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

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.