An Implicit Boundary Integral Method for Interfaces Evolving by Mullins-Sekerka Dynamics

Abstract

We present an algorithm for computing the nonlinear interface dynamics of the Mullins-Sekerka model for interfaces that are defined implicitly (e.g. by a level set function) using integral equations. The computation of the dynamics involves solving Laplace’s equation with Dirichlet boundary conditions on multiply connected and unbounded domains and propagating the interface using a normal velocity obtained from the solution of the PDE at each time step. Our method is based on a simple formulation for implicit interfaces, which rewrites boundary integrals as volume integrals over the entire space. The resulting algorithm thus inherits the benefits of both level set methods and boundary integral methods to simulate the nonlocal front propagation problem with possible topological changes. We present numerical results in both two and three dimensions to demonstrate the effectiveness of the algorithm.

Appendix

1.1 Connected Component Labeling

As explained in Section 2, the system of equations to solve depends on many properties including the number of connected components of the region, whether the region is bounded or not (exterior vs interior), and the orientation of the region. In our numerical simulations, we adapted a technique called connected component labeling (CCL), see e.g. [14], to find necessary topological information needed to obtain the correct set of equations. These include:

• 1. The boundedness of the connected component $C_{i}$. This decides which formulation (interior vs exterior) to use.

• 2. The orientation of $C_{i}$ which determines the normal direction.

• 3. The total number of connected boundary components of the boundary of $C_{i}$, denoted by $\varGamma ^{i} = \partial C_{i}$.

• 4. The selection of $\mathbf {z}_{i}$ in (9) and (11) for each component $\varGamma ^j$. This involves

  • the separation of $\varGamma ^{i}$ into boundary components $\varGamma ^{i}_{j}$, each bounding a hole in the region except for the most exterior one. We denote the exterior boundary as $\varGamma ^{i}_{0}$.

  • For each hole delimited by $\varGamma ^{i}_{j}~,j \ne 0$, find a point $\mathbf {z}_{i}$ inside the hole that gives the least singular value of $\Phi (\mathbf {x}-\mathbf {z}_{i})$. This means that $\mathbf {z_{i}}$ should be as far from the boundary as possible.

Since we deal with closed interfaces, every grid point belongs to a unique connected component and has a well-defined component label. Note that a connected component $C_{i}$ may belong to $\varOmega$ or $\mathbf R^m \setminus \varOmega$.

We use the algorithm as follows:

• We label the unbounded component as $C_{0}$. This is the only true exterior component for the boundary integral formulations.

• Each interior component ($d_{\varGamma }>0$, i.e. the solid phase) will have positive label i and each exterior component ($d_{\varGamma }<0$ i.e. the liquid phase) will take negative label $-i$ (except for the unbounded one).

• Each boundary piece of each component will have label j.

• The point $\mathbf {z_{i}}$ in $C_{i}$ is chosen to be the point with largest signed distance function in absolute value.

We adopt the two-pass CCL algorithm with 2m-connectivity in $\mathbb R^m$. This algorithm uses equivalence classes for labels: after the first pass, points in the same connected component may not have the same label, but the labels of points in the same connected component will be assigned to the same equivalent class in the second pass. The root of the equivalence class denotes the smallest label (in absolute value) in the equivalence class. The scanning process works as follows:

• 1. First Pass

  • a. Begins with label 0 which denotes the unbounded component $C_{0}$.

  • b. At the current point scanned, we look at the sign of the signed distance function of its 2m neighbors already visited (in 2D, west and north of the current point)

    • If none of the neighbors have the same sign as the current point, we create a new label (positive or negative, based on the sign of the distance function at the current point) and a new equivalence class.

    • If only one neighbor has the same sign, we pick the label for the current point to be the root of that neighboring point’s equivalence class.

    • If more than one neighbor has the same sign as the current point, we pick the label for the current point to be the smallest root of the neighbors that have the same sign’s equivalence classes. Furthermore, we combine the equivalence classes of the neighboring points that have the same sign since they are connected through the current point.

  • c. The largest root of all equivalence classes with a given sign (interior or exterior) will give the total number of connected components that have that sign. Thus, the sum is the total number of connected components.

• 2. Second Pass

  • a. At each point, we assign its label to be the root of its equivalence class.

  • b. We update and store the points with largest absolute distance within each equivalence class. These are the points $\mathbf {z_{i}}$.

  • c. For each point $\mathbf {x}$ within the $\varepsilon$ tubular neighborhood of the boundary $(|d_{\varGamma }|< \varepsilon )$, we look for the root of its equivalence class (say i) and the root of its projection point’s ($P_{\varGamma }(\mathbf {x})$) equivalence class (say j) that takes the opposite sign. To obtain j, we look at the vertices of the cell $P_{\varGamma }(\mathbf {x})$ falls in and scan for the label with opposite sign. This step identifies the boundary piece $\varGamma ^{i}_{j}$ and collects points within the $\varepsilon$ neighborhood of the boundary $\varGamma ^{i}_{j}$.

  • d. We store the total number of connected boundary pieces.