Generating Geometry Sensitivities

HyperVehicle also features the generation of geometry sensitivities via method of finite differences. This page outlines how to use this capability.

Nomenclature

Symbol

Description

\(\underline{\theta}\)

The set of design parameters defining a geometry

\(\mathcal{G}\)

The geometry mesh

Description of Implementation

  1. A HyperVehicle parameterised geometry Generator is defined, along with the parameters to be varied.

  2. Geometry meshes for the nominal geometry \(\mathcal{G}_\text{nominal}\) are generated.

  3. For each parameter \(\theta_i\) to be varied:

    • The perturbed parameter value is defined as \( \theta_i' = \theta_i + \delta\theta_i \).

    • Geometry meshes for the parameter set including \(\theta_i'\) are generated, yielding \(\mathcal{G}_i'\).

    • The sensitivity of each vertex of \(\mathcal{G}\) to the parmater perturbation \(\theta_i'\) is calculated by comparing the perturbed geometry \(\mathcal{G}_i'\) to the nominal geometry \(\mathcal{G}_\text{nominal}\). That is, \(\delta \mathcal{G}_i = \mathcal{G}_i' - \mathcal{G}_\text{nominal}\).

    • Finally, the sensitivity is approximated via:

\[ \frac{\partial\mathcal{G}}{\partial\theta} \approx \frac{\delta \mathcal{G}}{\delta \theta} \]

Working with Mutliple-Component Geometries

Merging components with PyMesh

If you have PyMesh installed, simply pass merge=True when calling Vehicle.to_stl(). This will merge all components of your Vehicle into a single file.

Cart3D intersected Components

Cart3D is a component based solver, meaning oftentimes, many individual components are stacked to form the final geometry. This final geometry is a single body, made up of many combined components. This body has a different mesh to the individual component STL meshes.

When working with multiple components, we need to map the component sensitivities onto the final combined mesh. The utility function append_sensitivities_to_tri contained within the hypervehicle.utilities module provides this capability. See the code below, which first finds the sensitivity files using glob, then runs append_sensitivities_to_tri. The Cart3D file, Components.i.tri will be overwritten, with the sensitivity data appended. A CSV file called all_components_sensitivity.csv will also be written, containing the sensitivity information as point data. This file is compatible with PySAGAS.

import glob
from hypervehicle.utilities import append_sensitivities_to_tri

sens_files = glob.glob("*sensitivity*")
append_sensitivities_to_tri(sens_files)

You can view this data of the Components.i.tri file in ParaView, by first converting the file into a Tecplot format using the Cart3D utility function trix:

trix Components.i.tri -T

Example

Interested in seeing this theory in application? Check out the parameter sensitivities tutorial.