Variational Dynamics of Free Triple Junctions

Abstract

We propose a level set framework for representing triple junctions either with or without free endpoints. For triple junctions without free endpoints, our method uses two level set functions to represent the three segments that constitute the structure. For free triple junctions, we extend our method using the free curve work of Schaeffer and Vese (J Math Imaging Vis, 1–17, 2013), Smereka (Phys D Nonlinear Phenom 138(3–4):282–301, 2000). For curves moving under length minimizing flows, it is well known that the endpoints either intersect perpendicularly to the boundary, do not intersect the boundary of the domain or the curve itself (free endpoints), or meet at triple junctions. Although many of these cases can be formulated within the level set framework, the case of free triple junctions does not appear in the literature. Therefore, the proposed free triple junction formulation completes the important curve structure representations within the level set framework. We derive an evolution equation for the dynamics of the triple junction under length and area minimizing flow. The resulting system of partial differential equations are both coupled and highly non-linear, so the system is solved numerically using the Sobolev preconditioned descent. Qualitative numerical experiments are presented on various triple junction and free triple junction configurations, as well as an example with a quadruple junction instability. Quantitative results show convergence of the preconditioned algorithm to the correct solutions.

Appendix: Derivation of the First Variation for Free Triple Junctions

Let $\phi ,\psi , \nu \in W^{1,\infty }$, recall the free triple junction Length function

$$\begin{aligned} L_{fT}(\phi ,\psi ,\nu ) = \int \left( l_{+} |\nabla H(\phi )| H(\psi ) + l_{-} |\nabla H(\phi )| H(-\psi ) + l_{0} |\nabla H(\psi )| H(\phi ) \right) H(\nu )dx \end{aligned}$$

To compute the first variation, let $w \in W^{1,\infty }$ and differentiate the energy with respect to perturbation using these functions. First, to find the minimizer $\phi$, we fix $\psi$ and $\nu$ and define $G(\varepsilon ):=L_{fT}(\phi +\varepsilon w,\psi ,\nu )$ and compute $G'(0)=0$.

$$\begin{aligned} \frac{d}{d \varepsilon }G(\varepsilon )&= \frac{d}{d \varepsilon } \int \left( l_{+} |\nabla H(\phi +\varepsilon w)| H(\psi ) + l_{-} |\nabla H(\phi +\varepsilon w)|\right. \\&\quad \quad \left. H(-\psi ) + l_{0} |\nabla H(\psi )| H(\phi +\varepsilon w) \right) H(\nu )dx\\&= \frac{d}{d \varepsilon } \int \left( |\nabla \phi +\varepsilon \nabla w| \delta (\phi + \varepsilon w) \chi + l_{0} |\nabla H(\psi )| H(\phi +\varepsilon w) \right) H(\nu )dx\\&= \int \left( \frac{\nabla \phi +\varepsilon \nabla w}{|\nabla \phi +\varepsilon \nabla w|} \cdot \nabla w \delta (\phi + \varepsilon w) \chi \right. \\&\quad \left. + |\nabla \phi +\varepsilon \nabla w| \delta '(\phi + \varepsilon w) w \chi + l_{0} |\nabla H(\psi )| \delta (\phi +\varepsilon w) w \right) H(\nu )dx \end{aligned}$$

Next setting $\varepsilon =0$ and integrating by parts (with $\frac{\partial \phi }{\partial n}=0$ on $\partial \varOmega$):

$$\begin{aligned} G'(0)&= \int \left( -\mathrm{div} \left( H(\nu ) \delta (\phi ) \chi \frac{\nabla \phi }{|\nabla \phi |} \right) + |\nabla \phi | \delta '(\phi )\chi + l_{0} |\nabla H(\psi )| \delta (\phi )H(\nu ) \right) wdx\\&= \int \left( -\delta (\phi ) \mathrm{div} \left( H(\nu ) \chi \frac{\nabla \phi }{|\nabla \phi |} \right) + l_{0} |\nabla H(\psi )| \delta (\phi ) H(\nu ) \right) wdx\\&= \int \delta (\phi ) \left( - \mathrm{div} \left( H(\nu ) \chi \frac{\nabla \phi }{|\nabla \phi |} \right) + l_{0} |\nabla H(\psi )| H(\nu ) \right) wdx \end{aligned}$$

Since this holds for all $w$, the first variation is:

$$\begin{aligned} \delta (\phi ) \left( - \mathrm{div} \left( H(\nu ) \chi \frac{\nabla \phi }{|\nabla \phi |} \right) + l_{0} |\nabla H(\psi )| H(\nu ) \right) =0 \end{aligned}$$

and can be embedded in a time-dependent descent:

$$\begin{aligned} \frac{\partial \phi }{\partial t}= \delta (\phi ) \left( \mathrm{div} \left( H(\nu ) \chi \frac{\nabla \phi }{|\nabla \phi |} \right) - l_{0} |\nabla H(\psi )| H(\nu ) \right) \end{aligned}$$

