Vol.:(0123456789) 1 3 Journal of the Brazilian Society of Mechanical Sciences and Engineering (2020) 42:462 https://doi.org/10.1007/s40430-020-02548-3 TECHNICAL PAPER Contributions to the adaptive Monte Carlo method José Eduardo Silveira Leal 1  · Joyce Antunes da Silva 1  · Rosenda Valdés Arencibia 1 Received: 2 March 2020 / Accepted: 3 August 2020 / Published online: 11 August 2020 © The Brazilian Society of Mechanical Sciences and Engineering 2020 Abstract The GUM Supplement 1 presented the adaptive Monte Carlo (AMC) method. A basic implementation of an AMC proce- dure involves carrying out an increasing number of Monte Carlo trials until four parameters have stabilized in a statistical sense. Although the AMC method has been successfully used for uncertainty evaluations, the amount of stored data was seen as signifcant, and even after achieving stability, it can be lost with the increase in the number of trials. To overcome these problems, two modifcations to the AMC method were proposed, implemented and validated. The frst is related to data storage, while the second consists of applying an alternative criterion to assess convergence. After modifcations, the AMC was named the modifed adaptive Monte Carlo (MAMC) method and was applied when estimating the uncertainty of measurements carried out with a micrometer. The MAMC efectiveness was validated through the comparison of the uncer- tainty values and those from the application of the GUM and AMC methods. Under the evaluated experimental conditions, the MAMC showed greater repeatability when compared to AMC, regarding the number of trials to be carried out. This factor contributes toward the higher reliability of this method. The amount of data to be stored and manipulated through the application of the MAMC method was decreased signifcantly. This fact may increase the adoption of the AMC method. Keywords Measurement uncertainty · GUM · Adaptive Monte Carlo method 1 Introduction The ‘Guide to the Expression of Uncertainty in Measure- ment’ (GUM) [1] provides concepts, recommendations and a procedure for the assessment of uncertainty. After the GUM publication, an increasing interest has arisen in evaluating the measurement uncertainty. This interest is motivated on the fact that many important decisions are based on measure- ment results, for example, to assess product conformity [2, 3], to check a material in relation to its specifcation limits [3] and to make therapeutic decisions. In such cases, it is essential to provide some indication of the result quality, and for this purpose, measurement uncertainty is the metrologi- cal parameter used. According to [4], measurement uncertainty is one of the most important concepts in geometrical product specif- cation. The study in Weckenmann et al. [5] showed how expanded uncertainty associated with the measurement result afects the limit values that defne dimensional toler- ance by reducing it, which gives rise to the compliance zone. These authors pointed out that it is only possible to make a reliable statement for assessment of a characteristic, if the value indicated is in the conformance zone or in the non- conformance zone. The measurement uncertainty declaration is as important as the reporting of the measurement result itself. A meas- urement result without an assessment of its reliability is completely useless [3]. The comparison between diferent measurements of the same measurand, as well as between a measurement result and a specifcation limit, is impossible to assess without knowing their uncertainty [1]. Moreover, the uncertainty assessment and reporting are essential for contributing to the required traceability of the result to the International System of Units imposed by the ISO 17025 [6] standard. Technical Editor: José Roberto de França Arruda. * José Eduardo Silveira Leal joseleal@ufu.br Joyce Antunes da Silva joyce.antunes@ltad.com.br Rosenda Valdés Arencibia rosenda.arencibia@ufu.br 1 Federal University of Uberlândia, 2121, João Naves de Ávila Avenue, Campus Santa Mônica, Uberlândia, MG 38400-902, Brazil