AreaMapping

This component is classified under the category of Mappings. It maps each triangle in a topology to a scalar value representing its area.

The inputs of the component are: - a State: it contains the list of coordinates of the triangles vertices - a BaseMeshTopology: it contains the list of triangles, typically defined by indices that reference their vertices.

The output is a State where the position field is the list of scalar values representing the area of the triangles. These values are output in the same order as the input triangle list, ensuring a direct correlation between each triangle and its corresponding area value.

Mapping function

Let us define 3 vertices forming a triangle: \(P_0\), \(P_1\), and \(P_2\).

Let us define the vector function \(N(x, y, z) = (y - x) \times (z - x)\), where \(\times\) is the cross product.

The area of the triangle is computed as:

\[ \text{Area}(P_0, P_1, P_2) = \frac{1}{2} \| N (P_0, P_1, P_2) \| \]

where \(\| \cdot \|\) is the magnitude of a vector.

\(n\) is the number of triangles in the topology and \(m\) is the number of vertices.

The mapping function of this mapping is \(f(x) = (f_0(x), \dots, f_{n-1}(x))\). For \(0 \le i < n\):

\[ f_i(x) = \text{Area}(v_{t_{i_0}}, v_{t_{i_1}}, v_{t_{i_2}}) \]

with \(t_{i_j}\) is the index of the \(j\)-th vertex in the \(i\)-th triangle. \(x \in \mathbb{R}^{3m}\) is the input vector, i.e. the concatenation of all vertices positions. \(v_i\) is the \(i\)-th vertex position, i.e. \(v_i = (x_{3i}, x_{3i+1}, x_{3i+2})\).

Jacobian Matrix

The Jacobian matrix of this mapping, which represents the first-order partial derivatives of the output with respect to the input, is not constant, and depends on the input.

The elements of the Jacobian matrix \(J \in \mathbb{R}^{n \times 3 m}\) are given by: for \(0 \leq i < n, 0 \leq j < 3m\),

\[ J_{ij} = \frac{\partial f_i}{\partial x_j} \]

Let us compute the \(3 \times 1\) sub-matrix \(D_{ij}\) such that \(j\) is a multiple of 3, and \(D_{ij_k} = J_{i,j+k}\) for \(0 \leq k < 3\):

\[ D_{ij} = \begin{cases} \frac{1}{2 \| N_i \|} l_j \times N_i, & \text{if}\ j \ \text{is a vertex in the triangle}\ i \\ 0, & \text{otherwise} \end{cases} \]

where \(N_i = N(v_{t_{i_0}}, v_{t_{i_1}}, v_{t_{i_1}})\) and \(l_i\) is the opposite segment to the vertex \(j\) in the triangle \(i\).

Hessian Tensor

The Hessian tensor represents the second-order partial derivatives of the output with respect to the input. For the AreaMapping component, the Hessian tensor is non-zero because the Jacobian matrix depends on the input \(x\).

The elements of the Hessian tensor are given by:

\[ H_{ijk} = \frac{\partial J_{ij}(x)}{\partial x_k} \]

Let us focus on the triangle \(i\), and call U the matrix such that \(U_{jk} = H_{ijk}\)

\[ U_{jk} = \begin{cases} \frac{1}{2 \| N_i \|^3}\left(-(N_i \times l_j) \otimes (N_i \times l_k) + \| N_i \|^2 (l_j \cdot l_k \ I - l_k \otimes l_j + s_{kj} [N]_\times) \right) , & \text{if}\ j \ \text{and}\ k \ \text{are vertices in the triangle}\ i \\ 0, & \text{otherwise} \end{cases} \]

where \([a]_\times\) refers to the skew-symmetric matrix such that

\[ [a]_\times = \begin{bmatrix} \,\,0&\!-a_3&\,\,\,a_2\\ \,\,\,a_3&0&\!-a_1\\ \!-a_2&\,\,a_1&\,\,0 \end{bmatrix} \]

\(\otimes\) is the outer product, and \(s = [(1,1,1)]_\times\)