application homotopic methods to static flexible pipe problems

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(ii) a ‘set’ of problems promoted by the International Ship and Offshore Structures

Congress (ISSC), focusing on the bending moment and torque interaction of a flexible

pipe, suspended in three dimensions between fixed and adjustable end points.

BACKGROUND TO SLENDER RODS AND CONTINUATION METHODS

Slender rods

A slender rod is a structure where the ratio of the length to a characteristic diameter of the perpendicular

cross section is large. Examples are threads, hoses, cables, umbilicals, flexible pipes and long spans of stiff

pipes. A geometric non-linearity arises from the finite displacement of the longitudinal axis and may be

further compounded if the flexural or torsional resistances are significant.

The classical equilibrium condition1 results in a set of non-linear ordinary differential equations for the

internal cross section force and moment, including terms for the load that are possibly displacement

dependent. Further relations used to form a closed set are the Hooke’s uniaxial, Bernoulli-Euler flexure and

Saint-Venant’s torsion constitutive models, the Eulerian angles between the principal flexure-torsion axes

and fixed reference axes, direction of the central-line, undisplaced and displaced distances along the central-

line and boundary conditions dependent on the unknown integration limits. The slender rod relations of

Kirchhoff 2 and Clebsch3 are briefly described in Appendix 1 and are the basis of the work herein.

The description of strain by directors, Erickson et al.4, allows for the inclusion of generalised constitutive

relations, elastic or otherwise, in slender rod models. This approach is a more formal introduction to the

classical equations5,6. Establishment of slender rod equations from direct considerations of three-

dimensional elasticity has, however, only recently been attained by Davi7.

Example boundary (multi-point) conditions, that may be specified at the ends or intermediate points, are

position, orientation, or fixity. More general are non-linear point conditions implicitly dependent on the state

of the rod. An example might be a touch-down point on a buoy or the seabed. Closed form analytic solution

methods are available for only a few load conditions, such as, the helix and the end-point loading in two

dimensions of the elastica. The helix geometric configuration a priori simplifies the problem considerably.

Numerical models using slender rods permitting the full six degrees of freedom of the cross section as in the

classical equations have been developed to simulate marine pipelaying, risers and flexible pipes 6,8-15.

Related models that impose some constraints are common. For example, combinations of two-dimensional

analysis16-19, moderate displacement and rotation20-23 or equal principal bending stiffnesses8-23 have all been

used. Most finite element models utilise linear and cubic interpolation on the tension/torsion and bending

respectively. Discretisations proceed either as a direct consequence of the classical equations or as

segmented beams11,12,15.

Application of homotopic methods to static flexible pipes and cable problemsR. J. Smith & C. J. MacFarlane 12 December 1995



A principal difficulty to obtaining solutions from numerical models lies with obtaining convergence over the

geometric non-linearity of non-small displacement and rotation. Standard iterative non-linear solution

procedures are known to be non-convergent in numerous cases if started from a naïve initial state. Cases of

interest are the coupling between bending moment and torque and non-linear boundary conditions. Coupling

between the bending moment and torque is inevitable in three-dimensional configurations especially with

non-circular cross sections.

Continuation methods

Irrespective of the choice of numerical model for the flexible pipe or cable model, solutions require the

determination of unknown parameters with non-linear influence on a set of simultaneous conditions. The

minimal set consist of the boundary conditions dependent on the unknown integration limits. The widely

used Newton-Raphson method is restricted by conditions of local convergence, requiring good initial

estimates that may be unrealisable in highly non-linear cases.

The domain of convergence can be expanded by using a continuation parameter. This may be an artificial

parameter that varies between, say [0, 1], or it may be a physical quantity such as a load factor, length or

material property, where the variation would be between a known solution and the final value. Where an

artificial parameter is used, the dependence of the simultaneous conditions on the parameter is established by

embedding the conditions and parameter in a set of equations of equal number. For a physical parameter,

although the dependence of the simultaneous conditions on the parameter is implicit to the problem, it may

still be advantageous to develop an equivalent embedding to yield an easier initial solution or avoid

intermediate bifurcations that may not be of interest. The continuation parameter is then transformed by

solving a series of closely related problems, starting from the known solution and leading to a final solution

that satisfies the governing simultaneous equations.

The transformation of the problem from n equations with n unknown parameters to one with the same

number of equations and (n+1) parameters, gives rise to a sequence of solutions forming a line in (n+1)

dimensional space. The transformation is a homotopy if it yields lines that connect two solutions with the

initial and final values of the continuation parameter. The lines of solutions are homotopy paths. These terms

are, however, commonly only applied to an embedding using an artificial parameter. A homotopy path may

meander and bifurcate and can be obtained by a variety of numerical methods

24-27

, some of which are morerobust than others.

