## 2 Materials and Methods
### Governing Equations
#### 2.2.3 Governing Equations for Lifting and Rotating Motion Computation  
In the proposed computation model, the motion of moving components can be decomposed into two modes, rotating motion and lifting motion, and we used two methods to set the governing equations of these two modes of motion separately. First, we noted that the transformation optics method has a good advantage in computing microwave heating processes with rotating motion [20]. Based on the method, the coordinate transformation is carried out according to the motion mode of the sample, and the tensor parameter of the transform optical region is computed to control the electromagnetic field in the transform optical region. Meanwhile, the implicit function and level set methods can accurately compute the microwave heating process with regular motion modes, such as lifting motion [22], using this method to solve the computation of lifting motion. In this research, the two methods are combined to solve the computation of spiral motion during microwave heating. To compute the rotating motion, we set up two transform optical regions outside the motion region, as shown in Figure 2a. The coordinate transformation of transform optical regions 1 and 2 is defined as:  
\[r^{\prime}=r \tag{6}\]  
\[\theta^{\prime}= \begin{cases}\theta+\theta_{0}&r<a\\ \theta+\frac{1}{2}f_{1}(r)&a<r<b\\ \theta+\frac{1}{2}f_{2}(r)&b<r<c\\ \theta&r>c\end{cases} \theta^{\prime}= \begin{cases}\theta+\theta_{0}\\ \theta+\frac{1}{2}f_{1}(r)\\ \theta+\frac{1}{2}f_{1}(r)\end{cases}  \tag{7}\]  
where:  
\[f_{1}(r)=(r-b)\frac{1}{a-b}\cdot\theta_{0} \tag{8}\]  
\[f_{2}(r)=(r-c)\frac{1}{b-c}\cdot\theta_{0} \tag{9}\]  
The above equations imply that the turntable region is rotated at the angle \(\theta_{0}\) and transforms optical region 1 and region 2, each bearing a rotating angle of 0.5\(\theta_{0}\). Based on the coordinate transformation equations, the transform optical region's tensor parameters can be obtained from [20]:  
\[\overline{\tilde{\epsilon}}=\epsilon\overline{\tilde{\eta}} \tag{10}\]  
\[\overline{\tilde{\mu}}=\mu\overline{\tilde{\eta}} \tag{11}\]  
where:  
\[\overline{\tilde{\eta}}=\overline{J}\cdot\overline{\tilde{J}}/\text{det} \overline{\tilde{J}} \tag{12}\]  
\(\overline{\tilde{J}}\) is the Jacobian tensor:  
\[\overline{\tilde{J}}= \begin{bmatrix}\partial x^{\prime}/\partial x&\partial x^{ \prime}/\partial y&\partial x^{\prime}/\partial z\\ \partial y^{\prime}/\partial x&\partial y^{\prime}/\partial y&\partial y^{ \prime}/\partial z\\ \partial z^{\prime}/\partial x&\partial z^{\prime}/\partial y&\partial z^{ \prime}/\partial z\end{bmatrix}  \tag{13}\]  
Combining Equations (6)-(13), the \(\overline{\tilde{\eta}}\) of regions 1 and 2 can be obtained:  
\[\overline{\tilde{\eta}}_{i}= \begin{bmatrix}\eta_{11i}&\eta_{12i}&0\\ \eta_{12i}&\eta_{22i}&0\\ 0&0&0\end{bmatrix} (i=1,2) \tag{14}\]  
where:  
\[\eta_{11i}=1+2\xi_{i}k+\xi_{i}{}^{2}n \tag{15}\]  
\[\eta_{12i}=-\xi_{i}^{2}k-\xi_{i}(m-n) \tag{16}\]  
and  
\[m=\frac{x^{\prime 2}}{r^{2}} \tag{17}\]  
\[n=\frac{y^{\prime 2}}{r^{2}} \tag{18}\]  
\[k=\frac{x^{\prime}y^{\prime}}{r^{2}} \tag{19}\]  
\[\xi_{i}=\frac{1}{2}\theta_{0}(t)f_{i}(r)(i=1,2) \tag{20}\]  
where \(\overline{\tilde{\eta}}_{1}\) and \(\overline{\tilde{\eta}}_{2}\) denote the transform tensor of regions 1 and 2, respectively, and \(\theta_{0}(t)\) is the rotation angle of the turntable. Figure 2: (**a**) The division of transform optical regions in a 3D microwave cavity; (**b**) cross-section diagram. Next, we set the implicit function of the lifting motion. As shown in Figure 3, the moving region can be divided into four regions, namely, the air region, the turntable region, the potato region and the rod region. Using the implicit function and level set methods, the lifting region's dielectric coefficient is set as [22]:  
\[ \begin{split}&\varepsilon=\varepsilon_{p}\cdot H\big{(}D_{p}(x,y,z,t) \big{)}+\varepsilon_{t}\cdot H(D_{t}(x,y,z,t))\\ &+\varepsilon_{r}\cdot H(D_{r}(x,y,z,t))+\varepsilon_{air}\cdot H (D_{air}(x,y,z,t))\end{split}