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Level set numerical approach to anisotropic mean curvature flow on obstacle

Gavhale, Siddharth Balu 京都大学 DOI:10.14989/doctor.k23677

2022.03.23

概要

The thesis investigates multiphase geometric evolutions with direction dependent energy densities and with global constraints on the enclosed volumes. This kind of problem has numerous applications in technology and natural science fields, while its mathematical understanding is still incomplete.

The exposition starts with a review of known results in order of increasing com- plexity. Thus, first, two-phase mean curvature flow model is introduced, together with its derivation and main results on well-posedness theory. This relates to the approaches to mathematically represent an evolving interface, which are described and the motivation is given for selecting the implicit level-set approach in view of possible topology changes in the investigated phenomenon. The overview then con- tinues with results on two-phase problems with anisotropic, i.e., normal direction- dependent, energy and / or with constraint on the enclosed volume. Finally, the multiphase problem with junctions is reviewed showing that for the anisotropic case mathematical theory is available only in the explicit approach without topology changes. Next, methods to numerically approximate the multiphase anisotropic problem are reviewed with the finding that although the anisotropic problem is treated explicitly [Wang et al., 2015] and the isotropic problem is treated implicitly [Xu et al., 2017], there is no method in the literature addressing possible topology changes in the anisotropic case.

The thesis then proceeds with a detailed comparative numerical analysis of convo- lution kernels that are the main tool to express anisotropy in the implicit approach. The output is a list of advantages and disadvantages of four main kernels proposed in the past, which is based on quantitative evaluation of their numerical properties, such as convergence order, required computational time, ability to deal with sharp corners, etc.

The main result of the thesis is then the construction and analysis of a numerical algorithm for solving a special type of a three-phase problem, namely the obstacle problem, where one of the phases is fixed throughout time. The algorithm is based on the linearization of the corresponding level-set partial differential equation in the fashion of Bence-Merriman-Osher algorithm, where the Gaussian kernel is replaced by a suitable convolution kernel to express the anisotropy and thresholding height is modified to preserve enclosed volume. Since the algorithm is designed in such a way that it decreases a nonlocal ”heat content” approximation of the interfacial energy, which Γ-converges to the perimeter functional, the author succeeded in proving its unconditional stability with respect to time step, under the condition that the Fourier transform of the adopted convolution kernel is positive.

Finally, numerical properties of the proposed scheme are studied through a series of numerical tests, including the convergence order, behavior of contact angles, and topology changes such as splitting and merging. Since the convergence at contact point is slower than at free interface, a modification of the algorithm is proposed to improve the convergence around triple junctions. The thesis concludes with a summary of its content and topics for possible future research.

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