inspection capabilities for enhanced ship safety · inspection capabilities for enhanced ship...

Inspection Capabilities for Enhanced Ship Safety

D3.3 (WP3): Hydrodynamic and structural analysis

Responsible Partner: RINA

Contributor(s): USG, LR, RINA, AES, DANAOS

Dissemination Level

PU Public x

PP Restricted to other programme participants (including the Commission Services)

RE Restricted to a group specified by the consortium (including the Commission Services)

CO Confidential, only for members of the consortium (including the Commission Services)

This document is produced by the INCASS Consortium. The INCASS project is funded by the European

Commission under the Seventh Framework Programme (FP7/2007-2013). Grant Agreement n°605200

D3.3 (WP3) – Hydrodynamic and structural analysis

This document is produced by the INCASS Consortium, funded by the European Commission (FP7/2007-2013).

Grant Agreement n° 605200.

of 94

Document information table

Contract number: 605200

Project acronym: INCASS

Project Coordinator: University of Strathclyde Glasgow (USG)

Document Responsible Partner: RINA Services SPA RINA

Deliverable Type: Report

Document Title : Hydrodynamic and structural analysis

Document ID: D3.3 Version: 3

Contractual Date of Delivery: 31/07/2014 Actual Date of Delivery: 31/07/2014

Filename: D3.3 Hydrodynamic and structural analysis

Status: Final version

Authoring & Approval

Prepared by

Author Date Modified

Page/Sections Version Comments

Adnan Kefal 05/06/2014 V0 Creation of the

document

Adnan Kefal 18/07/2014 All V1 Description of the

software

Erkan Oterkus 28/07/2014 All V2 Executive

summary

Adnan Kefal 29/07/2014 21,25,28 V3 Minor comments

Approved by

Name Role Partner Date

Document Manager Erkan Oterkus WP3 Leader USG 31/07/2014

Document Approval Iraklis Lazakis Project Coordinator USG 31/07/2014




of 94

Executive Summary

Current INCASS structural mechanics platform is based on previously EU funded

RISPECT methodology. RISPECT approach used two different commercial software

for performing structural and hydrodynamics analyses. These are MAESTRO software

for structural analysis calculations and FD-Wave Load software for hydrodynamic

analysis. Although commerial software is generally powerful and beneficial, it may not

be practical to use them as part of a more general platform due to connectivity issues.

Hence, as part Task 3.3 of the INCASS project, two in-house finite element and

hydrodynamics analyses software, ADFEM and ADPAN, have been developed by

using the object-oriented Java programming language.

The Java language has several advantages. These are an object-oriented paradigm,

multiplatform support, ease of development, reliability and stability, the ability to use

legacy C or C++ code, good documentation, development-tool availability, etc.

Moreover, Java programs are less susceptible to bugs and security flaws.

The newly developed in-house panel method code, ADPAN, is a frequency-domain

hydrodynamic software which can be used to predict the motions and wave loads of any

vessel. The software first calculates the velocity potentials, source strengths, and flow

velocities at the centroids of the hydrodynamic panels for requested speed, heading, and

frequency. By using this data, the hydrodynamic motions and loads can be computed.

The approach used to calculate the hydrodynamic forces is based on 3D potential theory

with zero speed Green’s function.

The newly developed in-house finite element code, ADFEM, contains a finite element

library including truss, beam, plane, plate, shell and solid element types. By using the

developed tool, it is also possible to combine beam and shell elements or truss and plane

elements (for simple cases) to build the finite element model of a ship. This feature will

result in significant computational efficiency.




of 94

Both the hydrodynamics tool and finite element analysis tool are validated extensively

by considering various problem cases. The numerical solutions obtained from in-house

software are compared against analytical solutions and the results generated by using

other available software including ANSYS, AQWA and PRECAL.

Utilization of ADPAN and ADFEM as an integrated tool allows solution of hydro-

elastic ship model in order to obtain both displacements and relevant stress distribution

of the hull structure. Then, these properties will be calibrated within INCASS platform

in order to calculate future conditions of the hull structure and predict the right

inspection intervals.




of 94

Table of Contents

1 INTRODUCTION ............................................................................................... 12

2 PANEL METHOD CODE ................................................................................... 14

2.1 SHIP MOTIONS ........................................................................................... 15

2.2 PANEL METHOD ........................................................................................ 20

2.3 ADPAN SOFTWARE .................................................................................. 25

2.4 APPLICATIONS AND VALIDATION OF ADPAN ............................................. 26

2.4.1 Radiation Problem of a Floating Hemisphere............................................... 27

2.4.2 Diffraction Problem of a Submerged Spheroid ............................................ 30

2.4.3 Rigid Body Motions of a Long Barge .......................................................... 32

2.4.4 Hydrodynamic Analysis of WIGLEY III ..................................................... 34

2.4.5 Hydrodynamics of S175 Type Container Ship ............................................. 39

3 FINITE ELEMENT CODE .................................................................................. 46

3.1 FINITE ELEMENT METHOD ......................................................................... 46

3.2 ELEMENT TYPES ........................................................................................ 48

3.2.1 Truss Element ............................................................................................. 48

3.2.2 Beam Element ............................................................................................. 49

3.2.3 Plane Element ............................................................................................. 49

3.2.4 Shell Element .............................................................................................. 50

3.2.5 Solid Element .............................................................................................. 52

3.3 ADFEM SOFTWARE .................................................................................. 53




of 94

3.4 APPLICATIONS AND VALIDATION OF ADFEM ............................................ 56

3.4.1 Truss Problem ............................................................................................. 56

3.4.2 Beam Problem ............................................................................................. 58

3.4.3 Frame Structure Problem ............................................................................. 60

3.4.4 Plane4 Problem ........................................................................................... 64

3.4.5 Plane8 Problem ........................................................................................... 67

3.4.6 Shell3 Element ............................................................................................ 69

3.4.7 Shell4 Element ............................................................................................ 73

3.4.8 Shell8 Element ............................................................................................ 78

3.5 GLOBAL FINITE ELEMENT ANALYSES OF SHIP AND COUPLING OF FINITE

ELEMENTS ............................................................................................................ 83

4 CONCLUSION .................................................................................................... 91

5 REFERENCES .................................................................................................... 92




of 94

List of Figures

Figure 1 Hemisphere Mesh ...................................................................................... 28

Figure 2 Hemisphere added mass for (a) surge and (b) heave motion ....................... 29

Figure 3 Hemisphere damping for (a) surge and (b) heave motion ........................... 30

Figure 4 Spheroid geometry and orientation in space ............................................... 31

Figure 5 Spheroid mesh ........................................................................................... 31

Figure 6 (a) Real and (b) imaginary component of excitation force .......................... 32

Figure 7 Barge mesh representation ......................................................................... 32

Figure 8 Barge motion amplitudes ........................................................................... 33

Figure 9 Barge motion phase angles ........................................................................ 34

Figure 10 WIGLEY III sectional view ....................................................................... 35

Figure 11 WIGLEY III mesh ..................................................................................... 35

Figure 12 WIGLEY III pitch added mass .................................................................. 37

Figure 13 WIGLEY III pitch damping ....................................................................... 37

Figure 14 WIGLEY III motion amplitudes ................................................................ 38

Figure 15 WIGLEY III motion phase angles ............................................................. 38

Figure 16 WIGLEY III oscillatory pressure distribution (Unit: Pa) ............................ 39

Figure 17 S175 container ship profile view ............................................................... 40

Figure 18 S175 container ship waterlines .................................................................. 41

Figure 19 S175 container ship body plan ................................................................... 41

Figure 20 S175 container ship isometric mesh view .................................................. 42

Figure 21 S175 container ship added mass ................................................................ 42

Figure 22 S175 container ship damping ..................................................................... 43

Figure 23 S175 container ship motion amplitudes ..................................................... 43

Figure 24 S175 container ship motion phase angles ................................................... 44

Figure 25 S175 container ship pressure distribution ................................................... 44

Figure 26 Space truss isometric view......................................................................... 57

Figure 27 Beam mesh and cross section .................................................................... 58




of 94

Figure 28 (a) Translational and (b) rotational displacement results of the beam problem

59

Figure 29 Key point locations and application of boundary conditions on frame

structure .............................................................................................................. 60

Figure 30 Frame structure translational displacements along (a) x- and (b) y- directions

of all nodes .......................................................................................................... 62

Figure 31 Frame structure (a) translational-z and (b) rotational-x displacements of all

nodes 62

Figure 32 Frame structure (a) rotational-y and (b) rotational-z displacements of all

nodes 63

Figure 33 Frame structure total translational displacement contour plot; .................... 63

Figure 34 Frame structure total rotational displacement contour plot; ........................ 64

Figure 35 A square plate mesh and boundary condition representation ...................... 64

Figure 36 Translational displacements along (a) x- and (b) y-directions of all nodes for

the Plane4 problem .............................................................................................. 65

Figure 37 Contour plot of translational displacement along x-direction for the Plane4

problem; (a) ADFEM and (b) ANSYS................................................................. 66

Figure 38 Contour plot of translational displacement along y-direction for the Plane4


Figure 39 A rectangular plate and its applied boundary conditions ............................ 67


the Plane8 problem .............................................................................................. 68





Figure 43 A clamped edge square plate and its discretization .................................... 70

Figure 44 Translational displacements along z-direction of all nodes for the Shell3

problem ............................................................................................................... 70

Figure 45 Rotational displacements along (a) x- and (b) y-directions of all nodes for

the Shell3 problem .............................................................................................. 71




of 94

Figure 46 Contour plot of translational displacement along z-direction for the Shell3


Figure 47 Contour plot of rotational displacement along x-direction for the Shell3

problem; .............................................................................................................. 72

Figure 48 Contour plot of rotational displacement along y-direction for Shell3

problem; .............................................................................................................. 72

Figure 49 T beam mesh and its boundary conditions ................................................. 73



Figure 51 (a) Translational-z and (b) rotational-x displacements of all nodes for the

Shell4 problem .................................................................................................... 74

Figure 52 Rotational displacements along (a) y- and (b) z-directions of all nodes for


Figure 53 Contour plot of translational displacement along x direction for the Shell4


Figure 54 Contour plot of translational displacement along y direction for the Shell4






Figure 57 Contour plot of rotational displacement along y-direction for the Shell4


Figure 58 Contour plot of rotational displacement along z-direction for the Shell4


Figure 59 A clamped square plate modelled with Shell8 ............................................ 79

Figure 60 Translational displacements along z-direction of all nodes for the Shell8

problem ............................................................................................................... 79

Figure 61 Rotational displacements along (a) x- and (b) y-directions of all nodes for





of 94





Figure 64 Contour plot of rotational displacement along y-direction for the Shell8

problem; .............................................................................................................. 82

Figure 65 Stiffened plate model ................................................................................. 84

Figure 66 Stiffened plate boundary conditions ........................................................... 85


the stiffened plate ................................................................................................ 86

Figure 68 (a) Translational-z and (b) rotational-x displacements of all nodes for the

stiffened plate ...................................................................................................... 86

Figure 69 Rotational displacements along y- and z- directions of all nodes for the

stiffened plate ...................................................................................................... 87

Figure 70 Contour plot of translational displacement along x-direction for the stiffened

plate; (a) ADFEM and (b) ANSYS ...................................................................... 87

Figure 71 Contour plot of translational displacement along y-direction for the stiffened


Figure 72 Contour plot of translational displacement along z-direction for the stiffened


Figure 73 Contour plot of rotational displacement along x-direction for the stiffened


Figure 74 Contour plot of rotational displacement along y-direction for the stiffened


Figure 75 Contour plot of rotational displacement along z-direction for stiffened plate;

(a) ADFEM and (b) ANSYS ............................................................................... 90




of 94

List of Tables

Table 1 WIGLEY III hull properties ...................................................................... 36

Table 2 S175 container ship properties ................................................................... 40

Table 3 Finite element library of ADFEM .............................................................. 55

Table 4 Displacement results of space truss problem .............................................. 57

Table 5 Strain and stress results of space truss problem .......................................... 58

Table 6 Key points of frame structure .................................................................... 61




of 94

1 INTRODUCTION

RISPECT uses pre-calculated hydrodynamic and structural analyses of the intact ship. These

are not adequate for the rapid emergency decision response required for a damaged ship

structure. A new efficient hydrodynamic and structural analysis system needs to be merged to

existing RISPECT system. This process is done within INCASS Task 3.3 by replacing the

two commercially available computational tools used within the RISPECT system

(MAESTRO software for structural analysis calculations and FD-Wave Load software for

hydrodynamic analysis), with in-house finite element and panel-method codes.

