A lunar relay carries its antenna and electronics on a light structural deck, beside reaction wheels and antenna mechanisms that shake at their own frequencies. Every change in payload mass, mount stiffness, or joint preload moves the structure’s modes, and a mode that lands in a disturbance band is a pointing problem. The usual check re-runs an eigenvalue analysis for every variant. Tensor Spectral certifies how many modes of the finite-element model fall below each threshold, for every variant, using prime-character reduction and integer-exact lattice algebra, and reuses the unchanged structure from one design change to the next.
Cold time per run: two thresholds, three preloads, every count certified. Lower is better.
One CPU core for each solver on the same desktop computer; cold time is the median of three runs, memory the largest whole-process peak observed. MUMPS 5.8.2 was given the same separation of the three displacement directions, so it factored a third of the unknowns (250,047 at the largest size), and both solvers reuse the joint’s attachment response across preloads. MUMPS used AMF ordering for the five smaller sizes and, for the largest, METIS, the fastest of the five orderings tried there. Tensor’s times include compiling the model, writing the output, and replaying the proof. 4 November 2026. Exact readings.
| degrees of freedom | Tensor time | Tensor memory | MUMPS time | MUMPS memory |
|---|---|---|---|---|
| 3 | 0.239 s | 32.5 MB | 0.954 s | 58.0 MB |
| 81 | 0.241 s | 32.4 MB | 0.928 s | 58.9 MB |
| 1,029 | 0.247 s | 32.3 MB | 0.920 s | 58.5 MB |
| 10,125 | 0.255 s | 32.3 MB | 0.965 s | 64.1 MB |
| 89,373 | 0.262 s | 32.3 MB | 1.819 s | 142.4 MB |
| 750,141 | 0.321 s | 32.3 MB | 22.222 s | 1,306.7 MB |
Which payload and mount variants keep the deck’s modes clear of the reaction-wheel and antenna-mechanism bands, and which need more analysis or a redesign. Each threshold and each preload is counted from the same compiled structure: the joint’s attachment response is computed once and reused, so a change at the mount does not start the analysis over.
A mode count is the number of natural frequencies below a threshold. Tensor Spectral certifies each count for the declared finite-element model with integer-exact lattice algebra, and an independent replay checks every one. Around the nonlinear joint, the counts describe small vibrations about its declared equilibrium, with the geometric stiffness of the prestress included.