  (Rephrased) Non-conformance decision-making processes in high-precision
manufacturing of engineering structures are often delayed due to numerical
simulations that are needed for analyzing the defective parts and assemblies.
Interfaces between parts of assemblies can only be simulated using the modeling
of contact. Thus, efficient parametric ROMs are necessary for performing
contact mechanics simulations in near real-time scenarios. Typical strategies
for reducing the cost of contact models use low-rank approximations.
Assumptions include the existence of a low-dimensional subspace for
displacement and a low-dimensional non-negative subcone for contact pressure.
However, the contact pressure exhibits a local nature, as the position of
contact can vary with parameters like loading or geometry. The adequacy of
low-rank approximations for contact mechanics is investigated and alternative
routes based on sparse regression techniques are explored. It is shown that the
local nature leads to loss of linear separability of contact pressure, thereby
limiting the accuracy of low-rank methods. The applicability of the low-rank
assumption to contact pressure is analyzed using 3 different criteria:
compactness, generalization and specificity. Subsequently, over-complete
dictionaries with a large number of snapshots to mitigate the inseparability
issues is investigated. Two strategies are devised: a greedy active-set method
where the dictionary elements are selected greedily and a convex hull
approximation method that eliminates the necessity of explicitly enforcing
non-penetration constraints in convex problems. Lastly, Dynamic Time Warping is
studied as a possible non-linear interpolation method that permits the
exploration of the non-linear manifoldm synthesising snapshots not computed in
the training set with low complexity; reducing the offline costs.