An in-house finite element code, called ADFEM, together with a panel-method code, called

ADPAN, are implemented by using Java computer programming language based on object-

oriented methodology. The Java language is selected for its numerous advantages: an object-

oriented paradigm, multiplatform support, ease of development, reliability and stability, the

ability to use legacy C or C++ code, good documentation, development-tool availability, etc.

The Java runtime environment always checks subscript legitimacy to ensure that each

subscript is equal to or greater than zero and less than the number of elements in the array.

Even this simple feature is very important for developers. As a result, Java programs are less

susceptible to bugs and security flaws.

The panel method code, ADPAN, provides the user with an integrated frequency-domain

computational tool to predict the motions and wave loads of any vessel. To compute these

hydrodynamic motions and loads, ADPAN first calculates the velocity potentials, source

strengths, and flow velocities at the centroids of the hydrodynamic panels for each speed,

heading, and frequency requested. The computation of these hydrodynamic forces is based on

3D potential theory using the zero speed Green’s function.

The finite element code, ADFEM, offers the user a finite element library that includes

formulation of truss, beam, plane, plate, shell and solid element types. Combination of beam

and shell elements or truss and plane elements (for simple cases) can be used to build the




of 94

finite element model of a ship. By applying the required boundary conditions, the solution of

hydro-elastic ship model can be performed in finite element code in order to obtain both

displacements and relevant stress distribution of the hull structure. Then, these properties will

be calibrated in INCASS software in order to calculate future conditions of the hull structure

and predict the right inspection intervals.




of 94

2 PANEL METHOD CODE

The design of marine structures such as ships, offshore and coastal structures is intensively

affected by wave-body dynamics; therefore hydrodynamic analysis of rigid bodies that are

freely oscillating under the free water surface turns out to be extremely important for today’s

naval architects and marine engineers. The fundamental principle of setting up a complete

linear hydrodynamic analysis can be described by stating two different types of hydrodynamic

problems. First kind of these problems is the interaction of regular surface waves with a rigid

body which is called “diffraction problem.” Second kind of these problems is the fluid motion

resulting from the forced harmonic oscillation of the body in still water which is called

“radiation problem.” Although these problems physically seem very distinctive, they are

mathematically very similar. In fact, they can be treated simultaneously by a solution

procedure or computer program. Essentially, all formulated boundary value problems are of

combined type and therefore, the radiation-diffraction problem can be solved by the same

theoretical numerical schemes.

It is not easy to apply pure numerical approaches like finite differences methods, finite

element method etc. to solve radiation-diffraction problem since the boundary regions of the

wave-body domain needs to be modelled. However, the boundary element method, which is

often called as panel method, could be more suitable for the solution of integral equations

presented in below in terms of flexibility, efficiency and computational time. The presented

formulations deal with three dimensional bodies of arbitrary shape being freely oscillating in

water of infinite depth. It is assumed that the body moves at zero forward speed and is excited

by regular surface waves. The problem is formulated as boundary value problem of potential

theory with mixed boundary conditions such as free water surface region condition. The

numerical solution procedure involves a discretization of the body surface by plane triangular

or quadrilateral elements as they better fit the given body surface. An efficient calculation of

the relevant influence matrices and the solution of a set of linear algebraic equations are

described by standard algorithms. Sample problems are solved and the results of these




of 94

problems are compared with those of other investigators, commercial software and analytical

results.

2.1 Ship Motions

The hydrodynamic forces acting on a ship in waves are solved using potential flow theory.

The present approach is similar to that is used by many other investigators including Beck and

Loken (1989), and Papanikolaou and Schellin (1992). In order to be able to compute all

responses of any vessel to regular waves, it is necessary to deal with six degrees of freedom

motion considering important couplings among them. It is desirable first to define an axis

system in order to calculate motions of ship at zero forward speed. The equation of motion is

solved with respect to the ship centre of gravity, although the defined coordinate system has

its origin in the still water plane aligned vertically with the ship centre of gravity. Global

coordinate system is fixed with respect to the earth having the origin at any desired point.

However it is better to choose the mid-ship as an origin of the system because one can easily

utilize the symmetry plane (x-z plane) of the ship. Moreover, x-y plane is coincident with

calm water level and z direction is positive upwards.

Deep water conditions are assumed for a ship moving with constant forward speed, 0U , at

any angle, , to regular sinusoidal waves of small amplitude. The frequency of the oscillation

will be shifted to the frequency of the wave encounter as stated in Equation (1) with

0 2 /k , where 0k is the wave number and is the wave length.

0 0 0U cose k (1)

Ship’s frequency of encounter is higher than the absolute frequency for waves coming from

ahead where direction of wave is 180deg . In stern seas ( 0deg ) the frequency of the

encounter is lower and may be equal to zero when the ship speed is equal to the phase

velocity of the waves. When direction of the wave propagation is 90deg , it is defined as

beam sea condition and waves come from starboard side of the ship. In case of ship has zero

forward speed, the frequency of the wave encounter equals to the regular waves of the

frequency. The ship is assumed to oscillate as a rigid body in six degree of freedom and the




of 94

resulting oscillatory motions of the ship are assumed to be linear and harmonic. The harmonic

and complex responses of the vessel is ( )j t where 1,2,3,4,5,6j refer to surge, sway,

heave, roll, pitch and yaw, respectively. These responses will be proportional to the amplitude

of the exciting forces and at the same frequency but with a phase shift. Consequently, the ship

motions will have the form given in Equation (2).

(t) cos( ) Re ei t

j j e j jt e (2)

The general formula of the basic linearized equations (Euler’s equation of motion) in six

degrees of freedom is given in Equations (3) to (5). jk are the components of the generalized

inertia matrix including all possible couplings for the ship in which zc and xc are the vertical

and longitudinal centre of gravity, and I represents the mass and moment of inertia terms.

( )k t are the accelerations in mode k and ( )jF t represents the total forces or moments acting

on the body in direction j .

6

1

( ) ( )jk k j

k

t F t

, 1,2...6j (3)

2(t) ei t

j e je (4)

44 46

55

46 66

0 0 0 0

0 0 0

0 0 0 0

0 0 0

0 0 0

0 0 0

c

c c

c

jk

c

c c

c

z

z x

x

z I I

z x I

x I I

(5)

The total forces and moments for each direction of motion can be written in terms of

gravitational ( G

jF ) and fluid force ( H

jF ) acting on the ship. The hydrostatic ( HS

jF ) and

hydrodynamic ( HD

jF ) forces acting on the ship are obtained by integrating the fluid pressure

over the underwater portion of the hull. Therefore, the actual components of the fluid force

are hydrostatic and hydrodynamic forces. The gravitational forces are simply due to the

weight of the vessel applied to the centre of gravity. Since the mean gravitational force cancel




of 94

the mean buoyant forces, they are usually combined with the hydrostatic part of the fluid

force to calculate the extended hydrostatic force ( EH

jF ). As a result, Equations (6) to (9) can

be written in order to define the relationship between the described forces.

G H

j j jF F F (6)

H HS HD

j j jF F F (7)

EH G HS

j j jF F F (8)

EH HD

j j jF F F (9)

To determine the hydrostatic forces, we must integrate the static pressure over the underwater

hull surface. The details of the integral evaluation can be found in Newman (1977). Although

this integration seems straightforward, it is a tedious process. Therefore, it is much easier to

directly define the extended hydrostatic force for each direction. The Equations (11) to (15)

can be used to calculate the extended hydrostatic force where is density, g is gravity,

( )B x is the full breadth of the water-plane at x, is the total mass of the ship, TGM is the

transverse metacentric height, LGM is the longitudinal metacentric height, LCF is the

longitudinal centre of flotation.

ei tEH

j jk kF C e (10)

33 35

44

35 55

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0

0 0 0 0 0

0 0 0 0

0 0 0 0 0 0

jk

C CC

C

C C

(11)

33 ( )L

C g B x dx (12)

35 ( )L

C g xB x dx (13)




of 94

44 TC g GM (14)

2

55 ( )L

L

C g GM gLCF B x dx (15)

The hydrodynamic pressure at a point in the fluid can be written by Bernoulli’s equation for

unsteady flow where is time-dependent velocity potential. Assuming the zero forward

speed condition and neglecting second and higher order term derivations of time-dependent

velocity potential, the oscillatory pressure acting on the ship hull is given in Equation (16).

The hydrodynamic force can then be evaluated by integration of defined oscillatory pressure

as stated in Equation (17). The components of the generalized unit normal for each motion

mode are expressed by Equations (18) and (19) in which , ,x y zn n n are the directional cosines

for the unit normal pointing outward from the hull and gz is the height of the ship centre of

gravity above the waterline.

Pt

(16)

HD

j j

S

F n dst

(17)

1 xn n , 2 yn n , 3 zn n (18)

4 ( )z g yn yn z z n , 5 ( )g x zn z z n xn , 6 y xn xn yn (19)

To accomplish calculating the hydrodynamic force acting on the ship hull, the total time-

dependent velocity potential for fluid flow needs to be calculated. The time-dependent total

velocity potential can be subdivided into a simple summation of the steady and unsteady

components. The steady part results from the steady forward speed of the vessel. Hence, it is

the combination of the free stream velocity and the steady perturbation potential of the ship

hull. The unsteady part of the time-dependent total velocity potential can be subdivided into

incident wave, the diffracted wave, and the radiation potentials due to the motion in each

degree of freedom. Note that incident wave, diffracted wave, and radiated wave potentials are

all independent of time and depend only on space variables, when the solution of these

potentials are done in frequency domain.




of 94

The steady part of the velocity potential is omitted for hydrodynamic force calculation since it

has been assumed that ship is travelling at zero forward speed. Therefore, the hydrodynamic

force can be now described in terms of time-independent unsteady velocity potential

components as represented in Equations (20) to (23).

HD I D R

j j j jF F F F (20)

e ei tI

j e I j

S

F i n ds

(21)

e ei tD

j e D j

S

F i n ds

(22)

6 6

2

1 1

e ( ) ee ei t i tR

j e k j k e jk e jk k

k kS

F i n ds A i B

(23)

Symbols of I , D , k denotes incident, diffracted, and radiated (for each motion mode k)

wave potentials whereas I

jF , D

jF , R

jF symbolizes the incident, diffracted and radiated wave

forces, respectively. Radiated wave force can be written in terms of added mass ( jkA ) and

damping ( jkB ) coefficients. Added mass and damping coefficients can be mathematically

expressed as given in Equations (24) and (25), respectively.

Im( )jk k j

e S

A n ds

(24)

Re( )jk k j

S

B n ds (25)

The incident wave force and diffracted wave force are the forces that excite the motion of the

vessel. Therefore, these forces are generally summed and called exciting forces ( EX

jF ). The

expressions for all different forces can now be substituted back into equation (3) in order to

obtain a compact form of oscillatory rigid body motion equation as formulated in equation

(28) where jk is the mass matrix, jkA is added mass matrix, jkB is the damping matrix,




of 94

jkC is the hydrostatic restoring force coefficient matrix, k is the unknown (complex)

vessel’s amplitudes, and EX

jF is the total exciting force, for subscripts of , 1,2,3,4,5,6j k .

EX I D

j j jF F F (26)

2 ei t EH EX R

e jk k j j j je F F F F (27)

2 EX

e jk jk e jk jk k jA i B C F (28)

2.2 Panel Method

The equation of the oscillatory motion can only be solved if radiation and diffraction velocity

potentials are known. Therefore, these potentials need to be determined by solving a well-

known radiation and diffraction problem of a body with zero translational velocity performing

small steady oscillations in the presence of a free surface. The mathematical formulation is

explicitly given by Wehausen and Laitone (1960) and results will simply be stated in this

report with no attempt to derive the formulation.

The fluid field in the half-space z < 0 is assumed to be homogeneous, inviscid, and

incompressible and the fluid motion is irrotational. Thus the velocity is equal to the negative

gradient of a scalar potential function, ( , , , )x y z t , which is a function of time t as well as the

position formulated by Wehausen and Laitone (1960) as given in Equations (29) to (32). In

these equations, r is distance between source and field points in three dimensional space, f

is the wave number for zero forward speed assumption, 0J is the first kind Bessel’s function,

and R is the horizontal distance between source and field point.

