## 2 Time-Domain Finite Integration Theory (TD-FIT)
### Discrete Material Relationship  
According to the constitutive relations of continuous fields, for non-dispersive and isotropic (or orthotropic) materials, material matrices in the Yee-cell are diagonal [32]:  
\[\left(\mathbf{D}_{\epsilon}\right)_{n,n}=\frac{\widehat{\widehat{d}}_{n}}{ \widehat{e}_{n}}=\frac{\int_{\widetilde{\mathcal{A}}_{n}}\mathbf{D}\cdot d \boldsymbol{A}}{\int_{\widetilde{\mathcal{L}}_{n}}\boldsymbol{E}\cdot d \boldsymbol{s}}\approx\overline{\epsilon}\frac{\widetilde{\mathcal{A}}_{n}}{L_ {n}}, \tag{12}\]  
\[\left(\mathbf{D}_{\sigma}\right)_{n,n}=\frac{\widehat{\widehat{i}}_{n}}{ \widehat{e}_{n}}=\frac{\int_{\widetilde{\mathcal{A}}_{n}}\boldsymbol{J}\cdot d \boldsymbol{A}}{\int_{\widetilde{\mathcal{L}}_{n}}\boldsymbol{E}\cdot d \boldsymbol{s}}\approx\overline{\sigma}\frac{\widetilde{\mathcal{A}}_{n}}{L_ {n}}, \tag{13}\]  
\[\left(\mathbf{D}_{\mu}\right)_{n,n}=\frac{\widehat{\widehat{b}}_{n}}{\widehat {h}_{n}}=\frac{\int_{\widetilde{\mathcal{A}}_{n}}\boldsymbol{B}\cdot d \boldsymbol{A}}{\int_{\widetilde{\mathcal{L}}_{n}}\boldsymbol{H}\cdot d \boldsymbol{s}}\approx\overline{\mu}\frac{A_{n}}{L_{n}}, \tag{14}\]  
where \(\mathbf{D}_{\epsilon}\), \(\mathbf{D}_{\sigma}\), and \(\mathbf{D}_{\mu}\) denote diagonal matrices of permittivity, conductivity, and permeability, respectively. Normally, matrix operators \(\mathbf{D}_{\epsilon}\) and \(\mathbf{D}_{\mu}\) are positive semi-definite and \(\mathbf{D}_{\sigma}\) is singular. \(A_{n}\) and \(\widetilde{\mathcal{A}}_{n}\) denote areas of the \(n\)-th primal facet and dual facet, respectively; \(L_{n}\) and \(\widetilde{L}_{n}\) denote lengths of the \(n\)-th primal edge and dual edge, respectively. \(\overline{\epsilon}\), \(\overline{\sigma}\), and \(\overline{\mu}\) denote material coefficients after properly weighted average.  
Figure 4 illustrates the sub-cell averaging scheme, where the averaged permittivity and conductivity parameters are defined by the dual grid facet \(\widetilde{A}_{i}\) contained in the four adjacent primary grids having a common edge \(L_{i}\) and the averaged permeability parameter is defined by the dual edge \(\widetilde{L}_{k}\) penetrating two adjacent primary grids having a common facet \(A_{k}\).  
The weighted average material coefficients \(\overline{\epsilon}\), \(\overline{\sigma}\), and \(\overline{\mu}\) can be obtained as:  
\[\overline{\epsilon}_{n}=\ \left(A_{u,n}\epsilon_{u,n}+A_{u,n-M_{v}}\epsilon_{u,n-M_{v }}+A_{u,n-M_{w}}\epsilon_{u,n-M_{w}}+A_{u,n-M_{w}-M_{w}}\epsilon_{u,n-M_{w}} \right)\left(4\widetilde{A}_{u,n}\right), \tag{15}\]  
\[\overline{\sigma}_{n}=\ \left(A_{u,n}\sigma_{u,n}+A_{u,n-M_{v}}\sigma_{u,n-M_{v }}+A_{u,n-M_{w}}\sigma_{u,n-M_{w}}+A_{u,n-M_{w}-M_{w}}\sigma_{u,n-M_{v}-M_{w}} \right)\left(4\widetilde{A}_{u,n}\right), \tag{16}\]  
\[\overline{\mu}_{n}=\ \widetilde{L}_{n}\left/\left(\frac{L_{n-M_{u}}}{2\mu_{u,n-M _{u}}}+\frac{L_{n}}{2\mu_{u,n}}\right), \tag{17}\]  
where \(u=x,\ y,\ z;\ v=y,\ z,\ x;\ w=z,\ x,\ y\).  
Obviously, discrete material relations are the only places where approximations are required in the spatial discretization process of [PERSON]'s equations [29].  
Figure 4: Construction of the material matrices. (**a**) The four cells involved in the average of permittivity and conductivity, (**b**) the two cells involved in the average of permeability.