Iterative method for solving the eikonal equation

0469375 - ÚFP 2017 RIV US eng C - Conference Paper (international conference)
Mokrý, Pavel
Iterative method for solving the eikonal equation.
Proceedings of SPIE 10151, Optics and Measurement International Conference 2016. Vol. 10151. Bellingham: SPIE, Society of Photo-Optical Instrumentation Engineers, 2016 - (Kovačičinová, J.), č. článku 101510Z. SPIE. ISBN 978-1-5106-0753-8. ISSN 0277-786X.
[OAM 2016, Optics and Measurement International Conference 2016. Liberec (CZ), 11.10.2016-14.10.2016]
R&D Projects: GA ČR GA13-10365S
Institutional support: RVO:61389021
Keywords : Iterative methods * Finite element methods * Data storage * Electromagnetic radiation * Geometrical optics * Numerical analysis * Partial differential equations * Refractive index * Wavefronts * Algorithms * eikonal equation
Subject RIV: BH - Optics, Masers, Lasers
http://dx.doi.org/10.1117/12.2257326

The paper present principles and derivation of the iterative method for solving the eikonal equation. The eikonal equation, which defines the relationship between the phase of the optical wave Φ(r) and the refractive index n(r), i.e. |grad Φ(r)|2 = n2(r), represents the fundamental equation in geometrical optics. It describes the evolution of the wavefront, which is given by the equation Φ (r) = C, of the electromagnetic wave in the limit of infinite frequency or zero wavelength. The eikonal equation is the nonlinear partial differential equation (PDE) of the first order. This classification makes the eikonal equation of rather diffcult to solve, both analytically and numerically. Several algorithms have been developed to solve the eikonal equation: Dijkstra's algorithm, fast marching method, fast sweeping method, label-correcting methods, etc. Major disadvantage of these methods is that their convergence puts rather high requirements on the density of the computing grid. It is known that finite element method (FEM) offers much more memory and time efficient approach to solve PDEs. Unfortunately, FEM cannot be applied to solve eikonal equation directly due to its first order. In order to provide the fast and memory efficient solution of the eikonal equation, it is suggested to solve a generalized version of the eikonal equation, which is of the second order and which can be solved using FEM
Permanent Link: http://hdl.handle.net/11104/0267286