Tolerancing decisions can profoundly impact the quality and cost of the mechanism. To evaluate the impact of tolerance on mechanism quality, designers need to simulate the influences of tolerances with respect to the functional requirements. This paper proposes a mathematical formulation of tolerance analysis which integrates the notion of quantifier: ''For all acceptable deviations (deviations which are inside tolerances), there exists a gap configuration such as the assembly requirements and the behavior constraints are verified'' & ''For all acceptable deviations (deviations which are inside tolerances), and for all admissible gap configurations, the assembly and functional requirements and the behavior constraints are verified''. The quantifiers provide a univocal expression of the condition corresponding to a geometrical product requirement. This opens a wide area for research in tolerance analysis. To solve the mechanical problem, an approach based on optimization is proposed. Monte Carlo simulation is implemented for the statistical analysis. The proposed approach is tested on an over-constrained mechanism.

h i g h l i g h t s • Gaps cannot be considered as random variables. • The tolerance analysis issue is formulated thanks to the quantifier notion. • Two defect probabilities are defined: functionality defect probability and assembly defect probability. • Defect probabilities are computed using a system reliability method: FORM system. One of the aims of statistical tolerance analysis is to evaluate a predicted quality level at the design stage. One method consists of computing the defect probability P D expressed in parts per million (ppm). It represents the probability that a functional requirement will not be satisfied in mass production. This paper focuses on the statistical tolerance analysis of over-constrained mechanisms containing gaps. In this case, the values of the functional characteristics depend on the gap situations and are not explicitly formulated with respect to part deviations. To compute P D , an innovative methodology using system reliability methods is presented. This new approach is compared with an existing one based on an optimization algorithm and Monte Carlo simulations. The whole approach is illustrated using two industrial mechanisms: one inspired by a producer of coaxial connectors and one prismatic pair. Its major advantage is to considerably reduce computation time.

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Turbomachinery components are designed to achieve high performances while being exposed to a complex flow environment with varying operating conditions. Whereas the purpose of a new design optimisation is straightforward — obtaining a better design than the already existing one — the actual process itself remains a challenging task, permanently confronted to the dual need to reduce the cycle time and to further integrate complexity and multiple physics. The extensive use of numerical simulations has contributed in a significant way to the design of state-of-the-art blade geometries. To deal with expensive high-fidelity computations, surrogate-based optimisation (SBO) has become an established and recognised approach. In order to be useful within an industrial context, it is crucial that this SBO process is capable of efficiently handling high-dimensional design spaces as well as managing highly constrained design problems. This work presents innovative auto-adaptive surrogates, exploiting a blend of interpolation/regression and classification, implemented in the integrated optimisation platform Minamo. As a demonstrator based on NASA Rotor 37, an aero-mechanical multi-point optimisation has been performed. For a design space with 60 parameters, significant performance gains have been obtained (+4% after 250 evaluations or less than a fortnight’s runtime) while considering over 30 constraints. The proposed SBO approach offers therefore many opportunities for turbomachinery applications tackling highly constrained design problems. Despite the unavoidable curse of dimensionality, the proposed approach is able to efficiently achieve reliable results at a cost that is in line with industrial needs and it provides a conclusive asset in the frame of design specifications evolving along the design cycle.

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