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

Abstract

TOGEARI constructs a compressed contact correction for condensed finite-element equations with incremental potential contact (IPC). A core-conditioned interaction matrix ranks contact combinations by their mechanical responses; a raw-factor proxy acquires only the retained responses. For a symmetric positive-definite core, the reference analysis gives the exact spectrum, scaled inverse error and optimal worst omitted interaction. The selected responses are maintained through signed contact changes using supported operator images and a small coarse matrix. A complementary Schur spectrum and perturbation bounds connect this construction to the full current linear equation, damped local Newton steps and compatible episode reuse. On a 325,260-coordinate finger/cup problem, eight selected directions reduce Krylov work from eleven columns to five, matching full contact. Two reversed-order load studies give matching anchored displacements and accepted normal reactions, with 28.8-29.3 percent less warm time and 5.1-5.3 percent less complete time than per-step direct refactorization. A completed twelve-episode run saves 18.8 percent of total time against direct and 2.8 percent against a held factor. Full response caching has comparable time with 7.5 times the response storage. A compressed repeat terminates at an energy line search; separate diagnostics identify constitutive cancellation and terminal-backtracking sensitivity. Larger-contact, held-out-load and frictional comparisons show where compression or preparation loses its advantage. The complete Newton equation remains the residual operator throughout; the convergence guarantees retain their stated reference and local hypotheses.