The modelling of flexible pipe interlocked carcasses is complicated when considering all the geometric complexity of their profile. A possible approach is to model them as cylindrical equivalent layers. To follow this path several alternatives can be considered in changing the geometrical and material properties. However, the thickness and the mean radius of those layers must not be changed to not interfere with the diameter of the other flexible pipe layers. In this paper, a model of an orthotropic cylindrical layer, with the same thickness and mean diameter of the original carcass layer is constructed and its material parameters are adjusted for axial loads using a finite element model of the real carcass profile.

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The layers of unbounded flexible pipes have relative movement, enhancing its capabilities to handle curvatures and moment loads. In a simplified approach, those pipes can be described using bonded elements; but to really capture this behavior, a frictional contact is utterly needed. In general, dealing with contact problems in computational mechanics is complicated, since it involves the constant evaluation of its status and can lead to convergence problems or simulation failure, due to intrinsically problematic and inefficient contact models or due to contact models that are insufficient to capture the desired details. The macroelement formulation, which was created to deal with flexible pipes in a simplified way, needed a frictional contact element to enhance the quality of results and closeness to real behavior. The major drawback for developing such element is the different nature of the nodal displacements descriptions. The first approach possible is the simplest contact model: it involves only the nodes in each contacting elements. The gap information and distances are evaluated using exclusively the nodal information. This kind of model provides good results with minimum computational effort, especially when considering small displacements. This paper proposes such element: a node-to-node contact formulation for macroelements. It considers that the nodal displacements of both nodes are in cylindrical coordinates with one of them using Fourier series to describe the displacements. To show model effectiveness, a case study with a cylinder using Fourier series and multiple helical elements connected with the contact element is done and shows great results.

Due to the number of layers and their respective geometrical complexities, finite element analyses of flexible pipes usually require large-scale schemes, with a high number of elements and degrees-of-freedom. If proper precautions are not taken, such as suitable algorithms and numerical methods, the computational costs of these analyses may become them unfeasible. Finite macroelements are finite elements formulated for the solution of a specific problem considering and taking advantage of its particularities, such as geometry patterns, to obtain computational advantages, as reduced number of degrees-of-freedom and ease of problem description. The element-by-element method (EBE) also fits very well in this context, since it is characterized by the elimination of the global stiffness matrix and its memory consumption grows linearly with the number of elements, besides being highly parallelizable. Over the last decades, several works regarding the EBE method were published in the literature, but none of them directly applied to flexible pipes. Due to the contact elements between the layers, problems with flexible pipes are usually characterized by very large matrix-bandwidth, making the implementation of EBE method more challenging, so that its efficiency and scalability are not compromised. Therefore, this work presents a parallelized implementation of an element-by-element architecture for structural analysis of flexible pipes using finite macroelements. Four synchronization algorithms were developed and analyzed in detail, including their scalability assessment, and comparisons were made with a well-stablished FEM software, with significant gains in simulation time and memory consumption.

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