reportfouverifikasjon 2012

REPORT

AQUASTRUCTURES

VERIFICATION AND BENCHMARKING OF AQUASIM,

A SOFTWARETOOL FOR SIMULATION OF FLEXIBLE

OFFSHORE FACILITIES EXPOSED TO ENVIRONMENTAL

AND OPERATIONAL LOADS

REPORT NO. 2012-1755-1 REVISION NO. 01

Head quarter: – Kjøpmannsgaten 21 - 7013 Trondheim, Norway

29 November 2012 , a/reportfouverifikasjon_2012_v_2.docx

Date of first issue: Project No.:

20 September 2012 1755 Approved by: Organisational unit:

Are Berstad

Project Manager

Research and development

Client: Client ref.:

Norges forskningsråd

Summary:

With support from Norges Forskningsråd, the simulation and analysis software tool Aquasim has been

developed for over a 10 year period. This report outlines several benchmarking tests verifying the

accuracy and capabilities of the software program.

Report No.: Subject Group:

2012-1755-1 Indexing terms

Report title:

Verification and benchmarking of AquaSim, a

software-tool for simulation of flexible offshore

facilities exposed to environmental and

operational loads

Finite Element Analysis

Flexible structures and systems

Sea loads

Work carried out by:

Are Johan Berstad, Line Heimstad No distribution without permission from the Client

or responsible organisational unit

Work verified by:

Ole Chr. Wroldsen

Limited distribution within

Aquastructures AS

Date of this revision: Rev. No.: Number of pages:

01 64 Unrestricted distribution

Page i

Aquastructures 2012 Reference to part of this report which may lead to misinterpretation is not permissible. 29 November 2012,

Table of Content Page

1. EXECUTIVE SUMMARY .......................................................................................... 1

2. SCOPE OF THIS REPORT ......................................................................................... 1

3. BENCHMARK CASES STATIC ELEMENT RESPONSE ....................................... 1

3.1. Bar elements 1

3.1.1. Bar element with clamped ends 1 3.1.2. Two crossing cables 3 3.1.3. Cable with axial tension load 4 3.1.4. Net built up by cable elements 5 3.1.5. Net built up with cable elements, 2

nd boundary condition 8

3.2. Beam elements 9

3.2.1. Beam element clamped at one end 9

3.3. Membrane elements 10 3.3.1. Membrane elements 10

3.4. Node to node springs 13

4. BENCHMARK CASES ELEMENT DYNAMICS ................................................... 14

4.1. Swinging spring case 14 4.1.1. Static displacement 15

4.1.2. Dynamic displacement 16

4.2. Swinging pendulum 17

5. BENCHMARK CASES MORRISON LOAD APPLICATION ............................... 18

5.1. Beam exposed to current 19

5.2. Beam exposed to wave loads 21

5.3. Beam exposed to wave and current 23

5.4. Horizontally hanging cable exposed to current loads. 23

5.5. Morison load application on membrane elements 27

6. BENCHMARK CASES HYDRODYNAMIC LOADS APPLICATION ................. 27

6.1. AquaSim results compared with the small body asymptote. 27

6.2. AquaSim results compared with the wave reflection asymptote. 31

6.3. Added mass and damping 34

6.4. Drift forces on hydrodynamic elements 35

7. LIFT LOAD APPLICATION .................................................................................... 39

7.1. Beam exposed to lift load 39

8. PROPERTIES ON NODES ....................................................................................... 40

8.1. Fixed nodes 40

Page ii

Aquastructures 2012 Reference to part of this report which may lead to misinterpretation is not permissible. 29 November 2012,

8.2. Linear “node to ground” spring 40 8.2.1. More springs, dampeners and mass 41

8.3. Local coordinates 42

8.4. Buoys 42

8.5. Prescribed displacements 44

8.6. RAO on nodes 47

8.7. Time domain fixed node motion and rotation 47

8.8. Node loads 49

9. WIND LOADS .......................................................................................................... 51

10. REFERENCES ........................................................................................................... 53

11. APPENDIX ................................................................................................................ 55

11.1. Loading when ropes gets stiff 55

Reference to part of this report which may lead to misinterpretation is not permissible. 29 November 2012, a/reportfouverifikasjon_2012_v_2.docx

1. EXECUTIVE SUMMARY Aquasim has been developed over a 10 year period with support from the Norwegian research

council. The program is capable of carrying out static as well as dynamic time domain

simulation of structures exposed to time varying loads and structure response such as current and

wave loads as well as operational conditions. The program accounts for the hydroelastic relation

between fluid and structure.

Several element types are included in the program. This report validates many of the different

elements by doing analysis for cases where analytic solutions can be derived. The loads from

waves and current are described and validated. Node properties are validated.

2. SCOPE OF THIS REPORT

This report is issued in order to verify the Aquastructures software-tool AquaSim to be used for

calculating structural response for systems and structures in a marine environment.

Calculations carried out by AquaSim are compared to handbook formulas and analytical

solutions for the elements used, and loads and boundary conditions.

Recalculation of the cases analysed in the 2006 validation report (Aquastructures 2006) with the

most recent version of AquaSim (2012 version) have been carried out.

New validation cases have been established and analysed.

The work in Section 3- 5 has been carried out by either Line Heimstad or Are Berstad The

analysis in Section 6 and outwards have been carried out by Are Berstad

3. BENCHMARK CASES STATIC ELEMENT RESPONSE

Case studies have been carried out to verify the results calculated by the program to analytical

results.

3.1. Bar elements

Bar elements can take only axial loads and are hence applicable for e.g mooring lines.

3.1.1. Bar element with clamped ends

Based on geometry considerations, the bar element used in the computer program has been tested

against analytic results. The geometry of Case 1 is shown in Figure 1.


Figure 1 Cable with both ends clamped.

Figure 2 presents the deformed geometry of the cable of Case 1. The material and load data used

in the calculation is given in

Table 1.

Figure 2 Cable in initial and deformed condition. U is the cable displacement under the

point load and Pc is the cable force.

Table 1 Structural data, bar with both ends clamped

Structural data Abbreviation Value

Length of cable from end to end L 10 m

Cross sectional area of cable A 100 mm2

Module of elasticity of cable E 2.1 * 109 N/m

3

Program calculated and analytical results are compared in Table 2.

Table 2 Results for cable. U is the displacement in meters and Pc in the cable force in

Newton as given in Figure 2.

Applied load AquaSim results Analytical results Relative difference

U (m) Pc (N) U Pc U Pc

L/2 L/2

PArea, A

Young’s modulus, E

L/2 L/2

PArea, A


Displacement, U

Cable force, Pc


100 N 0.391 641 0.391 641 1 1

10 000 N 1.873 14252 1.873 14252 1 1

1 000 000 N 16.69 521 900 16.69 521900 1 1

As seen from Table 2, the program calculation compares fully with analytical results. This means

that AquaSim calculates geometric nonlinearities of bar elements in a proper manner.

3.1.2. Two crossing cables

A case has been considered where two crossing cables have been modelled as shown in Figure 3.

Both cables have the same length, and the load is applied to the centre point.

Figure 3 Case with two cables crossing at 90o

Results for the geometry shown in Figure 3 were calculated with structural data as given in Table

3:

Table 3 Structural data, case with two cables crossed 90o

Structural data Abbreviation Value

Length of cable from end to end L 10 m

Cross sectional area of cable A 100 mm2

Module of elasticity of cable E 2.1 * 109 N/m

3

Program calculated and analytical results are compared in Table 4. Analytical results are derived

by considering the load situation in the case of Section 3.1.1. With two times the load applied to

the centre point, the displacement U of the crossing cables will be equal to the displacement

found for the case of a single cable in Section 3.1.1

L/2 L/2

PArea, A



Table 4 Results for crossing cables. U is the displacement in meters and Pc in the cable

force in Newton as given in Figure 2.

Applied load AquaSim results Analytical results Relative difference

U [m] Pc [N] U[m] Pc[N] U Pc

200 N 0.391 641 0.391 641 1 1

20 000 N 1.873 14252 1.873 14252 1 1

2 000 000 N 16.69 521 900 16.69 521900 1 1

As seen from Table 4, by multiplying the applied loads with 2, the calculated results for crossing

cables equals the results for a single cable described in Section 3.1.1, which again equals the

analytically calculated results.

