White paper · No. 03

WarpStack-MOR — localized design-space exploration

Method: physics-based compact model Package: 2nm CoWoS HBM4 + GPU Scope: localized exploration Series: WarpStack white papers

Abstract

Localized means something precise here: the compact model is built once over a bounded box around a nominal design — by default ±10% on each material property — and inside that box every answer carries a rigorous error bound with it. It is a physics-based reduction of the same 3D formulation, not a black-box fit to data. On a what-if deliberately chosen outside the box (ABF substrate stiffened 24.0→28.8 GPa, a change that moves peak-to-peak warpage by +40.9 µm) it tracks the full 3D solve to within 0.036 µm while running in 51 µs — 798,000× faster. This paper reports the method agreement, the eight-point validation across the material design space, and the cost model.

A what-if, resolved

Microseconds per design point, with a bound attached

The compact model is built over a bounded box around a nominal design — by default ±10% on each material property — and inside that box every answer carries a rigorous error bound with it. Ask for a point outside the box and it still answers, still reports its bound, and tells you that you have left the region it was built for. Change the floorplan and it refuses to load at all.

The nominal design, before any what-if

2nm CoWoS · 8 HBM stacks, 2 GPUs, 108 chiplets · active surface

51 µs

WarpStack-MOR warpage map of the nominal 2nm CoWoS package — a broad central dome falling away toward the package corners, with the GPU and HBM outlines drawn — WarpStack-MOR · nominal · µmThe nominal result. The warpage surface of the 2nm CoWoS package at its nominal material set — returned by WarpStack-MOR in 51 µs, matching the full 3D solve to within 0.04 µm. The what-if below starts from here.

A what-if the model was never built at

ABF substrate stiffened 24.0 → 28.8 GPa

798,000× faster

The WarpStack-MOR is a physics-based compact model built once per design from the same 3D formulation — not a black-box fit to data. Here the substrate is stiffened, a change that moves peak-to-peak warpage by +40.9 µm. The WarpStack-MOR resolves that shift to within 0.036 µm — in 51 µs rather than 40.6 s.

3D structure view of the 2nm CoWoS package — two GPU dies flanked by four HBM stacks each side on a silicon interposer over the substrate — Design · 3D structureThe stack. 108 chiplet instances, 237 mechanical layers, straight from IEEE 3Dblox.

Three-dimensional surface rendering of the warpage predicted by the full 3D solve for the stiffened-substrate variant — a broad dome peaking at the package centre and falling away steeply at the corners — Detailed 3D · referencefull 3D solve. 1372.7 µm peak-to-peak, solved from scratch in 40.6 s.

Three-dimensional surface rendering of the warpage predicted by WarpStack-MOR for the same stiffened-substrate variant — visually indistinguishable from the full 3D solve surface beside it — WarpStack-MOR · same pointWarpStack-MOR. 1372.6 µm peak-to-peak — returned in 51 µs.

Signed difference map between the full 3D solve and WarpStack-MOR surfaces, four orders of magnitude smaller than the warpage itself — 3D − WarpStack-MORThe difference. Signed residual, at most 0.036 µm — 0.003% of the warpage being predicted.

Both maps are the active surface at the same mesh and the same thermal load, so this compares the two methods like for like rather than against measurement. On this package the active and continuous package surfaces agree to within 0.1% of peak-to-peak — the bow is global, so the die-level and package-level views coincide.

How closely WarpStack-MOR tracks 3D

Stiffened-substrate what-if · same mesh, same load

within 0.036 µm

Quantity	Full 3D solve	WarpStack-MOR
Peak-to-peak warpage	1372.7 µm	1372.6 µm
Time per design point	40.6 s	51 µs
Peak-to-peak difference	reference	0.043 µm
Whole-map difference	—	0.003%
Largest local difference	—	0.036 µm

This point was deliberately chosen outside the set the model was built from. A further 8 design points drawn from across the material design space were each solved with a full 3D solve and compared: worst whole-map difference 0.006%.

What that buys you

Design points per hour, single core

798,000×

WarpStack-MOR

70.8M

full 3D solve

Building WarpStack-MOR for this design costs 17 min of 3D solves up front — about 25× a single 3D run — after which each additional design point is 51 µs. The investment pays for itself within roughly 25 evaluations, and everything past that is effectively free. That is what turns warpage from a sign-off check into something you sweep.

Read these numbers for speed and for agreement, not for absolute warpage. Mechanical properties for this reference design are illustrative values for the material classes involved, not calibrated 2nm, HBM4 or CoWoS process data, and its peak-to-peak figures are quoted at a fixed resolution rather than settled against refinement. The timings, the speedup, and the agreement between the 3D reference and WarpStack-MOR do not depend on either. For warpage that has been checked across resolutions, see white paper No. 01.

WarpStack-MOR — localized design-space exploration

Microseconds per design point, with a bound attached

The nominal design, before any what-if

A what-if the model was never built at

How closely WarpStack-MOR tracks 3D

What that buys you

Put warpage in the loop.