Figure 1 shows an idealised example of the development of a continuation/homotopy against some

(unknown) physical variable. The Davidenko and incremental methods employ a monotonic change of the

homotopy parameter using variants of the Newton-Raphson method to advance along the homotopy path in

differential or small steps. At a turning (fold or hysteresis) point, with respect to the homotopy parameter,

where the next solution in the monotonic sequence is not local - for example, point A in Figure 1 - the (n∗ n)

Jacobian (gradient) matrix of the equations is singular and the methods fail. The turning point represents a

fold bifurcation or instability of the system underlying the equations. Another parameterisation is necessary

for the computation to proceed along the homotopy path.




homotopy/continuation parameter λ

unknown variable

fold bifurcation

fold bifurcation

branch bifurcations

path of the homo topy/continuation

A

C

λa λb

incrementalNewton-Raphson

?

B

D

E

Typical failure points for co nventional solutions

Figure 1

It should be noted, however, that a perturbed incremental Newton Raphson method may jump to (iterate

towards) point C if the solution at A lies within the radius of convergence. The convergence of this jump is

independent of the incremental step (λb - λa) approaching zero. Similarily a perturbed Davidenko method

may push past the fold bifurcation and continue along the homotopy path towards point B.

Parameterisation of the homotopy, with respect to distance along the path, gives rise to a rectangular (n∗

n+1) Jacobian matrix that is of full rank at points that have a unique tangent, including fold bifurcations (A

or B in Figure 1). At a branch bifurcation point of the homotopy path (point D in Figure 1), the (n∗ n+1)

Jacobian matrix is rank deficient. Further analysis of the matrix will yield the directions of the emanating

branches28 and one will connect smoothly with the incoming homotopy path.

The probability of a branch bifurcation on a homotopy path parameterised with respect to distance is almost

zero. Either a small arbitrary change in the selection of the initial data will result in a shift onto a non-

bifurcating path, or, an adjustment of the step size or tolerance in a stable numerical method that calculates

the direction of the path will effect an advance across the point onto the smoothly connected emanating

branch.

Likely reasons for not connecting to a final solution are the tracking of a homotopy path attracted to a stable

limit cycle or that a solution of the governing simultaneous equations does not exist. Avoidance of a limit

cycle may require a significant change in the selection of the initial data or homotopy function. If a solution




of the governing simultaneous equations does not exist then the homotopy path will pass through an infinite

sequence of turning points or become unbounded.

An early review of standard continuation methods in which some still valid problems are discussed was

presented by Wasserstrom29. More recent discussions include Allgower26, Seydal27 Watson30. In a 1981

paper, Watson31, showed the non-convergence of various numerical continuation and more general

homotopic schemes, applied to an elastica. The method, clearly the most robust, was based on a rather well

disguised Newton homotopy24,32 which is utilised below. We consider the Newton homotopy parameterised

with respect to distance along the path, meets and solves many common difficulties and it is, perhaps, ironic

that after this discussion prompted by the non-convergence of standard non-linear equation solvers, we rather

paradoxically utilise a homotopy name after Sir Isaac Newton.

IMPLEMENTATION

Implementation of the homotopy method

The slender rod discretisation described below, generates a set of unknowns x = ( x1, x2, ... , xn) as the main

parameters of an initial value shooting scheme. The unknowns have a non-linear influence on boundary

conditions h = ( h1, h2, ... , hn) and it is required to find x = x* such that

h x 0( *) = . (1)

The smooth transformation of say x = a to x = x* is based on embedding h(x) in the Newton homotopy

( ) ( )=x a h(x) ( )h a 0, ,λ λ+ − =1 , for all x, a, λ , (2)

where λ is the continuation parameter and a = x at λ = 0. Methods to smoothly track a path that connects the

points (x = a, λ = 0) and (x = x*, λ= 1) are based on deriving ordinary differential equations from

d d d d=ρ ρ ρ∂

∂

∂

∂λλ

∂

∂xx

aa 0⋅ + + ⋅ = . (2a)

The Davidenko continuation method is based on holding a constant and the parameterisation of Equation 2a

with respect to λ:

A x h a 0⋅ ′ + =( ) , (3)

where

Ah

xx

xx a= ′ = =

∂

∂ λλ, , ( ) ,

d

d0 0 ≤ . (3a)

The integration of ′ x will succeed as long as the inverse of A exists. This is not the case at a turning point

with respect to λ, where det(A) = 0 and the slender rod is at a fold bifurcation or instability. Around these

points another parameterisation is necessary. For example, a similar parameterisation with respect to one of

the components of x or more generally, distance along the homotopy path.

We adopt the following parameterisation with respect to distance along the homotopy path for all points:

B y 0⋅ ′ = , (4)




where

Bh

xh a=

⎡

⎣⎢⎤

⎦⎥∂

∂( ) , ′ =

⎛

⎝ ⎜

⎞

⎠⎟y

xd

dα λ, ,y

a( )0 =

⎛

⎝ ⎜

⎞

⎠⎟

λλ'( )0 0> , , (4a)0 ≤ α

in which α is the distance coordinate, B is a rectangular (n∗ n+1) matrix and is the direction (tangent) of

the path. The non-trivial solution for lies in the kernel (null space) of B and it is readily shown to be a

linear combination of the eigenvectors for the zero eigenvalue of B

′ y

′ y

T B though the evaluation does not

proceed in this way. A set of basis vectors for the kernel is obtained from the factorisation B = LQ, where L

is a lower rectangular (n∗ n+1) matrix and Q is an orthonormal (n+1∗ n+1) matrix. Givens’ rotations are

used to obtain the factorisation and yield stable results. Alternative factorisations of B are Householder

reflections and LU methods. If the ith column of L is zero then the ith row of Q is part of the basis for the

kernel. At smooth points along the homotopy path, including turning points with respect to λ or components

of x, the kernel is spanned by a single basis vector qn+1, located in the last row of Q.