3.1.3. Cable with axial tension load

Axial force is applied to the cable end as shown in Figure 4.

Figure 4 Cable with tension force

The structural data for this case is given in Table 5.

Table 5 Values used in case study. Cable with tension force

Abbreviation Description Value

A Cross sectional area of cable 10 mm2

L0 Length of cable in original configuration 10 m

E Youngs module of the cable 1.0E08 N/m2

The results for a cable with tension force are given in Table 6.

Cable:

Young’s module E

Area A

Force F

z

x


Table 6 Calculated and analytic values case study with cable with axial tension load

Cable force Displacement calculated with AquaSim Analytical displacement

-100 N -1.0 -1.0

-1000 N -10 -10

-3000 N -30 -30

As seen from Table 6, analytic and numerical results are identical. The above analysis was

repeated using 4 elements. The results correspond perfectly.

3.1.4. Net built up by cable elements

A net structure as shown in Figure 5 has been established.

Figure 5 Net structure built up with cable elements

The structural data for the net built up with cable elements is given in Table 7.

Table 7 Values used in net structure built up with cable elements


A Cross sectional area of horizontal and vertical cables 10 mm2

L Length of cable in original configuration 10 m

E Young’s module of the cable 1.0E08 N/m2

L/4

L/4

L/4

L/4

L/4L/4L/4L/4

z

x

z

x

P


Node loads have been distributed on the 5 lowermost nodes as shown in Figure 5. The five upper

nodes are all fixed. The results for the net structure built up with cable elements are given in

Table 6.

Table 8 Calculated and analytic values case study with net structure built up with cable

elements

Node load, each node Vertical displacement lower side Horizontal displacement

-100 N -1.0 0.0

-1000 N -10 0.0

-3000 N -30 0.0

Figure 6 Applied loads are 5 times 1000 N downwards at each bottom node. The legend

shows the vertical displacement.


Figure 7 Applied loads 1000 N downwards at each weight. The legend shows the stress level

in each cable.

A second load case was applied including a load component in the horizontal direction. Load

values and calculated results are given in Table 9.

Table 9 Calculated and analytic values present case

Vertical node

loads

Horizontal node

loads

Vertical displacement

lower side

Horizontal displacement

lower side

-800 N 600 N -6.0 m 12.0 m


Figure 8 Vertical and horizontal loads at the bottom of the structure. The colours shown on

the cables reflect the axial forces found in the cables of the legend.

3.1.5. Net built up with cable elements, 2nd

boundary condition

The same structure of cables from Figure 5 is now fixed only at the two endpoints of the top, as

shown in Figure 9.

Figure 9 Geometry of present case. Cables are only fixed at two nodes at the top

The displacement for the structure of cables with the 2nd

boundary condition is shown in Figure

10. The results derived from this 2nd

boundary condition are not validated to handbook results.

L/4

L/4

L/4

L/4

L/4L/4L/4L/4

z

x

z

x

P


The results are only considered plausible. Cases for the validation of the membrane elements will

be compared to the results presented in Figure 10.

Figure 10 Structural displacements, case with cables only fixed at two nodes, 1000 N

vertical downwards load where weights are shown. The colour scale represents the axial

forces found in the cables.

3.2. Beam elements

3.2.1. Beam element clamped at one end

A beam element clamped at one end is considered. The case is shown in Figure 11.

Figure 11 Beam element clamed in one end. Point load has been applied at free end.

L

P

Area, A


Area moment of inertia

Iy and Iz

z

x

z

x


In this case, analytical linear displacements can be derived as

EI3

Plr

3

3

Equation 1

The structural data for this case is given in Table 10.

Table 10 Values used in beam case study


A Cross sectional area of horizontal and vertical cables 0.05 m2

L Length of beam in original configuration 10 m

Iy Area moment of inertia about local y- axis 0.001 (1/m4)

Iz Area moment of inertia about local z- axis 0.001 (1/m4)

IT Torsional area moment of inertia 0.002(1/m4)

G Torsional module 0.8E09 N/m2

E Youngs module of the beam 2.1E11 N/m2

Loads have been applied as shown in Figure 11. Results derived by AquaSim and analytical

results are given in Table 11.

Table 11 Calculated and analytic values using one element

Vertical node load Displacement calculated by

AquaSim

Analytic linear displacement

z- direction x- direction z-direction x- direction

-100 N -0.0002 0.0 -0.0002 0.0

-100 000N -0.1587 -0.0013 -0.1587 0.0

-1 000 000 N -1.5540 -0.1213 -1.587 0.0

-10 000 000 N -7.7361 -3.3487 -15.87 0.0

As seen from Table 11 results are exactly similar for small loads. This is expected since the

beam response is almost exactly linear in this case. As the load is increased, the nonlinear effect

becomes important and the analytical results are no longer valid. AquaSim accounts for the non-

linear effects.

3.3. Membrane elements

3.3.1. Membrane elements

One membrane elements mesh is shown in Figure 12. In this figure, the membrane element

represents a 7 x 7 mesh. Generally, membrane elements represent meshes of m x n, depending on

membrane element size and the established mesh size.


Figure 12 One membrane element representing 7*7 twines of flag shaped net.

The mesh structure made up with cable elements shown in Figure 5 and Figure 9 is now “rebuilt”

using membrane elements. Results will be compared for the two cases in order to validate the

results derived from using membrane elements. Each membrane element is 2.5 x 2.5 meters, and

the mesh size is assumed to be the same, meaning the the halfmesh size is 2.5 meters. This is not

a normal value for aquaculture nets, but is used to have a 1 to 1 relation between the net model

and the model built with bars. This means that each mesh will have a half thread along each side

of the mesh. In order to make the model similar to the models shown in Figure 5 and Figure 9, a

cable is arranged around the structure as shown in blue in Figure 13.

Dx

Dy


Figure 13 Model for test of the membrane elements. Each square is at membrane element.

The blue line corresponds to cable elements with cross sectional area of half the area of one

membrane element twine.

The present membrane model was tested with the same boundary condition as given in Figure 5.

Both load conditions described in Table 8 and Table 9 gave exactly the same results for the

present membrane case as for the case when modelled as bars.

Now consider the case which is only clamped in two upper nodes as shown in Figure 13. This

case was run for the same condition as the case shown in Figure 10. The results are shown in

Figure 14.

Figure 14 Membrane structure clamped at two nodes. Node loads have been applied at the

lower end of the net.

L/4

L/4

L/4

L/4

L/4L/4L/4L/4

z

x

z

x

P


As seen from Figure 14, the results correspond very well to the results in Figure 10. This means

that cable elements and membrane elements gives the same results for a case study where they

represent the same geometry.

3.4. Node to node springs

Element type 4 in AquaSim is the “node to node spring” element. For node to node springs, the

spring force is proportional to the difference in respective DOFs at the two nodes the spring is

connecting.

Figure 15 shows a test case where node to node springs have been tested. The beam is the same

as in Table 10 of section 3.2.

Figure 15 Geometry of test of node to node springs. Beam data not expressed explicitly in

this figure are the same as in the previous section in Table 21. The beam is very stiff

relative to the stiffness of springs

Table 12 shows results. As seen from this table, results using AquaSim corresponds well with

analytic results.

1 m

Node 2

Node 2

Node to node springs at

all degrees of freedom

between node 2 and 3

Value: 1000 N/M

1 m

Node 3

Node 1

Fixed all DOFs


Table 12 Results verifying node to node spring elements. Node numbers are referring to

Figure 15.

Force

magnitude

Force location Result

parameter

Analytical

result

Aquasim result

1000 Node 3 DOF 1 x/y/z-translation

node 4

1.0/0.0/0.0 1.0/0.0/0.0


node 4

0.0/1.0/0.0 0.0/1.0/0.0


node 4

0.0/0.0/1.0 0.0/0.0/1.0


node 4

0.0/-0.46/0.84 0.0/-0.46/0.84


node 4

0.0/0.0/0.0 0.0/0.0/0.0


node 4

-0.84/-0.46/0.0 -0.84/-0.46/0.0

4. BENCHMARK CASES ELEMENT DYNAMICS

4.1. Swinging spring case

The objective of this case is to verify that the load, mass and stiffness calculation give correct

results for a simple case where the eigenfrequency and amplitude can be found analytically.

