A Stable FDTD Subgridding Scheme with SBP-SAT for Transient Electromagnetic Analysis

Abstract

We proposed a provably stable FDTD subgridding method for accurate and efficient transient electromagnetic analysis. In the proposed method, several field components are properly added to the boundaries of Yee's grid to make sure that the discrete operators meet the summation-by-parts (SBP) property. Then, by incorporating the simultaneous approximation terms (SATs) into the finite-difference time-domain (FDTD) method, the proposed FDTD subgridding method mimics the energy estimate of the continuous Maxwell's equations at the semi-discrete level to guarantee its stability. Further, to couple multiple mesh blocks with different mesh sizes, the interpolation matrices are also derived. The proposed FDTD subgridding method is accurate, efficient, easy to implement and be integrated into the existing FDTD codes with only simple modifications. At last, three numerical examples with fine structures are carried out to validate the effectiveness of the proposed method.

A SBP-SAT FDTD Subgridding Method Using Staggered Yee's Grids Without Modifying Field Components

“…Since the Yee's grid is used in the proposed SBP-SAT FDTD method, the field locations in the grid are exactly the same as those in the traditional FDTD method, as shown in [31], where additional electric nodes are added on the boundaries to make the discrete operators satisfy the SBP property, field values on the boundaries are extrapolated when necessary.…”

“…However, one difficulty to meet the SBP property in (7) is that the magnetic fields on the boundaries are not defined in the Yee's grid. In [31], we added additional nodes to overcome this issue. In this paper, the extrapolation method, which is defined as the project operator [32], is used to calculate the magnetic fields on the boundaries.…”

“…However, four additional electric nodes are added at the corners of the computational domain, and several other electric nodes are added in the middle of edges of boundary cells. Compared with the existing and our previous works in [31], our work in this paper advances in the following three aspects.…”

“…In [30], a two-dimensional SBP operator with collocated grids and a one-dimensional SBP operator with staggered grids were proposed for the Maxwell-Duffing models. To further extend the capability in the Maxwell's equations, a provably stable SAT-SBP FDTD method with the staggered grid is proposed, and the interpolation matrices for stable coupling multiple mesh blocks are also investigated in [31].…”

“…Compared with the work in [30], the collocated grid is used for simulations, which shows significant different from the traditional FDTD method with the staggered Yee's grids. As mentioned before, the SBP-SAT FDTD method with the staggered grid was proposed in [31] to further extend its capability in solving the electromagnetic problems. However, four additional electric nodes are added at the corners of the computational domain, and several other electric nodes are added in the middle of edges of boundary cells.…”