Certified neural operators for frictional contact dynamics of jointed structures
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Mohammad Akram
Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah 42351, Saudi Arabia
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Umar Ishtiaq
Office of Research, Innovation and Commercialization, University of Management and Technology, Lahore 54770, Pakistan; Software Development Department, Biruni University, 34010 Istanbul, Turkey; Research Center of Applied Mathematics, Khazar University, Baku AZ1096, Azerbaijan
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Ioan-Lucian Popa
Department of Computing, Mathematics and Electronics, “1 Decembrie 1918” University of Alba Iulia, 510009 Alba Iulia, Romania; Faculty of Mathematics and Computer Science, Transilvania University of Brasov, 500091 Brasov, Romania
Abstract
Friction in mechanical joints dominates damping in assembled structures, and predicting it depends on set-valued contact across interfaces whose manufactured topography is only partially known. Topography-resolving simulation is the most predictive route, but its cost is governed by a non-smooth contact sub-problem at every implicit time step. We prove that this sub-problem is a strongly monotone variational inequality with a single-valued, Lipschitz continuous, piecewise real-analytic solution operator, and exploit this structure in an admissible, certified, structure-preserving learned solver. A neural operator is composed with a projection onto the Coulomb cone, so every prediction satisfies the contact law exactly. Training minimises the contact-law residual; a supervised guidance term on a small reference subset is annealed to zero after the opening epochs, so the deployed surrogate is a stationary point of the residual objective alone. It warm-starts a few certified projected-Jacobi sweeps that yield a computable error bound and preserve the negligible numerical damping of the base integrator. The bound’s pessimism and cost scale with the squared condition number of the condensed operator, and a cone-invariant per-point scaling removes most of that penalty on ill-conditioned interfaces. Modelling the topography as a Karhunen–Loève random field makes the solver the forward map of an uncertainty study whose global-sensitivity indices identify which topography length scales govern damping. Theorems with proofs, validation across preload, friction coefficient, mesh density, topography statistics, four vibration modes, and a held-out joint geometry and material, a stated range of validity, and a quantitative comparison with established methods support the framework.
Copyright (c) 2026 Mohammad Akram, Umar Ishtiaq, Ioan-Lucian Popa
This work is licensed under a Creative Commons Attribution 4.0 International License.
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