层次式直接边界元计算VLSI三维互连电容

Hierarchical Computation of 3-D Interconnect Capacitance Using Direct Boundary Element Method

文中将 Appel处理多体问题的层次式算法思想实现于直接边界元法 ,用以计算 VL SI三维互连寄生电容 .直接边界积分方程同时含有边界上的电势与法向电场强度 ,能比间接边界元法更方便地处理多介质及有限介质结构 .直接边界元法的层次式计算涉及对三种边界 (强加边界、自然边界与介质交界面 )及两种积分核 (1/r与 1/r3)的处理 ,显著区别于基于间接边界元法、仅处理强加边界与一种积分核的层次式算法 .文中以边界元的层次划分为基础 ,实现系数矩阵的层次式形成与存储 ,并在 GMRES迭代中实现层次式矩阵——向量乘法 ,显著提高了计算速度 .

The idea of Appel's hierarchical algorithm handling the many body problem is implemented in the direct boundary element method (BEM) for computation of 3 D VLSI parasitic capacitance. Both the electric potential and normal electric field intensity on the boundary are involved, so it can be much easier to handle problems with multiple dielectrics and finite dielectric structure than the indirect BEM. Three kinds of boundaries (forced boundary, natural boundary and dielectric interface) are treated. Two integral kernels with different singularity ( 1/r, 1/r 3 ) are involved while computing the interaction between the boundary elements. These features make it significantly distinct from the hierarchical algorithm based on the indirect BEM, which only handles single dielectric, one integral kernel and one forced boundary. The coefficient matrix is generated and stored hierarchically in this paper. As a result, computation cost of the matrix is reduced, and the matrix vector multiplication in the GMRES iteration is accelerated, so computation speed is improved significantly.