Conditions that complement Equation 4 are the tangent of the homotopy path is of unit magnitude and

changes in the direction are smooth:

′ =y2

1 , (4b)

detB

y′

⎛

⎝ ⎜

⎞

⎠⎟ >T 0 for all α, or for all α. (4c)det

B

y′

⎛

⎝ ⎜

⎞

⎠⎟ <T 0

If det(Q) = 1 as yielded by Givens’ rotations then

det' ,

B

y T i ii

n

l⎛

⎝ ⎜

⎞

⎠⎟ =

=∏

1

, (4d)

where li,i is the ith diagonal element of L and Equation 4c is satisfied by setting

y q'= + +n 1 for all α, or y q'= − +n 1 for all α. (4e)

At a branch bifurcation point on a homotopy path parameterised with respect to distance, the kernel is

spanned by two or sometimes more basis vectors and the determinant of the (n+1∗ n+1) row augmented

Jacobian matrix in Equation 4d is zero. In the main, there are two branch curves at a bifurcation point and

the tangents are the real solutions of a system of quadratic equations derived from B and its kernel28. The

choice of tangent will depend on a stability criterion which for a slender rod could be the minimal rate of

change of energy. As stated previously, the probability of tracking a homotopy path with a branch

bifurcation point is almost zero. A small change in the initial data will provide a shift onto a non-bifurcating

branch. A stable numerical method, such as Givens’ rotations, for calculating the tangent of the homotopy

path, will most likely step over the point and continue along the smoothly connected emanating branch. A

bifurcation point with two branches is detected by the change of sign of the determinant in Equation 4d, that

is, the direction of the tangent will flip back to the point. The change of sign is easily detected and the path

following is resumed by changing the conditions adhered to in Equations 4c and 4e.

Implementation of the slender rod method

The relations for the statical equilibrium, kinematics and strain energy of a slender rod are based on the

classical description of Love1 as given in Appendix 1. The relations are expressed as a core set of 15 non-




linear coupled ordinary differential equations. Additional ordinary differential equations are included for

certain unknown parameters and components of the matrix equation for the first variation, as explained

below. The rod is divided into multiple segments of non-dimensional unit length. The end-points of each

segment are chosen to coincide with the discontinuities or collection of data for the boundary (multi-point)

conditions. Discontinuities may occur in the load or geometry of the rod.

The independent variable of the set of ordinary differential equations is a non-dimensional distance along the

displaced central-line. The main dependent variables are the cross section stress resultants, kinematical

properties along the central-line, strain energy and certain unknown parameters. The cross section stress

resultants are the principal shears, compression or tension, principal bending moments and torque. The

kinematical properties along the central-line are the Eulerian angles between the principal flexure-torsion

axes and fixed reference axes, displaced position and longitudinal distances in the undisplaced and displaced

configurations. Unknown parameters which influence the boundary conditions, excluding integration limits

for the preceding list, are represented in the set by trivial differential equations as shown in Appendix 1.

This allows a consistent implementation of the non-linear equation solver for all of the unknown parameters.

The dependent variables from the matrix equation for the first variation are the partial derivatives with

respect to the unknown initial integration limits.

The resulting set of equations for the main dependent variables is of the form

( )d

d

qq

ug u

τ= , 0 1 1 2< < =τ , , , ... ,q Q

1

, (5)

u u u u discontinuities1 1 10 0 10

( ) , ( ) ( ) ( )= = + >−q q for q , (6)

where are respectively the number of segments, non-dimensional distance on the central-

line, main dependent variables on the qth segment, initial integration limits at an extreme end of the rod and

non-linear relations that couple the main dependent variables. The unknown initial integration limits in u ,

are denoted by x and are the independent variables of the boundary (multi-point) conditions

Q q, , , ,τ u u g10

10

h x u u u u u u 0( ; ( ), ( ), ( ), ( ),..., ( ), ( ))1 1 2 20 1 0 1 0 1Q Q = . (7)

where u1 (0), u2 (0),..., are the main dependent variables collected at the segmentation points of the rod.

