Modeling of Riser Contact-Friction Problems
Authors: Fontaine E. , Heurtier J.M. , Durville D. , Toumit S. , Prat C. 
Published: October 2008

Introduction
This paper is devoted to the mechanical analysis of contact friction problems met by slender offshore structures such as mooring lines, risers and pipes. Several types of interactions are considered since contact-friction arises either between multiple risers (internal and external contact) or between a single riser and a rigid surface. The later can either be fixed, or move freely under the influence of external forces. Non linear time domain simulations of the mechanical behavior of these structures undergoing large displacements and finite strains are presented based on the finite element method (FEM). To solve efficiently the geometrical nonlinearity associated with contact friction interactions, specific algorithms have been developed to localize automatically contact elements. The convergence of the iterative algorithms is sped-up through the use of a regularized Coulomb's law, which insure fast and reliable results even for complex systems, as demonstrated here. To illustrate the capabilities of the DeepLines?software, detailed modeling of realistic systems derived from industrial applications are presented, such as the layout of a riser on a sea floor arbitrarily shaped, the wave induced motion of a riser laying onto a freely moving sub-sea arch, a riser rubbing onto the moonpool or maintained within a bellmouth connected to an FPSO. Pipe in pipe configurations will also be considered, through applications linked to SCR. Finally, some results concerning the strains along the pipes or risers are given, demonstrating the necessity to model accurately these phenomena which can lead to high curvatures locally.

General Theoretical Framework
For the sake of completeness, we present here the basic theory sustaining the solution procedure used in the FEM code DeepLines?, following Fargues (1995) and Durville (1998a,b, 1999). More specific details can also be found in DeepLines? (2002,a,b).
Sea-bottom to surface links (mooring lines, risers...) are considered as slender structures. The structural problem for their dynamics is formulated as a principle of virtual work of interaction and can be stated as follow: find the cinematically admissible solution u, which verifies (Mathematical Equation-Available in full paper) for any cinematically admissible displacement field v. Here, S(u) is the second Piola-Kirchhoff stress tensor and E(u) is the non-linear Green Lagrange strain tensor. The first term corresponds to the virtual work of internal loads, the second and third terms represent the virtual work associated with contact-friction interactions, and the RHS corresponds to the virtual work of both volume and surface external loads which are respectively applied on the structure ? and part of its border ?F. ?C refers to the surface where contacts between structures effectively take place, with reaction forces R as defined in figure 1.
Figure 1: Definitions of contact-friction reaction force between a slender structure (?) and a rigid surface (obstacle).
The main difficulty in this formulation is to derive a precise expression for the virtual work Winteract of these contactfriction forces. Indeed, the location ?C where contacts between structures occur is a priori not known, therefore introducing a geometrical nonlinearity.