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Computer graphics applications in offshore hydrodynamics Hodgkinson, Derek Anthony Martin 1987

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COMPUTER GRAPHICS APPLICATIONS IN OFFSHORE HYDRODYNAMICS  by DEREK  ANTHONY  B.A.Sc. U n i v e r s i t y  MARTIN  HODGKINSON  of B r i t i s h Columbia, 1985  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  in FACULTY OF GRADUATE STUDIES Department of C i v i l  We accept t h i s  Engineering  t h e s i s as conforming  to the r e q u i r e d  standard  THE UNIVERSITY OF BRITISH COLUMBIA March 1987 ©  Derek  Anthony  Martin  Hodgkinson,  1987  ^1  In p r e s e n t i n g  this  thesis in  partial  fulfilment  requirements f o r an advanced degree at the The B r i t i s h Columbia, I  agree that  f r e e l y a v a i l a b l e for that permission scholarly  reference  for e x t e n s i v e  purposes  Department  or  by  may his  understood that copying f i n a n c i a l gain  shall  be or  the L i b r a r y and  copying of granted her  not  be  Engineering  1987  the  University s h a l l make  I further  the  Head  of it  agree  this thesis  representatives.  allowed  The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5  Date: March  by  or p u b l i c a t i o n of  permission.  Department of C i v i l  study.  of  of  for my  It  is  this thesis  for  without  my  written  Abstract The  r e s u l t s of  involving offshore  hydrodynamic analyses structures  are  of two  displayed  problems  graphically.  T h i s form of p r e s e n t a t i o n of the r e s u l t s and the l i b e r a l use of c o l o u r have been found  to s i g n i f i c a n t l y h e l p the ease  in  which the r e s u l t s a r e i n t e r p r e t e d . For the t r a n s f o r m a t i o n island, a  time h i s t o r y  u n i d i r e c t i o n a l wave  of waves  of the  breaking occurs technique  e v o l u t i o n of  f i e l d around  o b t a i n e d . Through the use of have been  around an the  an a r t i f i c i a l  regular,  island  c o l o u r , regions i n which  c l e a r l y defined.  used i s based on  artificial  The  is wave  numerical  the f i n i t e element method  using  e i g h t noded i s o p a r a m e t r i c elements. The d e t e r m i n a t i o n  of the  transformed  wave f i e l d takes wave breaking,  diffraction,  and  shoaling  into  g r a p h i c a l d i s p l a y i s achieved  by using  a plotting  developed The the RAO  reflection  wave r e f r a c t i o n ,  f o r the output  of f i n i t e element  motions of a semi-submersible curves  response i n a  of  the  r i g , used  random sea. The  and may be a n a l y s e d  using  The  program  analyses.  r i g a r e computed from to  obtain  numerical  the a n a l y s i s assumes that the  account.  i t s ' small  technique  v e r t i c a l members are  the Morison equation  used  in  slender  whereas  the  h u l l s are t r e a t e d as l a r g e members which a r e d i s c r e t i s e d and analysed  u s i n g d i f f r a c t i o n theory. The d i s c r e t i s a t i o n of the  c y l i n d e r s and h u l l s r i g ' s motions  together with the  are displayed  time h i s t o r y of the  g r a p h i c a l l y . Once  g r a p h i c a l d i s p l a y i s p l o t t e d using ii  again,  the  a program developed  for  the  output  elements.  of  finite  element  In t h i s case, a  finite  analyses  for  four  noded  element technique has not  been employed but the r e s u l t s were ordered to a c t as  though  t h i s i s the case. The s l e n d e r members ( c y l i n d e r s ) and  large  members  using  (hulls)  are  different colours. also c l e a r l y  shown. A video tape,  produced seen,  The elements  distinguishable used  in  by  the a n a l y s i s  are  shown.  The VAX 11/730  procedure,  clearly  system was used  to o b t a i n the  u s i n g the r e s u l t s  of a time  results stepping  was made by s u c c e s s i v e l y r e c o r d i n g the hardcopies  by the VAX p r i n t e r . The time s t e p p i n g c o u l d a l s o be  i n r e a l time, on the IRIS.  Table of Contents Abstract List  .  ii  of Figures  vi  Acknowledgements  viii  1.  Introduction  1  2.  Review o f Hydrodynamic Problems  5  2.1 Wave T r a n s f o r m a t i o n Around an A r t i f i c i a l  I s l a n d ..5  2.2 Wave E f f e c t s on Hybrid Offshore S t r u c t u r e s  3.  2.2.1 Large Diameter Members  10  2.2.2 Slender Members  12  2.2.3 T o t a l S t r u c t u r e  13  The D i s p l a y  Systems  15  3.1 VAX/VMS  "  3.2 I n t e g r a t e d Raster 4.  15  Imaging System  15  Computer G r a p h i c s C o n s i d e r a t i o n s 4.1 Wave T r a n s f o r m a t i o n Around an A r t i f i c i a l  5.  17 I s l a n d .17  4.2 Wave E f f e c t s on Hybrid Offshore S t r u c t u r e s  23  R e s u l t s and D i s c u s s i o n  29  5.1 Wave T r a n s f o r m a t i o n Around an A r t i f i c i a l  I s l a n d .29  5.2 Wave E f f e c t s on H y b r i d Offshore S t r u c t u r e s 6.  C o n c l u s i o n s and Recommendations 6.1 Wave T r a n s f o r m a t i o n Around an A r t i f i c i a l  7.  9  33 ...38  I s l a n d .38  6.2 Wave E f f e c t s on H y b r i d Offshore S t r u c t u r e s  40  References  42  APPENDIX A : THE ISLAND  45  APPENDIX  B : THE R I G  51  APPENDIX  C : TRANSFER VAX/VMS F I L E TO THE UBC/MTS SYSTEM .57  APPENDIX  D : PLOTTING  THE OUTPUT ON THE VAX/VMS SYSTEM ...59 iv  APPENDIX E : THE MOV  L i s t of F i g u r e s Fig.  1  Sketch of an a r t i f i c i a l  Fig.  2  Sketch of boundaries  66  Fig.  3  Sketch of a semi-submersible r i g  67  Fig.  4  An element f o r a slender member  68  Fig.  5  Undistorted  display  island  of  an  65  artificial  69  i sland Fig.  6  Scaled  island  (NODES-* 1 . 