Consider a weight with mass M hanging at the end of a truss element as shown in Figure 16.

The end node of the truss element, where the weight is located, is free to move in z- direction

only, i.e. one degree of freedom.

Cable:

Young’s module E

Area A

Mass: M

Weight: Mg

z

x


Figure 16 Test of dynamic loading. The weight is applied and left free to fall downwards.

The following values has been applied to the configuration shown in Figure 16

Table 13 Structural data in case study with a weight swinging freely.





M Mass of weight 305.81 kg

F The force of the weight =Mg -3000 N

4.1.1. Static displacement

Analytic static solution is as follows:

F = K x ΔL (Where F is the force in Newton and ΔL is the elongation of L0)

K= EA/L0 = 100 (Where K is the cross sectional stiffness of the cable)

ΔL = (L - L0) = F/K = -3000/100 = -30 m.

This means that the static vertical displacement of the weight when applied statically to this load

is 30 meters. This corresponds well to the values calculated by the program shown in Figure 17.


Figure 17 Static and dynamic solution for the cable with weight. The red line is static

displacement and the blue line is dynamic displacement as a function of time.

4.1.2. Dynamic displacement

In general, the motion of a system without damping can be describes by

)tcos()rr()tsin(r

)t(r s0

0

Equation 2

Applying loads in the present case, r will be 0 since there is no velocity of the weight at t = 0. r0 –

rs is the deviance from the position of static equilibrium at t = 0. In the present case this deviance

is 30 meter. This means that according to Equation 2 the amplitude of the harmonic motion will

be 30, and the motion can be described as

)tcos(30)tcos()rr()t(r s0

Where is the eigenfrequency of the motion being found as

M

k

The analytically calculated period T will in this case the be

99.10k

M2

2T

Compare this value with Figure 17, and a full match is observed. The results are summarized in

Table 14.

Weight at end of cable

-70

-60

-50

-40

-30

-20

-10

0

0 5 10 15 20 25 30 35 40 45 50

Time [Seconds]

Dis

pla

ce

me

nt[

me

ter]


Table 14 Calculated and analytic values in case study with weight swinging freely in the z-

direction.

Abbreviation Description Value calculated by Aquasim Analytic value

K Cable stiffness 100 100

rs Static displacement -30 -30

rd Dynamic amplitude 30 30

T Eigenperiod 10.99 10.99

4.2. Swinging pendulum

In this case a cable was modelled along the x-axis as shown in Figure 18. A weight was put on

one end of the cable, and the other end was fixed. This means that the structure will act as a

swinging pendulum.

Figure 18 Horizontal cable with weight at the end

The data used for this case is given in Table 15.

Table 15 Values used in case study with cable swinging sideways





M Mass of weight 305.81 kg

F The force of the weight =Mg -3000 N

The analytic results in this case will give a weight swinging from one side to the other with

minimum z- displacement being -10 meters. This corresponds to AquaSim results shown in

Figure 19. So does the horizontal displacement.

Cable:

Young’s module E

Area A

Mass: M

Weight: Mg

z

x

z

x


The period for the swinging pendulum can be found as

2/

0 22 sin14

k

d

g

LT

Equation 3

where T is the swinging period, L is the cable length and g is the acceleration of gravitation.

2sink

where is the angle of the pendulum in the original configuration relative to the vertical axis. In

the present study = 90 degrees. Introducing into Equation 3 the period T, the period is found

to be 7.49 s. Comparing with results found from AquaSim shown in Figure 19 it is seen the

period calculated by AquaSim is exactly the same.

Figure 19 Weight attached to cable end. Cable is swinging sideways. The red line shows

vertical displacement and the blue line shows horizontal displacement of weight

5. BENCHMARK CASES MORRISON LOAD APPLICATION

On beams and cables, the cross flow principle is used to derive the Morrison loads. Referring to

a local coordinate system where the beam or cable is located along the local x- axis, forces in the

local y- direction can be found as given in Equation 4.

Swinging pendulum

-25

-20

-15

-10

-5

0

5

0 5 10 15 20 25 30

Time

Dis

pla

ce

me

nt

y-displacement z-displacement


2020

2

33

2

2222

0

2

)1(

2

vCALaCAL

vuvuvuLDiamC

F

mwmw

Ndyw

Equation 4

Here Cdy is the drag coefficient in the local y- direction, DiamN is the diameter of the cross

section in the direction of the relative velocity 2

33

2

22 mm vuvu vector in the cross

sectional plane. currentwave uuu 222 where u2 wave is the fluid velocity due to waves and u2 current

is the current velocity in the local y- direction. The calculations are carried out at two separate

locations 20% of the length away from each node being applied as load for that particular node.

a2 is the fluid acceleration in the local y- direction and Cmy is the mass coefficient = Ca + 1 where

Ca is the added mass. A is the cross sectional area of the element. The expression will be similar

in the local z direction.

5.1. Beam exposed to current

Consider a case with a vertical beam exposed to uniform current as shown in Figure 20. The

beam has a circular cross section.

Figure 20 Beam exposed to uniform current

Structural data for the exposed beam is given in Table 16

Table 16 Values used in beam exposed to current load case study


A Cross sectional area 0.1 m2



Beam:

Young’s module E

Area A

Area moment of inertia I

z

x

Uniform current velocity u




G Torsional module 4E10 N/m2

E Young’s modulus of the beam 1.0E11 N/m2

The distributed load over the cross section can be found by using the Morison equation (e.g.

Equation 4.). Applied to this static case, this equation reads:

uuDC2

q D

Equation 5

where q is the uniformly distributed load. D is the diameter of the cross section. The

displacement r of the lower end of the beam and the shear force Vz and moment My at the upper

end can then be found as

qLVqL

MEI

qLr endUpperzendUpperyzend ,

2,

8

24

Equation 6

Introducing the following data

Table 17 Data for test of beam exposed to current loads

Description Abbreviation Value

Drag coefficient CD 1

Water density 1025

Diameter of cross section D 0.35 m

Current velocity u 1 m/s

Mass coefficient Cm 2

Analytical and computed results are compared in Table 18.

Table 18 Results for beam exposed to current in x direction

Response

parameter

Analytical results Computed results 10

elements

Computed results 100

elements

r lower end 2.24 mm 2.24 mm 2.24 mm

My upper end 8968.75 NM 7250 Nm 8790 Nm

Vz upper end 1793.75 N 1704 N 1785 N

As seen from this table the results correspond very well. The shear force is computed as constant

over each element meaning that the response at the clamped beam end will be “under predicted”


proportional to element size. For this circular cross section, the test is repeated by exposing the

beam to current 45 degrees this means that Ux = Uy = 0.70710678. All other data is the same.

Table 19 Results for beam expose to current 45 degrees relative to the x- and y- axis

Response

parameter

Analytical

results

Computed results 10

elements

Computed results 100

elements

Rx =Ry lower end 1.585 mm 1.585 mm 1.585 mm

|r| lower end 2.242 mm 2.242 mm 2.242 mm

My=Mz upper end 6342 Nm 5126 Nm 6216 Nm

|M| upper end 8969 Nm 7250 Nm 8790 Nm

Vy = Vz upper end 1268 N 1205 N 1262 N

|V| upper end 1794 N 1704 N 1785 N

This shows that response calculated by AquaSim corresponds well with analytical predictions

using the Morison formulae for load calculation as the drag load is quadratic with respect to

velocity.

Figure 21 Displacement at end of beam as function of current velocity in current direction

5.2. Beam exposed to wave loads

Wave loads are considered, using the Morison formula (see Equation 4). The same beam as

described in section 5.1 is considered (see Table 17 and Equation 6). In the present case the

beam is positioned horizontally, along the y- axis, 5 meters below wave surface. The beam

response is assumed static. Infinite wave depth is assumed.

0

5

10

15

20

25

0 0.5 1 1.5 2 2.5 3 3.5

Displacement

Displacement


Figure 22 Beam exposed to waves

The beam is exposed to waves in the x- direction with 5 meter wave height. The wave frequency,

is 1,0 second. As seen from Figure 23 the analytical and the program calculated results

correspond very well with analytical results calculated according to Equation 4.

