Mathematical Modelling of Moving Boundary Problems
| školitel: | prof. Dr. Ing. Michal Beneš |
| e-mail: | zobrazit e-mail |
| typ práce: | dizertační práce |
| zaměření: | MI_MM |
| klíčová slova: | nonlinear evolution problems; parametric method; levelset method; degenerate diffusion; surface diffusion; |
| odkaz: | http://geraldine.fjfi.cvut.cz/~benes/ |
| popis: | Investigation of free-boundary problems is impotrant in physical or biological context, in material science and fluid dynamics. Free boundaries are understood as hypersurfaces described by geometrical means, dynamically evolving due to the forces arising in governing partial differential equations. The entire problem becomes nonlinear and its mathematical treatment non-trivial. Closer analysis of these problems and their role in mathematical modelling lead to further improvements including incorporation of anisotropy as additional feature of the environment (crystalline lattice or surface energy density). In the Ph.D. project, a class of moving boundary problems will be investigated. Such problems are described by the law for the normal velocity of the interface incorporating mean curvature, Gaussian curvature and their differentials. The research in progress is carried out for the problems of pure advection used for fluid-component tracking, for the problems of curvature-dependent evolution up to the problems of surface diffusion. The scope of the law within the project is given by the following particular cases: mean-curvature flow relating the normal velocity of the interface to its mean curvature, constrained mean-curvature flow preserving the area enclosed by the interface, surface diffusion containing the surface Laplacian of curvature. The mathematical treatment of the motion law is based on methods directly tracking the position of or on methods yielding the position of the interface as a consequence of the solution of a higherdimensional problem. In any case, it leads to analysis of nonlinear degenerate partial differential equations, mostly of parabolic type. Applications of moving-boundary problems can be found in many areas of science. Variety of free-interface phenomena accompany processes in the material science - solidification processes, redistribution of the grain boundaries, elastic effects in materials, motion of material defects. The topic also finds rich application in computer image processing. e.g. in medical context. This project is designed for the Ph.D. degree in Mathematical Engineering and fits into the long-term international collaboration of the Department of Mathematics, e.g. with the Kanazawa University, Meiji University in Japan, Colorado School of Mines, Golden and TU Dresden. It is expected to lead to the impacted publications and to contributions in international conferences. |
| literatura: | [1] G.-Q. Chen , H. Shahgholian,J.-L. Vazquez: Free boundary problems: the forefront of current and future developments. Phil. Trans. R. Soc. A373: 20140285, 2015 [2] T. F. Banchoff and S. T. Lovett: Differential Geometry of Curves and Surfaces. CRC Press, New York, 2010. [3] S. Osher, R. Fedkiw: Level Set Methods and Dynamic Implicit Surfaces. Springer Verlag, New York, 2003. [4] M. Beneš, M. Kolář and D. Ševčovič: Curvature driven flow of a family of interacting curves with applications, Math. Meth. Appl. Sci. 43:4177–4190, 2020. [5] M. Beneš, M. Kolář and D. Ševčovič: Qualitative and Numerical Aspects of a Motion of a Family of Interacting Curves in Space, SIAM Journal on Applied Mathematics, Vol. 82, Iss. 2, 10.1137/21M1417181, 2022. |
| naposledy změněno: | 02.11.2022 11:04:28 |
za obsah této stránky zodpovídá:
Ľubomíra Dvořáková | naposledy změněno: 12.9.2011