A variational method for multiphase volume-preserving interface motions

Abstract

We develop a numerical method for realizing mean curvature motion of interfaces separating multiple phases, whose areas are preserved throughout time. The foundation of the method is a thresholding algorithm of the Bence-Merriman-Osher type. The original algorithm is reformulated in a vector setting, which allows for a natural inclusion of constraints, even in the multiphase case. Moreover, a new method for overcoming the inaccuracy of thresholding methods on non-adaptive grids is designed, since this inaccuracy becomes especially prominent in area-preserving motions. Formal analysis of the method and numerical tests are presented.

An efficient threshold dynamics method for wetting on rough surfaces

“…Recently, Esedoglu and Otto extended the threshold dynamics method to the multi-phase problems with arbitrary surface tension [12]. There have been many studies on the MBO threshold dynamics method, including some efficient implementations [26,27,30] and convergence analysis [3,5,14,17]. In particular, Laux and collaborators established the convergence of some computational algorithms including one with volume preservation [19,20].The generalization of MBO-type methods to the wetting problem where interfaces intersecting the boundary is not straightforward because of a lack of integral representation with a heat kernel for a general domain.In the original MBO scheme, when the interface does not intersect the solid boundary, one can solve the heat equation efficiently on a rectangular domain with a uniform grid using convolution of the heat kernel with the initial condition [26,27].…”

An Adaptive Threshold Dynamics Method for Three-Dimensional Wetting on Rough Surfaces

“…The other important step of the process in Section 3.1 is to preserve the volume of the fluid domain. This is to find a δ to make sure |D problem to compute δ and the standard method is by the Newton method [25,36,38]. In [47], a technique based on fast sorting scheme is developed to find δ efficiently in the case of uniform meshes.…”

Wave-type threshold dynamics and the hyperbolic mean curvature flow

“…where r(t) denotes the solution to (15), N e is the largest positive integer satisfying N e ∆t ≤ t e , and the value ofr n is zero for any iteration after which the approximate solution's radius becomes zero. As our final numerical test, we replace the steps 4 and 5 of the previous algorithm with the construction of the exact signed distance function according to the exact configuration of the finite element space's approximation of the interface.…”

“…This fact allows one to obtain the curvature of the interface by computing the Laplacian of (23). Moreover, as shown in [15], this setting introduces a multiple well potential which allows one to express multiphase volume constrained motions. Our approximation method for computing multiphase interfacial dynamics thus repeats the PDE step (19) and the thresholding step (22).…”

“…Here, in contrast to parabolic mean curvature flow, we remark that interfaces are not smoothed, but oscillate. As in the parabolic case [15], the method of minimizing movements can be used to inspect volume constrained interfacial motions as well. In particular, regarding volume-constrained motions, the minimization formulation of our algorithm can allow one to formally include constraints, via penalization.…”