The Jacobian matrix of the boundary conditions is

( )( )

( )( )∂

∂∂

∂∂

∂∂

∂∂

∂h

x

h

u

u

x

h

u

u

x= ⋅ + ⋅⎛

⎝ ⎜⎜

⎞ ⎠⎟⎟

=∑

q

q

q

q

q

Q

00

11

1

, (8)

where ∂ h/∂uq(.) is directly derived from Equation 7 and ∂uq(.)/∂ x is a matrix consisting of columns,

corresponding to x in the first variation. The first variation ∂ ∂u uq 10is obtained by the integration of the

first variational equation

d

d

q

q

q

τ

∂

∂

∂

∂

∂

∂

u

u

g

u

u

u1 10 0

= ⋅ , 0 1 1 2< < =τ , , , ... ,q Q , (9)

∂

∂

∂

∂

∂

∂

∂

∂

u

uI

u

uI

discontinuities

u

u

u1

1 1 1

1

1

0 0

1

11

0 0 0

( ),

( ) ( )

( )

( )

= = +⎛

⎝ ⎜⎜ ⎞

⎠⎟⎟ ⋅−

−q

q

q

for q > , (10)

in which I is the identity matrix.




Initial value shooting

The numerical method adopted, initial value shooting, is usually discounted in highly non-linear problems as

yielding Newton-Raphson iterations which diverge, oscillate or lead to numerical stiffness. We have

coupled the initial value integration of the rod with the homotopy method described above. This results in a

low initialisation over-head and small Jacobian matrices, providing rapid computation of the direction of the

homotopy path that is the argument of a second integration. The double integration is based on two

invocations of the Prince-Dormand33,34 variable step-size embedded Runge-Kutta scheme of eighth-order,

allowing the final results to be checked to a tolerance of 10-15.

A large gain of efficiency can be obtained from the minimisation of the arithmetic operations involving the

sparse coefficient matrix in the equation for the first variation and by only computing the necessary

components of the first variation relating to the unknown integration limits. The integrations of the main

dependent variables of the slender rod and the direction of the homotopy path are much less intensive.

Energy arguments

The slender rod model includes energy as an optional variable and provides a chance for it to be assessed at

any point on the homotopy path. We have noted above that energy minimisation might be a criterion on

which to select a path at a branch bifurcation. Schemes have been considered where a in Equation 2 is

defined as a variable in order to seek a minimum energy homotopy path. It was thought that this might lead

to faster convergence or selection of primary and secondary solutions. Practice, however, has shown

difficulties. A simple scheme to minimise energy was applied at each homotopy step. In the first stages of

the homotopy from an unloaded state, the solution path has to be unconstrained to avoid a form of resonance

near zero and in the later stages to allow a solution to be tracked efficiently. This suggests a filtered function

but the weighting to give the minimisation and its filters is entirely subjective - an undesirable property in a

general solution scheme.

This is a topic that might repay further study for particular sets of problems but it is considered at this stage

that energy arguments should only be mobilised at bifurcation points to select a preferred path.

CASE STUDIES

Helix

The problem selected here can be solved analytically but presents difficulties for standard numerical

techniques, therefore it demonstrates the potential of the homotopic technique to converge from a minimum

of prior knowledge of the solution state.

The task is to deform an initially straight pipe into a given circular helix using forces and moments only

applied at the ends. Analytically this has been reported, for example, by Love1. The properties of the pipe,

helix, end forces and moments are given in Appendix 2.




In the application of a finite element system, the analyst might use the end point boundary conditions to

search freely for solutions. It is not clear to us that the helical solution could be found unless the pipe was

formed around a cylinder of the required dimensions defined using contact elements - sometimes called an

inverse problem. Other approaches are possible but each will require some skill and effort from an

experienced finite element analyst. The problem is essentially one of convergence and can only be eased by

having available some preliminary estimate of the final load matrix. This can be demonstrated by

considering Figures 3 and 4, but first we consider Figure 2 which shows the evolution of the homotopy

parameter λ along the search path.

The homotopy parameter along the search path

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 2 4 6 8 10 12 14 16 1

distance along search path

h o m o t o p y p a r a m e t e r

solution

8

Figure 2

The boundary conditions are satisfied where the homotopy parameter is equal to unity and it can be seen that

five solutions have been identified that satisfy the boundary conditions. Importantly, however, only one

solution follows the shape of a helix because the boundary conditions in no way inferred the shape of the

pipe between the end points. In the finite element inverse approach discussed above, some of these solutions

could not be found by use of the shaping cylinder. In particular, it should be noted, that the inverse finite

element approach only works because the final helical configuration is known.

The determinant of the Jacobian matrix of the boundary conditions

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14 16 1


t a n h ( d e t e r m i n a n t )

8

Figure 3




Figure 3 plots the determinant of the Jacobian matrix of the boundary conditions along the search path using

the hyperbolic tangent to clarify the plot. In between each of the solutions is a turning point with respect to

the homotopy parameter, where the determinant is zero. At these points the Jacobian matrix of the boundary

conditions is singular and the equilibrium of the pipe is unstable.

The determinant of the row augmented Jacobian matrix of the homotopy

-1.01

-1

-0.99

-0.98

-0.97

-0.96

-0.95

0 2 4 6 8 10 12 14 16 1distance along search path

t a n h ( d e t e r m i n a n t )

solution points

8

Figure 4

By comparison, the determinant of the row augmented Jacobian matrix of the homotopy (Figure 4) does not

cross zero. Hence the path has no bifurcation points. The middle (third) solution is the desired helix but

because it and the adjacent solutions are very close to turning points, they are only weakly stable. The first

and fifth solutions are stable.

