Research on contact algorithm of unbonded flexible riser under axisymmetric load

https://doi.org/10.1016/j.ijpvp.2020.104248Get rights and content

Highlights

  • The Lagrange multiplier method and penalty function method can be used to introduce the layer relationship.

  • Lagrange multiplier method may cause the matrix singularity, which introduces instabilities in the solution.

  • The Penalty function method is more convenient, and the Lagrange multiplier method is more accurate.

  • The inner tensile armor layer is easier to be damaged than the outer tensile armor layer under the axial tension.

Abstract

The structure of unbonded flexible riser is very complex. It is very necessary to propose an efficient mechanical model. Applying the principle of minimum total potential energy, the theoretical model under axially symmetrical load is proposed in this paper. The geometric relationship between adjacent layers is introduced into the model by two methods (penalty function method and Lagrange multiplier method). In this paper, two additional matrices which can be directly added to the original stiffness matrix are derived by additional functions. Based on the experiment of the 2.5-inch 8-layer typical flexible riser, and combined with the finite element method, the value of penalty parameter in penalty function method is discussed and the accuracy of the two models is verified. The results show that: the value of penalty function can be consistent under the action of axisymmetric load; the Lagrange multiplier method will increase the number of variables and make the stiffness matrix singular, but the calculation results are more accurate and more sensitive to the relationship between adjacent layers and internal pressure; the penalty function method does not add extra variables, so it is more convenient to calculate, but the accuracy of the result is not as good as Lagrange multiplier method.

Introduction

Marine risers refer to the casing system used to connect the surface floating body and the seabed. It is the basic device used by the floating marine production system to transfer resources to (or from) the pipeline. The unbonded flexible riser is developed from the flexible pipe. It is a multi-layer composite pipe wall structure. When it bears external or internal loads, the walls of each layer can slide slightly with each other. Therefore, it has strong tensile strength and low bending rigidity and is widely used in the deep sea. Unbonded flexible risers are usually composed of many different structures. With the deepening of the application of water depth, the requirement of the flexible riser is higher and higher, so the analysis and design of flexible risers are particularly important.

A typical unbonded flexible riser consists of eight layers without adhesive agents between layers, as shown in Fig. 1.

There are few experiments on riser because of the high cost and time-consuming. In 1996, Witz [1] provided experiments on the 2.5-inch flexible riser to study the behaviour of the risers under cyclic loads. The experimental results show that the loading and unloading of the flexible riser in axial tension are linear, and the bending performance is nonlinear. Besides, the torsional stiffness is related to torsional direction. In 2013, Sousa [2] provided an experiment on the structural response of flexible riser under pure tension and combined load.

Theoretical method and the finite element methods are commonly used in riser analysis. For theoretical methods, Felippe [3] and Chung [4] use 3-D beam elements which can produce axial deformation, bending deformation and torsional deformation to simulate the pipeline model and analyze the nonlinear parallel configuration equation of marine riser under a static state. Based on the assumption that the spiral strip will not twist in the axial direction, Claydon [5] divided the unbonded flexible riser into a cylindrical shell and helical strip and obtains the balance equation by using the thin-walled cylinder theory and the force balance respectively. Guarracino [6] used the singular perturbation method to analyze the riser. Ramos [7] developed a theoretical model which considering the gap between the layers. Bai [8] used the principle of virtual work to develop the governing nonlinear equations of flexible riser. Yoo [9] derived the constitutive equation of each layer from the relation of principle of virtual works and the internal energy depending on the geometric characteristics of the layer and the solutions can be achieved by a direct calculation method. Liu [10] developed the model of composite tensile armour layer, introduced it into the whole model, and analyzed the tensile properties of the flexible riser under tensile action. As for the finite element analysis method, Merine [11] simplified carcass and pressure armour into the orthotropic shell to obtain a simplified model, and analyzed the structural response of the riser under the torsional load. Xiao Li [12] established the equivalent shell model of the carcass and pressure armour based on the strain energy method. Ren [13,14] and Zhang [15] used solid elements to establish a detailed model for each layer of the flexible riser and analyzed the structural response of the riser under axisymmetric and bending loads respectively.

