Finite element modelling of mooring lines - Semantic Scholar

Mathematics and Computers in Simulation 53 (2000) 415–422

Finite element modelling of mooring lines O.M. Aamo∗ , T.I. Fossen Department of Engineering Cybernetics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway Accepted 28 July 2000

Abstract In this paper, we develop a finite element model for a cable suspended in water. Global existence and uniqueness of solutions of the truncated system is shown for a slightly simplified equation describing the motion of a cable having negligible added mass and supported by fixed end-points. Based on this, along with well known results on local existence and uniqueness of solutions for symmetrizable hyperbolic systems, we conjecture a global result for the initial-boundary value problem. © 2000 IMACS. Published by Elsevier Science B.V. All rights reserved. Keywords: Finite element method; Symmetrizable hyperbolic systems; Mooring

1. Introduction Position mooring systems (PM) have been commercially available since the late 1980s, and have proven to be a cost-effective alternative to permanent platforms for offshore oil production. In traditional testing of the performance of PM systems by means of computer simulations, tabulated static solutions of the cable equation have been coupled to the vessel dynamics. This approach is adequate for shallow waters. However, in deeper waters, dynamic interactions between the vessel and mooring system render such a quasi-static approach inaccurate [3]. Software packages that solve the cable equation by means of the finite element method (FEM) are readily available. However, such general purpose FEM packages are not suited for control system design, and are usually slower than software tailored for a particular application. Moreover, the theoretical aspects, such as existence and uniqueness of solutions, are often taken for granted. In fact, FEM tools were developed and used, for instance, in structural engineering, decades before a sound theoretical foundation was established [4]. In this paper, a finite element model of a cable suspended in water is derived. The hydrodynamic loads on the cable are modelled according to Morison’s equation (see for instance, [2]). For a slightly simplified equation, describing the motion of a cable having negligible added mass and supported by two fixed ∗ Corresponding author. Tel.: +47-73591430. E-mail address: [email protected] (O.M. Aamo).

0378-4754/00/$20.00 © 2000 IMACS. Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 4 7 5 4 ( 0 0 ) 0 0 2 3 5 - 4

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end-points, we show global existence and uniqueness of solutions of the truncated system, and conjecture a global result for the initial-boundary value problem. The development of a full multi-cable mooring system that includes surface vessel dynamics, is not considered here, but is the topic of a forthcoming article [1].

2. PDE for the cable dynamics The equation of motion of a cable with negligible bending and torsional stiffness is given by (see for instance [6]) ρ0

∂ ∂ vE(t, s) = (T (t, s)tE(t, s)) + fE(t, s)(1 + e(t, s)) ∂t ∂s

where t is the time variable, and s ∈ [0, L], vE : [t0 , ∞) × [0, L] → R3 and tE : [t0 , ∞) × [0, L] → R3 are distance along the unstretched cable, velocity and tangential vector, respectively. L is the length of the unstretched cable, ρ 0 is mass per unit length of unstretched cable, T : [t0 , ∞) × [0, L] → R is tension, e : [t0 , ∞) × [0, L] → R is strain and fE : [t0 , ∞) × [0, L] → R3 is the sum of external forces (per unit length of unstreched cable) acting on the cable. By introducing the position vector rE : [t0 , ∞) × [0, L] → R3 , we get tE =

1 ∂ rE 1 + e ∂s

such that ∂ ∂ 2 rE ρ0 2 = ∂t ∂s



 T ∂ rE + fE(1 + e) 1 + e ∂s

Applying Hooke’s law yields   ∂ ∂ 2 rE e ∂ rE EA0 + fE(1 + e) ρ0 2 = ∂t ∂s 1 + e ∂s where E is Young’s modulus and A0 is the cross-sectional area of the unstretched cable. 2.1. External forces In addition to gravity, a submerged cable is subject to hydrostatic and hydrodynamic forces, i.e. fE = fE(hg) + fE(dt) + fE(dn) + fE(mn) where fE(hg) constitutes the bouyancy (gravity and hydrostatic) force per unit length of unstretched cable, fE(dt) and fE(dn) are tangential and normal hydrodynamic drag, respectively, per unit length of unstretched cable and fE(mn) is the hydrodynamic inertia force per unit length of unstretched cable.

