 5) technical reference guide NIST. Spec. Publ. 1018(5) (2008)

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  29. Monoši, M., Majlingová, A., Kapusniak, J.: Lesné požiare žilinská univerzita v žiline (2015)

  30. Nakamura, G., Potthast, R.: Inverse Modeling, pp. 2053–2563. IOP Publishing (2015). https://doi.org/10.1088/978-0-7503-1218-9

  31. Osher, S., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces. Applied Mathematical Sciences. Springer, New York (2002)

    MATH  Google Scholar 

  32. Pauš, P., Beneš, M.: Algorithm for topological changes of parametrically described curves. In: Proceedings of ALGORITMY, pp. 176–184 (2009)

  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)

  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)

  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)

  36. Sullivan, A.: A review of wildland fire spread modelling, 1990-present 3: Mathematical analogues and simulation models. arXiv:0706.4130(2007)

  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)

    Article  MATH  Google Scholar 

  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)

  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)

Download references

Funding

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

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Civil Engineering, Slovak University of Technology, Radlinského 11, 810 05, Bratislava, Slovak Republic

    Martin Ambroz, Martin Balažovjech, Matej Medl’a & Karol Mikula

  2. Algoritmy:SK s.r.o., Šulekova 6, 81106, Bratislava, Slovak Republic

    Karol Mikula

Authors
  1. Martin Ambroz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Martin Balažovjech
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Matej Medl’a
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Karol Mikula
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Martin Ambroz.

Additional information

Communicated by: Jean-Frédéric Gerbeau

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ambroz, M., Balažovjech, M., Medl’a, M. et al. Numerical modeling of wildland surface fire propagation by evolving surface curves. Adv Comput Math 45, 1067–1103 (2019). https://doi.org/10.1007/s10444-018-9650-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10444-018-9650-4

Keywords

Mathematics Subject Classification (2010)

Search

Navigation