Workshop on Very Large Floating Structures for the Future

Trondheim -- 28-29 October 2004

Efficient Hydrodynamic Analysis of VLFS

by J. N. Newman

jnn@mit.edu

Theoretical formulation and restrictions

• Linear (and 2nd order) potential theory (small amplitude waves and motions)

• Frequency-domain analysis (can be transformed to time domain if necessary)

• No considerations of current, slamming, viscous loads

Computational approach

• Boundary Integral Equation Method (BIEM) a.k.a. `Panel Method’ with free-surface Green function (source potential)

• General-purpose codes are applicable to a wide variety of structures and applications (different configurations, air-cushions, hydroelastic effects, multiple bodies)

• All relevant hydrodynamic parameters can be computed (loads, RAO’s, pressures, drift forces, free-surface elevations, etc.)

Computational tasks and restrictions

• Geometric representation of submerged surface

• Structural representation (if hydroelastic)

• Confirmation of results requires convergence tests

• Computing cost (time and memory) increases rapidly with number of unknowns

• Short wavelengths aggravate everything!

recent developments

• Higher-order panel method with B-spline basis functions for the solution

• pFFT Accelerated solver O(N log N)

• Exact geometry

• CAD coupling

Examples of Applications and Results

• Rigid barge 4km x 1km x 5m

• Same barge with hydroelasticity

• Hinged barges with hydroelasticity

• Hinged semi-subs with hydroelasticity

• Large arrays of cylinders

Drift forces on 4km x 1km x 5m rigid bargeWave incidence angle 45o -- Water depth = 50m

Red: Surge drift forceBlack: Sway drift force• higher-order method (727 unknowns) Solid lines: Accelerated pFFT (`precorrected Fast Fourier Transform’) method with 43,000 low-order panels (Order N log N)Both methods take about 0.4 hour/period on a 2GHz PCDashed lines: Asymptotic approximation for short wavelengths

Period10 15 20 25 30

0

500

1000

1500

2000

Methodology for hydroelastic analysis

• Structural deflections represented by generalized modes

• Uniform stiffness ratio EI/(mass*L^2) assumed for simplicity

• Extended vector of RAO’s gives the modal response

• Cf. `direct’ method where the structural and hydrodynamic analyses are integrated

4km x 1km x 5m barge, 27*16 generalized modesL/wavelength = 5,10,20 – depth = 20m – Period = 57,28,15 seconds

Head seas 45 degrees Head seas 45 degrees

5

10

20

5

From C.-H. Lee, `Wave interactions with huge floating structure’, BOSS `97

DEFLECTION PRESSURE

Array of 5 barges, each 300x80x6mjoined by simple hinges (red)

Symmetric/antisymmetric modes for a hinged structure with five elements

Modes j=7-11 and j=12-16 are 1st and 2nd Fourier modes j=17-96 are 3rd to 18th Fourier modes (not shown)

j=1,2 j=3-6 j=7-11 j=12-16

Array of five 300x80x6m barges with zero stiffnessRAO of each mode (j=1-96), period 12 seconds

Array of five barges with stiffness SMotion and shear force at the upwave hinge

Period6 8 10 12 14 16 18 20 22 24 26 28 30

0

0.5

1

1.5

2

2.5

3

3.5

4

rigid1E-31E-41E-6S=0

Amplitude of vertical motion

Period6 8 10 12 14 16 18 20 22 24 26 28 30

0

1000

2000

3000

4000

rigid1E-21E-31E-41E-6

Vertical shear force

Array of 5 hinged semi-subs, each 300m long(lower figure shows zoom of one element)

Array of five semi-subs with stiffness SMotion and shear force at the upwave hinge

Period6 8 10 12 14 16 18 20 22 24 26 28 30

0

500

1000

1500

2000

rigid1E-11E-21E-3

Vertical shear force

Period6 8 10 12 14 16 18 20 22 24 26 28 30

0

2

4

6

8

10

rigid1E-11E-21E-3

Amplitude of vertical motion

Array of 100 x 4 = 400 cylinders

Radius=11.5m, Draft=20m, Axial spacing=40m

V150

V25 0

V3

-20

-10

0

Mean drift forces on Array of 100 x 4 cylinders Wave incidence angle 45o -- Water depth = 50m

Based on Higher-order method with exact geometry.

NEQN=4200

2.3 hours per Wave periodOn 2GHz PC

Period8 12 16 20 24 28 32

0

50

100

150

200SwaySurge

Array of 100 x 25 = 2500 cylindersRadius=11.5m, Draft=20m, Axial spacing=40m

Drift forces on the 100x25 Cylinder Array Wave incidence angle 45o -- Water depth = 50m

240,000 low-order panelswere used here (96 on eachCylinder).

These computations wereperformed on a 2.6GHz PC using the accelerated pFFT method. The CPU time averaged about one hourper period.

The peak magnitudes aremuch larger below 8 seconds,where near-trapping occursbetween adjacent cylinders.

Period10 15 20 25 30

-20

0

20

40

60

80

100

120

140

SurgeSway

Conclusions

• Contemporary hardware/software can be used to analyze wave effects for many VLFS configurations with length scales of 1-4 km

• These capabilities will increase in the future

• Short wavelengths, bottom topography, breakwaters, etc, present special challenges

• Analytic/asymptotic theories are still required for >>1km length scales, e.g. ice fields, and to validate computations in the short-wavelength regime

Further information: www.wamit.com/Publications

Special Thanks:

Chang-Ho Lee, WAMIT, Inc.