0 , H=1.0, V=5.0)  70  Fig.  7  Scaled  island  (NODES=1.0, H=1.5, V=10.0)  71  Fig.  8  Scaled  island  (NODES=1.0, H=1.5, V=5.0)  72  Fig.  9  Scaled  island  (NODES=1.5, H=1.5, V=5.0)  73  Fig.  10  Mesh on e x t e r i o r of h u l l s  74  Fig.  1 1  O u t l i n e of h u l l s  75  Fig.  12  Arrangement of elements on c y l i n d e r  76  Fig.  13  Edges of elements on the c y l i n d e r s  77  Fig.  14  Dimensions of the i s l a n d  78  Fig.  15  I n t r u s i o n of yellow  79  Fig.  16  I n t r u s i o n of red elements  80  Fig.  17  Back of i s l a n d (t=0)  81  Fig.  18  Right  82  Fig.  19  Front of i s l a n d (t=0)  83  Fig.  20  Back of i s l a n d ( t = l )  84  Fig.  21  Back of i s l a n d (t=2)  85  Fig.  22  Master sheet  86  Fig.  23  Overlay  Fig.  24  Plan  elements  s i d e of i s l a n d (t=0)  for island  sheet  view  for island  of  island  waves vi  87 showing  breaking  88  Fig.  25  Dimensions  of the r i g  Fig.  26  Rig showing  Fig.  27  Plan view of semi-submersible r i g  91  Fig.  28  . Side view of semi-submersible r i g  92  Fig.  29  E l e v a t i o n of semi-submersible r i g  93  Fig.  30  R i g at.end of time s t e p #1  94  Fig.  31  Rig at end of time s t e p #2  95  the f l e x i b i l i t y  vii  89 of program  90  Acknowledgements T h i s work  would not  have  been p o s s i b l e  encouragement, guidance and h e l p f u l suggestions  without  the  of P r o f .  M.  de S t . Q. Isaacson. The c o n s t r u c t i v e c r i t i c i s m s of P r o f .  W.  F. C a s e l t o n  are  humbly  instrumental  i n h e l p i n g with the m o d i f i c a t i o n s of the  E n g i n e e r i n g Graphics R.  Nelson  and  complemented the video  his  A.  Martin  was Civil  Package f o r use i n these problems. staff  at  f o r the work that  Media  Services  they have done i n  should  Mr. be  preparing  tape.  The f i n a n c i a l support and  acknowledged. Mr.  p r o v i d e d by the N a t u r a l  Sciences  E n g i n e e r i n g Research C o u n c i l i s g r e a t l y a p p r e c i a t e d .  vi i i  1. INTRODUCTION Computer g r a p h i c s i s the use s t o r e , manipulate, (Mufti, the  i n t e r r o g a t e and present p i c t o r i a l  interpretation  interpretation  of  examining  and  high rate:  of  the  from  large The  amounts  finite  output  element  data  a system  from an i n s p e c t i o n of  the  becomes  the engineer  in  i n c r e a s e s at a  an e r r o r  very  increases  p i c t o r i a l d e s c r i p t i o n of the  r e v e a l a l o t about  in  increases,  of  time spent by  of making  use of a  a  problem  i n t e r p r e t i n g the r e s u l t s  the chance  r a p i d l y . The  results  complexity of the  increasingly d i f f i c u l t .  obvious  define,  1983). G r a p h i c a l d i s p l a y s p l a y an important p a r t  a n a l y s i s . As the  may  of a computer to  more  results  which would otherwise not the numerical data.  Pictures  make the computer a more e f f e c t i v e machine in a s s i s t i n g engineer  with  the  complex  human  processes  be  the  involved  in  meters  of  des i g n . One  picture  is  numerical output on the  worth  i f the  numerical  more  results.  Graphic output  interpretation  from  systems to the human  the  The computer  devices s h i f t human  visual  the  can  i n a form that  presence  and of  used  the burden and  of  visual  we  summarized  form of  the  to  data  o n l y . What  computational data that was r e q u i r e d .  1  be  based  i s more n a t u r a l  intellectual  system  highly structured, synthesized r e s u l t s without  several  engineer must make d e c i s i o n s  present the r e s u l t s v i s u a l l y , to man.  than  massive  see i s  amounts  a the of  2 The o f f s h o r e i n d u s t r y i s a r a p i d l y growing f i e l d and an i n c r e a s i n g amount of r e s p o n s i b i l i t y management's shoulders. D e c i s i o n s numerical procedures q u i t e o f t e n have  i s being p l a c e d on upper based on  must o f t e n  be  f a r reaching or  the r e s u l t s  made q u i c k l y  and  can  c o s t l y consequences.  Any  advances which can help i n the v i s u a l i z a t i o n of the of  analyses  performed on  offshore structures  There are s e v e r a l numerical  v i c i n i t y of  shape of  an  moorings, e t c . but  the  little  results in v i s u a l l y  motion of  has  two d i s t i n c t  a barge  and i t s '  to p r e s e n t  case, the necessary v a r i o u s computer so  that  the  programs. These programs is  direction.  displayed.  numerical output can  output  in  each  be o b t a i n e d  from  are then a  the  hydrodynamic  analyses  d i s c r e t i s a t i o n s of the flow f i e l d the  f i n i t e element method or  the  results  (runup,  motions  e t c . ) are  modified  form  which  based (either  element  i s u s e f u l to show  The c o o r d i n a t e s of  how  usually  the d i s c r e t i z a t i o n was  the vertex p o i n t s  on using  method), known  s p e c i f i c p o i n t s . In d i s p l a y i n g the r e s u l t s of the it  is  program.  are  or s t r u c t u r e  the boundary  on  In  compatible with the requirements of the p l o t t i n g Since  This  a n a l y s e s performed  s t r u c t u r e s are  necessary  these  i n f o r m a t i v e forms.  of hydrodrodynamic  offshore  field  deflected  been done  a p p e a l i n g and  to  the  t h e s i s attempts to make some p r o g r e s s i n t h i s The r e s u l t s  welcome.  p r o f i l e of the wave  offshore structure,  marine r i s e r s ,  is  results  methods p r e s e n t l y a v a i l a b l e  determine parameters such as the in the  of  at  analyses, performed.  (nodes), are  usually  3 known. I f the r e s u l t s have than  the vertex  been determined at p o i n t s  p o i n t s , they  must be  transposed  other  onto  the  nodes to f a c i l i t a t e use of the p l o t t i n g program. These nodes are the b u i l d i n g b l o c k s of each block  the p i c t u r e and a s s o c i a t e d  i s an a l g o r i t h m which e x p l a i n s how  connected. T h i s a l g o r i t h m  i s sometimes  with  the nodes  c a l l e d the  are  locator  matrix. Perhaps one  of the most commonly known a p p l i c a t i o n s  computer g r a p h i c s scanner.  It  i s the CAT  helps  the e n g i n e e r i n g  of  personnel  and p o t e n t i a l l y  field,  gaining insight  (computer - a x i a l - tomography)  medical  n e c e s s i t y of needless  graphics  i n t o concepts  the d i s p l a y s ,  Animation f o r e n g i n e e r i n g  neither  p r o d u c t i o n procedures the only  each separate  part  i d e n t i f i e d and  the  r e a l time,  for  to  reader  presented.  is  quite different  nor  the  The  from  the system  must  must be  the  engineering  money  for  to obtain r e a l i s t i c  the g r a p h i c a l  animation  tool  In  the  images;  d i s p l a y s are be  that  unambigiously  produced q u i c k l y ,  in  i f p o s s i b l e (Noma and K u n i i , 1985).  In the present FORTRAN and  of  a  speed at which the  time  required on  surgery.  as  f o r entertainment.  the  requirements  harmful  the  of c o l o u r c o n t r i b u t e s  and  a s s i m i l a t e s the i n f o r m a t i o n being  has  determine  i s used  use  the ease of r e a d i n g the output  industry  to  and a n a l y s i n g problems. In a l l  the l i b e r a l  more w e l l known animation  of  work, a l l the  the p l o t t i n g  programs are w r i t t e n  i s done using  the EUNICE o p e r a t i n g system i n s t a l l e d  a VAX  11/730  in  under  at The U n i v e r s i t y  of  4 British  Columbia.  (IRIS) was in r e a l  printer.  Integrated  used to d i s p l a y  time. A  r e s u l t s was  The  made  film using  Raster  the time h i s t o r y  showing  the  hardcopies  Imaging  System  of the  systems  time v a r i a t i o n produced  on  of  the  the  VAX  2. REVIEW OF HYDRODYNAMIC PROBLEMS  2. 1 WAVE TRANSFORMATION The  first  problem  considered  transformed wave f i e l d The  AROUND AN ARTIFICIAL ISLAND i s the  prediction  and an a r t i f i c i a l  s p e c i a l case of a c i r c u l a r a r t i f i c i a l  island  of the ( F i g . 1).  i s l a n d with a berm  of constant s l o p e w i l l be examined. For t h i s s i t u a t i o n , incident  wave f i e l d  diffraction,  be transformed by wave  and  energy  dissipation.  model c o n s i d e r e d  d i f f r a c t i o n and energy d i s s i p a t i o n  extended  wave-current  However,  some of these e f f e c t s are unimportant  n e g l e c t e d . In the  taken i n t o  refraction,  r e f l e c t i o n , wave n o n - l i n e a r i t i e s ,  interactions situations,  may  account. A  finite  here, wave  in  many  and can be refraction,  due to wave breaking  element  the  formulation  are  of the  m i l d s l o p e equation i s used.  Berkhoff  (1972) i n i t i a l l y  presented the l i n e a r  theory  of combined r e f r a c t i o n and d i f f r a c t i o n by assuming that waves a r e of s m a l l amplitude the seabed  the  ( l i n e a r approximation) and that  s l o p e i s g e n t l e and m i l d l y v a r y i n g . Consequently,  l i n e a r wave theory short d i s t a n c e s ;  f o r constant depth these assumptions  may be a p p l i e d  give r i s e  to the  over mild  slope equation: (1 ) where  wave c e l e r i t y  c c  9  group c e l e r i t y angular  frequency  a two-dimensional 5  surface p o t e n t i a l  at the  6  still  water  level  V = horizontal gradient The v e l o c i t y p o t e n t i a l , $(x,y,z,t) where  -igH  A  of the t h r e e - d i m e n s i o n a l  cosh  = Re [A  operator  k(z+d)  4(x,y) e  cosh kd  l w C  flow i s :  ]  (2)  (  2CJ  H  = i n c i d e n t wave height  0  g = a c c e l e r a t i o n due to g r a v i t y d = still  water depth  k = wave number = — c i = t In Eq.  1,  elevation. components  = time  both The  c  and c^  velocity  are  dependent  potential,  due t o the i n c i d e n t  <t>,  on is  the  made  seabed up  of  wave p o t e n t i a l , $j , and  the  unknown, s c a t t e r e d wave p o t e n t i a l , <t>g: * = + *  (3)  s  cosh k(z+d) where  = A  i(  k x  -  u t  )  e cosh kd cosh k(z+d)  <t>e = A  -icot 0_(x,y) e  cosh kd  a  The m i l d slope equation  13  i s subject  to boundary c o n d i t i o n s at  a s o l i d boundary, S,  and S , and  the  ( F i g . 2 ) . The  far  field,  S  3  2  c o n d i t i o n r e q u i r e s that there surface radiation  and  i s applied condition  a radiation condition solid  i s no flow  along  the  would  body  boundary  normal to the  island  contour.  require  in  body The that  7 0  g  decays  as the  o r i g i n . Eq. 1 may which  involves  point  of i n t e r e s t  be solved by a  moves away  the f i n i t e  two-dimensional  from  the  element  methods  discretization  of the  h o r i z o n t a l domain. The  d i s s i p a t i o n of  examained  by  Booij  energy by wave  (1981)  d i f f r a c t i o n by i n c l u d i n g  breaking  for linear  an extra  was  first  refraction  term i n  and  the m i l d  slope  equation: V(cc  c_ V0) + ( -3- co + icoW)0 = 0 " c  (4)  2  where W i s a damping  f a c t o r which  may be  p o s s i b l e e f f e c t s . However, i n the present energy d i s s i p a t i o n due to wave breaking  due to  various  formulation,  i s included.  only Dally,  Dean and Dalrymple  (1984) have developed  a model f o r wave  breaking  (based on  to the  and decay  jump) f o r waves e n t e r i n g Isaacson  an analogy  the surf zone  2  this  )  (5)  K' = wave decay f a c t o r r  = s t a b l e wave f a c t o r .  