The contact relationship between layers is very important in the theoretical model. Claydon [5] assumed that the radial deformation of each layer is consistent, but this is not consistent with the truth. Ramos [7] assumed that the adjacent layers are in contact with each other without any change, and derived the compatibility equation of the contact relationship between layers. Bai [8] used the deformation compatibility equation to express the contact relationship between layers and introduces the contact relationship between layers into the virtual work equation. Yoo [9] used the iterative method. First, he assumes that the layers are in contact with each other, and calculates the interlayer pressure and displacement of each layer, then plug this back in the original equation. Repeat several times until the pressure of each layer converges. There are many ways to calculate the contact relationship, but due to different assumptions, each of them has some limitations.

For contact problems, a variety of numerical methodologies have been proposed in the literature to be used. For example, Lagrange multiplier methods, penalty method and the augmented Lagrange approach. Lagrange multiplier method was first proposed by Hestenes [16] to solve mathematical programming problems. Simo and Laursen used this method to solve general contact problems [17]. In this method, the Lagrange multiplier satisfying the non penetrating condition is directly applied to the contact body, so the contact constraint conditions can be satisfied accurately. However, this method will add extra variables and increase the amount of calculation [18]. The penalty function method was first used by Yamazaki. K to deal with elastic contact problems [19]. In addition, the equations for the Lagrange multipliers introduce zeros in the diagonal of the system of equations, which makes the matrix singular [17]. In this method, the constrained variational problem is transformed into a penalty optimization problem, which does not increase the solution scale of the system, but it needs to assume a large penalty parameter, which may cause ill conditioned equation.

In mechanics, potential energy is the energy of a body due to its position or shape and consists of strain energy and kinetic energy [20]. The principle of minimum potential energy means that when the potential energy of a system is minimum, the system will be in equilibrium [20]. Kinetic energy is the energy that a body possesses due to its mechanical motion. And strain energy is the potential energy stored in an object in the form of strain and stress [21,22].

In this paper, based on the principle of minimum total potential energy, the stiffness matrix of each layer of the flexible riser is established. Two theoretical methods (Penalty function method and Lagrange multiplier method) are presented, which can introduce the relationship between adjacent layers into the overall stiffness matrix. According to the additional function, two additional matrices suitable for the two methods respectively are derived. The additional matrices represent the release relationship between layers and can be directly added to the global stiffness matrix. And this paper compares the two methods, and the accuracy of the model is proved by the finite element method and the experimental results in the literature.

Section snippets

Theoretical models

Under the action of axisymmetric load, the flexible riser can be divided into two parts: the helix layer and cylinder layer. The helical layer consists of tensile armour layer, compressive armour layer and carcass, which are made of metal materials and provide tensile and compressive capacity for the riser. The cylindrical layer consists of anti-wear layer, anti-leakage layer and sheath. They are made of polymer and used to reduce the friction damage between adjacent layers and provide

Model parameter

In this paper, the geometrical and material parameters of flexible risers are given in the experimental papers on non-bonded flexible risers published by de Sousa [2], as shown in Table 2, Table 3.

In order to avoid the influence of boundary conditions on the calculation results, a long enough length should be used, infinite model. According to the riser manufacturer's suggestion, the numerical model length of Unbonded flexible riser is usually taken as twice the thread pitch of the tensile

Tension

Axial tensile property is one of the main mechanical properties of unbonded flexible riser [30,31]. Fix one end of the riser, apply axial tension on the other end.

When the penalty function method is used, the penalty parameter will not become a new degree of freedom of the equation, which needs to be determined before calculation. Therefore, it is necessary to determine the value of the penalty function.

Conclusion

Based on the principle of minimum potential energy, the stiffness matrix of flexible riser under axisymmetric load is derived. In this paper, the penalty function method and Lagrange multiplier method are used to introduce the relationship between adjacent layers into the overall stiffness matrix. And two additional matrices which can be directly added to the original stiffness matrix are derived from additional functional. These matrices can make the calculation more convenient. The

Formatting funding sources

This research did not receive any specific grant from funding agencies in the public commercial, or not-for-profit sectors.

Author statement

I have made substantial contributions to the conception or design of the work; or the acquisition, analysis, or interpretation of data for the work; and I have drafted the work or revised it critically for important intellectual content; and I have approved the final version to be published; and I agree to be accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved.

All persons who

Declaration of competing interest

We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled.

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