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

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.

Appendix A: Routine used for numerical simulations

This appendix details the routine used for the numerical simulations presented in this paper. The constraint enforcement algorithm used is based on the application of the modular modeling methodology to produce a recursive form of Udwadia-Kalaba equation. In this case, the projection operator onto the kernel of $\mathbf {B}_{r+1}$ (see Sect. 1) is obtained by $\mathbf {C}_{r+1} = (\mathbf {I}- \mathbf {B}_{r+1}^{+} \mathbf {B}_{r+1})$, with $\mathbf {I}$ representing an identity operator and $(\cdot )^{+}$ standing for the Moore–Penrose pseudo-inverse. [12] proved that an algorithm based on such a procedure can be put in the explicit form detailed below, which is analogous to the algorithm of a Kalman filter for estimating the state of a static process from a set of noiseless measurements.

Consider that the rows and columns of the coefficient matrices in Eq. (35) are reordered according to the order of the generalized coordinates in the column-matrix $\mathbf{q}$ introduced in Eq. (37). Assuming U as a constant, the equations of motion of the system if neither compatibility nor boundary conditions are considered can be rewritten as a single matrix equation as follows:

$$\begin{aligned}&\mathbf {M}\ddot{\mathbf{q}}+ (\mathbf {B}- {\mathbf {B}}^{\mathrm {T}} + \mathbf {E}) U \dot{\mathbf{q}}+ ((\mathbf {W}+ \mathbf {F}) U^2 + \mathbf {K}) \mathbf{q}= \mathbf {g}. \end{aligned}$$

(65)

Let $\mathbf {N}= \mathbf {M}^{-1/2}$ and define $\mathbf {z}_0$ as:

$$\begin{aligned}&\mathbf {z}_0 = \mathbf {N}\big ( \mathbf {g}- (\mathbf {B}- {\mathbf {B}}^{\mathrm {T}} + \mathbf {E}) U \dot{\mathbf{q}}- ((\mathbf {W}+\mathbf {F}) U^2+ \mathbf {K}) \mathbf{q}\big ). \end{aligned}$$

(66)

It should be noticed that the values of $\ddot{\mathbf{q}}$ associated to the level 0 of the hierarchical representation of the problem shown in Fig. 1 are given by $\ddot{\mathbf{q}}= \mathbf {N}\mathbf {z}_{0}$. As shown in Orsino [12], the variables $\mathbf {z}_{r}$, from which the generalized accelerations at the level r of the hierarchy could be determined as $\ddot{\mathbf{q}}= \mathbf {N}\mathbf {z}_{r}$, are computed according to the following algorithm ($r=0,1,2$):

$$\begin{aligned}&\mathbf {G}_{r+1} = (\tilde{\mathbf {H}}_{r+1} \mathbf {P}_{r})^{+}, \end{aligned}$$

(67)

$$\begin{aligned}&\mathbf {z}_{r+1} = \mathbf {z}_{r} + \mathbf {G}_{r+1}(\tilde{\mathbf {b}}_{r+1} - \tilde{\mathbf {H}}_{r+1} \mathbf {z}_{r}), \end{aligned}$$

(68)

$$\begin{aligned}&\mathbf {P}_{r+1} = (\mathbf {I}- \mathbf {G}_{r+1} \tilde{\mathbf {H}}_{r+1}) \mathbf {P}_{r} (\mathbf {I}- \mathbf {G}_{r+1} \tilde{\mathbf {H}}_{r+1})^{\mathrm {T}}, \end{aligned}$$

(69)

In these equations $\tilde{\mathbf {H}}_{r} = \tilde{\mathbf {A}}_{r} \mathbf {N}$, and $\tilde{\mathbf {A}}_{r}$ and $\tilde{\mathbf {b}}_{r}$ correspond to coefficient matrices associated the second time derivatives of the constraint equations specifically enforced at level r. Indeed, denote by $\tilde{\mathbf {h}}_{r}(\mathbf{q}) = \mathbf {0}$ the aforementioned equations expressed in matrix form. Their first two time derivatives can be expressed as follows:

$$\begin{aligned}&\dot{\tilde{\mathbf {h}}}_{r} = \tilde{\mathbf {A}}_{r}(\mathbf{q}) \dot{\mathbf{q}}= \mathbf {0}, \end{aligned}$$

(70)

$$\begin{aligned}&\ddot{\tilde{\mathbf {h}}}_{r} = \tilde{\mathbf {A}}_{r}(\mathbf{q}) \ddot{\mathbf{q}}- \tilde{\mathbf {b}}_{r}(\mathbf{q}, \dot{\mathbf{q}}) = \mathbf {0}. \end{aligned}$$

(71)

Therefore, the computational routine adopted basically consists of performing a numerical integration using, at each step, the values of generalized accelerations computed by $\ddot{\mathbf{q}}= \mathbf {N}\mathbf {z}_{3}$. To apply Baumgarte’s technique [1] for stabilization of constraints, Eq. (68) can be replaced by:

$$\begin{aligned}&\mathbf {z}_{r+1} = \mathbf {z}_{r} + \mathbf {G}_{r+1}(\tilde{\mathbf {b}}_{r+1} - \tilde{\mathbf {H}}_{r+1} \mathbf {z}_{r} - 2 \zeta \omega \dot{\tilde{\mathbf {h}}}_{r} - \omega ^{2} {\tilde{\mathbf {h}}}_{r}). \end{aligned}$$

(72)

For the simulations shown, the following values for the stabilization constants were chosen: $\zeta = \sqrt{2}/2$ and $\omega = 10$.