These two f a c t o r s have found that  0.35 to  f o r W using  2  2  where  beach.  breaking:  c r d W = K' -3 (1 d H  K' l i e s  f o r a plane  (1985) has proposed an expression  model of wave  hydraulic  are determined e m p i r i c a l l y :  f o r beach slopes  between 0.10 0.48.  conditions wave height  i n the range 1/80 to  and 0.28 while  The a p p l i c a t i o n  D a l l y et  of  T  v a r i e s i n the  Eq. 5  depends  on  al  1/30, range the  f o r the onset and c e s s a t i o n of wave breaking. The is initially  unknown and i n order  to o b t a i n wave  8 h e i g h t s f o r use  i n the  wave b r e a k i n g  criterion,  height over the berm i s assumed to be due The e x p r e s s i o n s p r e s e n t e d calculating  since  the  other  al  can  are not b r e a k i n g , forms  of  wave  to s h o a l i n g  energy  only.  be used  the h e i g h t s of the d e c a y i n g waves once  has commenced. When waves zero  et  by D a l l y  the  for  breaking  W i s taken  as  dissipation  are  of the problem,  the  g e n e r a l l y s m a l l i n comparison. In  the f i n i t e  element  formulation  G a l e r k i n method i s used. The for  each element may  be w r i t t e n  [k] {0 } = e  e  1  where <j>^ which  are the  are  to  final  {  q i  }  (6)  of the The  c o o r d i n a t e system f o r each element  domain. The of  their  potential  function  isoparametric  can then be  g l o b a l c o o r d i n a t e system  local  transformed  of the  entire  system of e q u a t i o n s can then be s o l v e d .  ability  to a c c u r a t e l y  model  e i g h t node two-dimensional i s o p a r a m e t r i c  curved  different  Because  boundaries,  elements are  Once the numerical v a l u e s of the p o t e n t i a l at  equations  e  determined.  i n t o the C a r t e s i a n  element  i n the form:.  nodal v a l u e s  be  finite  used.  f u n c t i o n , <£(x,y),  l o c a t i o n s are known, the t o t a l p o t e n t i a l can be  o b t a i n e d from  Eq. 2.  The wave  o b t a i n e d from the l i n e a r i z e d  elevation,  77,  can then  dynamic f r e e s u r f a c e  be  boundary  c o n d i t ion. In  previous  work  1986), the f o r m u l a t i o n equations have  been  performed on  this  and s o l u t i o n of examined. The  sea  topic  (Talukdar  the f i n i t e state  element  around  the  9 artificial zero  i s l a n d was obtained  so that  comparison c o u l d  r e s u l t s . In the present for  at a time when t i s equal be made  to other  work, the t r a n s f o r m a t i o n  available of the sea  an extended p e r i o d of time around the a r t i f i c i a l  is represented  g r a p h i c a l l y . This representation  d e t e r m i n i n g the neighbourhood manipulation,  possible  of  the  maneuvering island  or  the sediment movement  of  island  i s useful in  vessels  with  to  some  i n the further  i n the v i c i n i t y of the  island.  2.2 WAVE EFFECTS ON HYBRID OFFSHORE STRUCTURES It  i s not uncommon  both  slender  f o r o f f s h o r e s t r u c t u r e s to be made up  and  large'  members.  As  an  example,  of a  semi-submersible r i g ( F i g . 3) may c o n s i s t of h u l l s which a c t as l a r g e members  and braces and  t r u s s e s which behave  like  small members. In the a n a l y s i s of wave-structure i n t e r a c t i o n problems, the two the  types of members  respond d i f f e r e n t l y  to  same wave c o n d i t i o n s and, consequently, give r i s e to two  different  approaches  t r e a t e d . For separation,  by  slender wave  which  structures  loading  and  g e n e r a l l y based on the Morison the body s i z e i s small the  i n c i d e n t flow  the  structure  wave  force  which  give  response  loading  behaviour.  i n c i d e n t wave  and For  flow  large  to  calculations  r e l a t i v e to the wave length  i . e . the  significantly  rise  equation which assumes  i s v i r t u a l l y unaffected  change  problems  so  are flow are that that  by the presence of kinematics  separation structures  do  not  dominates  the  which  span  a  10 significant  f r a c t i o n of  a wave l e n g t h ,  the i n c i d e n t  waves  undergo s i g n i f i c a n t s c a t t e r i n g or d i f f r a c t i o n and wave f o r c e c a l c u l a t i o n s should  account  cases, flow s e p a r a t i o n the problem  can be  f o r the  In  such  e f f e c t s can u s u a l l y be neglected  and  s o l v e d as  one of  complete s o l u t i o n using  potential  s o l u t i o n of  equations.  non-linear  scattering.  p o t e n t i a l flow.  The  flow theory r e q u i r e s  the  The problem  l i n e a r i z e d by assuming that the wave h e i g h t s  is  usually  are s m a l l .  2.2.1 LARGE DIAMETER MEMBERS The  fluid  (water)  i s assumed  i n v i s c i d , and the  flow i s assumed  f l u i d motion  therefore  potential, the  fluid  may  0, which region.  oscillating  in  be  described  flow  0 = [ where ^ The  ^  s  -iwH  Uo 0 ) 7  potentials,  r e p r e s e n t e d as due to a  as  associated  0 ) , the s c a t t e r e d  6  +  2 -ito$.0, ] e k=1 k k  point  waves due  1,...,6)  -\tsTw  0^,  (7)  c  (k=1,...,7) may  x = (x,y,z)  i n the f l u i d ,  motion. each  d i s t r i b u t i o n of p o i n t wave  over the immersed e q u i l i b r i u m body s u r f a c e , general  structure  the complex amplitude of each component  unknown  within  0 may be expressed as:  +  2  The  velocity  L a p l a c e equation  7) and the f o r c e d waves ( s u b s c r i p t s  to each mode of motion,  a  of components  with the i n c i d e n t waves ( s u b s c r i p t (subscript  by  p o t e n t i a l of a  waves i s made up  and  t o be i r r o t a t i o n a l .  