2 2( ) ( )R x a y b (29)

2 2( )r R z c (30)




of 94

2 2

00

ef kg g

(31)

(z ) ( )

0 0

0

1( , , , ) ( ) cos 2 ( )sink c f z c

e e

k fx y z t PV e J kR dk t fe J fR t

r k f

(32)

It is assumed that the fluid motion is harmonic with a single frequency e . Thus the time-

dependent potentials ( , , , )x y z t can be written as in equation (33) where the potential

(x, y,z) is independent of time and corresponds to the steady potential. It should be noted

that the potential that must be calculated for radiation, diffraction and incident waves. The

steady potential satisfies the boundary conditions described by equations from (34) to (38)

for z < 0:

( , , , ) Re{ (x, y,z)e }ei tx y z t

(33)

2 0 (34)

2

0

( , ,0)( , ,0) 0e

x yi U x y g

x z

(35)

lim 0z

(36)

2

lim 0R

dR i

dR g

(37)

( )n f Sn

on S (38)

The physical significance of the above equations is as follows. Equation (34) is the partial

differential equation for and expresses the conditions of incompressibility and

irrotationality. Equations (35), (36) and (37) are the auxiliary conditions on . Equation (35)

is the linearized free-surface condition, which requires that the pressure on the free-surface be

constant. Equation (36) expresses the vanishing of the disturbance at infinite depth, and

Equation (37) is the radiation condition that requires the disturbance to be an outgoing wave

at infinite horizontal distance. Equation (38) is the boundary condition on S. It expresses the




of 94

fact that the normal fluid velocity on S must be specified as a function f(S) of position on the

surface. Often the fluid normal velocity is specified as equal to the normal velocity of the

surface S. However, in the case of a known incident wave, denotes the disturbance

potential due to the body, and the boundary condition expresses the fact that the normal

velocity of the disturbance must cancel that of the incident wave on the body surface.

The method of solving the mathematical problem defined by equations (33) to (38), is based

on elementary solution or point source solution. The point source potential is defined as the

one that simultaneously satisfies the auxiliary conditions and the homogeneous partial

differential equation throughout space except at one point, where it is singular. The location

of the field point where the point source potential is singular is the location of the point

source. Expressions for the point source potential are given by Noblesse (1982) and Newman

(1985). A source distribution method (panel method or boundary element method) is used to

solve the radiation and diffraction potentials. The radiated or diffracted velocity potential at a

location in the fluid domain is expressed as follows:

1

, , ( , , ; , ,c) (a,b,c)dS4

S

x y z G x y z a b

(39)

where , ,ca b is the source location on the surface of the ship, and (a,b,c) is the strength of

the source at , ,ca b . ( , , ; , ,c)G x y z a b is the Green function, as given in Equation (40),

describing the flow at , ,x y z caused by a source of unit strength at , ,ca b ,

(z ) ( )

0 0

0

1, , ; , , ( ) 2 ( )k c f z ck f

G x y z a b c PV e J kR dk i fe J fRr k f

(40)

The Green function satisfies the continuity condition and all boundary conditions with the

exception of the normal velocity boundary condition on the hull surface given in equation

(41), where ( , , )nv x y z is the flow normal velocity on the hull surface. The source strengths

are solved such that the equation (42) is satisfied and the hull boundary condition for radiation

potential is given by equation (43) while the hull boundary condition for the wave diffraction

problem is stated by equation (44).




of 94

, ,

( , , )n

x y zv x y z

n

(41)

1 1 ( , , ; , ,c)

(a,b,c) (a,b,c)dS v ( , , )2 4

n

S

G x y z a bx y z

n

(42)

, ,k

e k

x y zi n

n

(43)

, , , ,D Ix y z x y z

n n

(44)

Solution of the three-dimensional radiation and diffraction potentials for zero forward speed is

discussed in detail by Hogben and Standing (1974), Faltinsen and Michelsen (1974), and

Garrison (1978). The ship hull should be discretized into flat panels, and the source strength is

taken as being constant over each panel. The normal velocity boundary condition should be

satisfied at the centroid of each panel. Panel source strengths for each radiation mode and for

wave diffraction needs to be solved by evaluation of the Green function and its derivatives

over each panel. Once the source strengths are solved, the potentials on the hull surface can be

calculated. When solving the radiation and diffraction potentials, evaluation of the Green

function is the most time consuming computational task. Telste and Noblesse (1986) and

Newman (1985) have developed efficient methods that are commonly used for evaluating the

Green function. Using an opposite sign convention with respect to the one provided by Telste

and Noblesse (1986), the composition of their Green function with Newman (1985)

formulation of the Green function can be written as Equation (45) to (47).

0 0

1

1 1, , ; , , 2 ( , ) 2 (h)vG x y z a b c fR h v i fe J

r r (45)

h fR , ( )v f z c , 2 2

1 ( )r R z c (46)

0 0 02 2

0

( , ) ( ) ( )2

v t vv e dt

R h v e E h Y hh t

(47)

where 1r is the distance between the image of source point with respect to the free surface

and field point, h is non-dimensional horizontal distance between source and field points, v




of 94

is the non-dimensional vertical distance between source and field points, 0 ( , )R h v is the

frequency dependent term that can be expressed by using Weber function, 0 ( )E h and second

kind Bessel’s function, 0 ( )Y h . The derivatives of the Green function can then be evaluated as

represented in Equations (48) to (52).

( , , ; , , ) x y z

G G G GG x y z a b c n n n

n x y z

(48)

2 2

1 13 3

1

( ) ( ) ( ) ( )2 ( , ) 2 (h)vG x a x a x a x a

f R h v i f e Jx r r R R

(49)

2 2

1 13 3

1

(y ) (y ) (y ) (y b)2 ( , ) 2 (h)vG b b b

f R h v i f e Jy r r R R

(50)

2 2

0 03 3

1 1

(z ) (z c) 12 2 ( , ) 2 (h)vG c

f f R h v i f e Jz r r r

(51)

1 1 1 32 20 2

( , ) ( ) ( )2

v t vv e dt

R h v e E h Y h h

h t

(52)

where 1(h)J is the second order of first kind Bessel’s function, 1( , )R h v is the derivative of

frequency dependent term with respect to non-dimensional horizontal distance that can be

expressed by using second order of Weber function, 1( )E h and second order of second kind

Bessel’s function, 1( )Y h .

Garrison (1978) gives a detailed discussion on evaluation of influence matrix terms to

determine velocity potentials using panel sources. If a field point is in close proximity to a

source panel, then the variation of the Green function term 1/r over the source panel must be

considered. The variation of the term 1/r1 over the image source panel can also be significant

if the field point is close to the image source; however, this situation is less common.

Fortunately, the frequency dependent term of Green function can usually be considered

constant with location over the source panel. Garrison gives a comprehensive overview on the

application of panel methods to offshore structures in waves and provides useful guidelines




of 94

for panelling of a body for computation of wave induced forces. For instance, there should be

sufficient number of panels to adequately model the geometry of the body. Adjacent panels

should have similar size, so that the source potential from a larger panel does not improperly

influence the adjacent smaller panels. Panel geometries should not be too elongated, with the

upper limit on the aspect ratio being approximately 5. In practice, the geometry of a ship hull

can often be adequately described with 200 panels on each side of the hull. The mesh size

should be based on the properties for the highest encounter frequency. Normally, the length of

the panels should be chosen such that one obtains about 5 panels per wave length for the

highest encounter frequency.

Hess and Smith (1964) is one of the earliest and most influential works on the application of

panel methods, since they describe the evaluation of panel normals and centroids. For a

triangular panel, the normal components are evaluated by taking the cross product of two of

its edges and then applying a normalization factor. The centroid coordinates are merely the

average of the coordinates of vertices. For a quadrilateral panel, the normal components are

evaluated by taking the cross product of the two diagonals, thus yielding the rotational

orientation of the panel plane. The location of the panel plane is fixed by specifying that it

passes through the mean of the four vertex coordinates. Vertex coordinates are then adjusted

to lie on the computed panel plane.

2.3 ADPAN Software

Based on the zero forward speed assumption and potential flow theory, a panel method

software, called ADPAN, has been generated in order to solve the hydrodynamic forces

acting on a ship in waves. The discretised body is one of the most important inputs for the

software. This discretization should be done by using either flat quadrilateral panels or flat

triangular panels. Once the nodes and corresponding panels are generated and written to an

input file, ADPAN reads these nodes and panels in order to calculate the panel properties.

Then, the radiation and diffraction source strengths of each panel are determined by using the

panel properties.




of 94

ADPAN has been implemented by using JAVA language based on object-oriented

programming methodology. Five different Java packages are built and twenty six different

classes have been implemented in these Java packages. An external package called Jama has

been used to solve the linear matrix equation systems. This package involves LU

decomposition and it solves the linear equation system by using the LU decomposition

method. The remaining packages are called adpan, model, util, influence and

oscillatorymotion respectively. The “adpan” package controls the orientation of the other

packages during the compilation and run processes. The “model” package is a general storage

for the input file variables. The “util” package is an auxiliary package that mainly contains

file reading and file writing classes.

The most important packages are essentially “influence” and “oscillatorymotion”,since they

are implemented to solve Green function and equation of oscillatory motion in six degree of

freedom, respectively. In addition to the solution of equation of motion, “oscillatorymotion”

package is used to calculate the hydrodynamic panel pressures for each panel. The equation of

motion is solved with respect to the ship centre of gravity. However, the coordinate system in

ADPAN has its origin in the still water plane aligned vertically with the ship centre of gravity

because this selection makes the solution of the three-dimensional velocity potentials for a

ship hull easier. It is assumed that the ship hull has lateral symmetry thus radiation potentials

are evaluated only on the port side of the hull, while diffraction potentials are evaluated on

both sides of the hull.

2.4 Applications and Validation of ADPAN

To verify implementation of the three-dimensional panel method, sample computations have

been performed for several different problems by using ADPAN. First of all, a radiation

problem of a floating hemisphere is considered because it is very important to calculate

correct added mass and damping coefficients. Secondly, a diffraction problem has been

solved over a submerged spheroid in order to find out the corresponding real and imaginary

part of the excitation force in surge direction. Both the floating hemisphere and submerged




of 94

spheroid problems can be solved analytically and the solutions are given by Hulme (1982)

and Wu and Eatock Taylor (1987), respectively. In addition to the analytical results, the

results generated by using AQWA, a commercial hydrodynamic software based on potential

flow theory, are used for validation purposes. For both problems, the numerical results

obtained by using ADPAN are in very good agreement with those found by using AQWA and

the analytical results. Once the results are validated for typical radiation and diffraction

problems, a simple long barge problem is solved in order to check the accuracy of the

implemented equation of rigid body motion. An analytical solution is not so simple for this

case, therefore an external tool, PRECAL developed by Lloyd’s Register, is used for

comparison purposes. Moreover, the hydrodynamic problems of more complicated

geometries such as WIGLEY III and S175 container ship are considered in both ADPAN and

PRECAL in order to demonstrate the capability of ADPAN. The problem descriptions,

results, and discussion of all these problems are given below.

2.4.1 Radiation Problem of a Floating Hemisphere

The radiation problem of a hemisphere with radius r = 1 m floating at zero forward speed in

head sea condition β = 180 degrees is considered to illustrate the capability of ADPAN in

obtaining the convergent values of the added mass and damping coefficient. The wave height

is H = 1 m and therefore the wave amplitude is a = 0.5 m. The vertical centre of the gravity of

hemisphere is selected at free surface zc = 0 and the longitudinal centre of gravity is located at

geometric centre of the hemisphere xc = 0.




of 94

Figure 1 Hemisphere Mesh

Full model of hemisphere is discretised by flat quadrilateral panels as shown in Figure 1. The

total number of panels is 768 for full model, however ADPAN uses just half of it for solution

process. The radiation problem of floating hemisphere is solved for 28 different wave

frequencies by using ADPAN and AQWA. The convergence of the computed added mass and

damping results in surge and heave direction obtained from ADPAN, AQWA and the

analytical results given by Hulme (1982) are shown in Figures 2 and 3, respectively. The

variation of the ADPAN results for surge added mass A11 are in good agreement with other

solutions while ADPAN and other solutions of heave added mass A33 are almost identical for

each frequency as presented in Figure 2.




of 94

(a) (b)

Figure 2 Hemisphere added mass for (a) surge and (b) heave motion

The numerical results of surge damping B11 and heave damping B33 are plotted together with

the other solutions as shown in Figure 3. It can be seen from these figures that ADPAN

provides very good results when low frequency range is considered, however there are some

slight differences in high frequency range. However, this doesn’t mean that results obtained

from ADPAN are incorrect, since the results of AQWA and ADPAN are similar. This

difference is related to the irregular frequency phenomenon due to the resonant frequencies at

which a fictitious fluid motion inside the body breaks down.