The strain energy of the along the serarch path

0

1

2

3

4

5

6

0 2 4 6 8 10 12 14 16


s t r a i n e n e r g y / k J

solution points

18

Figure 5

Figure 5 plots strain energy along the search path. It is surprising that the five solutions that satisfy the

boundary conditions are of almost equal strain energy, varying by 10 Joules around an average of 2520

Joules. We have not given this further consideration.

Figures 6 to 11 are plots of the forces and moments at the end S = 0 against the homotopy parameter. Theyare of interest in showing a characteristically flowing evolution of the homotopy and the way that it reverses.




The appearance of the homotopy path bifurcating is a function of two-dimensional plotting. The homotopy

was initiated from a 1 N lateral shear load.




Search for the f irst principal shear at end S = 0

-0.8

-0.7

-0.6

-0.5

-0.4-0.3

-0.2

-0.1

0

0 0.2 0.4 0.6 0.8 1 1.2

homotopy parameter

s h e a r 1 / k

N

Figure 6

Search for the second principal shear at end S = 0

-0.1

0.4

0.9

1.4

1.9

2.4

2.9

3.4

3.9

0 0.2 0.4 0.6 0.8 1 1

homotopy parameter

s h e a r 2 / k N

.2

Figure 7

Search for the tension at end S = 0

-2.5

-1.5

-0.5

0.5

1.5

2.5

3.5

-0.05 0.15 0.35 0.55 0.75 0.95 1.15

homotopy parameter

t e n s i o n / k N

Figure 8




Search for the first principal moment at S = 0

-0.8

-0.6

-0.4

-0.2

0

0.2

0 0.2 0.4 0.6 0.8 1 1.2

homotopy parameter

m o m e n t 1 / k N

m

Figure 9

Search for the second principal moment at S = 0

-1

-0.5

0

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6 0.8 1 1.2

homotopy parameter

m o m e n t 2 / k N m

Figure 10

Search for the torque at end S = 0

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 0.2 0.4 0.6 0.8 1 1.2

homotopy parameter

t o r q u e / k N m

Figure 11




Experimentally based case studies

This set of case studies allows comparison with experimental and calculated results reported by Kodaissi et

al.35 and the International Ship and Offshore Structures Congress36 (ISSC) as part of a series of studies37.

Kodaissi et al. set out two problems. The first is a cable hanging as a catenary, with a torque applied at the

ends to produce an out of plane displacement, leading to buckling. The other is described by Figures 12 and

13 and in Appendix 3. The torsioned cable is essentially the helical problem discussed above with which our

formulation has no difficulty.

α

x

y

z

double

articulation

about

fixed end

moving end

flexiblepipe

SIDE VIEW

β

horizontal axisof rotation

x and y

Figure 12: Experimental setup(from Kodaissi et al.35 and ISSC36 with amendments)

TOP VIEWy

x0

1

torque &

tensiongauges

fixed end

moving end

β

0 is initial position1,2,3,4 are displaced positionsby rotation around horizontal axes

32

4

these curves representdifferent α rotations

FIGURE 13: Moving end positions(from the ISSC36 case study with amendments)




The second case was adopted by the ISSC. It provided a comparison of alternative commercial programs

used to solve the cross section stress resultants and configuration of offshore flexible pipe. It focuses on the

bending and torsion interaction of a flexible pipe, suspended in three-dimensions between fixed and

adjustable end points. The results of the comparison are available from the ISSC. Only six of the 23 bodies

known to the ISSC to be involved with the development of flexible pipe numerical models completed the

study, suggesting that the prediction of taut three-dimensional problems may cause difficulties.

α

(deg)

β

(deg)

source

fixed

end

tension

(kN)

moving

end

tension

(kN)

sum

of

tensions

(kN)

moving

end

torque

(kNm2 /rad)

fixed

end

torque

(kNm2 /rad)

minimum

bending

radius

(m)

0 0

1

26

2.08

2.022.37

3.25

3.023.48

5.33

5.045.85

0.09

0.000.00

0.13

0.000.00

1.22

1.081.3

30 45

1

2

6

2.56

2.24

2.53

3.42

3.12

3.30

5.98

5.36

5.83

-0.88

-0.85

-0.74

-1.03

-0.58

-0.74

0.84

0.75

0.72

30 90

1

2

6

2.12

2.50

2.54

3.28

2.80

3.38

5.40

5.30

5.92

-1.60

-1.73

-1.37

-1.43

-0.52

-1.37

1.06

0.80

0.82

35 45 6 2.57 3.23 5.80 -0.79 -0.79 0.67

35 90

3

4

5

6

2.70

2.58 : 2.6

2.60

2.59

3.40

3.12 : 3.43

3.30

3.33

6.10

5.77 : 6.03

5.90

5.92

-1.70

-1.51 : -1.73

-1.56

-1.50

-1.50

-9.50 : -1.54

-1.43

-1.50

0.98

0.70 : 0.76

0.74

0.76

Table 1: Comparison of results with the Kodaissi et el. and ISSC case studies

sources: 1) Kodaissi et al.35 experiment, 2) Kodaissi et al.35 calculation, 3) ISSC36 experiment, 4) ISSC36 range of

participants’ calculations, 5) ISSC36 mean of participants’ calculations, 6) present results




The elements of the problem are defined in Appendix 3. Kodaissi et al. provide both experimental and

computational results for eight positions of the movable end of the pipe. The ISSC comparison used only

four end positions, not all comparable with Kodaissi et al. Table 1 provides a summary of the results for the

configurations used by the ISSC.

