A dynamic mesh algorithm for curvature dependent evolving interfaces

A new finite element method is discussed for approximating evolving interfaces\r\nin $\Rn$ whose normal velocity equals mean curvature plus a forcing function.\r\nThe method is insensitive to singularity formation and retains the \r\nlocal structure of the limit problem, and thus exhibits a computational\r\ncomplexity typical of $\R^{n-1}$ without having the drawbacks of \r\nfront-tracking strategies. \r\nA graded dynamic mesh around the propagating front is the sole partition \r\npresent at any time step and is significantly smaller than a full mesh.\r\nTime stepping is explicit, but stability\r\nconstraints force small time steps only when singularities develop,\r\nwhereas relatively large time steps are allowed before or past\r\nsingularities, when the evolution is smooth. \r\nThe explicit marching scheme also guarantees that\r\nat most one layer of elements has to be added or deleted per time step,\r\nthereby making mesh updating simple, and thus practical.\r\nPerformance and potentials are fully documented via a number\r\nof numerical simulations in 2D, 3D, 4D, and 8D, with axial symmetries.\r\nThey include tori and cones for the mean curvature flow,\r\nminimal and prescribed mean curvature surfaces with given boundary, \r\nfattening for smooth driving force, and volume constraint.