# Modular methodology applied to the nonlinear modeling of a pipe conveying fluid

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## Abstract

This paper proposes an extension of a modular modeling approach, originally developed for lumped parameter systems, to the derivation of FEM-discretized equations of motion of one-dimensional distributed parameter systems. This methodology is characterized by the use of a recursive algorithm based on projection operators that allows any constraint condition to be enforced a posteriori. This leads to a modular approach in which a system can be conceived as the top member of a hierarchy in which the increase of complexity from one level to the parent one is associated to the enforcement of constraints. For lumped parameter systems this allows the implementation of modeling procedures starting from already known mathematical models of subsystems. In the case of distributed parameter systems, such a novel methodology not only allows to explore subsystem-based modeling strategies, but also makes it possible to propose formulations in which compatibility and boundary conditions can be enforced *a posteriori*. The benchmark chosen to explore these further possibilities is the classical problem of a cantilevered pipe conveying fluid. Taking a pipe made of a linear-elastic material, allowing geometric nonlinearities and assuming an internal plug-flow, a Hamiltonian derivation of FEM-discretized equations of motion is performed according to this novel approach. Numerical simulations are carried out to address the nonlinear model obtained.

### Keywords

Analytical mechanics Mathematical modeling Finite element method Modular modeling Pipe conveying fluid## Notes

### Acknowledgements

Authors thank Dr. Guilherme Rosa Franzini and Prof. Luiz Bevilacqua for encouraging discussions. First author acknowledges the post-doctoral Grant #2016/09730-0, São Paulo Research Foundation (FAPESP). Second author acknowledges CNPq research Grant no. 308990/2014-5.

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