Conclusion

The level set method has been used for a wide variety of applications and continues to be a very popular tool. Since 2001, the method has been applied to multiphase flow [7-9,11,16,26,34,48,49,58,61,64,72,92,94,108-113,135-138], combustion [98], granular flow [36], surfactants [1], solid mechanics [90,119], crack propagation [53, 116, 117, 127], welding [65, 66], superconductor manufacturing [91], sintering [77], crystal growth [70, 71], Ostwald ripening and epitaxial growth [18, 37, 51, 89, 95], etching and deposition [59, 62, 63, 73, 96, 97,130, 132], inverse scattering and shape reconstruction [15,31,43-45], image processing [10, 13, 27, 54, 79, 93, 99, 125, 126, 128, 134], medical imaging [30, 87, 122], shape optimization and tomography [5, 60, 86, 131], grid generation [57], bacterial biofilms [33], tissue engineering [83], and string theory [56]. The breadth of the applications is a tribute to the level set method and its creators.

In addition, the fast marching method on its own has made a contribution to a number of areas including crack propagation [24,120], shape reconstruction [35], image processing [4,28,47,52,67,114], medical imaging [6,12,32,133], computer graphics and visualization [139], and robotic navigation [68,69].

Despite its tremendous popularity, the level set method is not suitable for every interface propagation problem. The implicit representation of the interface can be cumbersome at times, and if the more powerful features of the level set method are not required for a given problem, then simpler methods may be more appropriate. This is especially true if the alternative methods are also faster, which can often be the case. For this reason, it is important to remember the following key distinguishing features of the level set method:

1. topological changes are handled smoothly with no user intervention required,

2. corners and cusps in the interface are handled properly by using methods borrowed from hyperbolic conservation laws,

3. the method is easily extended to higher dimensions.

Any one of these reasons may be sufficient to employ the level set method, but not every problem requires these advantages. In that case, it would serve the practitioner to consider alternative numerical methods. It may or may not be the case that the level set method is still the best choice.

For a more comprehensive discussion on the level set method, the interested reader is directed to the books by Sethian [104] (which also includes the fast marching method) and Osher and Fedkiw [84].

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