Poisson Energy Formulation for Floorplanning: Variational Analysis and Mathematical Foundations

Arranging many modules within a bounded domain without overlap, central to the Electronic Design Automation (EDA) of very large-scale integrated (VLSI) circuits, represents a broad class of discrete geometric optimization problems with physical constraints. This paper develops a variational and spectral framework for Poisson energy-based floorplanning and placement in physical design. We show that the Poisson energy, defined via a Neumann Poisson equation, is exactly the squared H^{-1} Sobolev norm of the density residual, providing a functional-analytic interpretation of the classical electrostatic analogy. Through spectral analysis, we demonstrate that the energy acts as an intrinsic low-pass filter, suppressing high-frequency fluctuations while enforcing large-scale uniformity. Under a mild low-frequency dominance assumption, we establish a quantitative linear lower bound relating the Poisson energy to the geometric overlap area, thereby justifying its use as a smooth surrogate for the hard nonoverlap constraint. We further show that projected gradient descent converges globally to stationary points and exhibits local linear convergence near regular minima. Finally, we interpret the continuous-time dynamics as a Wasserstein-2 gradient flow, revealing the intrinsic nonlocality and global balancing behavior of the model. These results provide a mathematically principled foundation for PDE-regularized optimization in large-scale floorplanning and related geometric layout problems.