SUMMARYWe consider a recently introduced continuous data assimilation (CDA) approach for downscaling a coarse resolution configuration of the 2D Bénard convection equations into a finer grid. In this CDA, a nudging term, estimated as the misfit between some interpolants of the assimilated coarse grid measurements and the fine grid model solution, is added to the model equations to constrain the model. The main contribution of this study is a performance analysis of CDA for downscaling measurements of temperature and velocity. These measurements are assimilated either separately or simultaneously and the results are compared against those resulting from the standard point-to-point nudging approach (NA). Our numerical results suggest that the CDA solution outperforms that of NA, always converging to the true solution when the velocity is assimilated as has been theoretically proven. Assimilation of temperature measurements only may not always recover the true state as demonstrated in the case study. Various runs are conducted to evaluate the sensitivity of CDA to noise in the measurements, the size and the time frequency of the measured grid, suggesting a more robust behaviour of CDA compared to NA.
The accuracy of a machine tool is based on the direct position feedback from its built-in encoders which accurately measure displacements of motion axes by moiré patterns. However, the posture of an encoder is altered by errors from the machine tool, resulting in six geometric deviations. A comprehensive analysis and scientific understanding on the influences of these deviations on moiré patterns are necessary. To investigate the influences, a simulation model of a reflective encoder is constructed to obtain moiré patterns, and four new methods are developed to identify characteristic parameters of the patterns. Variations in characteristic parameters caused by four geometric deviations are simulated and experiments are designed and performed. The results prove that the approach is able to assess the influences of geometric deviations. The approach can be used as a powerful aid for estimating impacts of geometric deviations and external errors on encoders to improve the accuracy of a machine tool.
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