ngnet

This is an NGnet implementation. It is used for shape representation in the NGnet on/off method[3].

Overview

NGnet $y(\boldsymbol{w}, \boldsymbol{x})$ is defined as follows (for a two-dimensional coordinate space).

$$y(\boldsymbol{w}, \boldsymbol{x}) = \sum_{i=1}^{N} w_i b_i(\boldsymbol{x}) \\ b_i(\boldsymbol{x}) = \frac{G_i(\boldsymbol{x})}{\sum_{j=1}^N G_j(\boldsymbol{x})} \\ G_i(\boldsymbol{x}) = \frac{1}{2\pi \sigma_i^2}\exp(-\frac{\|\boldsymbol{x}-\boldsymbol{\mu_i}\|^2}{2\sigma_i^2}) \\ \boldsymbol{w} = \{w_i\}, \boldsymbol{x} = \{x, y\}$$

$\boldsymbol{w}$: Weight vector
$\boldsymbol{x}$: Coordinate vector
$N$: Number of Gaussian basis functions in the NGnet
$\mu_i$: Center vector of the $i$th Gaussian basis function $G_i(\boldsymbol{x})$
$\sigma_i$: Standard deviation of the $i$th Gaussian basis function $G_i(\boldsymbol{x})$

In the NGnet on/off method, optimizing the weight vector $\boldsymbol{w}$ changes the distribution of $y(\boldsymbol{w}, \boldsymbol{x})$, and the material at each position $\boldsymbol{x}$ in the design region is determined from its value.
The original paper defines locations where $y(\boldsymbol{w}, \boldsymbol{x})$ is greater than 0 as the ON-state material (magnetic core), and locations where it is 0 or less as the OFF-state material (air).

Configurable keyword arguments

• coordinate: str ... Coordinate system used to construct the NGnet: "Cartesian" or "Polar".
• sigma: float ... Standard deviation of the Gaussian basis functions in the NGnet. The default is 1.
• design_region: list[list[float, float], list[float, float]] ... Region used to construct the NGnet (the design region). For Cartesian, specify a rectangular region $[[x_1, x_2], [y_1, y_2]]$; for Polar, specify a sector region $[[r_1, r_2], [\theta_1, \theta_2]]$.

Gaussian basis-function arrangement

By default, the Gaussian basis functions that make up the NGnet are automatically arranged to fill design_region, as shown below.
Each circle in the figure represents the placement of a Gaussian basis function, and its radius represents the standard deviation. The figure is output as gaussian.png directly under the EMSOptimizer folder when shape optimization is run.

• distance_factor: float ... Determines the spacing between Gaussian basis functions. Values below 1 place the functions so that they overlap more. The default is 0.8.
• dimension: str ... Dimension in which to arrange the Gaussian basis functions: "2D" or "3D". The default is "2D".
• eliminate_bases_on_edge: bool ... Whether to remove Gaussian basis functions at the region boundaries. The default is False.
• normalize_output: bool ... Whether to normalize the NGnet output. When False,
  $y(\boldsymbol{w}, \boldsymbol{x}) = \sum_{i=1}^{N} w_i G_i(\boldsymbol{x}) \\$ is used. The default is True.
• fixed_design_region: list[list[float, float], list[float, float]] | null ... Force the output outside this range to 1. Specify it in the same way as design_region.
• check_basis_GUI: bool ... When True, launch a GUI at startup to inspect the arrangement of the Gaussian basis functions.