Figure 23 Horizontal beam exposed to waves. Wave elevation 5 meter, Wave frequency, ,

1 sec. Displacement at the beams free end is shown.

z

y

L = 10 m

-25.00

-20.00

-15.00

-10.00

-5.00

0.00

5.00

10.00

15.00

20.00

25.00

0.00E+00 2.00E+00 4.00E+00 6.00E+00 8.00E+00

Dis

pla

ce

me

nt

[mm

]

Time [Sec]

Displacement due to waves calculated by Morrissons formulae

Dx-analytical

Dz-analytical

Dx-Calculated

Dz-Calculated


5.3. Beam exposed to wave and current

A similar case is considered applying both current and waves. In this case the wave period is 10

seconds and a current velocity of one meter per second is applied in addition to the wave. It is

assumed that the waves are “riding” on top of the current field.

Figure 24 Horizontal beam exposed to waves. Wave elevation 5 meter, the wave period is

10 sec. Current velocity is 1 m/s in the x-direction. Displacement at the beams free end is

shown.

5.4. Horizontally hanging cable exposed to current loads.

This case considers a vertical cable with a point load of 5000N applied to the lower end, as

shown in Figure 25.

-30.00

-20.00

-10.00

0.00

10.00

20.00

30.00

40.00

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0

Dis

pla

ce

me

nt [m

m]

Time [seconds]

Displacement due to waves and current calculated by Morrissons formulae

Dx-analytical

Dz-analytical

Dx-Calculated

Dz-Calculated


Figure 25 Cable with a hanging weight is exposed to current flow

The E modulus and cross sectional area for this case is the same as in the above cases shown in

Table 16 and Table 17, E = 1.0*1011

N/m2 and A = 0.1 m

2. Displacements have been calculated

by a simplified formulae stating that there must be equilibrium at each cross section of the cable

as shown in Figure 26. In contrast to the AquaSim simulation program, the simplified formulae

does not account for the updated geometry of the cable when loads from current is derived. This

means that the simplified formulae will differ from the nonlinear results calculated by AquaSim

as the displacements increase. This is clearly seen in Figure 27, the results predicted by AquaSim

and the simplified formula corresponds exactly when the horizontal displacement is less than 0.2

meters. For this case the nonlinear geometry effect is not very important. Figure 28 shows the

same as Figure 27, but in Figure 28 the current velocity has been increased from 0.3 m/s to 1

m/s. This gives a displacement of the cable of almost two meters in the horizontal plane. As seen

from this figure there is a deviation between results predicted by AquaSim and the simplified

formulae which was expected since the simplified formulae does not account for the geometric

nonlinearities effect on the load.

Cable:

Youngs module E

Area A = 0.1 m2z

x

Uniform current velocity u

Downwards load Pz

5000 N


Figure 26 Sketch of how the simplified expression for cable displacement assumes

equilibrium at each vertical level but not accounting for the cable displacement, i.e. the

uniform force is calculated over the initial cable configuration.

Figure 27 Cable displacement predicted by the simplified formulae and the Aquasim

program respectively. The current velocity is 0.3 m/s. The red dots are the AquaSim results

whereas the blue line is the results predicted with the simplified formulae. There is no

deviation in this case.

z

x

Uniform force q caused by uniform flow u

Downwards load Pz

5000 N

Q(L-l)dx = pdz

L-l

Results compared to simplified formulae

-12

-10

-8

-6

-4

-2

0

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

Horizontal displacement [m]

Ve

rtic

al lo

ca

tio

n


Figure 28 Cable displacement predicted by the simplified formulae and the Aquasim

program respectively. The current velocity is 1.0 m/s. The red dots are the AquaSim results

whereas the blue line is the results predicted with the simplified formulae. The simplified

formulae is based on a linear consideration, and as nonlinear effects becomes more

predominant the deviation between this formulae and AquaSim increase.

Results compared to simplified formulae

-12

-10

-8

-6

-4

-2

0

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Horizontal displacement [m]

Ve

rtic

al lo

ca

tio

n


Figure 29 Cable with weight at the bottom. The legend gives the horizontal displacement in

the cable.

5.5. Morison load application on membrane elements

For Morison load application on membrane elements it is referred to Berstad et al (2012) for

documentation and validation of 3 alternative load formulations to nets and also Berstad et al

(2004).

6. BENCHMARK CASES HYDRODYNAMIC LOADS APPLICATION

This section shows results for calculations based on hydrodynamic loads. Case studies are used

to investigate how the program corresponds with results for cases where asymptotic solutions or

hand book solutions do exist.

In AquaSim added mass and hydrodynamic damping is established for elements based on linear

coefficients established at a user specified mean water line. Diffraction properties are established

in the same manner. During time domain simulation, the Froude Kriloff and diffraction of the

pressure is applied at the elements actual horizontal location.

6.1. AquaSim results compared with the small body asymptote.

When a submerged body is small relative to the wave length, the forces acting on the body can

be approximated as (see e.g. Faltinsen 1990 pp 59-60)

S

33i22i11iii aAaAaAdspnF

Equation 7


where p is the dynamic pressure in the undisturbed wave field, n = [n1,n2,n3] is the unit vector

normal to the body with positive direction into the fluid. a1,a2,a3 are the acceleration components

along the x-, y- and z- axis of the undisturbed wave field, which are to be evaluated at the

geometrical mass centre of the body. For a totally submerged body

S

ii Vadspn

Equation 8

where V is the volume of the body. For a body not totally submerged, the above equation is valid

only in the horizontal direction. A case as shown in Figure 30 is used as a case study. The beam

is located with its origin in the free surface (z = 0).

Figure 30 Geometry of case study for testing is the program results corresponds to results

derived from the long wave approximation. The diameter of the cylinder is 0.4 meters.

A wave with 1 meter amplitude and a period of 8 seconds is applied. Based on the long wave

approximation, the evenly distributed force amplitude due to Froude Kriloff and added mass can

be derived from Equation 7 and Equation 8 can be found as 79.45 N/m. The force will be a

sinusoidal for with this amplitude. Introducing this into the equation for free end displacement

for a beam clamped as shown in Figure 30

EI8

qLr

4

zend

Equation 9

Structural data for this case is given in Table 20

Table 20 Values used in beam exposed wave loads using numerical strip theory wave

diffraction theory.



Submerged part of circular cross sectional area 50%



z

y

L = 10 m





Cd Drag coefficient 0

Wave direction Beam seas

Using the beam properties for this case gives a sinusoidal response as shown in Figure 31. In this

figure, analytic results using the small body approximation is compared to the results predicted

by AquaSim which use a strip theory panel method. As seen from the figure predicted response

corresponds very well. Note that beam response is calculated statically.

Figure 31 Comparison of AquaSim results with results derived from calculating

analytically results using the long wave approximation. The results show horizontal

displacement at the beam end for the 10 meter long beam with hydrodynamic and

structural properties given in Figure 30, and Table 20 and with load formulation in

Equation 9. 1 meter high wave

Figure 32 shows visualization of the calculated displacement. Note that the results depend on the

numerical calculation of the hull. AquaSim generates straight lines between the given input

points, which indicate that with fewer points a true circle is not predicted. The shown case is for

a case with input points per 5 degrees, and number of hydrodynamic points are 40. In this case

there are no there is no viscous drag load added to the hydrodynamic loads.

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

8 10 12 14 16

x-

dis

pla

cem

en

t [m

m]

Seconds

Program results compared to small body approximation

Small body approximation

Calculated displacement


Figure 32 Calculated displacement, AquaSim. Case in Figure 31.

Figure 33 shows the same as Figure 32 but for a 10 meter high wave. As seen the results

compare well as this load component is linear with respect to wave height.


Figure 33 Calculated displacement, AquaSim. Case in Figure 31 but with 10 meter high

wave.

6.2. AquaSim results compared with the wave reflection asymptote.

In this section a cross section is established in order to compare calculated beam displacement

using the reflected wave asymptote with results using the AquaSim program. The reflected wave

asymptote is valid for waves approaching a wall. The wave will then be reflected causing at

oppositely directed wave of equal magnitude.

Figure 34 Data used for case study where data are compared to data using the reflected

wave asymptotic results. The depth below the water of the beam is 10 meters and the width

of the beam is 1 meter.

z

y

L = 10 m

D = 10 m


Structural data for this case is given in Table 21

Table 21 Beam properties in case study using a beam to test closeness to the wave reflection

asymptote.

