Analytical Solution of Poisson's Equation with Application to VLSI Global Placement

TL;DR

This paper presents an analytical solution to Poisson's equation for VLSI global placement, enabling faster and more accurate potential calculations compared to numerical methods, leading to improved placement algorithms.

Contribution

It introduces an exact analytical solution to Poisson's equation using separation of variables, facilitating efficient global placement in VLSI design.

Findings

Pplace achieves smaller wirelength than ePlace and NTUplace3.

The analytical solution enables faster computation of potential energy.

The method is robust and applicable to scientific fields involving Poisson's equation.

Abstract

Poisson's equation has been used in VLSI global placement for describing the potential field caused by a given charge density distribution. Unlike previous global placement methods that solve Poisson's equation numerically, in this paper, we provide an analytical solution of the equation to calculate the potential energy of an electrostatic system. The analytical solution is derived based on the separation of variables method and an exact density function to model the block distribution in the placement region, which is an infinite series and converges absolutely. Using the analytical solution, we give a fast computation scheme of Poisson's equation and develop an effective and efficient global placement algorithm called Pplace. Experimental results show that our Pplace achieves smaller placement wirelength than ePlace and NTUplace3. With the pervasive applications of Poisson's equation in scientific fields, in particular, our effective, efficient, and robust computation scheme for its analytical solution can provide substantial impacts on these fields.