Geometric evolution equations

Geometric evolution equations describe the motion of evolving hyperserfaces in the d-dimensional Euclidean space which are governed by its geometrical quantity, which is the mean curvature for instance. Here, we show some results of numerical computation for several geometric evolution equations. This page is a showcase for results of numerical experiments of our research.

Mullins-Sekerka flow equation

Charge simulation method

Here, we implement the charge simulation method (CSM) for the Mullins-Sekerka problem.

For the detail of this scheme, please see our paper.

Evolution of a Pi curve

The coordinate data of the initial curve is taken from Wolfram. The proposed scheme preserves the area surrounded by the evolving curve and decreases its length in time.

Evolution of an L-shaped curve

Note that the x-axis is not part of the evolving curve. In other words, the initial curve is open. We simulate a numerical solution to the Mullins-Sekerka problem with the 90-degree contact angle condition.

Mean curvature flow equation

Split Bregman method

For a piecewise constant contact angle, we implement the split Bregman method to solve the above problem:

The former movie shows the motion of the contour line of an approximate solution to the mean curvature flow with constant contact angle condition. The approximate solution was computed by the finite difference method for a PDE which was derived as the first variation of an energy functional. We were successful to simulate a soliton-like rigorous solution to the mean curvature flow with constant contact angle condition by the proposed scheme.

Meanwhile, the latter movie demonstrates the motion of the contour line in the case when it starts with a sine curve. It is known that the classical solution to the mean curvature flow starting with this curve converges to a straight line if the time tends to infinity. This fact was also confirmed by the proposed scheme.

For the detail of this scheme, please see our paper.

Level-set method

Deep learning approach

This idea is quite simple. We assume that the level-set function of an evolving interface can be represented as a Neural network (NN), and we optimize the NN so that it approximately satisfies the level-set equation of the mean curvature flow.

Here, we treat the case when β= 0. The first animation exhibits the evolution of the contour line of the level-set function which is approximated by a well-trained NN with the help of the level-set equation of the mean curvature flow. Whereas, the second one shows the evolution of the 3D graph of the level-set function.

This methodology allows us to compute a level-set function of an evolving interface without any inductive steps in temporal space, and we feel its feasibility for approximation of the motion of interfaces governed by its geometry.

Different contact angle case

Convergence to a straight line

Convergence to a soliton-like solution

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