Multi-fidelity error-estimate-based model management

Abstract

This paper presents a novel multi-fidelity model-management framework based on the estimated error between the low-fidelity and high-fidelity models. The optimization algorithm is similar to classical multi-fidelity trust-region model-management approaches, but it replaces the trust-radius constraint with a bound on the estimated error between the low- and high-fidelity models. This enables globalization without requiring the user to specify non-intuitive parameters such as the initial trust radius, which have a significant impact on the cost of the optimization yet can be hard to determine a priori. We demonstrate the framework on a simple one-dimensional optimization problem, a series of analytical benchmark problems, and a realistic electric-motor optimization. We show that for low-fidelity models that accurately capture the trends of the high-fidelity model, the developed framework can significantly improve the efficiency of obtaining high-fidelity optima compared to state-of-the-art multi-fidelity optimization methods and a direct high-fidelity optimization.

Appendix: Reluctivity model

We model the reluctivity, $\nu \left( {\varvec{B}}\right)$, as a piecewise-continuous function where each sub-function of $\nu$ is based on the material it is in. We use constant values for the reluctivity in the motor’s air-gap, magnets and windings. For the air-gap and motor windings, this function takes the value of the reluctivity of free space $\nu _{\text {0}} = \frac{1}{\mu _{\text {0}}} = \frac{1}{4 \pi \times 10^{-7}}$. For the magnets, we use the constant value $\nu _{\text {mag}} = \frac{1}{\mu _{\text {r}} \mu _{\text {0}}}$, where $\mu _{\text {r}}$ is the magnet’s relative permeability, a material-dependent value listed in a material data-sheets. We take $\mu _\text {r} = 1.04$ for the $\text {Nd}_2\text {Fe}_{14}\text {B}$ magnets considered in this work.

The reluctivity of the magnetic steel used in the motor’s stator and rotor is a nonlinear function of the magnetic flux density. We use the model given as follows:

$$\begin{aligned} \nu _{\text {Fe}} = \exp \left( f\left( \Vert {\varvec{B}}\Vert \right) \right) , \end{aligned}$$

where $f\left( \Vert {\varvec{B}}\Vert \right)$ is a cubic B-spline that represents the log-transformed reluctivity as a function of the magnitude of the magnetic flux density. The B-spline knot vector and control points are found by minimizing the least-squares error between the spline and discrete $B$-$\nu$ data points. The control points and knot vector for the Hiperco 50 magnetic steel used for the results presented in this work are listed in Table 8.

Table 8 Control points and knot vector for the cubic B-spline fit used in the Hiperco 50 reluctivity calculation