Abbreviation Parameter Value





IT Torsional area moment of inertia 2.0 (1/m4)



The wave data applied are given in Table 22

Table 22 Wave data

Description Abbreviation Value

Wave amplitude A 5

Water density 1025

Direction Beam seas (90o)

Current velocity U 0 m/s

The analytical results for this case have been derived by integrating the Froude Kriloff pressure

over the weather side of the beam. This leads to the following expression for the horizontally

distributed load q in this case

kDexp1*k

gAq fc

Equation 10

The total distributed force using the above expression is then

kDexp1*k

gA2q

Equation 11

These analytic values have been introduced to the above expressions, and compared with results

from the AquaSim program. Figure 35 shows this comparison for a wave period of 4 seconds. In

this case most of the wave will be reflected since the particle velocities are reduced downwards

proportional to exp(kz). In this case this means that at z = -10 the wave velocity will only be


approximately 8 % of the velocity at the surface. This means that due to continuity, most of the

wave will have to be reflected. As seen from Figure 35 this clearly happens.

Figure 35 Horizontal displacement at the end of beam. Calculated results are calculated by

Aquasim. Wave period is 4 seconds.

As seen from Figure 35 results calculated by AquaSim correspond very well with the analytic

expression based on the small body approximation. The deviation is results is due to the fact that

the small body approximation is not fully valid for this case. AquaSim reflects the actual beam in

a proper manner whereas the small body approximation is a slight simplification.

Figure 36 shows the same as Figure 35 but in this case the wave period is 10 seconds. This

means that the wave velocity at z = -10 meter is approximately 50 % of the wave velocity at the

surface. This means that much of the water is transported below the beam, and a smaller part is

reflected. Hence it is not expected to find results close to the reflected wave asymptote for this

case. As seen from Figure 36 this is also the case.

Program results compare to reflection asymptote

-15.0

-10.0

-5.0

0.0

5.0

10.0

15.0

0 1 1 2 2 3 3 4 4

Seconds

x-

dis

pla

ce

me

nt

[mm

]

Displacement by asyptpticexpression



Figure 36 Horizontal displacement at the end of beam. Calculated results are calculated by

Aquasim. Wave period is 10 seconds.

As seen from Figure 36 the response calculated by AquaSim is smaller than response predicted

by the small body asymptote. This is in good correspondence with the physics since AquaSim

accounts for the fact that much of the wave is not reflected in this case. The wave reflection

asymptote assumes that the whole wave is reflected.

6.3. Added mass and damping

Added mass and damping has been calculated for a cylinder with geometry as shown in Figure

22. The results have been normalized with respect to the wave frequency (shown as w in

Figure 37 ) the cross sectional area of the cylinder and the density of the fluid . The results are

shown in Figure 37. These values can be compared with results given in Faltinsen (1990). Good

correspondence is seen. This means that AquaSim calculates added mass and damping from

hydrodynamic loads in a proper manner.

Program results compare to reflection asymptote

-30.0

-20.0

-10.0

0.0

10.0

20.0

30.0

0 2 4 6 8 10

Seconds

x-

dis

pla

ce

me

nt

[mm

]

Displacement by asyptpticexpression



Figure 37 Normalized 2D added mass and damping for a cylinder. A is the cross sectional

area of the cylinder. The values in this figure can be compared with values in Faltinsen

(1990) pp 50. A22 is the added mass in sway (horizontal), B22 is the damping. A33 is the

added mass in heave and B33 is the damping in heave.

6.4. Drift forces on hydrodynamic elements

AquaSim has an option for calculating drift loads to elements where the hydrodynamic load

formulation is applied. This option calculates both an average drift force based on conservation

of momentum according for Maouro as well as a sum frequency load. This drift and sum

frequency calculation carried out in AquaSim works as follows:

Consider a 2 dimensional situation as assumed in strip theory.

The average drift force over a time period is assumed being according to Maoros formulae (See

e.g. Faltinsen 1990)

2

22

Arg

F

Equation 12

whereF2 is the second order force averaged over a wave period. The “normal” linearized forces

consisting of Froude Kriloff and diffraction as well as added mass and damping is then F1.

is the density of water.

g is the gravity constant.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.5 1 1.5 2

2R/g

Normalized added mass and damping

A22/A

B22/A

A33/A

B33/A


Ar is the peak value of the reflected wave. This wave originates from the first order solution of

the boundary value problem and is a sinusoidal wave with the same wave period as the incident

wave.

Using this information an alternative way to calculate F2 can then be

2

2 gArF

Equation 13

where Ar is the amplitude of the instantaneous reflected wave elevation. Comparing Equation 13

to Equation 12 it is seen that using Equation 13 will over one wave cycle give an average force

as given in Equation 12 which is the force according to Maouros formulae ( F2 always acts

normal to the hull in in the opposite direction of the reflected waves.) Ar is the reflected wave

calculated by the strip theory accounting for diffraction and radiation.

The reflected wave is derived by accounting for all components in the analysis contributing to

this wave. The instantaneous reflected wave elevation, Ar in the far field and then phase shifted

to along the ship side is traced caused by both diffraction and vessel motion. Then an

instantaneous force can be found from Equation 13.

The case study reported in Figure 34 and Table 21 is exposed to drift forces according to

Equation 13. The environmental data used for this case is given in Table 22 and Figure 35. The

resulting forces are seen in Figure 38 where;

Displacement from drift is calculated analytically from Equation 13.

Linear displacement is the displacement caused by the regular non-drift forces (the first

order terms, corresponding to the results seen in Figure 38.

Total is analytically summarized linear and drift forces analytically.

Total AquaSim are the forces calculated by AquaSim which should correspond to the

total forces calculated analytically.


Figure 38 Response from wave with wave amplitude 5 meter and wave period 4 seconds.

As seen from Figure 38 the results from AquaSim and the analytical results correspond very

well. Note that the “reverse Maoro” formulation used by AquaSim does not mean that the sum

frequency forces have the correct phase relative to the first order forces. The AquaSim

formulation is based on calculating the wave elevation of the reflected wave according to strip

theory and then, when this wave peaks at the location where the ship wall enters the water, the

calculated force peaks as seen in Figure 38.

Alternative drift forces can be introduced by the user. The reflected wave used as basis for the

drift force calculation according to Equation 13, Ar can be set by the user from 100% of the

incident wave down to 1%. Introducing this option to calculate drift forces means that Ar is in

phase with the incident wave and the peak force occurs when the incident wave peaks at the

upstream vessel side.

Figure 39 shows results for the case where the reflected wave amplitude is set to 100% of the

incident wave amplitude. As seen from the figure, results are as expected.

-15

-10

-5

0

5

10

15

20

25

0 1 2 3 4 5 6 7 8 9

End

dis

pla

cem

ent

[mm

]

Time [s]

Response including drift forces

Displacement from drift [mm]

Linear displacement

Total

Total Aquasim


Figure 39 Calculation of beam displacement including drift forces where the reflected wave

is set to 100% of the incident wave

AquaSim calculates the reflected wave based on a wave perpendicular to the hydrodynamic

object. In case waves are not perpendicular to the “upstream” line where the vessel side

intercepts the water, the drift force is corrected by

)(sin 2 cFac

Equation 14

where is the angle between the incident wave and the vessel water intersection line.

The current velocity is accounted for by formulae 5.22 in Faltinsen (1970), by adjusting the force

found in Equation 13 as

)1(22g

UCosFF

Equation 15

where is the wave frequency or peak frequency of the spectrum, U is the current velocity and

is the angle between the wave velocity and the current velocity. g is the gravity constant.

Figure 40 shows analysis for the same case as in Figure 38 but with an additional current

velocity of 3 m/s in the same direction as the waves. As seen from the results, the applied forces

to the beam have increased, increasing the maximum displacement of the beam Note that a

current velocity of 3 m/s is an extremely large value. It should also be noted that the waves used

in the present case is also way too steep with respect to physical criteria. Realistically, the

amplitude for this wave can at max be 1.8 meters which means that the drift forces will matter

only 1/3 relative to the first order forces compared to what is reported here.

-15

-10

-5

0

5

10

15

20

25

0 1 2 3 4 5 6 7 8 9

Dis

pla

cem

ent

[mm

]

Time [s]

Reflection 100% of incident wave


Linear displacement

Total

Total Aquasim


There are also first order viscous forces which may be much larger than the forces derived from

the current correction in Equation 15.