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2.1.1. Gravity and hydrostatic forces It is assumed that we can regard each element of the cable as completely surrounded by water so that ρc − ρw gE fE(hg) = ρ0 (1 + e)ρc where gE ∈ R3 is the gravitational acceleration, ρ c the density of the cable and ρ w the density of the ambient water. 2.1.2. Hydrodynamic forces From Morison’s equation, see for instance [2], we get the following expression for hydrodynamic drag per unit length of unstretched cable fE(dt) = − 21 CDT dρw |E v · tE|(E v · tE)tE = − 21 CDT dρw |E vt |E vt fE(dn) = − 21 CDN dρw |E v − (E v · tE)tE|(E v − (E v · tE)tE) = − 21 CDN dρw |E vn |E vn where CDT and CDN are tangential and normal drag coefficients for the cable, respectively, and d is the cable diameter. The hydrodynamic inertia force per unit length of unstretched cable is given by: πd πd a − (E a · tE)tE) = −CMN ρw (E ρw aEn fE(mn) = −CMN 4 4 2

2

where CMN is a hydrodynamic mass coefficient and aE : [t0 , ∞) × [0, L] → R3 is the acceleration. The subscripts ‘n’ and ‘t’ on vE and aE denote decompositions into the normal and tangential directions, respectively. 2.1.3. Formulation of the initial-boundary value problem We have the following initial-boundary value problem   ∂ ∂ 2 rE e ∂ rE ρ0 2 − EA0 − (1 + e)(fE(hg) + fE(dt) + fE(dn) + fE(mn) ) = 0 ∂t ∂s 1 + e ∂s

(1)

with boundary conditions rE(t, 0) = rE0 (0),

rE(t, L) = rE0 (L),

for all t ≥ t0

and initial conditions rE(t0 , s) = rE0 (s),

vE(t0 , s) = vE0 (s)

Here, rE0 : [0, L] → R3 and vE0 : [0, L] → R3 are initial cable configuration and initial cable velocity, respectively. 3. Discretization into finite elements Discretization of the initial-boundary value problem is performed using the Galerkin method and finite elements. This method consists of the following steps

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1. The initial-boundary value problem (1) is transformed into the corresponding generalized problem. This is done by multiplying the equation by the functions w E ∈ V , and then integrating by parts over [0, L]. V is a suitable space of functions in which to search for a solution. 2. Restriction of rE and w E to appropriate finite-dimensional subspaces Vn ⊂ V , yields the Galerkin method. 3. Choosing the finite-dimensional subspaces such that they are spanned by bases consisting of so-called finite elements, yields a particularly simple set of ordinary differential equations. This is the finite element method. The Galerkin equation resulting from (1) is given by   ρ0 l ek ek+1 (¨r¨ k−1 + 4¨r¨ k + r¨ k+1 ) + EA0 lk − l k+1 6 εk εk+1 Z L (fE(hg) + fE(dt) + fE(dn) + fE(mn) )(1 + e)ϕk ds, k = 1, 2, . . . , n − 1 (2) = 0

where l k = r k − r k−1 1 |ll k | ek = |rr k − r k−1 | − 1 = −1 l l εk = l(1 + ek ) = |ll k |  0 s < (i − 1)l       (1/ l)s − (i − 1) (i − 1)l ≤ s < il , ϕi (s) =  −(1/ l)s + i + 1 il ≤ s < (i + 1)l      0 (i + 1)l ≤ s

i = 0, 1, 2, . . . , n

The subscripts ‘hg’, ‘dt’, ‘dn’ and ‘mn’ stand for hydrostatic and gravity forces, tangential drag forces, normal drag forces, and hydrodynamic added inertia forces, respectively. n is the number of finite elements, and l = L/n is the unstretched length of each element. The triangular form of the ϕ i functions reflects the choice of a finite element basis for the subspaces Vn . Note that in this form, algebraic expressions for the drag forces cannot be found. However, in Section 5, approximations are introduced that eliminate the need for numerical integration of these terms. 4. Existence and uniqueness of solutions In this section, we show existence and uniqueness of solutions for a slightly simplified equation under the assumption of strictly positive strain. This is the main contribution of the paper. Assumption 1. There exists a constant c > 0, such that 1. e(t, s) ≥ c for s ∈ [0, L] and for all t ≥ t0 and; 2. ek (t) ≥ c for k = 1, 2, . . . , n and for all t ≥ t0 . Neglecting the added mass term fE(mn) which means that we assume drag dominant behaviour, and conside-