0.2

0.4

0.6

0.8

1

1.2

1.4

0 2 4 6 8 10

ADPAN - A11

ANALYTICAL - A11

AQWA - A11

AQWA_LID - A11

0.7

0.9

1.1

1.3

1.5

1.7

1.9

0 2 4 6 8 10

ADPAN - A33

ANALYTICAL - A33

AQWA - A33

AQWA_LID - A33




of 94

(a) (b)

Figure 3 Hemisphere damping for (a) surge and (b) heave motion

The definite improvement in reducing the irregular frequency behaviour is noticeable, if lid

panels are added to the discretised domain. As it can be seen from Figure 3 that when the

problem is solved by adding lid panels as in AQWA, the results are very promising in

comparison to the analytical results. This observation is similar to the one obtained from a

different type of the B-spline panel method proposed by Maniar (1995).

2.4.2 Diffraction Problem of a Submerged Spheroid

ADPAN is also verified for the diffraction problem of a submerged spheroid in head seas.

The solution of this problem illustrates the accuracy of ADPAN for obtaining the reliable

values of the excitation force. The submerged spheroid used by Wu and Eatock Taylor (1987)

is chosen for comparison purposes. The radius and length of the submerged spheroid is 1 m

and 6 m, respectively. The submergence depth of the spheroid, which is the vertical distance

between the global coordinate system and mass centre of the spheroid, is 2 m. Figure 4 shows

0

0.5

1

1.5

2

2.5

3

3.5

0 2 4 6 8 10

ADPAN - B11

ANALYTICAL - B11

AQWA - B11

AQWA_LID - B11

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 2 4 6 8 10

ADPAN - B33

ANALYTICAL - B33

AQWA - B33

AQWA _LID - B33




of 94

the general geometrical aspect ratio and orientation of the spheroid according to global

coordinate system.

Figure 4 Spheroid geometry and orientation in space

The spheroid is assumed to move at zero forward speed with the unit wave height, 1 m.

Hence, the wave amplitude is 0.5 m. The vertical centre of the gravity is at zc = -2 m and the

longitudinal centre of gravity is at xc = 0 according to global coordinate system. Although the

full model is discretised by 1176 different flat quadrilateral panels as shown in Figure 5,

ADPAN uses xz-plane symmetry condition to solve the diffraction problem.

Figure 5 Spheroid mesh

The variation of the diffraction potential over the spheroid body surface is calculated for 18

different wave frequencies. Figure 6 represents the computed real and imaginary parts of the

wave exciting forces in surge direction, respectively. The numerical results are in excellent

agreement with the analytical solutions obtained by Wu and Eatock Taylor (1987).




of 94

(a) (b)

Figure 6 (a) Real and (b) imaginary component of excitation force

2.4.3 Rigid Body Motions of a Long Barge

The accuracy of the implemented equations related to the rigid body motion are confirmed by

solving a long barge problem in head seas condition with β = 180 degrees. Length, breadth,

and draft of the barge is L = 100 m, B = 20 m, and T = 5 m, respectively.

Figure 7 Barge mesh representation

-3

-2.5

-2

-1.5

-1

-0.5

0

0.4 0.9 1.4 1.9 2.4

Analytic Real Excitation

ADPAN Real Excitation

-20

-15

-10

-5

0

5

0.4 0.9 1.4 1.9 2.4

Analytic Imaginary Excitation

ADPAN Imaginary Excitation




of 94

Although the geometry presented in Figure 7 seems quite simple, generating an analytical

solution for this problem is very tedious and complex. Both ADPAN and PRECAL are used

to solve the barge problem for comparison purposes. The barge is assumed to move at zero

forward speed U = 0 m/s condition with the unit wave height H = 1 m. Hence, the wave

amplitude is a = 0.5 m. The vertical centre of the gravity is chosen as zc = -1 m, while the

longitudinal centre of gravity is set at xc = 0 according to the global coordinate system. The

full model is discretised by 1080 flat quadrilateral panels as shown in Figure 7; however

ADPAN uses 504 of them when xz-plane symmetry condition is taken account during the

solution process.

Figure 8 Barge motion amplitudes

First, the radiation - diffraction problem is solved over the body surface for 30 different wave

frequencies. Added mass and damping matrices are obtained from the solution of the radiation

problem, while excitation force vector is calculated by solving the diffraction problem. Then,

the corresponding hydrodynamic coefficient matrix together with the mass and inertia matrix

of the body are defined by using the surface panels. Once the required parameters are derived

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00

PRECAL - SURGE AMPLITUDE ADPAN - SURGE AMPLITUDE

PRECAL - HEAVE AMPLITUDE ADPAN - HEAVE AMPLITUDE

PRECAL - PITCH AMPLITUDE ADPAN - PITCH AMPLITUDE




of 94

for the solution of the rigid body motion equation, motion amplitudes and phase angles are

determined for the directions of six degree of freedom.

Figure 9 Barge motion phase angles

The numerical results generated by using ADPAN and PRECAL are plotted in Figures 8 and

9. Since the barge is assumed to move in head sea condition, the numerical motion amplitudes

generated by both software in sway, roll and yaw directions can be negligibly small. In fact,

these result are theoretically zero. Therefore, there is no need to present these results.

According to the Figures 8 and 9, the variation of the motion amplitudes and phase angles in

surge, heave and pitch directions obtained from ADPAN are in excellent agreement with

those generated by using PRECAL.

2.4.4 Hydrodynamic Analysis of WIGLEY III

The body surfaces that are used to present the results so far are not as complex as a ship hull

surface. In addition to the simplicity of the body surface, these results are used to validate

subclasses of the whole software in a compact mode. Therefore, the accuracy of ADPAN in

-200.00

-150.00

-100.00

-50.00

0.00

50.00

100.00

150.00

200.00

0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00

PRECAL - SURGE PHASE ADPAN - SURGE PHASE

PRECAL - HEAVE PHASE ADPAN - HEAVE PHASE

PRECAL - PITCH PHASE ADPAN - PITCH PHASE




of 94

compact mode needs to be tested by solving hydrodynamics of a more complex surface which

looks more like a ship type. Such a validation of ADPAN can be done by solving the

hydrodynamics of WIGLEY III. The frame/cross section lines of the hull are shown in Figure

10.

Figure 10 WIGLEY III sectional view

WIGLEY III hull floating at zero forward speed U = 0 m/s in head sea condition, β = 180

degrees, properties are given in Table 1. The wave height is H = 2 m and therefore the wave

amplitude is a = 1 m. The vertical centre of the gravity of hull is located at zc = -1.250 m

while the longitudinal centre of gravity is set at xc = 0.

Figure 11 WIGLEY III mesh




of 94

In addition to ADPAN, PRECAL is used to model the problem for comparison purposes.

Figure 11 shows the hull surface represented by flat quadrilateral panels and the orientation of

the hull with respect global coordinate system. Total number of panels is 2000 for full model

but ADPAN uses just half of the model to solve the hydrodynamic responses.

Table 1 WIGLEY III hull properties

Property Value Unit

LOA 100 m

LBP 100 m

B 20 m

T 6.25 m

VOL 2843.8 m3

AWP 691.32 m2

LCF 0 m

GML 125.42 m

GMT 0.333 m

KX 4.087 m

KY 25 m

KZ 25.332 m

KXZ 0 m

The radiation and diffraction problem of floating WIGLEY III hull is solved for 51 different

wave frequencies. Six degree of freedom motion amplitudes and phase angles are calculated

by using the parameters generated from radiation and diffraction problem. The numerical

results generated by using ADPAN are compared with those calculated by using PRECAL.

The variation of the heave added mass A33, pitch added mass A55, heave damping B33, and

pitch damping B55, with respect to wave frequencies are plotted in Figures 12 and 13,

respectively. The numerical results of ADPAN are in good agreement with those obtained

from PRECAL. Since the hull is assumed to move in head sea condition, the numerical

motion amplitudes generated by both software in sway, roll and yaw directions are

theoretically zero. Therefore, there is no need to present these results.




of 94

Figure 12 WIGLEY III pitch added mass

The surge, heave and pitch motion amplitudes and motion phase angles for corresponding

wave frequencies are shown in Figure 14 and 15, respectively. The numerical results of

ADPAN match very well with those obtained from PRECAL.

Figure 13 WIGLEY III pitch damping

1000

1500

2000

2500

3000

3500

4000

4500

5000

5500

6000

0.00 0.50 1.00 1.50

ADPAN - A33 PRECAL - A33

400000

600000

800000

1000000

1200000

1400000

1600000

0.00 0.50 1.00 1.50


0

500

1000

1500

2000

0 0.5 1 1.5

ADPAN - B33 PRECAL - B33

0

100000

200000

300000

400000

500000

600000

700000

800000

900000

0 0.5 1 1.5





of 94

Figure 14 WIGLEY III motion amplitudes

Figure 15 WIGLEY III motion phase angles

0.000

0.500

1.000

1.500

2.000

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40




-200.0

-150.0

-100.0

-50.0

0.0

50.0

100.0

150.0

200.0

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40







of 94

Figure 16 WIGLEY III oscillatory pressure distribution (Unit: Pa)

Once the ship motions are determined, the variation of panel pressures for each frequency is

calculated. In order to demonstrate the dynamic pressure variation, each panel pressure for the

frequency of 0.32 rad/sec are exported from ADPAN. The resultant dynamic pressure

variation is plotted in Figure 16. Since the geometry of WIGLEY hull has two symmetry

planes and oscillatory waves are in head sea direction, the expected variation of dynamic

pressure is symmetric with respect to longitudinal centre of gravity which aligns with the

solution presented in Figure 16.

2.4.5 Hydrodynamics of S175 Type Container Ship

A final validation case is conducted by solving the hydrodynamic response of S175 container

ship. ADPAN results are compared with those obtained from PRECAL. S175 container ship

hull properties are given in Table 2. The profile view, waterlines, body plan, and isometric

view of hull surface represented by flat triangular and quadrilateral panels can be seen at

Figures 17 to 20, respectively. As it can be seen from the presented figures, the geometry of




of 94

S175 container ship is more complex in comparison to the geometries used in previous

problems.

Table 2 S175 container ship properties

Property Value Unit

LOA 178 m

LBP 175 m

B 50.8 m

T 9.5 m

VOL 24067 m3

AWP 3155.9 m2

LCF -7.048 m

GML 205.29 m

GMT 1.022 m

KX 9.167 m

KY 43.75 m

KZ 43.752 m

KXZ 0 m

A finer mesh and some triangular panels are needed to be used to overcome such a

complexity of the body surface during pre-processing stage. As a consequence, total number

of panels used for the full model is 2946 from which 2930 of them are quadrilateral panels

and the remaining 16 are triangular panels.

Figure 17 S175 container ship profile view




of 94

ADPAN takes the advantage of the lateral symmetry condition and generates the results by

using 1473 panels. It is assumed that ship floats at zero forward speed in head sea condition.

The wave height is H = 2 m and therefore the wave amplitude is a = 1 m. The vertical centre

of the gravity is located at zc = 0 and the longitudinal centre of gravity is set at xc = -2.556.

Figure 18 S175 container ship waterlines

The radiation and diffraction problem of floating S175 container ship is solved for 37

different wave frequencies. Six degree of freedom motion amplitudes and phase angles are

calculated by using the parameters generated from radiation and diffraction problem.

Figure 19 S175 container ship body plan

The numerical results generated by using ADPAN are compared with those calculated by

using PRECAL. The variation of the heave added mass A33, pitch added mass A55, heave

damping B33, and pitch damping B55, with respect to wave frequencies are plotted in Figure




of 94

21 and 22, respectively. The numerical results from ADPAN are in well agreement with those

obtained from PRECAL.

Figure 20 S175 container ship isometric mesh view

Since the hull is assumed to move in head sea condition, the numerical motion amplitudes

generated by both software in sway, roll and yaw direction are zero. Therefore there is no

need to present these results.

Figure 21 S175 container ship added mass

The surge, heave and pitch motion amplitudes and motion phase angles for corresponding

wave frequencies are shown in Figures 22 and 23, respectively. According to the plotted

15000

20000

25000

30000

35000

40000

45000

50000

55000

0.3 0.8 1.3


15000000

20000000

25000000

30000000

35000000

40000000

45000000

50000000

55000000

60000000

0.3 0.8 1.3





of 94

motion results, the numerical results of ADPAN are in very good agreement when they are

compared with those obtained from PRECAL.

