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.

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泊松方程解析解及其在VLSI全局布局中的应用

泊松方程已在 VLSI 全局布局中使用，用于描述由给定电荷密度分布引起的势场。与之前以数值方式求解泊松方程的全局放置方法不同，在本文中，我们提供了方程的解析解来计算静电系统的势能。基于变量分离法和精确密度函数导出解析解，以对放置区域中的块分布进行建模，该块分布是无限级数且绝对收敛。利用解析解，我们给出了泊松方程的快速计算方案，并开发了一种有效且高效的全局放置算法，称为 Pplace。实验结果表明，我们的 Pplace 实现了比 ePlace 和 NTUplace3 更小的贴装线长。