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

Poisson's equation has been used in VLSI global placement for describing the potential field induced 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 a 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, two leading wirelength-driven placers. 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 to these fields.