An interface-fitted finite element based level set method : algorithm, implementation, analysis and applications
Simulation of problems involving different media that are flowing or deforming requires tracking the boundary between the media. These are called moving interface problems. Often the velocity of the moving interface is determined by some underlying physical model. Thus, moving interface problems usually require separate methods to track the interface and to determine the velocity of the interface. The level set method has become a popular way of numerically tracking a moving interface. This method represents the interface implicitly as the zero level set of a higher dimensional continuous function, and this function is evolved in time by a partial differential equation, known as the level set equation. The interface normal velocity must be determined concurrently by some separate method depending on the specific problem. Finite difference methods are well established for implementing the level set method, but finite difference methods are not ideally suited for solving problems involving arbitrarily shaped regions such as those that occur in moving interface problems. This is because they require a regular spaced grid that does not explicitly locate the interface. Thus, they cannot easily solve problems that depend on knowing the exact location of the interface. To address these difficulties, an interface-fitted finite element level set method is considered. This method uses a base mesh to solve the level set equation and track the interface. The base mesh is refined at each time step to explicitly locate the interface and separate regions defined by the interface. A finite element method can then be used to solve field equations on the refined mesh. Using the interface-fitted mesh, a new method of reinitializing of the level set function, a new method of extending the velocity away from the interface, and a new method of calculating curvature are proposed. The interface-fitted finite element level set method is applied to modeling solidification problems and determining optimal solute-solvent interfaces of solvation systems.