s a t i s f i e s the The t o t a l  incompressible  sources  S^, and, at  may be  be  any  represented  11  1  0 (x)  =  t  /„ b  —  where  =  f. (£) G(x,£) dS  b  47T  K  s  o  u  r  c  k = 1,...,7 ( 8 )  strength d i s t r i b u t i o n function  e  I  = ( f r S) on the body  G  (x,J_) =  Green's f u n c t i o n f o r the p o i n t x due to a source of u n i t s t r e n g t h  located  at i The the  source strength d i s t r i b u t i o n f u n c t i o n body  surface  boundary  condition  i s chosen so  that  is satisfied.  The  Green's f u n c t i o n i s chosen to s a t i s f y the Laplace the  l i n e a r i z e d f r e e surface c o n d i t i o n , the seabed  and  the r a d i a t i o n  condition  p o t e n t i a l s , 0^,  c o n d i t i o n . Once the  known, the hydrodynamic pressure  equation,  on the body s u r f a c e may  determined from the l i n e a r i z e d B e r n o u l l i equation. on  are be  The loads  the body may then be c a l c u l a t e d : F.  where  = -icop /<, 0 n. dS b F  1  f  F  2  i F  =  3  j =  f o r c e components  1,...,6  i n the  (9)  x,y and  z  directions respectively ^ni^si^s  ~  moment components about the x,y and z directions respectively  Each component of  f o r c e , Fy (e)  the f l u i d  i n t o an e x c i t i n g force component,  Fj  may be  decomposed  , and a f o r c e d  force  component, F ^ ^: f  Fj  -  where  [ F ^  e  ) +  F J  f  )  ]  e-  i w t  F . ' = - - pHco 3 2  2  j =  J" b b  (0  O  + 0 ) 7  n. dS 3  1,..,6 ( 1 0 )  1 2  and  = . J / ^ j k  F< > £  with  +  i  " jk>5 X  k  = added-mass c o e f f i c i e n t X j ^ = damping c o e f f i c i e n t  The  added-mass and damping c o e f f i c i e n t s and e x c i t i n g  forces  may be obtained by a  d i s c r e t i s a t i o n procedure i n which  the  source s t r e n g t h s a r e  first  each  facet  o b t a i n e d at  the centre of  on the submerged s u r f a c e of the s t r u c t u r e .  potentials  may  then  be  evaluated.  The  c o e f f i c i e n t s a r e determined by s u b s t i t u t i n g  The unknown hydrodynamic  Eq. 9 and Eq. 10  i n t o Eq. 7 and c o l l e c t i n g c o r r e s p o n d i n g terms.  2.2.2  SLENDER MEMBERS The  fluid  arbitrary  on  orientation  the member velocity the  force  axis  a slender  acts i n  and  may  per  unit  in a direction 7TD F  n  =  expressed  by  f o r m u l a t i o n of the Morison equation  force  length,  * —  F^,  the  to  relative  ( F i g . 4 ) . Thus,  (prime used  for  slender  a t a p o i n t x on the member  axis  n is: 7TD  2  m*n "  c  at an  a d i r e c t i o n perpendicular  be  member c o e f f i c i e n t s ) , a c t i n g and  member i n c l i n e d  ~  p  2  (  C  nf  1  )  %  • J * d n- n>l n- nl  +  where  DC  (u  v  u  v  p = d e n s i t y of f l u i d u v  n n  = fluid =  v e l o c i t y at x i n the d i r e c t i o n n —  member  velocity  direction n  component at  x  i n the  13 = d e r i v a t i v e with respect  to time  D = diameter of member C = inertia m = drag The  coefficient  drag term i s u s u a l l y l i n e a r i z e d by t a k i n g (u -v )|u -v n n ' n n where  a  n  harmonically formulation  1  I  a  =  (u -v ) n n n  = drag l i n e a r i z a t i o n  In a d d i t i o n , the  flow v e l o c i t y and  i n time  (12)  factor. s t r u c t u r e motions  vary  co. The  with angular frequency,  above  can be a p p l i e d to two orthogonal d i r e c t i o n s f o r  each element of  a slender  then be used to o b t a i n the  coefficient  the  member. total  S u i t a b l e summations  can  f o r c e s and moments on a l l  slender members. As with the l a r g e diameter members, the  total  force and  slender  moment components,  members may  (e) F^  potentials  and  of  .  with  the  components  potentials, F j ^ ^ , terms  be decomposed i n t o .  , associated  F ^, a c t i n g  which  added-mass  on a l l the  an e x c i t i n g  force,  .  incident  and  associated  with  themselves can coefficients,  scattered  be  M--,,  the  wave forced  expressed and  in  damping  T  coefficients,  X ., . jk  2.2.3 TOTAL STRUCTURE Once a l l the hydrodynamic equations of motion of the the  s i x components of  c o e f f i c i e n t s a r e known,  body may be s o l v e d to  the body  motion may be w r i t t e n as f o l l o w s :  motion. The  the  determine  equations  of  14  U  -« <m 2  =  where  F  j k +  (e)  M  j k +  +  Mj > k  p  i"(*  "  +  j k  V  *(e)  •  =  ]  j  f  k  m  +  m  C j ) ] k  t  f  6  (  $  1  k  3  )  m.. = mass matrix Ilk Cj  k  Cj  k  = hydrostatic =  s t i f f n e s s matrix  additional  stiffness  matrix  moorings which may be The  ( c  +  response amplitude o p e r a t o r s ,  be determined by  s o l v i n g Eq.  defined  due  present. as 2^/H,  13 (these  to  describe  may then the  body  responses f o r i n c i d e n t waves of u n i t amplitude and d i f f e r e n t possible  frequencies).  waterline  of the s t r u c t u r e can a l s o be  R -  co  2  = —  H For  a  fixed  excluded.  The  6  100  +  <t>i +  2g  R,  around  the  obtained:  0 I„ _ k  = i (H/2) the  runup,  Si,  £ k  structure,  wave  k  forced  2  -  (14)  n  0  potential  terms  are  3. THE DISPLAY SYSTEMS  3.1 VAX/VMS The VAX/VMS  system i s used t o with  plotting  i s performed u s i n g DI-3000, an i n t e g r a t e d system of  computer  graphics  system) or primitives  3D  The a p p l i c a t i o n  objects  using  draws,  d e f i n e s the mapping from  e t c . ) . The  i s c a l l e d the viewing taken  of  a  program  defines  graphics  application  output program  system  system (a v i e w p o r t ) .  coordinate  to get  object,  hardcopies  hardcopies  (a This  i n which a "photograph" with  scaling,  r o t a t i o n s , t r a n s l a t i o n s e t c . a l r e a d y performed. T h i s allows the user  The  coordinate  the world c o o r d i n a t e  transformation  world  resolution.  (virtual  system)  window) i n t o a v i r t u a l c o o r d i n a t e  is  2D  (world c o o r d i n a t e  (moves,  pixel  graphics  display  tools.  780  and 3D  using a  graphics software  1024 x  o b t a i n 2D  of the  recording  successive  on  film,  impression  of the changes may be observed.  views a  system and by  realistic  3.2 INTEGRATED RASTER IMAGING SYSTEM The  .Integrated  Raster  Imaging System  produce high  r e s o l u t i o n 2D  heart of the  system i s the Geometry  p o i n t s , v e c t o r s , polygons, d e f i n e d c o o r d i n a t e system. screen  i n screen  s c a l i n g and  and 3D  These are with  other t r a n s f o r m a t i o n s 15  i s used  computer g r a p h i c s .  characters  coordinates  (IRIS)  Engine which and curves transformed arbitrary  p o s s i b l e . In  to The  accepts  in a onto  user the  rotations, excess  of  16 65000 c o o r d i n a t e s screen  i s 1024 x  bottom l e f t a  wide  variety  be processed.  The  i s located in  IRIS the  of the d i s p l a y . T h i s system i s u s e f u l f o r of  graphics  drawback of  p o r t a b l e . However, r e s u l t s may  can  768 p i x e l s . The o r i g i n  corner  s i m u l a t i o n . One  per second  with the  be s t o r e d  applications,  t h i s system i s use of  including  that  a video  i n a compact and p o r t a b l e  i t is  machine, form.  not the  4. COMPUTER GRAPHICS CONSIDERATIONS  4.1  WAVE TRANSFORMATION AROUND AN ARTIFICIAL ISLAND  The s i z e of the elements used i n the a n a l y s i s , t o a  large  e x t e n t d i c t a t e s the accuracy of the r e s u l t s . Where  there  are  short  large  variations  distances,  i.e.  island, smaller  i n the wave  near the v e r t i c a l elements  profile  face  a r e needed.  over  of the  artificial  Neighbouring  diameters should not d i f f e r by more than about  facet  50% s i n c e any  improvement made by the presence of the small f a c e t s w i l l be l o s t due t o the computational i n e f f i c i e n c i e s a s s o c i a t e d with the l a r g e r diameter mathematical  f a c e t s . However, there  i s no  explicit  r e l a t i o n s h i p between the s i z e of an element and  the accuracy of the s o l u t i o n . Consequently, the l i m i t s i z e of an  element  i s d i c t a t e d by  the n u m e r i c a l  r e q u i r e d and the computational e f f o r t convergence  accuracy  necessary t o  t o the a c c u r a t e s o l u t i o n . No a d d i t i o n a l  are needed f o r the p l o t t i n g of the s u r f a c e An e i g h t  noded  obtain  elements  profile. i s used.  The  c o o r d i n a t e s of each node a r e determined a t the end of  each  time i n t e r v a l . Each  isoparametric  on the  node i s f i x e d  element  i n the h o r i z o n t a l (x-y)  plane. I t i s f r e e t o move v e r t i c a l l y . Each node i s a s s i g n e d a number and the o u t l i n e of each element numbers of consists  the nodes  of  four  touching  components:  i s d e s c r i b e d by the  the element. the b r e a k i n g  The  system  region,  non-breaking r e g i o n , the berm and the v e r t i c a l w a l l . component i s u n i q u e l y c o l o u r e d 17  so that  i t may  easily  the Each be  18  distinguished  from the other p a r t s of the system. Dark  i s used  non-breaking  for  distinguishable is coloured  title,  these  are  clearly  from the breaking (yellow) r e g i o n s . The berm  green while  c o l o u r s enable each of readily  regions and  blue  identified.  the v e r t i c a l  wall i s  the components of  The output  red.  These  the system to  c o n s i s t s of  a  descriptive  the numbers of nodes and elements i n the p i c t u r e  number of nodes r e q u i r e d to d i s p l a y the i s l a n d , of each node, the element  and  coordinates  number with i t s a s s o c i a t e d  nodes  and an a s s i g n e d c o l o u r parameter. When breaking o c c u r s , normally expects to see whitecaps.  This i s c l e a r l y  on the VAX/VMS t e r m i n a l screen but when .a hardcopy these  areas  show  distinguishable yellow  up  as  dark  be  one  visible is  blue/black,  made, barely  from the non-breaking r e g i o n s . Consequently,  i s used f o r these a r e a s .  There are two ways i n which the c o l o u r f o r the elements may  be s p e c i f i e d :  i t may  as an e x t r a column when the colour  plotting  i s included  be i n c l u d e d i n the l o c a t o r  or i t may is  be  determined by the  being performed.  In t h i s  matrix analyst  case,  the  i n the l o c a t o r matrix. Consequently,  the  l o c a t o r m a t r i x , f o r the eight noded element, i s  dimensioned  (number of  within  analytical  nodes,  9).  The c o l o u r  i s assigned  program. The advantage of t h i s procedure i s  elements which s a t i s f y c e r t a i n c o n d i t i o n s sloping surface,  f o r example) may  (wave b r e a k i n g  be a s s i g n e d t h e i r  correspondence  with  the  definition  that or  colour  d i r e c t l y . The number a s s i g n e d f o r the c o l o u r need not be a one-to-one  the  of  in the  19  c o l o u r code i n the p l o t t i n g program but component should breaking,  f o r example, may  i f breaking and  occurs,  be  been read  executable  assigned  within  program c o l o u r code may  code may  a boolean  Wave  variable:  c o l o u r code i s zero.  the  plotting  one  After  ( f i v e may  i f the c o l o u r code  the a n a l y s t to keep the two be generated on one  program  be  processes  separate.  