Figure 40 Analysis with a current of 3 m/s.

7. LIFT LOAD APPLICATION

Lift loads have been used for a wide range of cases and compared to measurements and analytic

considerations. The lift load model works such that a lift, drag and moment coefficient is applied

where these coefficients in general are functions of the relative angle of the inflow of the element

and the component. Lift elements are typically used in towing operations such as seismic

acquisition, see e.g Berstad and Tronstad (2008).

There are a set of options which need to be chosen carefully in the input to obtain correct input.

7.1. Beam exposed to lift load AquaSim allows for lift loads applied on beam and bar elements. This has been considered in

Berstad and Tronstad (2008). Consider the case with results presented in Figure 21. The case

shown in Figure 41 has been designed such that the vertical displacement at the tip shall be the

same as the horizontal displacement in Figure 21 for the same horizontal current. As seen in

Figure 41 the results compare.

-15

-10

-5

0

5

10

15

20

25

0 1 2 3 4 5 6

End

dis

pla

cem

en

t [m

m]

Time [s]

Response including drift forces with current


Linear displacement

Total

Total Aquasim


Figure 41 Vertical displacement at horizontal current velocity 3 m/s.

8. PROPERTIES ON NODES

Several types of properties may be introduced to nodes in AquaSim. Basically nodes can be free

or have prescribed properties. Otherwise node loads or springs may be attached to nodes

8.1. Fixed nodes

Several papers have been issued and several analyses have been carried out where nodes have

been fixed including references, Berstad et. al. (2003-2012). This validates this property.

8.2. Linear “node to ground” spring

Node to ground springs may be attached to nodes in 6 degrees of freedom. Translation along the

x- y- and z- axis, as well as rotation about the x- y- and z- axis as possible input. Using DOF as

abbreviation for degree of freedom means that x- translation is DOF 1, y- translation DOF 2 and

z- translation DOF 3. Respectively, rotation about the x- axis is DOF 4, y- axis DOF 5 and z-axis

DOF 6.

In the present case a very stiff beam is considered. The beam consists of ten elements with one

element as shown in Figure 42. The spring resistance is in this case 1000 N/mm. The springs are

applied to node 1 the other nodes are free. Loads are also applied to node 1. The displacement of

the beam is calculated by AquaSim and compared to analytical results. Conservative node loads

are applied to various nodes and DOFs are given in Table 23. As seen from this table the results

correspond very well. Note that node to ground springs are conservative meaning that a spring


attached to a node will not rotate proportional to other elements attached to the nodes. Using

local coordinates such effects can be introduced

Figure 42 Geometry of the test case node to ground springs. For beam data not expressed

explicitly in this figure they are the same as in the previous section in Table 21. The beam is

very stiff relative to the stiffness of springs

Table 23 Results verifying node to ground spring elements

Force

magnitude

Force location Result

parameter

Analytical

result

AquaSim result


node 2

1.0/0.0/0.0 1.0/0.0/0.0


node 2

0.0/1.0/0.0 0.0/1.0/0.0


node 2

0.0/0.0/1.0 0.0/0.0/1.0


node 2

0.0/-0.46/0.84 0.0/-0.46/0.84


node 2

0.0/0.0/0.0 0.0/0.0/0.0


node 2

-0.84/-0.46/0.0 -0.84/-0.46/0.0

As seen from Table 23 results predicted by AquaSim are equal to analytic results.

8.2.1. More springs, dampeners and mass

AquaSim allows for a wide range of springs, dampeners and point masses that may be

introduced. Thi type of node load applicationshave been validated through extended usage in

projects. To avoid misinterpretations and possible errors, the AquaSim user manual should be

carefully studied and the various node applications should be introduced to simple test cases and

validated by the end user before implementation into larger, more complex analysis models.

1 m

Node 1 Node 2

Springs at all degrees

Of freedom at node 1

Value: 1000 N/M


8.3. Local coordinates

AquaSim allows for the introduction of a local coordinate system at any node. The advantage of

this is that one may introduce for example hinges in any direction by specifying which nodes that

are coupled and which that are not in any. The local coordinate system at a node may “follow”

node rotations such that the location of the local coordinate system always follows rotations. In

this case the elements the local coordinate system shall rotate in proportion to must be specified

A totally fixed hinge from node 2 to 3 with a local coordinate system was introduced to the case

study above. This was applied by making node 4 having a local coordinate system. The results

were the same as the results presented in Table 23.

8.4. Buoys

Consider a buoy located at the free surface. The buoy will then act as a spring relative to the sea

surface in the z direction with a spring force of

wgA

where Aw is the cross sectional area of the buoy in the horizontal plane at the water surface.

Consider a case with a beam located at the free surface. Assume the beam have no water plane

area or weight itself, but that there are one buoy connected to the beam at each side as shown in

Figure 43.

Figure 43 Beam located at the water line with a buoy at each end. Beam data not expressed

explicitly in this figure are the same as in Table 21. The buoy force is 10,000

Figure 44 shows the AquaSim analysis model. It is seen that the beam follows the wave

elevation. Wave amplitude is 5 meters. Wave period 8 seconds.

L = 10 m

Beam located at water lineNo buoyancy in beam. Buoys at both sides


Figure 44 Beam element seen in wave. The beam follows the wave elevation

Figure 45 shows the wave elevation compared to the vertical z elevation of the buoys for this

case. As seen from this figure, the buoy follows the vertical wave elevation as one will assume.

Forces acting on the buoy in the horizontal direction can be added.

Figure 45 Buoy elevation compared to wave elevation

-6

-4

-2

0

2

4

6

0 5 10 15 20 25

Ver

tica

l dis

pla

cem

en [

m]

Time [s]

buoy

Displacement by analytic expression Calculated displacement Aquasim


8.5. Prescribed displacements

Prescribed displacements may be used in AquaSim. A test case study has been established. The

structural data for this case is given in Table 24.

Table 24 Values used in beam case study testing predescribed displacements

Abbreviation Parameter Value








The beam is exposed to a set of different boundary conditions at Node 2 whereas at Node 1 the

beam is fixed as shown in Figure 46.

Figure 46 Beam for testing prescribed displacements

Different boundary conditions have been applied for Node 2. The status of the DOF’s and results

are given in Table 25. As seen from this table AquaSim and analytical results corresponds very

well. AquaSim results for load case 1, 3 and 4 is shown in Figure 47, Figure 48 and Figure 49.

Table 25 Results applying prescribed displacements

Case Node2 status Bending moment Node 1

AquaSim results Analytical results

1 DOF 1 = 0.1 Mx = 0 kN Mx = 0 kN

z

y

L = 10 m

Node 1 Node 2


DOF 2-6 = Free My = 0 kN

Mz = -300 kN

My = 0 kN

Mz = -300 kN

2 DOF 3 = 0.1

DOF 1-2, 3-6 =

Free

Mx = 0 kN

My = -300 kN

Mz = 0 kN

Mx = 0 kN

My = -300 kN

Mz = 0 kN

3 DOF 3 = 0.0

DOF 4 = 0.1

Others = Free

Mx = 0 kN

My = 2000 kN

Mz = 0 kN

Mx = 0 kN

My = 2000 kN

Mz = 0 kN

4 DOF 1 = 0.1

DOF 6 = 0.0

DOF 2-5 = Free

Mx = 0 kN

My = -600 kN

Mz = 0 kN

Mx = 0 kN

My = -600 kN

Mz = 0 kN

Figure 47 Load case 1. Bending moment about vertical (z-) axis


Figure 48 Load case 3. Bending moment about horizontal axis

Figure 49 Load case 4


8.6. RAO on nodes

In AquaSim there is a possibility for applying RAO. There is a large variety in possibilities on

how to arrange the input data. In the case shown in Figure 50 a vessel was exposed to sinusoidal

waves, then the response displacements and rotations was compared to the input values in the

RAO tables and correspondence was seen. To avoid misinterpretations and possible errors, the

AquaSim user manual should be carefully studied and the RAO applications should be

introduced to simple test cases and validated by the end user before implementation into larger,

more complex analysis models.

