# Efficient Parallel FDTD Method Based on Non-Uniform Conformal Mesh
## 2 Theory and Methods
### The Non-Uniform Conformal FDTD Algorithm  
#### 2.1.1 Internal Field Update  
As shown in Figure 1, conformal grids are applied at the curved surface boundary of the object, where \(l_{x}\) and \(l_{y}\) represent the grid cell dimensions, excluding the perfect electric conductor (PEC) parts, and \(A_{z}\) denotes the grid cell area, excluding the PEC parts. The spatial distribution of the electromagnetic field components is consistent with the conventional Yee grid.  
The core idea of the conformal FDTD method is to replace the differential discretization equations with integral discretization equations based on [PERSON]'s electromagnetic induction law at the conformal grid boundaries of the model [18], aiming to reduce errors caused by the discrete approximation of differential equations. The integral equation of [PERSON]'s electromagnetic induction law is as follows:  
\[\int_{S}\mu_{0}\frac{\partial\overset{\rightarrow}{H}}{\partial t}\cdot d \overset{\rightarrow}{s}=-\oint_{I}\overset{\rightarrow}{E}d\overset{ \rightarrow}{l} \tag{1}\]  
Since the electromagnetic field inside the ideal metal is zero, the discrete loop integral of the electric field equals the electric field multiplied by the grid cell dimensions, excluding the ideal metal parts, and the integral of the magnetic field over the grid face equals the magnetic field multiplied by the area of the grid face, excluding the ideal metal parts. Taking the z-component as an example (other components follow a similar logic), the discrete formula for the magnetic field can be expressed as the following:  
\[\begin{split} H_{z}^{n+\frac{1}{2}}(i+\frac{1}{2},j+\frac{1}{2},k )=& H_{z}^{n-\frac{1}{2}}(i+\frac{1}{2},j+\frac{1}{2},k)\\ &+\frac{\Delta t}{\mu_{0}}\cdot\frac{1}{A_{z}(i+\frac{1}{2},j+ \frac{1}{2},k)}\cdot\left[\begin{array}{l}-E_{x}^{n}(i+\frac{1}{2},j,k)\cdot l _{x}(i+\frac{1}{2},j,k)\\ +E_{x}^{n}(i+\frac{1}{2},j+1,k)\cdot l_{x}(i+\frac{1}{2},j+1,k)\\ -E_{x}^{n}(i+1,j+\frac{1}{2},k)\cdot l_{y}(i+1,j+\frac{1}{2},k)\\ +E_{y}^{n}(i,j+\frac{1}{2},k)\cdot l_{y}(i,j+\frac{1}{2},k)\end{array}\right] \end{split} \tag{2}\]  
The discrete update formula for the electric field component is consistent with the formula used in the FDTD method with non-uniform grids. Specifically,  
\[\begin{split} E_{z}^{n+1}\Big{(}i,j,k+\frac{1}{2}\Big{)}& =E_{z}^{n}\Big{(}i,j,k+\frac{1}{2}\Big{)}-\frac{\Delta t}{\varepsilon_{0}} \cdot\left[\begin{array}{l}H_{y}^{n+1/2}\big{(}i+\frac{1}{2},j,k+\frac{1}{2 }\big{)}-H_{y}^{n+1/2}\big{(}i-\frac{1}{2},j,k+\frac{1}{2}\big{)}\\ \hbar_{x_{i}}\end{array}\right.\\ &\left.-\frac{H_{x}^{n+1/2}\big{(}ij+\frac{1}{2},k+\frac{1}{2}\big{)}-H_{ y}^{n+1/2}\big{(}ij-\frac{1}{2},k+\frac{1}{2}\big{)}}{\hbar_{y_{j}}}\right] \end{split} \tag{3}\]  
where the edge length between non-uniform grid points is given by the following:  
\[\left\{\begin{array}{l}\Delta x_{i}=x_{i+1}-x_{i},i=1,\cdots,N_{x}-1\\ \Delta y_{j}=y_{j+1}-y_{j},j=1,\cdots,N_{y}-1\\ \Delta z_{k}=z_{k+1}-z_{k},k=1,\cdots,N_{z}-1\end{array}\right. \tag{4}\]  
Figure 1: Schematic diagram of surface boundary conformal mesh.  
The edge length in Formula (3) is given by the following:  
\[\left\{\begin{array}{l}h_{x_{i}}=\left(\Delta x_{i}-\Delta x_{i-1}\right)/2,i=2, \cdots,N_{x}-1\\ h_{y_{j}}=\left(\Delta y_{j}-\Delta y_{j-1}\right)/2,j=2,\cdots,N_{y}-1\\ h_{z_{k}}=\left(\Delta z_{k}-\Delta z_{k-1}\right)/2,k=2,\cdots,N_{z}-1\end{array}\right. \tag{5}\]