system and  been suggested,  the  red). This  results  in a  be p l o t t e d on  other d i s p l a y systems, each of which may  colour enables  The  when o r g a n i z e d  they may  the  correspond  i s three,  ( t h i s may  may  which  c o l o u r code i s zero, then  be set to f i v e  be r e s e t to seven  such as has  code.  i n by the p l o t t i n g program, there  steps  to l i g h t blue) or  same c o l o u r  the c o l o u r code f o r t h a t element i s  r e - a s s i g n the c o l o u r : i f the  may  the  f o r non-breaking waves, the  the data has be  be a s s i g n e d  the elements of each  form  several  have i t s own  colour  code d e f i n i t i o n . To put is included  the whole p i c t u r e  i n the d i s p l a y . Since  about the x - a x i s , only one The  coordinates  prior  to  of  analysis.  The  the  other  side.  i s l a n d enables the wave runup wall i s coloured and  analysed.  t h i s s i d e were  determined  half  of  the  island  was  of  the  around  the  to be seen more c l e a r l y .  The  A  the y - c o o r d i n a t e v e r t i c a l wall  (red)' to c o n t r a s t the s l o p i n g berm  the f r e e s u r f a c e  island  i s l a n d was  the s i g n of  analysed  the  the s t r u c t u r e i s symmetric  s i d e of the  the nodes on  generated by changing nodes on  into perspective,  (dark blue and  yellow).  (green)  20 Once  the  analysis  mainframe, the system  has  been  r e s u l t s must  (Appendix  C).  coordinates  elements  a  takes  be t r a n s f e r e d  Using  c o n t a i n i n g the  little  performed  on  the  to the  UBC  VAX/VMS  3200  baud  lines,  a  file  of over  2800  nodes  and  1100  more  than h a l f  an  hour  to  be  t r a n s f e r e d . I f one p e r i o d i s d i v i d e d i n t o s i x t e e n equal time s t e p s , the take over  output produced eight  systems. T h i s  be  t r a n s f e r e d between  process can  be  quickened i f  made:  be  i s l a n d and one  f o r the  the  of  nodes  needed  i n t o two  describe  c a t e g o r i e s , one  of the  each time s t e p .  for  nodes  subsequent  which  time  for  describe  step  are  the  wave  nodes  the  at  there  will  than h a l f of the  the wave is  coordinates  profile  transfered,  c o n s i d e r a b l e time saved. More  the  l o c a t o r matrix  same f o r a l l time s t e p s . Hence, i f only the the  the  of time. On  nodes The  two  following  waves. The c o o r d i n a t e s of the  coordinates  will  the  the  to  i s l a n d are independent  p r o f i l e change with the  the  divided  which d e s c r i b e the other hand,  numerial a n a l y s i s  hours to  observation i s s t r u c t u r e may  from a  each be  coordinates  and a l l of the l o c a t o r matrix w i l l have been d i s c a r d e d . With the it  i n f o r m a t i o n from the f i r s t  time s t e p a l r e a d y t r a n s f e r e d ,  takes a s k i l l e d operator l e s s than one minute  f i l e s on the  VAX  to get each  input f o r the p l o t t i n g program. copy of the i n i t i a l time s t e p . The  time  l i n e s which  of them ready  to e d i t  the  to be used  as  T h i s i s done by making  stepping f i l e  f o r each  one  subsequent  c o n t a i n the c o o r d i n a t e s f o r  wave p r o f i l e are replaced with those f o r the p a r t i c u l a r  the time  21 step. The  time taken  steps i s cut i n steps are  to t r a n s f e r  h a l f and  added.  Just  the  i s reduced over  ten  f i l e s f o r three  time  l i n e a r l y as more  time  minutes  are  needed  to  t r a n s f e r the r e s u l t s of each time s t e p . D e c i s i o n s on changes i n the  numerical  savings i n time  program may  be  made more  and expense (computer  rapidly  time and  with  personnel)  being q u i t e s u b s t a n t i a l . Because of the s i z e model being  of the elements  displayed, i f  the e n t i r e  unsealed, there would be l i t t l e  and s c a l e of  s t r u c t u r e was  ( i f any)  the shown  knowledge gained of  the p r o f i l e of the wave s u r f a c e ( F i g . 5 ) . The wave e l e v a t i o n may  be a m p l i f i e d to f u r t h e r emphasize any  which may  be p r e s e n t .  parameter i n the  output: the  d i s p l a y the i s l a n d . the output  are  must be  the  which last  in  the  wave  surface.  nodes  is  performed  program; each  l e s s expensive and time consuming than having to  vertical  program. I f  wall's the  waves  height  factor change  r e l o a d i n g . T h i s process  the numerical a n a l y s i s program. I t the  a f t e r these nodes  of  the p l o t t i n g  r e q u i r e s i t s ' r e c o m p i l a t i o n and far  to  a f t e r they are read i n . The a m p l i f i c a t i o n  changed w i t h i n  extra  nodes r e q u i r e d  describe group  variations  by having an  number of  The nodes l i s t e d  those  A m p l i f i c a t i o n of immediately  This i s achieved  subtle  is  re-run  must be remembered fixed  are a m p l f i e d  in  too  the  much,  is  that  numerical they  may  overtop the s t r u c t u r e . Initially, i s l a n d and  wave  the waves were not a m p l i f i e d but the s u r f a c e were  scaled.  A trial  and  entire error  22 p r o c e s s has been best d i s p l a y  used ( F i g . 6  and F i g . 7)  to a c h i e v e  ( F i g . 8 ) . The v e r t i c a l dimensions  small compared  to  Consequently, the  the  horizontal  vertical  (x  and  (z) are  y)  very  dimensions.  amplified  five  times while the h o r i z o n t a l dimensions are a m p l i f i e d by  50%.  T h i s s t r e t c h e s the while s t i l l limits. (Fig.  