Emulation and Simulation of
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A Calculating Filling Fractions from Level Set Values
Consider a single cell in a grid. In its centre, there is a point of a level set grid, \vec {O} = (0.5, 0.5, 0.5) . In order to approximate a surface corresponding to this level set point, a plane is constructed using \hat {n} , the
normalised normal vector of the level set at \vec {O} . A point on the plane, \vec {P} is then found by shifting \vec {O} in the direction of \hat {n} by the level set value at \Phi (\vec {O}) :
\seteqsection {A}
\begin{equation} \vec {P} = \vec {O} - \hat {n} \frac {\Phi (\vec {O})}{\mid \nabla \Phi (\vec {O}) \mid } \end{equation}
The plane approximating the surface inside the cell, p(\vec {x}) is therefore defined as
\seteqsection {A} \seteqnumber {2}
\begin{equation} \hat {n} \cdot (\vec {x} - \vec {P}) = 0 \end{equation}
Due to the symmetry of the cubic cell, the normal vector can always be mapped into one half/quarter of the first quadrant/octant, bounding the polar angles by 0 < \theta \le \frac {\pi }{4} and 0 < \phi \le \frac {\pi
}{4} .
A.1 2D Problem
In two dimensions the filling fraction is the area below the line describing the surface within the cell shown in Fig. A.1 . This line is given by
\seteqsection {A} \seteqnumber {3}
\begin{equation} p = \frac {\hat {n} \cdot \vec {P}}{n_y} - \frac {n_x}{n_y} x = q - \frac {n_x}{n_y} x \end{equation}
Therefore, the integral is bound to (0, 0) \le \vec {x} \le (1, 1) . However, the line might intersect the x-aligned edge of the cell at other points, for y = 0 and y = 1 . These intersections, a and b , are defined by:
\seteqsection {A} \seteqnumber {4}
\begin{equation} a = p(y = 0) = \frac {n_y q}{n_x} = \frac {\hat {n} \cdot \vec {P}}{n_x} \quad , \end{equation}
\seteqsection {A} \seteqnumber {5}
\begin{equation} b = p(y = 1) = \frac {n_y}{n_x}q - \frac {n_y}{n_x} = \frac {\hat {n} \cdot \vec {P} - n_y}{n_x} \quad , \end{equation}
where both a and b are bound to the interval [0, 1]. The area A is then given by
\seteqsection {A} \seteqnumber {6}
\begin{equation} A = \int _{0}^{b} dx + \int _{b}^{a} q - \frac {n_x}{n_y} x dx \end{equation}
Solving this integral gives an expression for the area under curve within the cell, i.e. the filling fraction:
\seteqsection {A} \seteqnumber {7}
\begin{equation} A = b + \frac {\hat {n} \cdot \vec {P}}{n_y}(a-b) - \frac {n_x}{2n_y}(a^2 - b^2) \end{equation}
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