Case α = 35o , β = 90o

First discussing the tensions, where we would expect accurate experimental results, it is interesting to

note that the weight of the pipe and end fittings in the experiment is 6125 N. A simple summation of end

tensions from the experiment is 6100 N. The mean from the original participants is 5900 N and our result

of 5920 N is within the range of calculated values. It is not clear why the calculations should differ from

the experimental values unless the instrumentation is measuring strain in slightly different axes.

A change in torque along the pipe is most likely induced by either friction or a non-uniform torsional

stiffness. The frictional resistance to the relative movement of the unbonded layers in a flexible pipe mayeffectively induce a large localised increase of the torsional stiffness, leading to a lock-in of the torsion.

The bending radius is obviously also affected by internal effects and it is important to note how much this

seems to be dependent on friction in the experimental results.

Our technique has the capability to deal with asymmetric cross sections, prescribed bending moment and

torque load distributions and non-uniform stiffnesses. The stiffnesses may depend, for example, on the

curvature about the principal flexure-torsion axes and parameters of the load. The effect of friction and

other internal effects can be modelled as bending moment and torque load distributions. This is a critical

area for development in the solution of practical problems. Reliance on radii derived without internal

effects may lead to systems that lock-in and suffer compression failures in service.

Cases α = 30o

Again there are clear differences in the torque due, as we suppose, to locking in by friction and this

affects the bending radius. One case provides fair agreement on tensions at the ends of the pipe but the

other shows a significant difference. If we assume as before, that the sum of the tensions is a reasonable

comparator, we find that the experimental values jump considerably between one case and the other.

This seems strange as the rotation of the pipe is not so great as to justify a significant redistribution of stress. The results of the calculations presented by Kodaissi et al. do not show a large change in the sum

of the tensions. Thus we (rather arrogantly) suggest that it may be some unsteadiness in the experimental

results that causes the differences in outputs.

The homotopy found only one solution for each end configuration which is likely to be unique given the

relative short length and axial stiffness of the pipe. Multiple solutions for each end condition exist upon

lengthening the pipe though only one is stable.




The result of the helix and flexible pipe case studies demonstrate that the homotopic technique can perform

calculations equivalent to existing schemes. It should be noted, that it requires no special discretisation near

the terminations to provide almost continuous information in these difficult areas. It should also be noted,

that the solution was initiated from a of a one Newton lateral load at the fixed end and zero mass distribution.

Inconsistencies in information

We have to note that we have found inconsistencies in the information that we have concerning the ISSC and

Kodaissi et al. studies. The variables in question are the length of the pipe used and the offset of the initial

experimental position of free end of the pipe. We cannot obtain consistent length/weight figures for the

system. For example, if we compare the experimental and calculated results from Kodaissi et al. for the two

dimensional initial configuration, dividing the summation of the end tensions by the given mass distribution

(and gravity) gives a length of 8.53 metres (experimental) and 7.6 metres (calculated) as against the 10.1

metres that we were given as an ISSC official length. Because in this initial two-dimensional case the ends

are vertically embedded these results should be equivalent. It would appear from the published results that

the parameters we have used are consistent with those used by other participants in the ISSC study and we

look forward to the problem being resolved.

CONCLUSIONS

A technique using homotopic mapping to solve six degree of freedom cable and pipe problems has been

developed. It does not require prior knowledge of the solution hence the solution may be initiated from an

effectively unloaded state. It is also very robust and unlike conventional continuation schemes, does not fail

at turning points in the homotopy parameter. The combination of homotopic mapping, initial value shooting

and classical slender rod equations segmented for discontinuities and multi-point conditions, allows complex

state dependent boundary conditions to be considered. We are developing boundary conditions such as

intermediate contact with a contoured surface and a mid-length linkage of a number of pipes. Further scope

exists for modelling the resultants of the internal loads as bending moment and torque distributions. It is

considered that the technique represents an advance on conventional methods because it converges where

others do not, may be initiated from an effectively unloaded state and solves for boundary conditions not

easily addressed by conventional schemes. Furthermore the computations are compact and efficient allowing

very fine resolutions and tolerances.

ACKNOWLEDGEMENTS

This work was developed over a number of years at the University of Strathclyde using funds scraped

together from a number of sources. It has greatly benefited from support in 1993 and 1994 by the UK

Engineering and Physical Sciences Research Council (EPSRC) and we acknowledge that funding with

thanks.




