Parallel algorithms for inductance extraction

Abstract

In VLSI circuits, signal delays play an important role in design, timing verification and signal integrity checks. These delays are attributed to the presence of parasitic resistance, capacitance and inductance. With increasing clock speed and reducing feature sizes, these delays will be dominated by parasitic inductance. In the next generation VLSI circuits, with more than millions of components and interconnect segments, fast and accurate inductance estimation becomes a crucial step. A generalized approach for inductance extraction requires the solution of a large, dense, complex linear system that models mutual inductive effects among circuit elements. Iterative methods are used to solve the system without explicit computation of the system matrix itself. Fast hierarchical techniques are used to compute approximate matrix-vector products with the dense system matrix in a matrix-free way. Due to unavailability of system matrix, constructing a preconditioner to accelerate the convergence of the iterative method becomes a challenging task. This work presents a class of parallel algorithms for fast and accurate inductance extraction of VLSI circuits. We use the solenoidal basis approach that converts the linear system into a reduced system. The reduced system of equations is solved by a preconditioned iterative solver that uses fast hierarchical methods to compute products with the dense coefficient matrix. A Green’s function based preconditioner is proposed that achieves near-optimal convergence rates in several cases. By formulating the preconditioner as a dense matrix similar to the coefficient matrix, we are able to use fast hierarchical methods for the preconditioning step as well. Experiments on a number of benchmark problems highlight the efficient preconditioning scheme and its advantages over FastHenry. To further reduce the solution time of the software, we have developed a parallel implementation. The parallel software package is capable of analyzing interconnects con- figurations involving several conductors within reasonable time. A two-tier parallelization scheme enables mixed mode parallelization, which uses both OpenMP and MPI directives. The parallel performance of the software is demonstrated through experiments on the IBM p690 and AMD Linux clusters. These experiments highlight the portability and efficiency of the software on multiprocessors with shared, distributed, and distributed-shared memory architectures.