Timestep-dependent element interpolation functions for axial FSI analysis in pipelines

This work is a continuation of a series of works by the authors, where dynamic problems (time derivatives) are first modified using a central difference scheme. Then the main function divided by the time step becomes part of a differential equation for an element, that is solved analytically (exactly in the case of one-dimensional problems, and approximately in the case of multidimensional problems). For Skalak's problem, this reduces to an ordinary fourthorder differential equation with a right-hand where the time step is a simple multiplier for the unknown function. First, the eigenfunctions of the homogeneous equation are found, which are represented as generalized Krylov functions with remarkable properties regarding their values and the values of their derivatives at the initial point of each element. All solutions (general homogeneous and partial inhomogeneous) are represented in an original way through these Krylov functions (introduced here for the first time) and are given in a form convenient for application of the transfer matrix method, TMM. The properties of the solution are presented in detail for the well-known Delft Hydraulics Benchmark Problem A. To verify the accuracy, we also implement the method of separation of variables (Fourier). A comparison of the results showed the high efficiency of the proposed solution scheme, where it is possible to understand in advance what accuracy can be achieved for a given time moment and the selected time step.