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analysis of spatial beams. Trans. Canadian Society of Mechanical Engineers, Vol. 18, No. 1,

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Verlag, Berlin, 1990.

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32. Keller, H. B., Global homotopies and Newton methods. In de Boor, C., and Golub, G. H.,

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Academic Press, New York, 1978, pp. 73-94.




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Computational and Applied Mathematics, Vol. 7, No. 1, 1981, pp. 67-75.

34. Hairer, E., N∅rsett, S. P. and Wanner, G., Solving ordinary differential equations 1: non-stiff

problems. Springer-Verlag, Berlin, 1987.

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behaviour of flexible risers. Proceedings of the Eleventh International Conference on Offshore

Mechanics and Arctic Engineering (OMAE 1992), Vol. 1-A, Offshore Technology. American

Society of Mechanical Engineers, New York, 1992, pp. 257-263.

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Committee V.7, Slender Marine Structures. The information provided is partly reproduced in

Figures 12 and 13, Table 1 and Appendix 3.

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Structures, Vol. 5, 1992, pp. 103-119.




APPENDIX 1: THE SLENDER ROD EQUATIONS

Slender rod statical equilibrium is expressed in terms of the cross section stress resultants of tension, shear,

bending moment and torque as first exemplified by Kirchhoff 2. Coupled with the Hooke’s uni-axial,

Bernoulli-Euler flexure and Saint-Venant’s torsion constitutive models yields Love’s classic description1 and

is the basis of the governing equations presented here.

Equilibrium at a cross section in a rod that is initially straight and unloaded and then displaced by a static

load is described by

dN

ds GJN H

EITM n1

22

21 1

= − 1− ,dN

ds EITM

GJN H n2

11 1

1 1= − − 2 , (A1.1, 2)

dT

ds EIN M

EIN M t= −

1 1

2

1 2

1

2 1 − ,dM

ds GJ EIM H N m1

2

2 21 1

= −⎛

⎝ ⎜

⎞

⎠⎟ + − 1 , (A1.3, 4)

dM

ds EI GJM H N m2

11 1

1 1= −

⎛

⎝ ⎜

⎞

⎠⎟ − − 2 , ,

dH

ds EI EIM M h= −

⎛

⎝ ⎜

⎞

⎠⎟ −

1 1

2 11 2 , (A1.5, 6)

in which, s is the distance coordinate along the central-line, EI 1 , EI 2 , GJ, are the elastic stiffnesses about the

cross section principal flexure-torsion axes and (N 1 , n1), (N 2 , n2), (M 1 , m1), (M 2 , m2 ) (T, t), (H, h) are the

cross section stress resultants and load distributions of shear, bending moment, tension and torque resolved

along the respective principal axes. The directions of the central-line and principal torsion axis are incident.

All quantities are referred with respect to the loaded configuration.

The orientation of the cross section principal flexure-torsion axes of x1 , x2 , x3 , with respect to the fixed

orthogonal axes of X, Y, Z , is described via the Eulerian angles. The imaginary orthogonal axes of x, y, z

have the reference orientation of X, Y, Z and are rotated to the directions of x1 , x2 , x3 by the ordered right

hand rotations of ψ , θ , φ about z, y, z. The angles change at rates

d

ds

M

EI

M

EIψφ φ

θ=

− +1

1

2

2

cos sin

sin,

d

ds

M

EI

M

EI

θφ φ= +1

1

2

2

sin cos ,d

ds

H

GJ

d

ds

φ ψθ= − cos . (A1.7, 8, 9)

These equations have a singularity at sinθ = 0 but this can be numerically integrated across without loss of accuracy by the local approximation

1 10

2sin sinsin , sin sin

θ εθ θ ε≅ ≤ ≤ 1<< , (A1.10)

where in practice experience has found setting sinε = 0.01 is sufficient.

The direction cosines between the cross section principal flexure-torsion axes and the fixed axes are shown

in the following table and they are mainly used to transform loads between the two coordinate systems.




X Y Z

x1 -sinφ sinψ + cosφ cosθ cosψ sinφ cosψ + cosφ cosθ sinψ -cosφ sinθ

x2 -cosφ sinψ - sinφ cosθ cosψ cosφ cosψ - sinφ cosθ sinψ sinφ sinθ

x3 sinθ cosψ sinθ sinψ cosθ

Table A1.1 Cosines between the cross section principal axes x1 x2 x3 and the fixed axes XYZ .

The directions of the cross section principal torsion axis of x3 and the central-line are incident hence the last

row of Table 1 gives differential relations for the position of the rod with respect to the axes of X, Y, Z :

dX

ds= sin cosθ ψ ,

dY

ds= sin sinθ ψ ,

dZ

ds= cosθ . (A1.11, 12, 13)

Other useful relations are

dS

ds

EA

EA T=

+,

ds

ds= 1 ,

dU

ds

M

EI

M

EI

H

GJ

T

EA= + + +

⎛

⎝ ⎜⎜

⎞

⎠⎟⎟

1

212

1

22

2

2 2

,dp

ds

j = 0 (A1.14, ... ,17)

in which S, EA, U , p j are the distance coordinate along the central-line of the unloaded rod, axial stiffness,

strain energy and the jth element from a set of problem dependent unknown constants.

