Figure 2

Coordinate transformation method. The function \(\chi(s)= \frac{cs}{(1-s^{2})}\) maps the bounded domain \([0,1 ] \times [-1,1 ]\) (left) with coordinates \((\xi,\eta)\) into the unbounded domain \([0,+\infty ] \times [-\infty, +\infty ]\) (right, truncated in this plot at a distance of 12 km) with coordinates \((r,z)\) by the coordinate transformation (13). A uniform Cartesian grid for the bounded domain (left) automatically results in a quasi-uniform grid for the unbounded domain (right). In this plot, an example of grids obtained by using a spatial step \(h=\Delta\xi=\Delta\eta= 1/25\) in \(\Omega_{M}^{b}\) is represented (note that, for representation purpose, we have chosen a larger spatial step than that adopted in numerical tests). The original domain \(\Omega_{M}\) with surface \(\Gamma_{M}\) (right) is therefore associated to a domain \(\Omega_{M}^{b}\) with surface \(\Gamma_{M}^{b}\) in the bounded domain (left). Equations (8) in \(\Omega _{M}\) (as well as the stress-free boundary condition (9) on \(\Gamma_{M}\)) are transformed according to (14) into a new set of equations in \(\Omega_{M}^{b}\) (with stress-free boundary conditions on \(\Gamma_{M}^{b}\)), solved by the finite-difference method described in Section 2.3.2. A level-set function ϕ is used to discriminate between internal points (\(\phi<0\), represented by black dots) and external points (see (15)). Among the latter, grid points close to the surface (ghost points, represented by red circles) are included to the discretized problem and a suitable value is assigned to them by enforcing boundary conditions (by the ghost-point method described in Section 2.3.2). The coordinate transformation method is used also to solve Eq. (12) in the domain \(\mathbb {R}^{2}\) (then without a ghost-point level-set method).