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.

• The paper introduces TOGEARI, a low-rank Woodbury preconditioner that selects important IPC contact-interaction combinations while leaving the Newton system and residual checks unchanged.
• The method reduced Krylov vectors from 11 to 5 and warm median solve time from 2.849 to 1.629 seconds on a 325,260-coordinate finite-element contact system using an eight-dimensional subspace.
• The results show that interaction content matters more than rank alone, but evidence remains limited to frozen, frictionless, quasistatic cases without demonstrated nonlinear, frictional, or cross-state performance.

Overview

TOGEARI (Truncated Operator-Gram Eigenspace Approximation of Relevant Interactions) addresses a narrow but well-posed linear-algebra question arising in condensed finite-element contact simulation: given a reusable sparse core solve for the material–volumetric operator, how much of the factorized IPC contact update must be represented explicitly to precondition the complete Newton equation effectively? The paper's answer is to select a low-dimensional subspace of contact interaction combinations and install its contribution as a Woodbury right preconditioner, while leaving the complete Newton operator, the right-hand side, the FGMRES recurrence, and the independently recomputed residual acceptance test entirely unchanged. The work is deliberately scoped: all empirical claims concern frozen, frictionless, quasistatic linearizations, and the authors are explicit that nonlinear speedup, transfer across Newton states, and scaling behavior remain untested (2608.14162).

Operator structure and the interaction-space construction

The target system arises from a Tet10/P0 formulation—ten-node quadratic tetrahedra with one elementwise-constant latent pressure per element—after exact local pressure condensation and boundary reduction. With $n$ free displacement coordinates, the complete boundary-reduced Newton matrix admits the split

$A = M + U^T U + N, \qquad M = K_0 + B^T D^{-1} B,$

where $M$ is the reusable material–volumetric core, $U \in \mathbb{R}^{m \times n}$ is a row factor of the projected positive-semidefinite IPC normal-contact tangent, and $N$ collects the signed pressure-geometric curvature, follower-load tangent, and factor-reconstruction remainder. The paper is careful to separate $K_{\mathrm{vol}} = B^T D^{-1} B$ (positive) from the signed pressure-geometric term $G_e = p_e \nabla^2 c_e$, noting that conflating them as a single "pressure block" would obscure the metric argument.

The core-response selector, Togeari-M, ranks interaction combinations through the interaction-response matrix $G_M = U M^{-1} U^T$, which under an SPD core measures the core-energy of the displacement response induced by each contact combination. The cheaper proxy Togeari-R selects the leading eigenspace of $UU^T$ before any core response is formed. Both retain an orthonormal factor-space basis $Q_r$, form $A = M + U^T U + N, \qquad M = K_0 + B^T D^{-1} B,$0, and apply the rank-$A = M + U^T U + N, \qquad M = K_0 + B^T D^{-1} B,$1 Woodbury correction $A = M + U^T U + N, \qquad M = K_0 + B^T D^{-1} B,$2 as a right preconditioner. The correction itself requires only a nonsingular core and a nonsingular reduced matrix $A = M + U^T U + N, \qquad M = K_0 + B^T D^{-1} B,$3; symmetry and definiteness enter only the selection and certification layers.

Theoretical results

Three results constitute the paper's mathematical contribution, all conditional on a fixed SPD core $A = M + U^T U + N, \qquad M = K_0 + B^T D^{-1} B,$4. First, the metric-spectrum specialization gives the exact generalized spectrum of the pair $A = M + U^T U + N, \qquad M = K_0 + B^T D^{-1} B,$5 with $A = M + U^T U + N, \qquad M = K_0 + B^T D^{-1} B,$6: it consists of $A = M + U^T U + N, \qquad M = K_0 + B^T D^{-1} B,$7 unit eigenvalues together with the omitted values $A = M + U^T U + N, \qquad M = K_0 + B^T D^{-1} B,$8 for $A = M + U^T U + N, \qquad M = K_0 + B^T D^{-1} B,$9, where $M$0. Consequently the preconditioned condition number is exactly $M$1 when $M$2 and exactly $M$3 at $M$4. Second, a scaled inverse-error identity shows that $M$5, attained by the first omitted mode. Third, a selector-optimality proposition establishes that the metric projector minimizes the worst core-normalized contact energy left outside any rank-$M$6 factor-space projection, with value $M$7; any other projector satisfies $M$8. The paper also proves a frozen-algebra monotonicity result: strengthening the condensed volumetric term by a scalar $M$9 cannot increase any ordered eigenvalue of $U \in \mathbb{R}^{m \times n}$0, a consequence of Loewner-order reversal under inversion.

These certificates are conditional, and the paper says so plainly. The reported large systems used a cuDSS general (not Cholesky) factorization, and global inertia was never recorded, so the SPD status of the core remains unresolved. The exact spectral statements therefore describe an idealized preconditioner, while the FGMRES observations on $U \in \mathbb{R}^{m \times n}$1 are direct but empirical. An appendix extends the analysis to approximate SPD cores via a two-sided equivalence bound, and notes that a nonsymmetric core places core-response selection outside the present theory altogether.

