Controllable Curvature Smoothing of the Pipeline Positions by 2D Corotational Beam Spline

Abstract

An inline inertial measurement unit (IMU) gives X-Y-Z positions of pipeline points, which can then be used to assess the bending deformation along the pipeline axis. Mathematical treatment of these results is complicated by two factors. The first is related to the measurement errors arising, for example, due to vibration of an IMU inside the pipeline and local out-of-roundness or misalignment of the pipe surface. The second is related to embedded curvature of the axis resulting from factory or field-installed continuous and mitered bends. This paper elaborates a smoothing technique based on corotational beam spline (CBS), and contains several ideas specific to the pipeline application. The points of measurement are treated as compliant springs, i.e., the lower their compliance, the smoother the calculated pipeline axis, and the smaller the calculated deformations. We suggest two alternative approaches to find the necessary “spring” compliances in order to provide some controllable level of smoothing, both of which are based on the assumed or known averaged error of measurement in the close vicinity of a considered point. Another unique spline feature is the possibility to account for jumps in the curvature or the tangent angle between two pipe sections due to fabricated bends, as these jumps produce no additional deformation. Several examples and challenges related to practical calculations are presented. First, artificial examples of a geometrically ideal pipeline with manufactured circular and mitered bends are considered. It is demonstrated that CBS is able to give no artificial strains. Second, a real example of “restoring” the lengths and radius of an unknown fabricated bend based on the calculated strains is analyzed. Third, a detailed comparison of two consequent in-line inspection runs is performed. It is shown how the increased level of smoothing (spring compliances) impacts the calculated curvatures: from a very different noisy behavior to almost identical results. The recommended level of smoothing based on the admissible level of false stress (strain) is suggested. The problem of smoothing raw data is urgent in many fields such as signal processing, geometrical measurements and tracking, pattern recognition. However, the inevitable noise distorts useful information. Further, often the goal of the analysis is second derivatives (which are proportional to the outer forces, curvatures); however, during the differentiation, the short-wave (high frequency) noise is amplified. It is widely understood that the smoothing process is always the trade-off between suppressing noise and preserving useful data. But criteria for keeping such a balance are very rarely discussed in literature. This paper is devoted to determination of the curvature (proportional to additional axial stresses caused by the ground) of a pipeline when given discrete points of centerline measurements. As such, our work has two objectives. First, we give an instrument for smoothing based on our original corotational spline technique, which in its turn uses the findings of the theory of a beam on an elastic foundation. The pipeline is virtually laid on elastic supports, whose compliances predetermine the degree of smoothing. The second objective is to provide the required balance by establishing the notion of acceptable noise and therefore to properly manage the process of smoothing.