Figure 22 S175 container ship damping

Figure 23 S175 container ship motion amplitudes

3000

5000

7000

9000

11000

13000

15000

17000

0.3 0.8 1.3


0

5000000

10000000

15000000

20000000

0.3 0.8 1.3


0

0.2

0.4

0.6

0.8

1

1.2

0.3 0.5 0.7 0.9 1.1 1.3 1.5







of 94

Figure 24 S175 container ship motion phase angles

Figure 25 S175 container ship pressure distribution

-200

-150

-100

-50

0

50

100

150

200

0.3 0.5 0.7 0.9 1.1 1.3 1.5







of 94

In order to show the dynamic pressure variation, each panel pressure for the frequency of 0.4

rad/sec are exported from ADPAN. The resultant dynamic pressure variation is plotted in

Figure 25. Since the geometry of S175 has only one symmetry plane and oscillatory waves

are in head sea direction, the expected variation of dynamic pressure should be symmetric

with respect to xz-plane but should be non-symmetric with respect to yz-plane which aligns

with the solution presented in Figure 25.




of 94

3 FINITE ELEMENT CODE

Designing ships too strong makes them heavy, slow and very costly to build and operate since

their cargo space is decreased. On the other hand, structural failures, hull damage, weather

conditions can easily cause a big injury or in extreme cases a catastrophic failure and sinking

of ships which are designed weak. Therefore, the structural strength of ships is a key topic

that affects the safety of crew, economic costs, and the pollution of the environment in which

ships are trading. The required structural management and safety of ship can be achieved by

performing appropriate inspections at the right intervals and repairing defects that are

identified. Structural management tools of INCASS project can achieve better prediction of

the right interval of ship inspections, if an in-house finite element code exists.

3.1 Finite Element Method

The finite element method (FEM) is a computational technique for solving problems that are

described by partial differential equations or can be formulated as functional minimization. A

domain of interest is represented as an assembly of finite elements. Interpolation functions in

finite elements are determined in terms of nodal values of a physical field that is sought. A

continuous physical problem is transformed into a discretized finite element problem with

unknown nodal values. For a linear problem, a system of linear algebraic equations should be

assembled and solved. Values within finite elements can be recovered by interpolating nodal

values. Piece-wise approximation of physical fields (a finer mesh of domain) on finite

elements provides good precision even with simple interpolation (shape) functions. An

arbitrary precision of results can be achieved by increasing the number of elements and nodes.

Theory, practice, and programming of the finite element method are described in many

textbooks, such has comprehensive books of Bathe (2006), Fish and Belytschko (2007),

Logan (2011), and Oñate (2013).




of 94

Finite element procedure can be described by several steps. The first step is the discretization

of the domain by dividing a solution region (domain) into finite elements that are connected at

nodes. The finite element mesh is typically generated by a pre-processor program because of

the large amount of data. The description of a mesh consists of several arrays, the main of

which are nodal coordinates and element connectivity. Secondly, the logical determination of

interpolation functions is required for the finite element procedure. Interpolation functions are

used to interpolate the field variables over the element. Usually, polynomials are selected as

interpolation functions. The degree of the polynomial depends on the number of nodes

belonging to the element. Interpolation functions are commonly called shape functions since

they are also used for the definition of the element shape.

The third step is the computation of the element matrices and vectors. The matrix equation for

the finite element is established that relates the nodal values of the unknown function to

known parameters. For this task different approaches can be used. The most convenients are:

the variational approach and the Galerkin method. The fourth step is the assembly process of

the element equations. To find the global equation system for the whole solution region, one

must assemble all the element equations. In other words, local element equations must be

combined properly for all elements used for discretization. Element connectivity matrix is

used for the assembly process. Before starting the solution process, boundary conditions

(which are not accounted for in the element equations) should be imposed.

Solving the global equation system is the fifth step. The finite element global equation system

is typically sparse, symmetric and positive-definite. Direct and iterative methods can be used

for solution. The nodal values of the sought function are produced as a result of the solution.

The sixth and the last step is the computation of the additional results such as strains, stresses

etc. In many cases, we need to calculate additional parameters. For example, in mechanical

problems, strains and stresses are of interest in addition to displacements, which are obtained

after solution of the global equation system. It is important to mention that the displacements,

which represent the primary result function, are continuous, however its derivatives (strains

and stresses) have discontinuities at element boundaries. Therefore, the most reliable results

of the finite element method is the displacements. The stress and strain distribution might be




of 94

slightly different depending on the assumption of the stress interpolation used to calculate

these values for the same problem, even though the displacement results are completely same.

3.2 Element Types

A finite element is the basic building block of finite element analysis. There are several basic

types of elements. Type of the finite element that is used to conduct a finite elements analysis

depends on the type of object that is to be modelled and the type of analysis that is going to be

performed. An element is a mathematical relation that defines how the degrees of freedom of

a node relate to the next. These elements can be in the form of lines (trusses or beams), areas

(2-D or 3-D plates and membranes) or volumes (brick or tetrahedral). It also relates how the

deflections create stresses. The following content will describe element types in more detail

which could possibly be used to perform a finite element analysis of an arbitrary problem.

3.2.1 Truss Element

Truss elements are two node members which allow arbitrary orientation in a rectangular

Cartesian coordinate system. In fact, the truss transmits only axial force and truss element has

three degree of freedom (three global translation components) at each node. Trusses are used

to model structures such as towers, bridges and buildings. The three-dimensional truss

element is assumed to have a constant cross-sectional area and can be used in linear elastic

analysis. Linear elastic material behaviour is defined only by the modulus of elasticity. Truss

elements cannot be subjected to a boundary condition that includes a rotational degree of

freedom, since they don’t have any rotational degrees of freedom. In order to model a

problem, truss elements can be used when the length of the element is much greater than the

width or depth (approximately 8-10 times). A truss is connected to the rest of the model with

hinges that do not transfer moments. The external applied forces can be only at joints. When

using truss elements, the axial cross-sectional area of the truss elements must be specified.




of 94

3.2.2 Beam Element

A beam element is a slender structural member that offers resistance to forces and bending

under applied loads. A beam element differs from a truss element because a beam resists

moments (twisting and bending) at the connections. The classical beam theories based on

Bernoulli-Euler and Timoshenko beam kinematics can be used to formulate beam elements.

These two node elements are formulated in three-dimensional space and the nodes are

specified by the element geometry, for example nodes located at end of the element. A

maximum of three translational degrees of freedom and three rotational degrees-of-freedom

can be defined for beam elements. Three orthogonal forces (one axial and two shear) and

three orthogonal moments (one torsion and two bending) are calculated at each end of the

element. Optionally, the maximum normal stresses produced by combined axial and bending

loads are calculated. Uniform inertia loads in three directions, fixed-end forces, and

intermediate loads are the basic element based loadings. Beam element is used to model

structures when the length of the element is much greater than the width or depth. The

element has constant cross-sectional properties along its axial direction and the element must

be able to transfer moments. The element must also be able to handle a load distributed across

its length. Beam elements can be used for finite-element analysis of elastic spatial frame

structures.

3.2.3 Plane Element

Plane elements can be used for two dimensional modelling of solid structures. Plane elements

can have either triangular, rectangular or quadrilateral shapes. In fact, the development of a

quadrilateral element is very much the same as the rectangular element, except for an

additional procedure for coordinate mapping. The elements are connected at common nodes

and/or along common edges to form continuous structures. A quadrilateral element can be

defined by four nodes or eight nodes whereas a triangular shape plane element can be defined

by using either three nodes or six nodes. All of these nodes have two degrees of freedom at

each node: translations in the nodal x and y directions in plane element local coordinate

system. The formulation of these elements can differ depending on the problem state. In solid




of 94

mechanics, plane elasticity can be referred as the set of mathematical models which describe

the behaviour of a body using only displacements on a plane. Two types of plane elasticity

needs to described, namely plane stress and plane strain. Plane stress is defined to be a state of

stress in which the normal stress and the shear stresses perpendicular to the plane are assumed

to be zero. Plane strain is defined to be a state of strain in which the strain normal to the x-y

plane and the shear strains except the shear strain in x-y plane are assumed to be zero.

Nodal displacement compatibility is enforced during the formulation of the nodal equilibrium

equations for two-dimensional elements. If proper displacement functions are chosen,

compatibility along common edges is also obtained. The two-dimensional element is

extremely important for (1) plane stress analysis, which includes problems such as plates with

holes, fillets, or other changes in geometry that are loaded in their plane resulting in local

stress concentrations and (2) plane strain analysis, which includes problems such as a long

underground box culvert subjected to a uniform load acting constantly over its length, pipes

subjected to loads that remain constant over their lengths.

3.2.4 Shell Element

Before discovering a shell type of element, a plate type of element need to be described since

a plate element is a simplified case of a shell element. Physically, a plate can be considered as

a two-dimensional extension of a beam in simple bending because both beams and plates

support loads transverse or perpendicular to their plane and through bending action. However,

a beam has a single bending moment resistance, while a plate resists bending about two axes

and has a twisting moment. Plate structures are geometrically similar to the structure of the

solid plane element for plane stress problem because both plane and plate structures are flat

(if a plate were curved, it would become a shell). As for the solid plane element, a plate

element can also be triangular, rectangular or quadrilateral in shape. Therefore, a plate

element can have different number of nodes depending on the shape as discussed in solid

plane elements description. In general, each node has three degrees of freedom, a translation

out of plane direction with two rotations in plane direction. The out-of-plane rotational degree

of freedom is not considered for plate elements.




of 94

These elements are used to model and analyse objects such as pressure vessels, or structures

such as automobile body parts. To illustrate a plate example, consider the horizontal boards

on a bookshelf that support the books. Those boards can be approximated as a plate structure,

and the transverse loads are the weight of the books. The plate structure can be schematically

represented by its middle plane laying on the x–y plane. The deformation caused by the

transverse loading on a plate is represented by the deflection and rotation of the normals of

the middle plane of the plate, and they will be independent of z and a function of only x and y.

It is assumed that the element has a uniform thickness. If the plate structure has a varying

thickness, the structure can be divided into small elements, each of uniform thickness, to

approximate the overall variation in thickness.

There are a number of theories that govern the deformation of plates. In particular, Kirchhoff

plate theory that works for thin plates and Reissner–Mindlin plate theory that works for

thin/thick plates are well known. While many of the assumptions of Kirchhoff plate theory are

analogous to the classical beam theory or Euler–Bernoulli beam theory, Reissner-Mindlin’s

assumptions are analogous to the Timoshenko beam theory. Many structures may not be

considered as a “thin plate,” or rather their transverse shear strains cannot be ignored.

Chapelle and Bathe (2011) propose that Reissner–Mindlin plate theory is more suitable in

general, and the elements developed based on the Reissner–Mindlin plate theory are more

practical and useful.

Shell elements can be in the form of four or eight node quadrilaterals and three or six node

triangular elements in any three dimensional orientation. The four node elements require a

much finer mesh than the eight node elements to give convergent displacements and stresses

in models involving out-of-plane bending. A shell can be seen, in essence, as the extension of

a plate to a non-planar surface. The non-planarity introduces axial (membrane) forces in

addition to flexural (bending and shear) forces, thus providing a higher overall structural

strength. Shell-type structures are common in many engineering constructions such as roofs,

domes, bridges, containment walls, water and oil tanks and silos, as well as in airplane and

spacecraft fuselages, ship hulls, automobile bodies, mechanical parts, etc. The way in which a

shell supports external loads by the combined action of axial and flexural effects is similar to

that of arches and frame structures. Thus, while a beam and a plate typically resist the external




of 94

forces by flexural effects only, frames, arches and shells offer a higher resistance to load due

to the coupled action of axial and flexural forces.

Shells are typically classified by the shape of their middle surface. The governing equations

of a curved shell (equilibrium and kinematic equations, etc.) are quite complex due to the

curvature of the middle surface. A way of overcoming this problem is considering the shell as

formed by a number of flat shells. Oñate (2013) states that a shell formulation doesn’t need to

be considered as curved shell, if a good number of flat shell elements are used to model the

structure. The flexural and in-plane states are typically decoupled at the element level when

the shell element is flat. This decoupling extends to the element stiffness matrix which is

formed by a simple superposition of the flexural and membrane contributions.

