On the direct numerical computation of Hopf bifurcations to assess the dynamic stability of fluid-conveying cantilevered pipes

In the structural analysis of fluid-conveying pipes, dynamic instabilities can occur at specific values of the flow velocity, depending on the geometry as well as the material parameters of the pipe and the interior fluid. These critical points fall into the broader category of Hopf bifurcations. Typical numerical models of this problem employ a one-dimensional weighted residual method, leading to a velocity-dependent eigenvalue problem. The solutions form eigencurves, and the critical points are characterized by eigenvalues with vanishing real parts. In this paper, we show that critical points can be computed directly as solutions to a single three-parameter eigenvalue problem. In addition, we employ a recently developed method for computing individual eigencurves, based on the concept of exponential residual relaxation. For the discretization of the weak form, we use a finite element method with a particular version of C 1 -continuous high-order spectral elements, suited for fourth-order differential equations, and we discuss the differences compared to the more commonly used weighted residual method based on the basis functions of a linear Euler-Bernoulli beam. Four numerical examples demonstrate the effectiveness of the implemented algorithms. For verification, we provide a detailed derivation of analytical solutions for special cases.