Level Set Methods for Multiphase Motion by Mean Curvature

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

This project will develop new algorithms for simulating on the computer a class of mathematical models that describe the evolution in time of a network of surfaces. These models play a prominent role in many applications. A very important example that will receive particular attention is the evolution of the internal structure (microstructure) of polycrystalline materials, such as most metals and ceramics, during manufacturing processes such as heat treatment (annealing). Polycrystalline materials are very common. They are composed of tiny crystallites, known as grains, stuck together. During annealing, the boundaries between the grains, described by a network of surfaces, start to move as some grains get larger, while others shrink and disappear. The shapes and sizes of the grains making up these materials are known to have profound implications for their physical properties, such as their strength and conductivity. Materials scientists have long had mathematical models that describe the motion of the network of grains; what has been lacking is accurate, efficient, reliable, and flexible numerical methods that would allow them to compare large scale simulations of their models against experimental measurements. In recent years, as experimental measurements of time evolution of the three dimensional internal structure of materials have become available, the need for algorithms to simulate the relevant models have become increasingly acute. The project will take steps to address this need. Resulting algorithms will be implemented in software, which will be made available to the broader scientific community. The project will also support the training and research of one graduate student working towards a Ph.D. in mathematics. The project will take a new approach to designing level set methods for multiphase geometric motions such as motion by mean curvature of networks of surfaces. It will exploit a precise, mathematical connection between a particular discretization of the level set formulation of motion by mean curvature, known as the median filter scheme, and another class of algorithms known as threshold dynamics. This will allow extending advantages of one method to the other. The advantage of threshold dynamics is its generality and highly developed theory of stability and convergence. In particular, recent advances in our theoretical understanding of threshold dynamics enabled its extension to the more elaborate microstructure evolution models of interest to materials scientists. Via its precise connection to median filter schemes, elements of this theory will be carried over to level set methods. The new level set methods will allow arbitrary, normal dependent (anisotropic) surface tensions and mobilities to be assigned to any interface in the network of surfaces — a level of generality that cannot even be attempted by most existing techniques. They will also allow subgrid accuracy in locating the interface even when implemented on uniform grids — a distinct advantage of the level set method over threshold dynamics. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.