The full flexural-membrane coupling appears when flat elements meeting at different angles

are assembled in the global stiffness matrix. The superposition leads each nodes of a shell

element has five degrees of freedom: two translational in-plane deformations from in plane

(membrane) element and one translational out of plane deformation with two rotational

deformations from plate element. However, having five degrees of freedom might not be

practical enough to capture a logical deformed shape when a complex structure is tried to be

modelled in rectangular Cartesian coordinate system.

As a result, a rotational degree of freedom in out of plane direction, drilling rotation, needs to

be added to the general flat shell element formulation. Adding this drilling rotation not only

increases the robustness of the element, but also provides such a logical transformation

scheme when the superposition of the elements is done in global coordinates. As a

consequence, a general flat shell element has six degrees of freedom: three translations and

three rotations in x, y, z directions.

3.2.5 Solid Element

Solid elements are three-dimensional finite elements and can be used to model the structure

without any a priori geometric simplification. They are suitable for the stress analysis of




of 94

general three-dimensional bodies that require more precise analysis than is possible through

two-dimensional analysis. Examples of three-dimensional problems are arch dams, thick-

walled pressure vessels, and solid forging parts as used, for instance, in the heavy equipment

and automotive industries.

Finite element models of this type of element have the advantage of directness. Geometric,

constitutive and loading assumptions required to effect dimensionality reduction, for example

to planar or axisymmetric behaviour, are avoided. Boundary conditions on both forces and

displacements can be more realistically treated. However, utilization of solid elements is

costly. In terms of modelling, mesh preparation, computing and post processing effort. The

rapid increase in computer time as the mesh is refined should be noted. In fact, use of solid

elements should be restricted to problem and analysis stages, such as verification, where the

generality and flexibility of full three dimensional models is warranted. They should be

avoided during design stages. Furthermore, they should also be avoided in thin-wall

structures, since solid elements tend to perform poorly because of locking problems.

The tetrahedron shape is applicable for basic three-dimensional solid element and the

hexahedron shape is another option, but it requires a more complex formulation in

comparison to tetrahedron. The number of nodes used to formulate a solid element can be

different. Six or ten node tetrahedron shape, eight or twenty node hexahedron shape

formulations are commonly used. Each node of these solid element types has three degrees of

freedom: translation in the nodal x, y, and z directions.

3.3 ADFEM Software

A finite element software, called ADFEM, is developed by using Java language based on

object-oriented methodology. During program development, three different tasks of the finite

element analysis (pre-processing, processing, and post processing) are often implemented as

three separate computer programs. Since these tasks have many common data structures and

methods, the three modules contain duplicated or similar code fragments which complicates

support and modification. In Java language, it is possible to have several main methods. The




of 94

code (classes) can be organized into packages. A package is a named collection of classes

providing encapsulation and modularity, which can eliminate code duplication and provide a

means for easy code reuse. ADFEM is organized into six class packages, namely adfem, util,

model, solver, element including and external package called Jama that is useful for matrix

operations.

From the data point of view the finite element solution is transformation of input data into

output data. Since data describing a finite element mesh is too large, input is usually

generated by a pre-processor. Currently, ADFEM does not have a mesh generation capability.

Once the mesh of a problem is generated by an external tool, the nodal coordinates and the

element connectivity should be set up as an input file for ADFEM. There is a robust scanner

class, called fescanner, implemented in order to read very large input data. ADFEM is

designed to solve elastic and static problems with displacement and force boundary

conditions. The data can be divided into the data related to the finite element model and the

data describing loading conditions. The finite element model data and loading conditions

don’t change during problem solution since we suppose that the model shape, material

properties, and boundary conditions are constant. The description of the finite element model

is listed below:

Program initiator

Element types

Scalar parameters such as section types, section properties etc.

Material properties

Nodal data such as coordinates of nodal points

Element data such as element materials and connectivity

Description of displacement boundary conditions

Surface and concentrated loads

Solution command




of 94

Table 3 Finite element library of ADFEM

Element

Name

Element

Type

Total Number of

Nodes

Nodal Degree of

Freedom

Element Degree of

Freedom

Truss Truss 2 Ux, Uy, Uz 6

Beam Beam 2 Ux, Uy, Uz, Rotx,

Roty, Rotz

12

Plane3 Plane 3 Ux, Uy 6

Plane4 Solid Plane 4 Ux, Uy 8



Shell3 Shell 3 Ux, Uy, Uz, Rotx,

Roty, Rotz

18


Roty, Rotz

24


Roty, Rotz

48

Solid8 Solid 8 Ux, Uy, Uz 24

Finite element types are considered as main objects in both mathematical and algorithmic

senses. In order to implement the main methods for a finite element, an abstract class Element

is designed. The class holds element data, methods common to all element types, and empty

methods specific to particular element types. Overriding of the parent methods allows one to

create new element types using standard procedures. Formulation of truss, beam, plane, plate,

shell and solid element types has been implemented into the ADFEM. The formulation of

these element types has already been discussed and purely generated by many authors such as

Logan (2011), Bathe (2006) and Oñate (2013). Therefore, instead of providing the details of

the finite element formulation for each element type, only the capability parameters are given

in Table 3 for each element implemented into ADFEM.

A math library called SuanShu is used to solve linear matrix equation system in ADFEM.

SuanShu is an object-oriented, high performance, extensively tested, and professionally




of 94

documented library of numerical methods. A CPU parallelized Cholesky decomposition

implemented in SuanShu is the current solver method used in ADFEM.

3.4 Applications and Validation of ADFEM

To verify implementation of the finite element code, sample problems have been solved by

using ADFEM. In fact, the selection of the problem type is completely dependent on the

selection of the element type that will be used to model the problem. Hence, at least one

problem needs to be described for each element type. First of all, a space truss structure is

considered by using Truss element. Secondly, a basic I cross section beam problem is

modelled in order to validate Beam element formulation in ADFEM. Thirdly, solution of a

complex frame structure is discussed in order to illustrate the capability of Beam element in

more detail. Next, two different plate problems under axial loading condition are considered

to confirm the accuracy of both Plane4 and Plane8. Since Plane3 and Plane6 are

simplification case of Plane4 and Plane8, respectively, there is no need to model a problem

with these element types. Once the plane element formulations used in ADFEM is validated,

transverse loading case of these problems is considered in order to demonstrate the capability

of the Shell3, Shell4 and Shell8 element types. The numerical results obtained by using

ADFEM are compared with the results that are found by using a commercial finite element

code, ANSYS. Based on the validation cases, the results obtained by using ADFEM are very

reasonable because they are in very good agreement with those found by using ANSYS. The

problem descriptions, results, and discussion of all these problems are given below.

3.4.1 Truss Problem

The space truss problem as shown in Figure 26 is considered for validation of truss element

formulation in ADFEM. Cross section area is 2 cm2 and elastic modulus is 210000 MPa for

all elements. The coordinates of each node, in centimetres, are shown in the figure. Nodes 1–4




of 94

are supported by ball-and-socket joints (fixed supports). The space truss is subjected to a 1000

N load in the x direction at node 5 and the aim is to determine the displacement of node 5.

Figure 26 Space truss isometric view

Table 4 Displacement results of space truss problem

ADFEM ADFEM ADFEM ANSYS ANSYS ANSYS

NODE UX UY UZ UX UY UZ

5 0.040908 0.000000 -0.011979 0.040908 0.000000 -0.011979

Once the unknown displacement is found, the strain and the stresses in each element are

calculated. For comparison purposes, the same problem is modelled with a space truss

element, Link180, in order to solve the described problem in ANSYS. The displacement,

strain and stress results that are obtained from ADFEM and ANSYS are tabulated in Table 4

and 5, respectively. Displacement results are listed in millimetres, while the corresponding

stress results has unit of MPa. As it can be seen from both tables, the results completely agree

with each other. Therefore, it is should be indicated that Truss element in ADFEM is

analogous to Link180 element in ANSYS.




of 94

Table 5 Strain and stress results of space truss problem

ADFEM ADFEM ANSYS ANSYS

ELEMENT EXX SXX EXX SXX

1 1.786E-05 3.750E+00 1.786E-05 3.750E+00

2 -1.331E-05 -2.795E+00 -1.331E-05 -2.795E+00

3 -1.331E-05 -2.795E+00 -1.331E-05 -2.795E+00

4 1.786E-05 3.750E+00 1.786E-05 3.750E+00

3.4.2 Beam Problem

A console beam problem with I type cross section as shown in Figure 27 is considered for

validation of Beam element implemented in ADFEM. According to Figure 27, I cross section

properties are b =200 mm, h = 200 mm, tf = 15 mm, and tw = 15 mm. The selected material’s

elastic modulus is 210000 MPa with the Poisson’s ratio of v = 0.3. Length of the beam is L =

2 m and it is divided into 20 finite elements. In order to apply displacement boundary

conditions, one end node of the beam is fixed for all degrees of freedom while the other nodes

are free to translate and rotate. Beam is subjected to -1000 N concentrated load in the z

direction at all nodes except the fixed node.

Figure 27 Beam mesh and cross section

The aim of this problem is to determine the unknown translational and rotational

displacement for each node. The same problem is also modelled in ANSYS by using

Beam188 which is a beam element working based on Timoshenko Beam Theory. Moreover,

it is also solved analytically based on Euler Beam Theory for comparison purposes. Euler




of 94

Beam Theory as well as Timoshenko Beam Theory is implemented into the formulation of

beam element used in ADFEM. There is a specific key option that is defined to select the

appropriate theoretical approach when the input file is prepared. This problem is solved for

both cases and the both results are provided in below. The translational and rotational

displacement results that are obtained from ADFEM, ANSYS and analytical solution are

plotted in Figure 28 in order to emphasize the difference between Euler and Timoshenko

beam theory.

(a) (b)

Figure 28 (a) Translational and (b) rotational displacement results of the beam problem

As shown in Figure 28 (a), the translational displacements obtained from ADFEM and

ANSYS are in good agreement, while those obtained from ADFEM and analytical solution

agree well as well. However, the results from Euler and Timoshenko Beam Theory are

slightly different since Timoshenko Beam Theory uses shear correction parameter in order to

indicate the transverse shear effects, which is especially important when a beam becomes

thicker. Hence, the definition of shear correction parameter affects the displacements in

translational degree of freedom. Moreover, this affect is negligibly small in rotational degree

-2.00

-1.80

-1.60

-1.40

-1.20

-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0 5 10 15 20

ADFEM Euler Beam

Analytical Result

ADFEM Timoshenko Beam

ANSYS Beam 188

0.00E+00

2.00E-04

4.00E-04

6.00E-04

8.00E-04

1.00E-03

1.20E-03

1.40E-03

0 5 10 15 20

ADFEM Euler Beam

Analytical Result

ADFEM Timoshenko Beam

ANSYS Beam 188




of 94

of freedom for this problem, since all the rotational displacement results are in very well

agreement as shown in right hand side of Figure 28 (b).

3.4.3 Frame Structure Problem

A frame structure is modelled in ADFEM by using a beam element based on Timoshenko

Beam Theory. The same model is built in ANSYS with Beam188 element as well in order to

validate the results. Key points of the frame structure are represented in Figure 29 (a). The

coordinates of key points are listed in Table 6. Circular solid cross section is selected for each

beam with a radius of r = 10 mm. The selected material’s elastic modulus is 210000 MPa with

the Poisson’s ratio of v = 0.3. The frame structure divided into beam elements and the length

of each beam element is 50 mm. Therefore, the total number of the nodes used to discretise

this problem becomes 466.

(a) (b)

Figure 29 Key point locations and application of boundary conditions on frame structure

The applied displacement and loading boundary conditions for this problem including the

direction of the concentrated forces are shown in Figure 29 (b). The magnitude of all the

applied concentrated loads is 1000 N. The displacement results generated by using ADFEM




of 94

and ANSYS are compared for each degree of freedom, namely translations and rotations in x,

y, z directions as shown in Figures 30-32, respectively.

