  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.