Figure 50 RAO applied to node on vessel Results shows pitch [DEG]

8.7. Time domain fixed node motion and rotation

A test case as shown in Figure 51 was established. A time domain RAO was applied to the

towing vessel. A time domain RAO means that node motions and/or rotations are parametrically

introduced as a function of time. In this case the aim was to see how the system laid out when the

vessel did a 360 degrees turn with a given radius. In order to obtain this, both translation and

rotation to a node on the vessel was specified.


Figure 51 Towed system analysis

Figure 52 shows the system at the start of the circling motion.

Figure 52 Towed system at the beginning of circle motion

Figure 53 shows the system towards the end of the circling motion. Note that the colours

indicating rotation are only shown on vessel and door as these components are modelled with

beam elements which have rotation degrees of freedom.

Door /Deflector

Towing vessel

Streamers


Figure 53 Towed system towards the end of circle

By comparing the AquaSim results to what the results expected from the input, the possibility for

prescribing a time series for node motions and rotations in AquaSim are exactly as they should.

Hence, the AquaSim calculations are validated.

8.8. Node loads

AquaSim holds several possibilities for node load application. Depending on type of node load,

the node loads are treated differently.

Conservative node loads are conserving the same magnitude and direction throughout the full

analysis. This can be used with or without automatic introduction of mass corresponding to

negative vertical force (weight) (type 0 or 100)

Input value are multiplied with Vxr*abs(Vxr) where Vxr is the relative velocity between the

fluid and the node. This means that for flow along the x- axis, this can be used to introduced

drag, lift or moment in any direction. Corresponding effect in y- and z- direction (type 1,2 and 3

respectively)

Input value is multiplied with Vi*abs(Vtot) where Vi is the relative velocity between the fluid

and the node in direction x-, y- and z- respectively. Vtot is the total velocity vector (x-,y-,z-). If

the node has rotational DOFs spring stiffness values 4-6 is multiplied with velocities 1-3

respectively (type 4).

Input value is multiplied with Vi*abs(Vxy) where Vi is the relative velocity between the fluid

and the node in direction x- and y- direction. Vxy is the velocity vector in the horizontal plane

(x-,y-). If the node has rotational DOFs spring stiffness values 6 is multiplied with

abs(Vxy)*abs(Vxy) (type 5)

Input value is 6 means lift to the direction such that flow along the positive y- axis leads to force

along the positive y- axis. #=6 means that node Fx is found at -Vy*abs(Vxy) where Vy is the

relative velocity between the fluid and the node in the y- direction. Vxy is the velocity vector in


the horizontal plane (x-,y-). Fy is found as Vx*abs(Vxy) where Vx is the relative velocity

between the fluid and the node in the x- direction.

Input value is 7 means lift to the direction such that flow along the positive x- axis leads to force

along the negative y- axis. #=7 means that node Fy is found at -Vv*abs(Vxy) where Vy is the

relative velocity between the fluid and the node in the y- direction. Vxy is the velocity vector in

the horizontal plane (x-,y-). Fy is found as -Vx*abs(Vxy) where Vx is the relative velocity

between the fluid and the node in the x- direction.

Consider the test case with data given in Table 26.

Table 26 Data for test case

Node load at tip P 10000 N

Length L 10 m

Youngs module E 1.00E+11

Bending moment of inertia I 0.001

Beam theory tip displacement 33.33 mm

Displacement calculated by AquaSim is shown in Figure 54. As seen by comparing Figure 54

and Table 26 it is seen that results compare well.

Figure 54 Displacement in x direction from AquaSim analysis model. 10 elements

Figure 55 shows results calculated with analytic formulae compared to AquaSim results. Load

model 1 is the load model where node load is proportional to the relative velocity squared as

outlined above. As seen from the figure result compare perfectly.


Figure 55 Results calculated with analytic formulae compared to AquaSim results. Load

model 1 is the load model where node load is proportional to the relative velocity squared

as outlined above.

9. WIND LOADS

AquaSim may account for wind loads. Wind loads are calculated in the same manner as drag

loads on an element. It is specified the wind area of the element. There are two alternatives for

application of wind loads. Both have the same type of input wind formulation. The following

wind velocity is applied:

113.0

1010

z*UU

Equation 16

U10 is the wind velocity 10 meters above the surface and z is the distance upwards from the

surface. The force on a surface caused by the wind is then calculated by the following expression

)t(AU2

CF 2Dair

D

Equation 17

air is the density of the air = 1.21 kg/m3 CD is the drag coefficient of the surface an A is the area

of the surface. As a simplification, the wind velocity is averaged over the surface. And the

averaged value is used as the wind value over the full surface. A case study is investigated where

wind is applied on a beam which as shown in Figure 56. Beam data not expressed explicitly in

this figure are the same as in Table 21.

0.00

5.00

10.00

15.00

20.00

25.00

30.00

35.00

0 0.2 0.4 0.6 0.8 1

Tip

dis

pla

cem

en

t [m

m]

Current velocity [m/s]

Response from node load

Analytic displacementconservatice node load

Aquasim displacementconservatice node load

Analytic displacement loadmodel 1

Aquasim displacement loadmodel 1


Figure 56 Case study for testing wind loads. Beam data not expressed explicitly in this

figure are the same as in the previous section in Table 21.

Input and analytic results are shown in Table 27 and the AquaSim calculated responding moment

about the vertical axis is seen in Figure 57. Also the other results correspond well.

Table 27 input data and analytic results wind load case Drag coefficient, Cd 1

Length of beam to mid point last element 9.5 m

Length of beam full length 10 m

Height of wind area 5 m

Lower part of wind catch area 0 m

Transverse drag area 47.5

Half of air desity

Avarage wind at 10 meters height 10 m/s

Average wind used in calculation 8.30785 m/s

Rho_air*Cd/2 0.605

windforce, this case 1983.47 N

Aquasim shear force 1983.5 N

Torsion moment 4958.68 Nm

Aquasim Torsion moment 4959.8 Nm

Moment about vertical axis 10439.3 Nm

Azuasim moment about vertical axis 10422 Nm

z

y

L = 10 m

D = 5 m


Figure 57 Bending moment about the vertical axis

10. REFERENCES

Aquastructures (2006) “Verification and benchmarking of AquaSim, a softwaretool for safety

simulation of flexible offshore facilities exposed to environmental and operational loads“,

Aquastructures report 2003-002.

Berstad, A. J., Tronstad, H., Ytterland, A. (2004)” Design Rules for Marine Fish Farms in

Norway. Calculation of the Structural Response of such Flexible Structures to Verify

Structural Integrity.” Proceedings of OMAE2004 23rd International Conference on Offshore

Mechanics and Arctic Engineering June 2004, Vancouver, Canada. OMAE2004-51577

Berstad, A. J. and H. Tronstad (2007) “Development and design verification of a floating tidal

power unit” OMAE 2007, The 26th International Conference on Offshore Mechanics and

Arctic Engineering San Diego, California, 10-15 June, 2007. Paper 29052. ISBN #: .

Berstad, A. J. and H. Tronstad (2005a) “Response from current and regular/irregular waves on

a typical polyethylene fish farm”Maritime Transportation and Exploitation of Ocean and

Coastal Resources. Eds. C. Guedes Soares, Y. Garbatov, N. Fonseca. 2005 Taylor & Francis

Group London. ISBN #: 0 415 39036 2.

Berstad, A. J., H. Tronstad, S. A. Sivertsen and E. Leite. (2005b) “Enhancement of Design

Criteria for Fish Farm Facilities Including Operations” OMAE 2005, The 24th International

Conference on Offshore Mechanics and Arctic Engineering Halkidiki, Greece, 12-17 June,

2005. Paper 67451. ISBN #: 0791837599.

Berstad, A.J. and H. Tronstad (OMAE 2008)"Use of Hydroelastic Analysis for Verification of


Towed equipment for Aquisition of Seismic Data" Proceedings of OMAE2008 27th

International Conference on Offshore Mechanics and Arctic Engineering June 15-20 2008

Estoril, Portugal OMAE2008-57850

Berstad, A.J., J. Walaunet and L. F. Heimstad (2012) "Loads From Currents and Waves on

Net Structures" Proceedings of the ASME 2012 31st International Conference on Ocean,

Offshore and Arctic Engineering OMAE2012 July 1-6, 2012, Rio de Janeiro, Brazil

OMAE2012-83757

Faltinsen, Odd M. (1990) “Sea loads on ships and offshore structures.” Cambridge university

press ISBN 0 521 37285 2.

