Fig. 1.  (Color online) Schematic diagram of the (a) interlock primal mesh and (c) dual mesh, (b) and (d) the corresponding equivalent capacitance networks, when taking two dimensional (2D) space as an example. The unknown physical quantities utilized by the dual DGMs are located, respectively, on the vertexes of the primal and dual meshes, which are denoted by hollow circles.

Fig. 2.  Relation of the primal-dual pair of sequences and meshes

Fig. 3.  Equivalent capacitances along the connections among vertexes, i.e., edges, of the primal mesh and the dual mesh, and the elementary contributions of the neighbour elements. As shown in (a), $\sigma$ is the length of the edge shared by the element ${e_i}$ and element ${e_j}$ , $ \sigma^{*}=\sigma^{*}_{i}+\sigma^{*}_{j}$ is the length of the edge of the dual mesh. $\sigma^{*}_{i}$ and $\sigma^{*}_{j}$ belong to ${e_i}$ and ${e_j}$ , respectively, as shown in (b). The relative dielectric constants of ${e_i}$ and ${e_j}$ are $\varepsilon_i$ and $\varepsilon_j$ , respectively. (c) and (e) show the capacitances C and $C^*$ imposed on edges of the primal mesh and the dual mesh, respectively. (d) and (f) show the elementary contributions ${C_i}$ and ${C_j}$ for C, and $C^*_i$ and $C^*_j$ for $C^*$ , respectively. As a note, ${C_i}$ and ${C_j}$ are in parallel connection, and $C^*_i$ and $C^*_j$ are in series connection.

Fig. 4.  (a) Triangular element and elemental contributions to the (b) primal matrix and (c) dual matrix, where O is the circumcenter of the triangle ijk, O’is the foot of the perpendicular on the edge jk, and also the midpoint on jk. ${\alpha _s}$ is the interior angle at node s where s=i, j, and k. The three capacitances in (b) are in delta ( $\Delta $ ) connection, and (c) star (Y) connection.

Fig. 5.  Treatment of boundaries, such as boundary of truncated computational domain denoted by $\mathit{\Gamma}_{e}$ in (a), and Dirichlet boundary denoted by $\mathit{\Gamma}_{c}$ in (b). For $\mathit{\Gamma}_{e}$ , extra vertexes, such as ${v_1}$ and ${v_2}$ , are appended as unknowns. For $\mathit{\Gamma}_{c}$ , since the conductor is a equipotential body, the elementary contribution of element ${e_j}$ , ${C_j}$ for the primal net and $C_j^*$ for the dual net are eliminated. That is, ${C_j}$ in (c) is set to zero (0) and $C_j^*$ in (d) is set to infinity ( $\infty$ ). For the dual net, ${v_j}$ is moved to ${v_{\left({ij} \right)}}$ , which is the intersection point of edge ${v_i}$ ${v_j}$ and boundary $\mathit{\Gamma}_{c}$ .

Fig. 6.  (Color online) Schematic diagram of boundary condition. Only part of the meshes are presented. The Dirichlet boundary conditions are imposed on $\mathit{\Gamma_{ei}}$ and $\mathit{\Gamma}_{ej}$ , i.e., the surfaces of conductor i and j. The Dirichlet nodes along the surfaces of conductor i and j are denoted by empty circles in black, and unknown nodes are denoted by dots in blue.

Fig. 7.  (Color online) Integration path { $\mathit{\Gamma} _{p}$ } and { $\mathit{\Gamma} _{d}$ } for electric flux computation in the case of (a) primal solution and (b) dual solution, respectively. Obviously, the flux can be obtained easily by summing up the flux inside each equivalent capacitance that connects to the boundary of the objective conductor.

Fig. 8.  Example model (not to scale) for interconnect capacitances with multiple conductors. In nominal condition, conductors 1, 2 and 3 have the same dimensions of 4 (line width W) $\times$ 4 μm² (line thickness T), and the conductor 0 24 $\times$ 2 μm². The distance between every two conductors is 2 μm, including the spacing S and D₂₀. The dimensions of rectangle computation domain are 40 $\times$ 30 μm², where the four conductors are centered. The relative permittivity of dielectric material surrounding the conductors is 3.70.

Fig. 9.  (Color online) Primal mesh (up) and corresponding dual mesh (down) of a 4-conductor electrostatic system. The primal mesh demonstrated consists of 883 nodes and 1668 triangular elements. The dual mesh is constructed through circumcenter from the primal mesh. Only part of the computation domain is shown.

Fig. 10.  (Color online) (a) The primal and dual results and (b) the average results of capacitance calculated by field solver directly and estimated by first order sensitivities with respect to the variation of the line width W.

Fig. 11.  (Color online) (a) The primal and dual results and (b) the average results of capacitance calculated by field solver directly and estimated by first order sensitivities with respect to the variation of the line thickness $\Delta $ T.

Fig. 12.  (Color online) Absolute magnitude of relative errors of capacitance sensitivity of $C_{22}$ between results calculated by field solver and by sensitivities with respect to the variation of the (a) line width $\Delta W$ and (b) line thickness $\Delta T$ , in case of the primal, dual and average solutions.

Fig. 13.  (Color online) Absolute magnitude of relative errors of capacitance sensitivity of ${C_{20}}$ , ${C_{21}}$ , and ${C_{22}}$ between results calculated by field solver and by sensitivities with respect to the variation of the (a) line width $\Delta W$ and (b) line thickness $\Delta T$ , in case of the average solution.