Next, to find the minimizer $\psi$, we fix $\phi$ and $\nu$ and define $G(\varepsilon ):=L_{fT}(\phi , \psi +\varepsilon w,\nu )$ and compute $G'(0)=0$ again.

$$\begin{aligned} \frac{d}{d \varepsilon }G(\varepsilon )&= \frac{d}{d \varepsilon } \int \left( l_{+} |\nabla H(\phi )| H(\psi +\varepsilon w) + l_{-} |\nabla H(\phi )| H(-\psi +\varepsilon w)\right. \\&\quad \left. + l_{0} |\nabla H(\psi +\varepsilon w)| H(\phi ) \right) H(\nu )dx\\&= \frac{d}{d \varepsilon } \int \left( l_{+} |\nabla H(\phi )| H(\psi +\varepsilon w) + l_{-} |\nabla H(\phi )| H(-\psi +\varepsilon w)\right. \\&\quad \left. + l_{0} |\nabla \psi +\varepsilon \nabla w| \delta (\psi +\varepsilon w) H(\phi ) \right) H(\nu )dx\\&= \int \left( \left( l_{+}-l_{-} \right) |\nabla H(\phi )| \delta (\psi +\varepsilon w) w\right. \\&\quad \left. + l_{0} \frac{\nabla \psi +\varepsilon \nabla w}{|\nabla \psi +\varepsilon \nabla w|} \cdot \nabla w \ \delta (\psi +\varepsilon w) H(\phi )\right. \\&\quad \left. + l_{0} |\nabla \psi +\varepsilon \nabla w| \delta '(\psi +\varepsilon w) H(\phi ) w \right) H(\nu )dx \end{aligned}$$

Next setting $\varepsilon =0$ and integrating by parts (with $\frac{\partial \psi }{\partial n}=0$ on $\partial \varOmega$)

$$\begin{aligned} G'(0)&= \int \left( \left( l_{+}-l_{-} \right) |\nabla H(\phi )| H(\nu ) \delta (\psi ) - l_{0} \mathrm{div} \left( \delta (\psi ) H(\phi ) H(\nu ) \frac{\nabla \psi }{|\nabla \psi |} \right) \right. \nonumber \\&\quad \left. + l_{0} |\nabla \psi | \delta '(\psi ) H(\phi ) \right) w dx\\&= \int \delta (\psi ) \left( \left( l_{+}-l_{-} \right) |\nabla H(\phi )| H(\nu )- l_{0} \mathrm{div} \left( H(\phi ) H(\nu ) \frac{\nabla \psi }{|\nabla \psi |} \right) \right) w dx \end{aligned}$$

Since this holds for all $w$, the first variation with respect to $\psi$ is:

$$\begin{aligned} \delta (\psi ) \left( \left( l_{+}-l_{-} \right) |\nabla H(\phi )| H(\nu )- l_{0} \mathrm{div} \left( H(\phi ) H(\nu ) \frac{\nabla \psi }{|\nabla \psi |} \right) \right) =0 \end{aligned}$$

and can be embedded in a time-dependent descent:

$$\begin{aligned} \frac{\partial \psi }{\partial t}= \delta (\psi ) \left( l_{0} \mathrm{div} \left( H(\phi ) H(\nu ) \frac{\nabla \psi }{|\nabla \psi |} \right) -\left( l_{+}-l_{-} \right) |\nabla H(\phi )| H(\nu ) \right) \end{aligned}$$

Lastly, to find the minimizer $\nu$, we may repeat the previous procedure. Note that since in terms of $\nu$ the functional is:

$$\begin{aligned} L_{fT}(\phi ,\psi ,\nu ) = \int F(\phi ,\psi ) H(\nu )dx =\left< F(\phi ,\psi ), H(\nu )\right>_{L^2} \end{aligned}$$

with $F(\phi ,\psi )=l_{+} |\nabla H(\phi )| H(\psi ) + l_{-} |\nabla H(\phi )| H(-\psi ) + l_{0} |\nabla H(\psi )| H(\phi )$, then the first variation is $\langle F(\phi ,\psi ), \delta (\nu )\rangle _{L^2}=0$. Embedding the equation in a descent differential equation yields:

$$\begin{aligned} \frac{d\psi }{d t} = -\delta (\nu ) \left( l_{+} |\nabla H(\phi )| H(\psi ) + l_{-} |\nabla H(\phi )| H(-\psi ) + l_{0} |\nabla H(\psi )| H(\phi ) \right) \end{aligned}$$