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ring a damping term in the form v |E v fE(d) = − 21 CD dρw |E yield the following slightly modified initial-boundary value problem   e ∂ rE ρc − ρw 1 ∂ 2 rE EA0 ∂ − − gE + (1 + e)CD dρw |E v |E v=0 2 ∂t ρ0 ∂s 1 + e ∂s ρc 2ρ0

(3)

with boundary conditions rE(t, 0) = rE0 (0),

rE(t, L) = rE0 (L),

∀t ≥ t0

and initial conditions rE(t0 , s) = rE0 (s),

vE(t0 , s) = vE0 (s)

Our goal is to apply Proposition 2.1 in ([5], p. 370), which states local existence and uniqueness of solutions for symmetrizable hyperbolic systems. Thus, we need to show that Eq. (3) is symmetrizable. Define u (t, s) as follows   rE    ∂ rE  u0       u (t, s) =  u 1  ,  ∂s     ∂ rE  u2 ∂t Carrying out the differentiation in the second term in (3), the equation, in terms of u , can be written as A0 where

u u ∂u ∂u = A1 +g ∂t ∂s 

(4)

I

  A 0 (t, s, u ) =  0  0  0   0 A 1 (t, s, u ) =     0

EA0 ρ0



0 u1uT1 e + I (1 + e)3 1 + e 0



0



  0  I

0 0 EA0 ρ0



  g (t, s, u ) =    ρ c − ρw ρc



u 1u T1 e + I 3 (1 + e) 1+e



EA0 ρ0

u2 0 gE −

1 u2 |u u2 (1 + e)CD dρw |u 2ρ0



0 u 1u T1 e + I 3 (1 + e) 1+e 0

     

       

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In (4), we have multiplied the equation by the matrix A 0 , which is symmetric positive definite for s ∈ [0, L] and for all t ≥ t0 under Assumption 1 (in fact, there exists a constant c, such that A 0 ≥ cII > 0, s ∈ [0, L], ∀t ≥ t0 ). Notice that the matrix A 1 is rendered symmetric, by means of the symmetrizer A 0 . Thus, Proposition 2.1 in [5] (p. 370), provides local existence of a unique solution to (4). However, based on the following arguments, we will conjecture that the solution can be continued for all time. For the Galerkin equation corresponding to (3), which is given by   ek+1 ek ρ0 l lk − l k+1 (¨r¨ k−1 + 4¨r¨ k + r¨ k+1 ) + EA0 6 εk εk+1 Z L (fE(hg) + fE(d) )(1 + e)ϕk ds, k = 1, 2, . . . , n − 1 (5) = 0

we can state the following result. Theorem 1. For any n ∈ {2, 3, 4, . . . }, let the initial state (rr (t0 ), v (t0 )) = (rr 0 , v 0 ) be given. If Assumption 1 holds, then there exists a unique solution of (5) for all t ≥ t0 . 

Proof. See [1]. k

k

Theorem 1 implies that u n ∈ H ([0, L]), for t ≥ t0 , where H ([0, L]) denotes the Sobolev space defined by l   ∂ u k 2 2 H ([0, L]) = u ∈ L ([0, L]) l ∈ L ([0, L]), 0 < l ≤ k ∂s with the natural norm ukH k ([0,L]) ku

l=k l X

∂ u

=

∂s l l=0

L2 ([0,L])

and un is given in terms of the finite element basis defined in Section 3, that is   r i (t)ϕi (s) n   X   u n (t, s) =  r i (t) ∂ϕi (s)   ∂s  i=0 v i (t)ϕi (s)

(6)

In fact, Theorem 1 implies that there exists a constant c, independent of k and n, such that un kH k ([0,L]) ≤ c, ku

∀t ≥ t0 , n = 2, 3, . . .

