Thermal stress impacts various engineering fields significantly, such as aerospace and precision instruments. This adverse effect can be greatly reduced, if not eliminated, by the application of micro-thermal expansion materials, and bi-material beams are widely utilized in the design of micro-thermal expansion structures, thereby exhibiting great application potentials. The elasticity solution of bi-material curved beam under free thermal expansion has been proposed by scholars. Based on this solution, the simplified form is proposed in this paper, and extended to the case where the rotation angles at both ends of the circular arc are constrained under thermal loads. Besides, the geometric parameters and the nonlinear problems of the thermal expansion of bi-material curved beam are analyzed. In addition, a novel type of negative thermal expansion material has been designed by applying the bi-material curved beam to the tetra chiral and anti-tetra chiral materials. The proposed material has greater negative thermal expansion effect than the traditional tetra and anti-tetra chiral materials that are with straight beams.

Deep neural networks (DNNs) have achieved superior performance in diagnosing process faults, but they lack robustness when encountering novel fault types absent from training sets. Such unknown faults commonly appear in industrial settings, and conventional DNNs often misclassify them as one of the known fault types. To address this limitation, we formulate the concept of open-set fault diagnosis (OSFD), which seeks to distinguish unknown faults from known ones while correctly classifying the known faults. The primary challenge in OSFD lies in minimizing both the empirical classification risk associated with known faults and the open space risk without access to training data for unknown faults. In order to mitigate these risks, we introduce a novel approach called hypersphere-guided reciprocal point learning (SRPL). Specifically, SRPL preserves a DNN for feature extraction while constraining features to lie on a unit hypersphere. To reduce empirical classification risk, it applies an angular-margin penalty that explicitly increases intra-class compactness and inter-class separation for known faults on the hypersphere, thereby improving discriminability among known faults. Additionally, SRPL introduces reciprocal points on the hypersphere, with each point acting as a classifier by occupying the extra-class region associated with a particular known fault. The interactions among multiple reciprocal points, together with the deliberate synthesis of unknown fault features on the hypersphere, serve to lower open-space risk: the reciprocal-point interactions provide an indirect estimate of unknowns, and the synthesized unknowns provide a direct estimate, both of which enhance distinguishability between known and unknown faults. Extensive experimental results on the Tennessee Eastman process confirm the superiority of the proposed method compared to state-of-the-art OSR algorithms, e.g., an 82.32% AUROC score and a 71.50% OSFDR score.