TOGEARI: Interaction-Space Preconditioning for Condensed Finite-Element Systems with IPC Contact

Abstract

Condensed finite-element contact systems combine a large material-volumetric core with a thin factorized contact update. We formulate Truncated Operator-Gram Eigenspace Approximation of Relevant Interactions (TOGEARI), a factor-space compression of that update. It selects a low-dimensional space of contact combinations and installs their contribution as a Woodbury right preconditioner. The assembled Newton equation and independent residual test remain fixed. The inverse action requires an invertible core and reduced Woodbury system. Core-response selection additionally assumes a symmetric core and remains available when that core is indefinite. For a positive definite core, the response values have an energy ordering. The same analysis then gives the exact generalized spectrum, a scaled inverse-error identity, and the optimal worst omitted interaction at each dimension. A raw contact-factor selector provides a lower-setup empirical alternative. We derive the split for a quadratic-tetrahedral displacement formulation with locally condensed elementwise-constant pressure and a factorized positive-semidefinite IPC normal tangent. In a frozen system with 325,260 free coordinates, an eight-dimensional subspace of a 94-row contact factor with registered numerical rank 42 reduces the Arnoldi basis from 11 vectors to 5. The core-response and raw spaces are closely aligned on the primary and near-repeat extractions; row-norm and deterministic-random controls require 12 vectors. The retained-dimension sweep reduces the warm median from 2.849 s to 1.629 s. Post-load setup plus first solve changes from 53.885 s to 61.958 s. These results identify a compact interaction correction in one frozen, frictionless regime. Distinct Newton states, evolving contact, contact-rank and mesh scaling, friction, and nested pressure-space performance remain open experimental questions.

23 pages, 4 figures. Develops factor-space compression for a condensed finite-element IPC operator, with conditional spectral analysis, a frozen-equation experiment, matched selector controls, and a prospective nested pressure-space extension