Unknown constants, for example, are a load or geometric parameter with an effect on the boundary

conditions measured via the first variation. It is convenient to differentiate the above relations with respect

to a non-dimensional distance coordinate along the central-line instead of by s and express the result as thevector system

d

d

T

EAL

yf y

τ= +

⎛ ⎝ ⎜

⎞ ⎠⎟( ) 1 , 0 1≤ ≤τ , (A1.18)

in which y = ( N1, N2, T, M1, M2, H, ψ, θ, φ, X, Y, Z, S, s, U, p1, p2, ...)T, L is the undisplaced length of the

rod and components of f are the matching right sides of Equations A1.1 to A1.9 and Equations A1.11 to

A1.17.




APPENDIX 2 INFORMATION FOR THE HELIX CASE STUDY

This case study asks the analyst to determine the undisplaced length and cross section stress resultants for a

pipe displaced into a helix, given the configuration in terms of a two-point boundary condition and material

properties. Loads may only be applied at the ends. The theoretical results are derived from Love1.

Helix configuration

The position of the central-line of the helix in terms of the coordinates on the fixed axes of X, Y, Z and

Eulerian angles of ψ , θ , φ as defined in Appendix 1 is

X = a sinψ , Y = -a cosψ , Z = tan(π /2 − θ ),

where a is the radius from the axis of the helix, that is the axis of Z , to the central-line, ψ the angular

position about the axis of the helix and π /2 − θ the constant pitch angle α of the central-line. The angles

ψ and α are measured with respect to the plane formed by the axes of X, Y and the line of reference for ψ

is the axis of -Y . The components of the tangent t, normal n and binormal b direction vectors, resolved

along the axes of X, Y, Z are:

t = = = =( , , ), cos cos , cos sin , sint t t t t tX Y Z X Y Zα ψ α ψ α ;

n = = − = =( , , ), sin , cos ,n n n n n nX Y Z X Y Zψ ψ 0 ;

b = = − = =( , , ), sin cos , sin sin , cosb b b b b bX Y Z X Y Zα ψ α ψ α .

The configurational parameters of interest are:

deformed length over one pitch lP = 2πa/cos(α), curvature of the central-line 1/ ρ = cos2(α)/a,

torsion of the central-line 1/ Σ = sin(α)cos(α)/a and total torsion σ = dφ /ds + 1/ Σ,

where d φ /ds is the torsion of the cross section.

Boundary conditions

In the displaced configuration the following are specified at the ends s = 0 and s = lP:

s = 0, X = 0, Y =-a, Z = 0, ψ = 0, θ = π /2 − α, φ = 0;

s = lP, X = 0, Y = -a, Z = 2πa tanα , ψ = 2π, θ = π /2 - α, φ = (σ − 1/ Σ) lP.

The boundary conditions in no way infer the constancy of the pitch angle α , or linear variation of ψ with

respect to s between the end points. Hence, non-helical solutions may satisfy the boundary conditions.

Material properties

The cross section is doubly symmetric, EI1 = EI2 = EI.

Principal results

The parameters of the solution for the helix are the constants:

undeformed length LP = lp EA/(EA + T), total shear N = (GJτ - EI/ Σ)/ ρ along the binormal,

tension T = (GJτ - EI/ Σ)/ Σ, total moment M = EI/ ρ about the binormal, torque H = GJ/ Σ.

The principal shears and bending moment are:




N1 = -N cosφ , N2 = N sinφ , M1 = -M cosφ , M2 = M sinφ ,

where φ = (σ− 1/ Σ) s.

The discussion of the numerical solution in the main text is based on the following configurational and

material parameters:

a = 0.5 m, α = 45o, σ = 10-2 rad/m, EA = 1.2 x 108 N, EI = 1.09 x 103 Nm, GJ = 459 x 103 Nm.

The twist σ was selected such that the helix configuration is weakly stable. The material stiffness are equal

to those of the case study in Appendix Three.




APPENDIX 3: THE KODAISSI ET AL .35AND ISSC

36CASE STUDIES

The configuration of the problem is given by Figures 12, and 13 in the main text. A flexible pipe is fixed at

one end and the other end can be rotated about axes in the horizontal plane so that the pipe takes up a range

of positions in three dimensions. Such a system might be used for fatigue testing of a flexible pipe with the

incorporation of rotating drives. The properties of the flexible pipe and end-fittings are shown in Table

A3.1.

undeformed

length

(m)

mass per

length

(kg/m)

outside

diameter

(m)

axial

stiffness

(N)

bending

stiffness

(Nm2 /rad)

torsional

stiffness

(Nm2 /rad)

end-fitting

length

(m)

end-fitting

mass

(kg)

10.1 33.7 0.116 120 x 106

1090

clockwise

459 x 103

anticlockwise14.9 x 103

0.8

fixed

142

moving114

Table A3.1: Properties of the flexible pipe and end-fittings

______________________________

application homotopic methods to static flexible pipe problems

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