Table 6 Key points of frame structure

Key

Point

X Y Z

(mm)

1 0 -500 0

2 1500 -500 0

3 3000 -500 0

4 0 0 0

5 1500 0 0

6 3000 0 0

7 0 500 0

8 1500 500 0

9 3000 500 0

10 0 -500 500

11 1500 -500 500

12 3000 -500 500

13 0 0 500

14 1500 0 500

15 3000 0 500

16 0 500 500

17 1500 500 500

18 3000 500 500

19 0 0 750

20 1500 0 750

21 3000 0 750

According to the figures, the results are perfectly coincident. Also, the variation of the total

translational and rotational displacement magnitudes over the structure obtained from

ADFEM and ANSYS are presented in Figure 33 and 34, respectively. According to the

comparison of these results, it can be concluded that the ADFEM beam element is equivalent

to the ANSYS Beam 188 element.




of 94

(a) (b)

Figure 30 Frame structure translational displacements along (a) x- and (b) y- directions of

all nodes

(a) (b)

Figure 31 Frame structure (a) translational-z and (b) rotational-x displacements of all nodes

-1.00

0.00

1.00

2.00

3.00

4.00

5.00

6.00

0 100 200 300 400 500

ADFEM UX ANSYS UX

-0.50

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0 100 200 300 400 500

ADFEM UY ANSYS UY

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

0 100 200 300 400 500

ADFEM UZ ANSYS UZ

-9.00E-03

-8.00E-03

-7.00E-03

-6.00E-03

-5.00E-03

-4.00E-03

-3.00E-03

-2.00E-03

-1.00E-03

0.00E+00

1.00E-03

2.00E-03

0 100 200 300 400 500

ADFEM ROTX ANSYS ROTX




of 94

(a) (b)

Figure 32 Frame structure (a) rotational-y and (b) rotational-z displacements of all nodes

(a) (b)

Figure 33 Frame structure total translational displacement contour plot;

(a) ADFEM and (b) ANSYS

-4.00E-03

-2.00E-03

0.00E+00

2.00E-03

4.00E-03

6.00E-03

8.00E-03

1.00E-02

1.20E-02

0 100 200 300 400 500

ADFEM ROTY ANSYS ROTY

-5.00E-04

-4.00E-04

-3.00E-04

-2.00E-04

-1.00E-04

0.00E+00

1.00E-04

2.00E-04

3.00E-04

4.00E-04

5.00E-04

0 100 200 300 400 500

ADFEM ROTZ ANSYS ROTZ




of 94

(a) (b)

Figure 34 Frame structure total rotational displacement contour plot;


3.4.4 Plane4 Problem

A square plate structure with 1 m edge length as shown in Figure 35 is considered to be

solved under tension loading by using ADPAN’s Plane4 element which is implemented based

on isoparametric formulation.

Figure 35 A square plate mesh and boundary condition representation




of 94

The same problem is solved in ANSYS by using Plane182 as well for comparison purposes. It

is assumed that the plate will deform based on plane stress condition and the plate thickness is

10 mm. The selected material’s elastic modulus is 210000 MPa with the Poisson’s ratio of v =

0.3.The geometry is discretised by using 4489 elements and 4624 nodes as presented in

Figure 35. All nodes at the left edge of the plate are considered to be fixed and a nodal force

of 1000 N in x-direction is applied to each node at the right edge of the plate.

(a) (b)

Figure 36 Translational displacements along (a) x- and (b) y-directions of all nodes for the

Plane4 problem

-0.01

0.00

0.01

0.01

0.02

0.02

0.03

0.03

0.04

0.04

0 1000 2000 3000 4000 5000

ADFEM UX ANSYS UX

-0.01

-0.01

0.00

0.00

0.00

0.00

0.00

0.01

0.01

0 1000 2000 3000 4000 5000

ADFEM UY ANSYS UY




of 94

(a) (b)


problem; (a) ADFEM and (b) ANSYS

The displacement results generated by using ADFEM and ANSYS are compared for each

degree of freedom, namely translations in x, y directions, as shown in Figure 36, respectively.

The displacement results are perfectly matching with each other according the figures. The

variation of the nodal translational displacement in x- and y-directions over the plate obtained

from ADFEM and ANSYS are presented in Figures 37 and 38. According to the comparison

of these variations, it can be concluded that the ADFEM Plane4 element is similar to the

ANSYS Plane182 element.

(a) (b)






of 94

3.4.5 Plane8 Problem

A rectangular plate structure with 1 m length and 0.5 m height is considered to be solved

under a loading condition as shown in Figure 39.

Figure 39 A rectangular plate and its applied boundary conditions

This problem is not just only modelled by using ADFEM’s Plane8 element, but also solved

with Plane183 element in ANSYS for validation purposes. It is assumed that the plate will

deform based on plane stress condition and the plate thickness is 10 mm. The selected

material’s elastic modulus is 210000 MPa with the Poisson’s ratio of v = 0.3. According to

the Figure 39, the plate is meshed by using 1250 elements and 3901 nodes and all nodes at

left edge of the plate are considered to be fixed.




of 94

(a) (b)


Plane8 problem

(a) (b)



In addition to the displacement boundary condition, Figure 39 illustrates that nodal force of

1000 N in x-direction together with a nodal force of -1000 N in y-direction is applied to each

node at the right and upper edge of the plate, respectively. The displacement results generated

by using ADFEM and ANSYS are individually compared for each degree of freedom, namely

translations in x, y directions (see Figure 40). The displacement results are perfectly matching

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0 1000 2000 3000 4000

ANSYS UX ADFEM UX

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0 1000 2000 3000 4000

ANSYS UY ADFEM UY




of 94

each other according the figures. The variation of the nodal translational displacement in x

and y direction over the plate obtained from ADFEM and ANSYS are respectively

demonstrated in Figure 41 and 42. According to the comparison of these variations, it can be

indicated that the ADFEM Plane8 element is good agreement with ANSYS Plane183.



3.4.6 Shell3 Element

Shell3 element has three nodes on the edges of the element and bending capability of the

elements is implemented based on Mindlin Plate Theory. Isoparametric shape functions are

used to formulate the element and the mathematical foundation that describes the flexural

capability of the element is provided by Tessler (1985) and Tessler and Hughes (1985).

Membrane stiffness of the element is identical to the Plane3 (constant strain triangle)

formulation in ADFEM.


solved under a distributed out of plane loading condition by using ADFEM’s Shell3 element.

For comparison purposes, the same problem is solved with Shell 181 which is a three or four

node shell element in ANSYS. The plate thickness is 10 mm. The selected material’s elastic

modulus is 210000 MPa with the Poisson’s ratio of v = 0.3.The geometry is discretised by

using 3200 elements and 1681 nodes as presented in Figure 43.




of 94

Figure 43 A clamped edge square plate and its discretization

All the nodes at right, left, upper, and bottom edges of the plate are considered to be fixed,

and therefore, the clamped edge boundary condition is considered. Nodal force of -100 N in z

direction is applied to all nodes of the plate except the nodes that are already fixed.

Figure 44 Translational displacements along z-direction of all nodes for the Shell3 problem

-12.000

-10.000

-8.000

-6.000

-4.000

-2.000

0.000

0 500 1000 1500

ADFEM UZ ANSYS UZ




of 94

(a) (b)

Figure 45 Rotational displacements along (a) x- and (b) y-directions of all nodes for the

Shell3 problem

(a) (b)



The non-zero displacement results generated by using ADFEM and ANSYS are compared for

each degree of freedom, namely translation in z-direction and rotations in x and y directions,

respectively, as shown in Figure 44 and 45. The displacement results are perfectly matching

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

0 500 1000 1500


-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

0 500 1000 1500





of 94

with each other according the figures. The variation of the nodal translational displacement in

z-direction over the plate obtained from ADFEM and ANSYS are shown in Figure 46.

Furthermore, the variation of the nodal rotational displacement in x- and y-directions over the

plate are illustrated in Figures 47 and 48. According to the comparison of these variations, it

can be concluded that the ADFEM Shell3 element is working very well.

(a) (b)

Figure 47 Contour plot of rotational displacement along x-direction for the Shell3 problem;


(a) (b)

Figure 48 Contour plot of rotational displacement along y-direction for Shell3 problem;





of 94


Shell4 element is a quadrilateral element with four nodes at its corners. The bending

capability of the element is implemented based on the Mindlin Plate Theory. Isoparametric

shape functions are used to formulate the element and the mathematical foundation that

describes the flexural capability of the element is provided by Tessler and Hughes (1983).

Membrane stiffness of the element is identical to the Plane4 formulation in ADFEM. In

addition to the bending capability and membrane stiffness, drilling rotation degree of freedom

is added to Shell4 formulation based on the proposed formulation by Cook, R. D. (1994). It

should be noted Shell4 becomes very robust to solve complex problems after adding all these

capabilities.

Figure 49 T beam mesh and its boundary conditions

A beam problem which has a T type cross section is solved by using Shell4 as shown in

Figure 49. Two different plates are attached together in order to create the geometry of the

structure. These plates have the same length of 2000 mm together with different heights, 200

mm and 400 mm.




of 94

(a) (b)


Shell4 problem

The same problem is modelled with Shell181 in ANSYS in order to show the capability of

Shell4 with respect to Shell181. Each plate has thickness of 10 mm. The selected material’s

elastic modulus is 210000 MPa with the Poisson’s ratio of v = 0.3.The geometry is discretised

by using 1920 elements and 2025 nodes as presented in Figure 49.

(a) (b)

Figure 51 (a) Translational-z and (b) rotational-x displacements of all nodes for the Shell4

problem

-0.200

0.000

0.200

0.400

0.600

0.800

0 500 1000 1500 2000

ADFEM UX ANSYS UX

-8.000

-6.000

-4.000

-2.000

0.000

2.000

0 500 1000 1500 2000

ADFEM UY ANSYS UY

-0.010

-0.005

0.000

0.005

0.010

0 500 1000 1500 2000

ADFEM UZ ANSYS UZ

-3.00E-04

-2.00E-04

-1.00E-04

0.00E+00

1.00E-04

2.00E-04

3.00E-04

0 500 1000 1500 2000





of 94

(a) (b)

Figure 52 Rotational displacements along (a) y- and (b) z-directions of all nodes for the

Shell4 problem

One end of the beam is fixed for all degrees of freedoms and a nodal force of -1000 N in y-

direction is applied to other end nodes which are on symmetry plane of the structure as shown

in Figure 49. The unknown displacement results of ADFEM and ANSYS are compared for all

degree of freedom as shown in Figure 50-52. The variation of nodal translational and

rotational displacements over the plate are respectively presented in Figures 53-58.

(a) (b)

Figure 53 Contour plot of translational displacement along x direction for the Shell4


-1.50E-04

-1.00E-04

-5.00E-05

0.00E+00

5.00E-05

1.00E-04

1.50E-04

0 500 1000 1500 2000


-5.00E-03

-4.00E-03

-3.00E-03

-2.00E-03

-1.00E-03

0.00E+00

0 500 1000 1500 2000

ADFEM ROTZ ANSYS ROTZ




of 94

(a) (b)

Figure 54 Contour plot of translational displacement along y direction for the Shell4


As it is shown in the figures that the displacement results are in perfect agreement for all

translational displacement and rotational displacement in x- and z-directions, although there is

a difference exists for the rotational displacement in y-direction. Actually, this difference is

due to the drilling rotation formulation used for Shell4. When the structure is considered to be

modelled by beam elements, it can be seen that drilling rotation degree of freedom is

rotational degree of freedom in y direction. Since the direction of load applied to the structure

is same as the direction of the drilling rotation degree of freedom, a very small contribution is

expected from rotational displacement in y-direction to the total rotational displacement.




of 94

(a) (b)



(a) (b)



In ADFEM results, range of rotational displacements in y-direction is very small in

comparison to other rotational displacements, while corresponding ANSYS results are much

higher than ADFEM as shown in Figure 52. Furthermore, the variation of this displacement

over the structure for ADFEM is more reasonable than ANSYS as it can be seen from Figure

57. As a consequence, it can be concluded that the implemented drilling rotational stiffness

proposed by Cook (1994) is very promising and Shell4 element is robust enough to model

complex structures.




of 94

(a) (b)

Figure 57 Contour plot of rotational displacement along y-direction for the Shell4 problem;


(a) (b)

Figure 58 Contour plot of rotational displacement along z-direction for the Shell4 problem;



Shell8 element has eight nodes on the edges of the element in total. Four of these nodes

located at the corner of the element while the remaining four of them located at the mid-point




of 94

of the edges. Isoparametric shape functions are used to formulate both bending and membrane

stiffness of the element. Formulation of the bending capability is based on the Mindlin Plate

Theory while membrane capability of the element is equal to the Plane8 formulation in

ADFEM.

Figure 59 A clamped square plate modelled with Shell8


solved by using Shell8 when out of plane loading is applied to the structure. The same

problem is solved with Shell281 which is a six or eight node shell element in ANSYS for

validation process.

