VolumeMapping

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

The inputs of the component are: - a State: it contains the list of coordinates of the tetrahedra vertices - a BaseMeshTopology: it contains the list of tetrahedra, 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 volume of tetrahedra. These values are output in the same order as the input tetrahedron list, ensuring a direct correlation between each tetrahedron and its corresponding volume value.

Mapping function

Let us define 4 vertices forming a tetrahedron: \(P_0\), \(P_1\), \(P_2\), and \(P_3\).

The volume of a tetrahedron is computed as:

\[ \text{Volume}(P_0, P_1, P_2, P_3) = \frac{1}{6} | (P_1-P_0) \cdot \left( (P_2-P_0)\times(P_3-P_0) \right) | \]

Here, the cross product \(\times\) and the dot product \(\cdot\) are combined in the scalar triple product to compute the volume.

Given \(n\) tetrahedra in the topology and \(m\) vertices, the mapping function of this mapping is \(f(x) = (f_0(x), \dots, f_{n-1}(x))\). For \(0 \le i < n\):

\[ f_i(x) = \text{Volume}(v_{t_{i_0}}, v_{t_{i_1}}, v_{t_{i_2}}, v_{t_{i_3}}) \]

with \(t_{i_j}\) is the index of the \(j\)-th vertex in the \(i\)-th tetrahedron. \(x \in \mathbb{R}^{3m}\) 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 \(0 \leq k < 3\).

\[ D_{iv_{t_{i_1}}} = \frac{1}{6} (v_{t_{i_2}}-v_{t_{i_0}})\times(v_{t_{i_3}}-v_{t_{i_0}}) \]

\[ D_{iv_{t_{i_2}}} = \frac{1}{6} (v_{t_{i_3}}-v_{t_{i_0}})\times(v_{t_{i_1}}-v_{t_{i_0}}) \]

\[ D_{iv_{t_{i_3}}} = \frac{1}{6} (v_{t_{i_1}}-v_{t_{i_0}})\times(v_{t_{i_2}}-v_{t_{i_0}}) \]

\[ D_{iv_{t_{i_0}}} = -\frac{1}{6} (D_{iv_{t_{i_1}}} + D_{iv_{t_{i_2}}} + D_{iv_{t_{i_3}}}) \]

The computation uses the triple product property that it remains unchanged under a circular shift. That is why \(D_{iv_{t_{i_1}}}\), \(D_{iv_{t_{i_2}}}\), and \(D_{iv_{t_{i_3}}}\) are very similar. \(D_{iv_{t_{i_0}}}\) is more complex because \(v_{t_{i_0}}\) appears in all terms.

Hessian Tensor

The Hessian tensor represents the second-order partial derivatives of the output with respect to the input. For the VolumeMapping 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:

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

It can be noticed that for \(0 \le i < n\) and \(0 \le j < 4\):

\[ \frac{\partial D_{iv_{t_{i_j}}}}{\partial v_{t_{i_j}}} = 0 \]

Then,

\[ \frac{\partial D_{iv_{t_{i_0}}}}{\partial v_{t_{i_1}}} = \frac{1}{6} [v_{t_{i_2}} - v_{t_{i_3}}]_\times \]

\[ \frac{\partial D_{iv_{t_{i_0}}}}{\partial v_{t_{i_2}}} = \frac{1}{6} [v_{t_{i_3}} - v_{t_{i_1}}]_\times \]

\[ \frac{\partial D_{iv_{t_{i_0}}}}{\partial v_{t_{i_3}}} = \frac{1}{6} [v_{t_{i_1}} - v_{t_{i_2}}]_\times \]

\[ \frac{\partial D_{iv_{t_{i_1}}}}{\partial v_{t_{i_2}}} = \frac{1}{6} [v_{t_{i_0}} - v_{t_{i_3}}]_\times \]

\[ \frac{\partial D_{iv_{t_{i_1}}}}{\partial v_{t_{i_3}}} = \frac{1}{6} [v_{t_{i_2}} - v_{t_{i_0}}]_\times \]

\[ \frac{\partial D_{iv_{t_{i_2}}}}{\partial v_{t_{i_3}}} = \frac{1}{6} [v_{t_{i_0}} - v_{t_{i_1}}]_\times \]

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

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

The other elements of the matrix are obtained by symmetry of the Hessian matrix.