(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.

翻译：工程结构高精度制造中的不合格决策过程常因分析缺陷部件与装配体所需的数值模拟而延迟。装配体部件间的界面仅能通过接触建模进行模拟。因此，高效的参数化降阶模型对于在近实时场景中执行接触力学模拟至关重要。降低接触模型成本的典型策略采用低秩逼近，其假设包括位移存在低维子空间及接触压力存在低维非负子锥。然而，接触压力具有局部特性，因为接触位置可能随载荷或几何等参数变化。本文研究了低秩逼近在接触力学中的适用性，并探索了基于稀疏回归技术的替代路径。研究表明，局部特性会导致接触压力线性可分离性的丧失，从而限制了低秩方法的精度。本文通过三个不同准则——紧致性、泛化性与特异性——分析了低秩假设对接触压力的适用性。随后，研究了采用大量快照的超完备字典以缓解不可分离性问题，并提出了两种策略：一种是贪婪选择字典元素的贪婪主动集方法；另一种是凸包逼近方法，该方法消除了凸问题中显式执行非穿透约束的必要性。最后，探讨了动态时间规整作为一种可能的非线性插值方法，该方法能以较低复杂度探索非线性流形，合成训练集中未计算的新快照，从而降低离线计算成本。