Based on the above considerations, along with the results of Chapter 16, Sections 1 and 2 in ([5], pp. 359–372), we conjecture the following. Conjecture 1. Suppose Assumption 1 holds, and that u (t0 , s) ∈ H k ([0, L]), with k ≥ 2. Then there exists a unique solution u ∈ C([t0 , ∞]), H k ([0, L]), to the initial-boundary value problem (4). Moreover, the sequence of solutions u n (as given in (6)) of the Galerkin Eq. (5), converges to u in the following sense u − u n kH k ([0,L]) → 0 as n → ∞ ku

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Remark 1. We stress the fact that since Theorem 1 and Conjecture 1 are stated under Assumption 1, global solutions are not guaranteed for all initial conditions. The problem of finding conditions on the initial data under which Assumption 1 holds (for all t ≥ t0 ), is outside the scope of this work. 5. Implementation It is desirable to apply certain approximations to the terms of Eq. (2) in order to simplify implementation. Looking at the kth node, we see by inspection of Eq. (2), that it takes an advantageous form if the following approximations are applied r˙ k−1 ≈ r˙ k ,

r˙ k+1 ≈ r˙ k

(7)

r¨ k−1 ≈ r¨ k ,

r¨ k+1 ≈ r¨ k

(8)

With these approximations, Eq. (2) reduces to the following: " !#  l k+1l Tk+1 C1 C1 l kl Tk ρ0 l + + (εk + εk+1 ) I 3×3 − r¨ k 2 2 εk εk+1 = f k(hg) + f k(dt) + f k(dn) + f k(r) , where



f k(r) = EA0

ek ek+1 l k+1 − l k εk+1 εk

k = 1, 2, . . . , n − 1

(9)



ρc − ρw [0 0 g]T ρc # " l k+1l Tk+1 C2 l kl Tk =− + r˙ k |˙r˙ k · l k | 2 + |˙r˙ k · l k+1 | 2 2 εk εk+1

f k(hg) = lρ0 f k(dt)

f k(dn)

" ! ! C3 l kl Tk l kl Tk =− εk I 3×3 − 2 r˙ k I 3×3 − 2 2 εk εk ! !# l k+1l Tk+1 l k+1l Tk+1 +εk+1 I 3×3 − r˙ k I 3×3 − r˙ k 2 2 εk+1 εk+1

C1 = CMN

π d2 ρw , 4

C2 = 21 CDT dρw ,

C3 = 21 CDN dρw

I 3×3 is the 3 × 3 identity matrix, and the subscript ‘r’ stands for internal reaction forces. Clearly, in the limit as n → ∞, (2) and (9) are identical. However, simulations suggest that (8) imposes fundamental restrictions on the accuracy of the model, so (8) may be an undesirable approximation in certain applications. The effect of omitting (8) is a fuller mass matrix, that requires more computations to invert. Modelling a moored vessel is now a matter of assembling the above equations for each mooring line. The details of this procedure are available in [1].

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6. Conclusions In this paper, we have developed a finite element model for a cable suspended in water. Global existence and uniqueness of solutions of the truncated system is shown for a slightly simplified equation describing the motion of a cable with negligible added mass and supported by fixed end-points. Based on this, along with well known results on local existence and uniqueness of solutions for symmetrizable hyperbolic systems, we conjecture a global result for the initial-boundary value problem. Acknowledgements This work was supported by the Research Council of Norway, which is gratefully acknowledged. The first author would also like to thank Professor Helge Holden, Department of Mathematical Sciences, Norwegian University of Science and Technology, for his helpful comments to Section 4, and Dr. Jann Peter Strand, ABB Industri AS, for his general comments, and support of the project. References [1] O.M. Aamo, T.I. Fossen, Finite element modelling of moored vessels, J. Math. Comput. Model. Dyn. Syst., accepted for publication. [2] O.M. Faltinsen, Sea Loads on Ships and Offshore Structures, Cambridge University Press, Cambridge, 1990. [3] H. Ormberg, I.J. Fylling, K. Larsen, N. Sødal, Coupled analysis of vessel motions and mooring and riser system dynamics, in: Proceedings of the 16th International Conference on Offshore Mechanics and Arctic Engineering, New York, 1997, pp. 91–100. [4] G. Strang, G.J. Fix, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, 1973. [5] M.E. Taylor, Partial Differential Equations III, Springer, New York, 1996. [6] M.S. Triantafyllou, Cable mechanics with marine applications, lecture notes, Department of Ocean Engineering, Massachussetts Institute of Technology, Cambridge, MA 02139, USA, May 1990.