Finite-element and contact realization

The derivation connects the abstract split to the implemented operator with reconstruction errors at float64 level: $U \in \mathbb{R}^{m \times n}$2 relative Frobenius error for the complete matrix and below $U \in \mathbb{R}^{m \times n}$3 for the local IPC factor reconstruction ($U \in \mathbb{R}^{m \times n}$4 for point–triangle and $U \in \mathbb{R}^{m \times n}$5 for edge–edge blocks). The contact factor is built by symmetrized eigendecomposition of each pulled-back pair block, PSD projection, and stacking of local row factors. On the primary state, 36 active pairs (10 point–triangle, 26 edge–edge) yield $U \in \mathbb{R}^{m \times n}$6 stored factor rows, though the registered numerical rank of $U \in \mathbb{R}^{m \times n}$7 is only 42 under a relative threshold, with a wide spectral gap between $U \in \mathbb{R}^{m \times n}$8 and $U \in \mathbb{R}^{m \times n}$9.

Experimental results

The principal system is a frozen Newton linearization from the final equilibrium of a pneumatic finger in contact with a deformable cup: 325,260 free coordinates, 66,315 quadratic tetrahedra, three Neo-Hookean material groups with Poisson ratios up to approximately 0.495, and nine follower-pressure chambers. All comparisons run on a single Tesla V100S in float64 with a strict same-equation contract: only the retained interaction subspace changes.

The central result is that an eight-dimensional interaction subspace of the 94-row factor reduces the complete-system Krylov basis from 11 Arnoldi vectors to 5, with warm median solve time falling from 2.849 s to 1.629 s. Critically, dimension-matched controls do not reproduce this: row-norm and deterministic-random selectors of the same rank require 12 vectors. The content of the retained subspace, not its dimension, is decisive. A numerically near-repeat extraction (displacement differing by $N$0 in relative norm) reproduces the reduction, and a contact-free system recovers the exact rank-zero path with no performance ordering. A frozen volumetric-scaling sweep over $N$1 holds the 11-to-5 reduction with first omitted eigenvalue varying only from 0.07832 to 0.07716; at $N$2 the counts rise to 19 and 12.

The selector comparison carries a notable and honestly stated tension: the raw and metric eight-dimensional subspaces are closely aligned (principal angles from $N$3 to 0.232 rad, median 0.0205 rad), and both yield five Krylov vectors on both contact-active extractions. Togeari-R is also cheaper in setup, since selection precedes the core responses. The authors conclude that raw Euclidean factor geometry recovers the useful space as effectively as core-conditioned selection in this regime, leaving the practical advantage of the metric selector unresolved; only Togeari-M carries the a priori SPD certificate.

The timing picture is mixed and the paper does not overstate it. Post-load setup plus first solve rises from 53.885 s to 61.958 s in the sweep execution: sparse construction, cuDSS factorization, and the 94 sequential core responses (one response column occupies 2.6 MB; the full response matrix 244.6 MB) dominate, while the reduced eigensolve takes only 4.27 ms. The measured benefit is therefore confined to repeated warm solves. Every contact-active case peaks at 6150 MiB device memory, so no memory reduction is demonstrated.

Limitations and open questions

The paper's scope restrictions are stated with unusual explicitness. The evidence comprises one contact-active extraction, one near-repeat, and one contact-free system—all frozen, frictionless, quasistatic. The complete-system condition number is left unquantified because a reliable smallest admissible singular value was unavailable; the reported block norms serve only as scale diagnostics. Dimension seven in the retained sweep was untested, so minimality of the dimension-eight transition is open. The near-identical metric and raw subspaces leave the cause of the shared five-vector behavior unexplained. The nested pressure-space extension in the appendix—selecting 443 pressure modes via randomized Ritz with a maximum Ritz residual of 0.4472—is explicitly prospective, with sketch-dependent dimension and no nonlinear-trajectory measurement. The authors identify the most informative next tests as a contact-rank sweep across meshes and active sets, a consecutive-Newton sequence measuring basis refresh and reuse, and a friction-active nonsymmetric equation.

Conclusion

TOGEARI provides a mathematically precise, carefully scoped study of low-rank interaction-space preconditioning for condensed finite-element contact systems. Its exact spectral, inverse-error, and selector-optimality results specialize established rank-structured SPD theory to the row-factor orientation of a verified Tet10/P0–IPC split, and its frozen-equation experiments demonstrate that a compact, content-specific eight-dimensional contact subspace—rather than retained dimension alone—drives an 11-to-5 Krylov reduction with roughly 43% warm-solve time savings. Whether core-conditioned selection offers practical value beyond its certificate, and whether the correction survives evolving contact, friction, and scaling, remain open and are correctly left as measured questions rather than claims.