## 2 Multiphysics Design Process for HSPMM  
Due to the high-speed, high-power density and difficult heat dissipation of HSPMM, rotor stress and rotor dynamics, electromagnetic losses and temperature rise of HSPMM need to be considered [17; 18]. In rotor stress and rotor dynamics, sleeve stress of rotor, stress of PM and first-order critical speed of rotor need to be considered. The stress of sleeve and PM of HSPMM should be less than the allowable stress of corresponding material [19]. The first-order critical speed of the rotor structure should be greater with a certain margin than the rated speed, to work below the first-order critical speed. In the multiphysics optimization of electromagnetic torque and losses, the no-load back electromotive force (EMF), output torque, and PM eddy current loss need to be considered. According to the motor design theory, the no-load back EMF is slightly less than the rated load voltage to ensure that the PM can establish a self-excitation magnetic field so that the motor works in a demagnetization state. With the increase of the speed of HSPMM, in order to keep the motor working in the constant torque region under the condition of constant power, it is necessary to consider the torque above the rated torque. In order to ensure that the motor has high efficiency and the temperature rise of the motor is within the controllable range, in addition to the design of the cooling system, the loss shall be small. The core loss and winding copper loss of the motor are mainly related to the loading power frequency, higher harmonics and material parameters of the motor, so it is necessary to consider the eddy current loss and thermal load of the PM. According to the general mathematical model of multi-objective optimization problem [20], the optimization mathematical model of the HSPMM can be established as follows:  
\[\min:\quad\left\{ \begin{array}{l}f_{1}\Big{(}\mathbf{x})=\ -T_{\text{average}}\\ f_{2}(\mathbf{x})=\ -\eta\\ f_{3}(\mathbf{x})=P_{\text{PMloss}}\\ f_{4}(\mathbf{x})=AJ\end{array} \right.\]  
\[\text{s.t. }\quad\left\{ \begin{array}{l}g_{1}(\mathbf{x})=F_{\text{sleeve}}-F _{\text{CF}}\leq 0\\ g_{2}(\mathbf{x})=F_{\text{PM}}-F_{\text{NaFeB}}\leq 0\\ g_{3}(\mathbf{x})=n_{s}-n_{1\text{st}}\times S\leq 0\\ g_{4}(\mathbf{x})=E_{0}-U_{a}\leq 0\\ g_{5}(\mathbf{x})=T_{e}-T_{\text{average}}\leq 0\end{array} \right. \tag{1}\]  
\[x_{l}\leq\mathbf{x}\leq x_{u}\]  
where, \(T_{\text{average}}\), \(\eta\), \(P_{\text{PMloss}}\) represent average output torque, motor efficiency, permanent magnet eddy current loss, respectively and motor thermal load; \(F_{\text{sleeve}}\) and \(F_{\text{CF}}\) denote sleeve stress and stress intensity of carbon fiber material, respectively; \(F_{PM}\) and \(F_{\text{NaFeB}}\) represent the stress of PM and the stress intensity of NdFeB material respectively; \(n_{s}\) and \(n_{1 st}\) represent rated speed of HSPMM and first-order critical speed corresponding to rotor structure respectively, \(S\) represents safety factor; \(E_{0}\) and \(U_{a}\) represent the no-load line back EMF and the loading terminal voltage of the HSPMM respectively; \(T_{e}\) indicates rated torque. Based on the aforementioned analysis, this study incorporates a multi-physics optimization design methodology for HPMSM, which leverages a subspace and sequential optimization strategy. Firstly, the rotor performance of HPMSM motor is considered, the rotor stress intensity, such as PM stress distribution and sleeve stress distribution, is analyzed, and the value range of rotor main parameters is defined [21]. On this basis, the electromagnetic performance and loss characteristics of the motor for multi-objective optimization design. Finally, through the heat dissipation system design and temperature field verification [22; 23], the final multiphysics optimization design scheme is improved. The overall design flow chart is shown in Figure 1.  
Step 1: Determine an initial design scheme for that motor according to a conventional motor design theory and a structural size equation of the motor.