A highly efficient beam-in-beam large sliding contact method for flexible multibody dynamics

Abstract

Beam-in-beam large sliding contact exists in simulations of drillstring-riser systems, offshore pipe-in-pipe systems, medical catheter-guidewire-stent systems, among others. Large sliding between the pipes may lead to discontinuity of contact force and non-sparsity of system Jacobian in numeric simulations, which then may result in non-physical disturbances and low efficiency. To overcome this phenomenon, a consistent-contact-force algorithm and a highly efficient implicit-integration scheme are proposed in this paper. Geometric treatments are applied in gap and overlap areas to obtain continuous contact force, in addition to contact-history-based fast detection. The contact force Jacobian is ignored as the flexible multibody system is solved with an inexact Newton iteration and an empirical formula is used to estimate the convergence conditions. Four numeric examples are computed to verify the presented method, including a buckling benchmark and a riser-recoil industrial application. The numeric results show that the proposed method is accurate and highly efficient.

Appendix A: Magnitude analysis of \(s_{c}\), \(s_{m}\) and \(s_{k}\) in the empirical estimation

Begin with Eq. 49:

$$\begin{aligned} s_{c}&=1+\beta _{0}a_{c}\nonumber \\ s_{m}&=h^{2}\left( \gamma _{0}+\beta _{0}a_{m}\right) \nonumber \\ s_{k}&=1+\beta _{0}a_{k} \end{aligned}$$

(58)

where the magnitude of $\beta _{0}$ and $\gamma _{0}$ are $O\left( h^{-1}\right)$ and $O\left( h^{-2}\right)$, respectively, as presented in Yang et al. [32].

When $\beta _{0}=A_{\beta }h^{-1}$ and $\gamma _{0}=A_{\gamma }h^{-2}$—where $A_{\beta }=1.5,1.83,2.08\text { and }2.28$; and $A_{\gamma }=1,2,2.92\text { and }3.75$, respectively, for BDF order $p=2,3,4,5$ in the fixed-step-size integration—we have:

$$\begin{aligned} s_{c}&=1+A_{\beta }\left( h^{-1}a_{c}\right) \nonumber \\ s_{m}&=h^{2}\left( A_{\gamma }h^{-2}+A_{\beta }h^{-1}a_{m}\right) =A_{\gamma }+A_{\beta }\left( ha_{m}\right) \nonumber \\ s_{k}&=1+A_{\beta }\left( h^{-1}a_{k}\right) \end{aligned}$$

(59)

If all damping is ignored (i.e., $a_{c}=a_{k}=a_{m}=0$), we have:

$$\begin{aligned} s_{c}&=1\nonumber \\ s_{m}&=A_{\gamma }\nonumber \\ s_{k}&=1 \end{aligned}$$

(60)

If the step size takes the same magnitude as the system damping, i.e., $h=a_{m}=a_{c}=a_{k}=o\left( 1\right)$, we have:

$$\begin{aligned} s_{c}&=1+A_{\beta }\nonumber \\ s_{m}&=A_{\gamma }+o\left( 1\right) \nonumber \\ s_{k}&=1+A_{\beta } \end{aligned}$$

(61)