Figure 60 Translational displacements along z-direction of all nodes for the Shell8 problem

-6.00

-4.00

-2.00

0.00

2.00

4.00

6.00

0 500 1000 1500 2000

ADFEM UZ ANSYS UZ




of 94

The plate thickness is 10 mm. The selected material’s elastic modulus is 210000 MPa with

the Poisson’s ratio of v = 0.3.The geometry is discretised by using 625 elements and 1976

nodes. Figure 59 shows that all nodes at right, left, upper, and bottom edges of the plate are

considered to be fixed, and therefore the clamped edge boundary condition is considered.

Nodal force of -100 N in z direction is applied to all nodes below the centreline along x

direction of the plate, while the reverse of the same loading condition is applied to the nodes

above the centreline along x direction of the plate. The direction of the applied loading

condition can be seen in Figure 59 in more detail.

(a) (b)

Figure 61 Rotational displacements along (a) x- and (b) y-directions of all nodes for the

Shell8 problem

-0.20

-0.10

0.00

0.10

0.20

0.30

0 500 1000 1500 2000


-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0 500 1000 1500 2000





of 94

(a) (b)



The unknown displacement results found in ADFEM and ANSYS are compared for each

degree of freedom. The displacement results are perfectly identical to each other according to

Figures 60 and 61. The variation of the nodal translational displacement in z direction,

rotational displacement in x- and y- directions over the plate are demonstrated in Figures 63

and 64, respectively. According to the comparison of these variations, it can be concluded that

the Shell8 element gives reasonable results and can be used to build complex structures as

well as Shell4 element.




of 94

(a) (b)



(a) (b)

Figure 64 Contour plot of rotational displacement along y-direction for the Shell8 problem;





of 94

3.5 Global Finite Element Analyses of Ship and Coupling of Finite

Elements

The global three dimensional finite element model is a representative of the hull structures of

three cargo holds with the middle cargo hold within 0.4L amidships. It is used to determine

both the global response of the hull girder and local behaviour of the main supporting

structures. The stress results from such models must be suitable for strength evaluation of the

watertight boundaries of cargo holds and non-tight main supporting structures. The global

three dimensional finite element analyses establish the scantling requirements of plates and

stiffeners, and they are sufficient for establishing the steel weight estimate. Structural details

are evaluated by the subsequent local three dimensional finite element analyses.

To evaluate the vessel’s structures within 0.4L amidships with reasonable accuracy, the finite

element models ideally place the target cargo hold in the middle and extend approximately the

length of the adjacent holds fore and aft. In addition, there is a short extension beyond the

transverse bulkheads at both ends. Even though finite element analysis can be done by using

both full and half-width models, it is recommended that the finite element models should be

created with both the port and starboard sides of cargo hold structures, that are symmetrical

with respect to the centreline, for easier review, result analysis and subsequent strength In

general, the ship structural finite element model consists of four types of elements:

For stiffeners:

Truss element

Beam element

For plates:

Membrane plate element (simplified case of Shell Element).

Shell Element.




of 94

Figure 65 Stiffened plate model

These four simple types of elements are considered sufficient to represent the hull structures

even though higher order element types exist. Ship structures consist of various stiffened

plates. These stiffened plates can be represented by a combination of membrane plates and

rod elements as long as only in-plane stress is calculated from the model. Combined use of

membrane plates and rod elements may simplify the modelling processes and reduce the total

number of degrees of freedom in the model. However, additional operations, such as shifting

load, may result in less accurate results for some elements.

As a consequence, combination of shell and beam elements is preferable in order to obtain

more realistic results. Appropriate properties of stiffeners are assigned by considering

equivalent beams model. Also, the required properties of plates are attached to the equivalent

shell model. The combination of beam model and shell model can be done by defining an

appropriate rigid link between them. Since ADFEM have the Beam and Shell element types,

portion of the side of a typical longitudinally and transverse framed (complex frame system)

ballast tank as shown in left hand side of Figure 65 can be modelled to demonstrate the

capability of the element combination. A stiffened square plate with 1 m edge length as

shown on right hand side of Figure 65 is considered to be solved under out of plane loading.




of 94

Figure 66 Stiffened plate boundary conditions

The plate is modelled with Shell4 element, while longitudinal and transverse stiffeners are

modelled with Beam element based on Timoshenko Beam Theory. The required rigid links

between Shell4 and Beam elements are defined by using Beam element based on Euler Beam

Theory. The same problem is designed to be solved with Shell181 and Beam181 elements in

ANSYS for validation purposes.

(a) (b)

-1.50E-02

-1.00E-02

-5.00E-03

0.00E+00

5.00E-03

1.00E-02

1.50E-02

0 500 1000 1500 2000

ADFEM - UX ANSYS - UX

-1.50E-02

-1.00E-02

-5.00E-03

0.00E+00

5.00E-03

1.00E-02

1.50E-02

0 500 1000 1500 2000

ADFEM - UY ANSYS - UY




of 94


stiffened plate

(a) (b)

Figure 68 (a) Translational-z and (b) rotational-x displacements of all nodes for the stiffened

plate

The plate thickness is 10 mm. Rectangular solid cross section is selected for each beam with

thickness of 10 mm and height of 100 mm The selected material’s elastic modulus is 210000

MPa with the Poisson’s ratio of v = 0.3.The geometry is discretised by using 1993 nodes

together with 2232 elements from which 1920 of them are Shell4 elements and 312 of them

are Beam elements.

0.00E+00

2.00E-02

4.00E-02

6.00E-02

8.00E-02

1.00E-01

1.20E-01

1.40E-01

1.60E-01

0 500 1000 1500 2000

ADFEM - UZ ANSYS - UZ

-8.00E-04

-6.00E-04

-4.00E-04

-2.00E-04

0.00E+00

2.00E-04

4.00E-04

6.00E-04

8.00E-04

0 500 1000 1500 2000

ADFEM - ROTX ANSYS - ROTX




of 94

(a) (b)

Figure 69 Rotational displacements along y- and z- directions of all nodes for the stiffened

plate

(a) (b)

Figure 70 Contour plot of translational displacement along x-direction for the stiffened

plate; (a) ADFEM and (b) ANSYS

As shown in Figure 66, all nodes at the right, left, upper, and bottom edges of the plate are

considered to be fixed, and therefore the clamped edge boundary condition is considered.

Nodal force of 100 N in z direction is applied to nodes attached to the mid-plane of the plate.

-6.00E-04

-4.00E-04

-2.00E-04

0.00E+00

2.00E-04

4.00E-04

6.00E-04

0 500 1000 1500 2000

ADFEM - ROTY ANSYS - ROTY

-6.00E-05

-4.00E-05

-2.00E-05

0.00E+00

2.00E-05

4.00E-05

6.00E-05

0 500 1000 1500 2000

ADFEM - ROTZ ANSYS - ROTZ




of 94

(a) (b)

Figure 71 Contour plot of translational displacement along y-direction for the stiffened


The unknown displacement results of ADFEM and ANSYS are compared for each degree of

freedom in Figures 67-69. The variation of the nodal translational and rotational

displacements over the plate are demonstrated in Figures 70-75.

(a) (b)

Figure 72 Contour plot of translational displacement along z-direction for the stiffened





of 94

As it can be seen from the figures that the results are in perfect agreement for all translational

displacement. The results are very similar to each other as well for rotational displacements in

x- and y-directions, while a difference exists for the rotational displacement in z-direction due

to the drilling rotation formulation used for Shell4 which is clearly described earlier in Shell4

Problem section.

(a) (b)

Figure 73 Contour plot of rotational displacement along x-direction for the stiffened plate;





of 94

Figure 74 Contour plot of rotational displacement along y-direction for the stiffened plate;


As a summary, current finite elements implemented into ADFEM enables the coupling of

elements in finite element analysis sense. Therefore, this capability of ADFEM provides the

opportunity to conduct a three dimensional global finite element analysis of any type of ship

in a more computationally efficient manner.

Figure 75 Contour plot of rotational displacement along z-direction for stiffened plate; (a)

ADFEM and (b) ANSYS




of 94

4 CONCLUSION

An in-house finite element code, ADFEM, together with a panel-method code, ADPAN, are

implemented by using Java language based on object-oriented methodology. The theoretical

background used to develop both software is described in detail. The finite element code is

examined and verified by solving a problem for each element type implemented into its finite

element library.

The procedures how to build a global three dimensional finite element model of the hull

structures are described. One of the most effective techniques for global finite element

analysis of ship structures is coupling suitable element types together. A portion of the side of

a typical longitudinally and transverse framed (complex frame system) ballast tank is

analysed in ADFEM to demonstrate the capability of finite element coupling.

Moreover, ADPAN is used to conduct hydrodynamic analysis of floating hemisphere,

submerged spheroid, long barge, WIGLEY III and S175 container ship in order to validate

and demonstrate the capability of ADPAN. For analysis of WIGLEY III and S175 container

ship, hydrodynamic pressure are calculated and exported from ADPAN for each panel that is

used to model the body surface. The variation of the hydrodynamic pressure is demonstrated

over the hull surfaces. The pressure calculation allows the user of the codes to perform a

hydro-elastic response analysis of the ship structure.




of 94

5 REFERENCES

Bathe, K. J. (2006). Finite Element Procedures. Prentice-Hall.

Beck, R. F. and Loken, A. E. (1989). Three-dimensional effects in ship relative-motion

problems. Journal of Ship Research, 33(4), 261-268.

Chapelle, D. and Bathe, K. J. (2011). The finite element analysis of shells: Fundamentals

(Vol. 1). Berlin: Springer.

Cook, R. D. (1994). Four-node ‘flat’shell element: Drilling degrees of freedom, membrane-

bending coupling, warped geometry, and behavior. Computers & Structures, 50(4), 549-555.

Faltinsen, O.M. and Michelsen, F.C. (1974). Motions of large structures in waves at zero

Froude number. In The Dynamics of Marine Vehicles and Structures in Waves, London.

Fish, J. and Belytschko, T. (2007). A First Course in Finite Elements. John Wiley & Sons.

Garrison, C.J. (1978). Hydrodynamic Loading of Large Offshore Structures. In Zienkiewicz,

O.C., Lewis, O.C., and Stagg, K.G., (Eds.), Numerical Methods in Offshore Engineering, pp.

87–139. Chichester, England: John Wiley & Sons.

Hess, J. L. and Smith, A. M. O. (1964) Calculation of nonlifting potential flow about arbitrary

three-dimensional bodies. Journal of Ship Research, 8, 2, 22–44.

Hogben, N. and Standing, R.G. (1974). Wave Loads on Large Bodies. In The Dynamics of

Marine Vehicles and Structures in Waves, London.

Hulme, A. (1982). The wave forces acting on a floating hemisphere undergoing forced

periodic oscillations. Journal of Fluid Mechanics, 121, 443–463.




of 94

Logan, D. (2011). A first course in the finite element method. Cengage Learning.

Maniar , H. D. (1995). A three dimensional higher order panel method based on B-splines,

Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA.

Newman, J. N. (1977). Marine hydrodynamics. MIT press.

Newman, J.N. (1985). The evaluation of free-surface Green functions. In Fourth International

Conference on Numerical Ship Hydrodynamics, Washington.

Noblesse, F. (1982). The Green function in the theory of radiation and diffraction of regular

water waves by a body. Journal of Engineering Mathematics, 16(2), 137-169.

Oñate, E. (2013). Structural Analysis with the Finite Element Method. Linear Statics: Volume

2: Beams, Plates and Shells (Vol. 2). Springer.

Papanikolaou, A. D. and Schellin, T. E. (1992). A three-dimensional panel method for

motions and loads of ships with forward speed.

Telste, J. G. and Noblesse, F. (1986). Numerical evaluation of the Green function of water-

wave radiation and diffraction. Journal of Ship Research, 30(2), 69-84.

Tessler, A. and Hughes, T. J. (1983). An improved treatment of transverse shear in the

Mindlin-type four-node quadrilateral element. Computer Methods in Applied Mechanics and

Engineering, 39(3), 311-335.

Tessler, A. (1985). A priori identification of shear locking and stiffening in triangular Mindlin

elements. Computer Methods in Applied Mechanics and Engineering, 53(2), 183-200.

Tessler, A. and Hughes, T. J. (1985). A three-node Mindlin plate element with improved

transverse shear. Computer Methods in Applied Mechanics and Engineering, 50(1), 71-101.




of 94

Wehausen, J. V. and Laitone, E. V. (1960). Surface waves (pp. 446-778). Springer Berlin

Heidelberg.

Wu, G. X., and Eatock Taylor, R. (1987) The exciting forces on a submerged spheroid in

regular waves. Journal of Fluid Mechanics, 182, 411–426.

inspection capabilities for enhanced ship safety · inspection capabilities for enhanced ship...

Documents