PySAGAS Definitions
This page defines all symbols, terms and data types used by PySAGAS.
Nomenclature
Some of the nomenclature used by PySAGAS is defined in the table below.
Symbol |
Description |
|---|---|
\(\mathcal{\underline{v}}\) |
Vertex vector |
\(\mathcal{\underline{v}}\) |
Normal vector |
\(\underline{\theta}\) |
Design parameters vector |
\(A\) |
Area |
Cell Definition
A Cell in PySAGAS is a triangular element, defined by three
unique vectors pointing to the cell’s vertices,
\(\underline{v}_0, \underline{v}_1\) and \(\underline{v}_2\). These vertices
form the corners of the triangular cell, and also form the face
of the cell. This face has both an area \(A\) and a normal vector
\(\underline{n}\) associated with it. These properties are defined in
the figure below.
See also
The Cell definition shown below is consistent with the
Cell object.
Cell Normal and Area Sensitivities
Given the definition of a cell above, the sensitivities of the cell normal and the cell area to the cell’s vertices can be determined. That is, \(\frac{\partial\underline{n}}{\partial\underline{v}}\) and \(\frac{dA}{\partial\underline{v}}\), respectively. The figure below exemplifies how a cell’s normal vector and area changes with variations in one of its vertices, \(\underline{v}_2\) to \(\underline{v}_2'\).
These sensitivities can be calculated using analytical derivitives.
See also
The calculations of \(\frac{\partial\underline{n}}{\partial\underline{v}}\) and
\(\frac{dA}{\partial\underline{v}}\) are implemented in
n_sensitivity() and A_sensitivity(),
respectively.
Geometric Parameter Sensitivities
Although users of PySAGAS are required to provide their own geometric parameter sensitivities, the figure below may be insightful. To be clear, a user must provide the sensitivity of each vertex defining a geometry to the design parameters, that is, \(\frac{\partial\underline{v}}{\partial\underline{\theta}}\). The diagram in the figure below illustrates a cell’s vertices changing as a result of a change in a parameter \(\theta_1\).
Given \(\frac{\partial\underline{v}}{\partial\underline{\theta}}\), the sensitivity of both the cell normals and cell areas to the deisgn parameters can be calculated using the chain rule, as per the equations below.