Analytical and Force-Directed Floorplanning

The idea

Drop combinatorics entirely. Let every block's center be a continuous (x, y), model each net as springs pulling its pins together (a quadratic wirelength objective, minimized by solving a sparse linear system), and handle the one thing springs get catastrophically wrong, everything collapsing onto one point, with a counterforce: repulsion computed from local density, added each iteration until blocks spread into an overlap-light arrangement. Iterate solve-spread until stable, then snap to a legal floorplan.

This is force-directed placement mathematics (Eisenmann and Johannes' formulation, and the quadratic engines of the Placement catalog) applied at block granularity, and the two fields genuinely merged here: a floorplan with a thousand macros and a placement with a thousand clusters are the same computation.

Trade-offs

Analytical methods are the only ones indifferent to object count: thousands of blocks cost a few sparse solves. Wirelength, being the actual objective rather than a side effect, tends to beat annealed packings on connected designs. The weaknesses are geometric: smooth models ignore exact non-overlap, so a legalization step must convert the fuzzy result into disjoint rectangles (constraint-graph adjustment, or handing the topology to a packer), and hard fixed-outline plus odd aspect ratios can strain the density model.

Where you meet it today

Everywhere the block count is large or blocks mix with standard cells: mixed-size placement engines treat macros as huge cells inside the analytical solve, which is how most modern flows actually floorplan. The electrostatics refinement of this idea (ePlace and RePlAce in the Placement catalog) is the current state of the art, and OpenROAD's macro-aware global placement is exactly this lineage in production.