The 2D Spring Splines Procedure Application With Prescribed Accuracy for Determination of the Global (Pipe Centerline) as Well as the Local (Dent) Curvatures

The problem of smoothing the spatial line based on position measurements of discrete points exists in cases where a) the positions of points are determined with some errors, b) the goal of smoothing is not a continuous position itself but the higher derivatives of it. It is a very common problem in many engineering applications. With respect to the pipeline industry this problem is very prominent at least in two cases but regretfully many researchers do not pay due attention to it at all. First, the Geopigs are widely used for the determination of spatial position of the pipe centerline points. This information inter alia may be (and in fact are widely) used for the calculation of the global centerline curvatures which are proportional to the global bending strains. Second, the maximum strain levels of the dents are calculated based on the local geometry of the dent as determined by radial sensor measurements from the in line inspection survey. Note, that in both cases mathematically the curvatures are the second derivatives of the function of global (pipeline) or local (dent) positions. The input information about the global X–Y–Z position of each consecutive point of axis line as well as the local radial position of the dent points are given with some error. This leads to a huge noise in predicted curvatures which can overrun the useful information. The amplitude of errors of calculation is inversely proportional to the squared distance between the points of measurement. The application of any smoothing procedure may lead to the loss of the useful information about real curvatures. Thus tradeoff between the smoothing of the noise and the loss of accuracy presents a big problem in the pipeline industry. Two quantitative parameters are introduced here to allow performing such a tradeoff. First parameter characterizes the standard deviation (also referred to as standard in the following) of the random value of the position measurement accuracy by the devices, ρ. Second parameter is the requested accuracy of the curvature determination and is defined in terms of the standard deviation of the bending stress, σ or strain, ε. The spatial beam on elastic foundation model is used to fit the measured point positions to the spatial curve. Its main characteristic is the specific compliance of the foundation α which is determined based on two above root-mean-square errors ρ and σ. The corresponding formulas and tables based on the solution for the elastic beam are obtained. The bigger the allowed error in bending stress σ the lesser is required compliance of the foundation, α. In turn this leads to the smaller value of characteristic wave length of solution and the possibility to retain more useful information about the actual short length stresses in the pipeline. Some practical examples of applications of the procedure are given.