## 3 Description of the interface through level set
The relevant evolution equation is then given in the Hamilton-Jacobi form by requiring the material derivative of the level set to vanish,  
\[\frac{D\phi}{Dt}=\frac{\partial\phi}{\partial t}+\
abla\phi\cdot\mathbf{v}=0\,. \tag{13}\]  
In particular, Eq. (13) requires that \(\phi\) be constant on the interface. The field \(\mathbf{v}\left(\mathbf{x},t\right)\) represents an extension of the interface velocity, which is physically defined only on \(\Gamma_{\text{\tiny{max}}}\), to the whole domain \(\Omega\). In fact, whereas in a Lagrangian approach, where the position of the interface is tracked explicitly, the velocity on \(\Gamma_{\mbox{\tiny tran}}\) suffices for its evolution, in an Eulerian approach, it is necessary to extend the velocity field to the entire domain so that the level set function, which controls the interface position and is defined everywhere in \(\Omega\), can be advected by such extended velocity field. It is convenient to introduce the scalar normal velocity  
\[F\left(\mathbf{x},t\right)=\mathbf{v}\cdot\frac{\
abla\phi}{\left\|\
abla\phi \right\|}, \tag{14}\]  
that is the component of \(\mathbf{v}\) in the direction of \(\
abla\phi\); Eq. (13) rewrites then as  
\[\frac{\partial\phi}{\partial t}+\left\|\
abla\phi\right\|F=0\,. \tag{15}\]  
The field \(F\) is the extension of the normal interface velocity to the entire domain, and it is known only on the interface, where it is physically meaningful and can be obtained by Eq. (5),  
\[F\left(\mathbf{x},t\right)=v,\ \ \ \mathbf{x}\in\Gamma_{\mbox{\tiny tran}} \left(t\right). \tag{16}\]  
Since the velocity field in \(\Omega\setminus\Gamma_{\mbox{\tiny tran}}\) is artificial, there is a certain freedom in its construction, the only requirements being that \(F\) be continuous through the interface and satisfy Eq. (16). It can be demonstrated [22] that, if \(F\) and \(\phi\) are smooth and their gradients are orthogonal,  
\[\
abla\phi\cdot\
abla F=0\,, \tag{17}\]  
then \(F\) tends to preserve the initial signed-distance properties of the level set. The evolution of \(\phi\left(\mathbf{x},t\right)\) is controlled by Eq. (15), with the initial condition  
\[\phi\left(\mathbf{x},0\right)=\phi_{0}\left(\mathbf{x}\right). \tag{18}\]  
Practically, in the initial time instant, \(\phi_{0}\) is constructed by the knowledge of \(\Gamma_{\mbox{\tiny tran}}\left(0\right)\) by using the definition in Eq. (9), whereas in the subsequent time instants, the updated position of the interface is obtained from \(\phi\left(t\right)\) by Eq. (10). The field \(F\) is constructed by solving Eq. (17) with the essential boundary conditions provided by Eq. (16). The problem can be casted in a discrete form by using standard finite element techniques developed from the weak forms of Eqs (15) and (17),  
\[\int_{\Omega}\delta\phi\!\left(\frac{\partial\phi}{\partial t}+\left\|\
abla \phi\right\|\!\right|F\right)=0,\ \ \ \int_{\Omega}\delta F\left(\
abla F\cdot\
abla\phi\right)=0\,, \tag{19}\]  
where \(\delta\phi\) and \(\delta F\) are test fields belonging to \(H^{1}\left(\Omega\right)\). Essential boundary conditions are provided by a discretised version of Eq. (16), which can be obtained, for instance, by considering the projection of the interface velocity onto the finite element nodes whose support is intersected by the interface. The complete procedure is thoroughly derived in Reference [16], where the numerical issues arising in the integration of first order hyperbolic equations are solved by adding stabilisation and shock-capturing terms; further stabilisation techniques for the update of the level set function are described in Reference [21]. In the numerical analyses, the discretised field \(\phi\) progressively diverges from a signed distance function, so that every few steps a reinitialisation must be performed; some relevant procedures based on the fast marching algorithm are described in References [22] and [17].