Morison, J. R., M.P. O'Brien, J.W. Johnson and S.A. Schaaf (1950), "The Force Exerted by

Surface Waves on Piles," Petroleum Transactions, AIME. Vol. bold 189, 1950, 149-154


11. APPENDIX

11.1. Loading when ropes gets stiff

A typically occurring load condition for moored offshore structures is that the system moves and

moorings gets stiff leading to large accelerations and forces in the system. Due to the ability to

handle large nonlinear effects, AquaSim may be used to investigate the occurrence and

magnitude of such loading. This effect occur typically for any kind of moored structure including

fish farms, barges, ship shaped structures and offshore platforms. It also typically occurs to

bouys and similar surface penetrating floats.

In order to demonstrate the effect and validate the AquaSim analysis capabilities, a case study

with a float connected to a fixed point by a rope as shown in Figure 58 has been established.

Consider the particular time instant when the distance rope goes from slack (seen in Figure 59)

to straight as seen in Figure 1.

Figure 58 Float attached to bottom with rope

In Figure 58, k is the (spring) stiffness of the rope, E is the Young modulus, A is the (nominal)

cross sectional area of the rope and L is the rope length. m = mf + ma is the mass of the float

where mf is the mass of the float itself and ma is the added mass which is the hydrodynamic load

proportional to the float acceleration. Assume that the mass and added mass of the rope itself is

much lower than for the float and hence can be neglected.

k = EA/L

m = mf + ma

L


Figure 59 Float with slack rope

Consider a time instant when the float rope has become slack as seen in Figure 59. Assume that

the float moves upwards with the wave elevation. At some point, the distance rope goes from

slack to straight as seen in Figure 58. As a simplification one can assume that the vertical

stiffness of the rope is 0 when the rope is slack and that it is EA/L when it gets stiff and the

condition in Figure 58 applies. At that moment, an impact load is introduced to the float.

Assume that the response from this impact load can be described with the classic impulse

response function:

0 kzzm

Equation 18

where z is the vertical displacement. As a simplification a one dimensional system where the

motion is this case is assumed to be vertical in direction parallel with the distance rope is

considered. Define the coordinate system such that vertical motion z = 0 when t = 0 as shown in

Figure 60.


Figure 60 Initial value condition at moment rope goes stiff

Neglect damping in the system (as has been done in Equation 18). This means the system can be

solved by applying the classic impulse response equation solution for the motion:

)sin()( tatz

Equation 19

where ω2 = k/m and a is the amplitude of the response. k is the stiffness of the system caused by

the rope holding the float back like a spring. k = EA/L. This is a linear solution where the

stiffness and mass is assumed to be time invariant. The velocity is the time derivative of the

displacement:

)cos()( tatz

Equation 20

where )(tz is the vertical velocity of the system. In our case we have an initial velocity v0 which

is the velocity at the initial time (t = 0 (exactly when the ropes gets stiff). Assume that the

velocity of the float follows the vertical velocity of the wave elevation as the wave is built up.

Then v0 can be found from the velocity of the vertical wave elevation at the moment the rope

snap. Assume a regular wave with amplitude . The wave elevation can be expressed as

)sin( te

Equation 21

where ωe is the wave frequency of encounter, this frequency has nothing to do with the

eigenfrequency of the rope and float. Applying Equation 19 to timestep t= 0 we get

)cos( 00 tetev

Equation 22

Now assume that the float rope snaps when v0 is at its max possible value, v0 = ωe:

zm = mf + ma

Z = 0 k (z-direction) = EA/L

Vz (vertical velocity)


aazev )0cos()0()0(

Equation 23

This means the amplitude, a is found as

)0(va

Equation 24

Now the relation between eigenperiod and mass and stiffness is introduced (ω2 = k/m) to

Equation 24:

EAmLvkmvmk

vva /)0()/)(0(

)/(

)0()0( )2/1(

2/1

Equation 25

This means that an impact as described above will introduce an harmonic impact response with

amplitude as given in Equation 25.

From the maximum response amplitude, the maximum force can be derives as

LmEAvEAmLvL

EAkaF /)0(/)0(max

Equation 26

This means the maximum force is proportional to the initial velocity and the square root of the

mass and stiffness. From Equation 22 it is seen that the initial velocity is proportional to the

wave amplitude and the wave frequency of encounter, e. Introducing v0 = ωe to Equation

26 Fmax can be expressed as

LmEAeF /max

Equation 27

Consider a case with parameters shown in Table 28.

Table 28 Main data for system and analysis

Float data

Float length [m] 5.4

Float volume [l] 5132

Float circular diameter [m] 1.1

Float weight [kg] 2907

Rope data

Length [m] 10

Cross sectional area [mm2] 1000


E-modulus [Mpa] 10000

Environment data

Wave period [s] 6

Current velocity n[m/s] 1

Added mass [kg] 4737

A model has been established in AquaSim for the case presented in Table 28. The model is

shown in Figure 61.

Figure 61 Analysis model, impact load test case

In the analysis waves and current is from the left to the right along the positive x- direction as

shown in Figure 62.


Figure 62 Analysis model, wave and current direction is along positive x- axis.

Analysis has been carried out with varying wave amplitudes and compared to Equation 27. This

is shown in Figure 63. In this figure, the labels mean:

Analytic formulae: Max load calculated from input from Equation 27.

Peak load AquaSim 1: Max load calculated with AquaSim with the analysis model taking

in and out of water into account.

Peak load AquaSim 2: Max load calculated with AquaSim with the analysis model not

taking in and out of water into account.


Figure 63 Maximum axial load in rope

As seen from Figure 63, the results by Equation 27 and the AquaSim 2 model shows very good

correspondence for all wave heights. Figure 64, Figure 65 and Figure 66 shows how the impact

load strikes the rope as the float is moved from slack rope to the rope getting stiff.

Figure 64 Float when rope is slack. There are no forces in the rope

0.00

100.00

200.00

300.00

400.00

500.00

600.00

700.00

0 1 2 3 4 5 6 7

Max

load

in r

op

e [

kN]

Wave height [m]

Rope shock response from wave load

Analytic Formulae [kN]

Peak load AquaSim 1

Peak load AquaSim 2


Figure 65 Forces commencing in rope as float is moved upwards by wave motion


Figure 66 Impact load to rope as the float has been moved so much upwards that the rope

gets stiff

The time series response for the axial force in the rope is in the AquaSim analysis is shown in

Figure 67 and Figure 68.

Figure 67 Time series for the AquaSim analysis of the axial force

-50

0

50

100

150

200

250

300

350

0 2 4 6 8 10 12 14 16 18 20

Forc

e [

kN]

Time [s]

Peak load response as function of time

Axial force


Figure 68 Excerpt of time series for the AquaSim analysis of the axial force

As seen from Figure 67 and Figure 68 the load is at 0 and then increase sharply due to the rope

going from slack to stiff. As seen from the figures, a harmonic response is then decaying over

time. As seen from Table 29 the natural period of a swinging system with mass and added mass

of the float and stiffness of the rope is approximately 0.55 s.

Table 29 Key data for the natural period of the rope

Length L 10

Youngs module E 1E+10

Cross sectional area A 0.001

Stiffness K 1000000

Total mass m 7644

omega 11.4378

Period s 0.5493

The natural period of 0.55 seconds corresponds very well to the response seen in Figure 68 apart

from the first succeeding cycles where the time between succeeding peaks are longer. That is

plausible as when inspecting Figure 68 the load gets to 0 between the first and the second

response cycle after the impact, in that case the stiffness decrease and the natural period increase.

This analysis case shows that AquaSim manage to calculate the peak loads occurring in mooring

lines as they goes from slack to stiff. This is an important design criteria for a wide range of

moored structures and equipment. Hence the ability of complex nonlinear dynamic response

calculations of AquaSim is validated.

- o0o -

-50

0

50

100

150

200

250

300

350

6 6.5 7 7.5 8 8.5 9 9.5 10

Forc

e [

kN]

Time [s]

Peak load response as function of time

Axial force

reportfouverifikasjon 2012

Documents

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cable elements

membrane elements

operational loads report

hydrodynamic elements

bar elements

current loads

benchmarking of aquasim