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Bayesian cubature (BC) has emerged to be one of most competitive approach for estimating the multi-dimensional integral especially when the integrand is expensive to evaluate, and alternative acquisition functions, such as the Posterior Variance Contribution (PVC) function, have been developed for adaptive experiment design of the integration points. However, those sequential design strategies also prevent BC from being implemented in a parallel scheme. Therefore, this paper aims at developing a parallelized adaptive BC method to further improve the computational efficiency.

By theoretically examining the multimodal behavior of the PVC function, it is concluded that the multiple local maxima all have important contribution to the integration accuracy as can be selected as design points, providing a practical way for parallelization of the adaptive BC. Inspired by the above finding, four multimodal optimization algorithms, including one newly developed in this work, are then introduced for finding multiple local maxima of the PVC function in one run, and further for parallel implementation of the adaptive BC.

The superiority of the parallel schemes and the performance of the four multimodal optimization algorithms are then demonstrated and compared with the k-means clustering method by using two numerical benchmarks and two engineering examples.

Multimodal behavior of acquisition function for BC is comprehensively investigated. All the local maxima of the acquisition function contribute to adaptive BC accuracy. Parallelization of adaptive BC is realized with four multimodal optimization methods.

The purpose of this study is to investigate the application of non-destructive testing methods in measuring bearing oil film thickness to ensure that bearings are in a normal lubrication state. The oil film thickness is a crucial parameter reflecting the lubrication status of bearings, directly influencing the operational state of bearing transmission systems. However, it is challenging to accurately measure the oil film thickness under traditional disassembly conditions due to factors such as bearing structure and working conditions. Therefore, there is an urgent need for a nondestructive testing method to measure the oil film thickness and its status.

This paper introduces methods for optically, electrically and acoustically measuring the oil film thickness and status of bearings. It discusses the adaptability and measurement accuracy of different bearing oil film measurement methods and the impact of varying measurement conditions on accuracy. In addition, it compares the application scenarios of other techniques and the influence of the environment on detection results.

Ultrasonic measurement stands out due to its widespread adaptability, making it suitable for oil film thickness detection in various states and monitoring continuous changes in oil film thickness. Different methods can be selected depending on the measurement environment to compensate for measurement accuracy and enhance detection effectiveness.

This paper reviews the basic principles and latest applications of optical, electrical and acoustic measurement of oil film thickness and status. It analyzes applicable measurement methods for oil film under different conditions. It discusses the future trends of detection methods, providing possible solutions for bearing oil film thickness detection in complex engineering environments.