The 9):  island across  keeping  the  slope  transformed  wave  within field  coordinates  of the  screen  acceptable  visual  is  then  of  the  amplified were  i n c r e a s e d by 50%. These two sets of a m p l i f i c a t i o n s are  done  of the  vertical  the width  nodes  as p a r t  the  dimensions are  the  plotting  process and  nodes. The r e s u l t i n g p i c t u r e i s important parameter easily  the  somewhat d i s t o r t e d but  the  ( v a r i a t i o n of the wave p r o f i l e )  seen. C r e s t s and troughs are more c l e a r l y  In an attempt to o b t a i n is  i n v o l v e a l l of  really  happening,  the  a more r e a l i s t i c time  stepping  is  more  seen.  idea of  what  procedure  was  performed on the IRIS. S e v e r a l d i f f i c u l t i e s were encountered when using the IRIS. Because  of the " l a r g e " number of  nodes  and elements, the system aborted on s e v e r a l attempts to l o a d the data f o r the time stepping d i s p l a y . Hidden c o u l d not be  performed. Though there  programs to perform  an e f f i c i e n t  use of  are s e v e r a l  t h i s procedure, newer  system c o n t a i n t h i s f e a t u r e and i t was time to t r y  line  felt  removal packaged  v e r s i o n s of  the  that  not  i t was  to a c t i v a t e t h i s  feature.  The system could produce p i c t u r e s l i k e those produced on the VAX  only f o r the i n i t i a l  h a l f megabyte memory was  c o n d i t i o n . A l a r g e p o r t i o n of used to s t o r e  the code needed  the to  23 produce  the f i l l  f o r the elements.  In an attempt  to  harness  some p o s i t i v e r e s u l t s from the e x p e r i e n c e , a l l elements  were  made t r a n s p a r e n t ; t h e i r  with the  same  c o l o u r s with which they would normally be f i l l e d . Using  this  process,  i t was  combination  of  possible these  element f i l l i n g )  problem  to  two  show  the  program was  l i m i t a t i o n s of  combining  two  processes  would have produced  s e p a r a t i n g them, w i t h i n the  o u t l i n e s were l i n e d  time  (time  steps.  stepping  when a more powerful c o u l d be o b t a i n e d .  and  the d e s i r e d r e s u l t s . By shown  the system.  to  be  effective  There would  these p r o c e s s e s on a more powerful  The computer program has  be  i s a v a i l a b l e , the proper  The e x t r a  steps have  no  system.  been c a r e f u l l y documented so  system  A  that images  been i n c l u d e d  as  comments i n t h i s v e r s i o n of the program.  4.2 WAVE EFFECTS ON HYBRID OFFSHORE STRUCTURES Hybrid s t r u c t u r e s , as the name i m p l i e s , are made up two d i f f e r e n t exploit  the  types  of members: s m a l l  capabilities  c l a s s i f i c a t i o n s of  of  and l a r g e . To  the  members a r e  system,  h i g h l i g h t e d with  of  fully  the  two  different  c o l o u r s . The l a r g e members ( h u l l s ) a r e c o l o u r e d p u r p l e while the small members ( v e r t i c a l c y l i n d e r s ) are c o l o u r e d The elements  which have been used  i n the numerical  yellow. procedure  are shown on the members. The  rectangular  analysis  of  procedure  in  large  h u l l s are members  which the  plotted  in  involves  a  submerged body  two s t e p s .  The  discretization  surface i s  divided  24 i n t o a number of small f a c e t s . These elements may by the i n t e r s e c t i o n of a planes.  Each  profile  of  these  which  elements  d i s t r i b u t i o n technique to  s e r i e s of h o r i z o n t a l and  crosssection the  for  the  Next, the c o r n e r s  elements  defined vertical  the  surface  wave  source  i s p l o t t e d . There i s no f i l l  are connected, i n an appropiate These  represents  used  elements ( F i g . 1 0 ) .  the h u l l .  be  assigned  of the  hull  manner, to form the edges of  are f i l l e d  with  a  transparent  c o l o u r so that the c r o s s s e c t i o n s which were p r e v i o u s l y drawn can be e a s i l y seen on the s u r f a c e The  c y l i n d e r s are  members are analysed  (Fig. 1 1 ) .  a l s o p l o t t e d in  two  steps.  using Morison's equation  Slender  a p p l i e d to  c e n t r e of elements formed by t a k i n g t r a n s v e r s e s l i c e s the l e n g t h of the member. I n i t i a l l y ,  the  along  the columns are p l o t t e d  as long, slender r e c t a n g l e s with a r e c t a n g l e going along l e n g t h of each element. By was  found  that  s i x t e e n short  in order chords  a trial to  and  e r r o r procedure,  produce a  are s u f f i c i e n t  the  for  circular  it  profile,  cylinders.  Arcs  c o u l d not be used since the p l o t t i n g program i s only a b l e draw s t r a i g h t first  draws  l i n e s . When the  fill  p l o t t i n g elements,  pattern  for  the  the  interior  to  program of  the  element, i n c l u d i n g the boundary. Then, the boundary i s drawn on the edge of the f i l l present. fill  area  the f i l l  already  In p l o t t i n g these elements, the o u t l i n e c o l o u r  c o l o u r for each element  result  to overwrite  is a solid  However, to show  are the same: y e l l o w . The  yellow mass in the the arrangement of  shape of a the computer  and net  cylinder. graphics  25 elements of the c y l i n d e r , they are o u t l i n e d blue Next, the  top  and  bottom  edges  s l e n d e r member are o u t l i n e d by in  another  colour:  black.  of each  (Fig.  element  of  p l o t t i n g the c r o s s  This i s  done  by  12). the  sections  joining  the  s i x t e e n chords i n a head to t a i l manner ( F i g . 13). In the d i f f r a c t i o n are d i s c r e t i z e d and is applied  to  the  theory,  the  wave source d i s t r i b u t i o n  the c e n t r e s  of  discretize  the h u l l s .  needed. Part of the output of  the elements. The  g r a p h i c a l procedure,  the  each of these elements are  not  i s the c o o r d i n a t e s of the  centres  c o o r d i n a t e s of the four corners of  the elements  c o o r d i n a t e s of the  facet used  s l i c e s are determined by adding the width of  The  1986a), has been  In the  c o o r d i n a t e s of the nodes of  (hulls) technique  these elements.  g e n e r a t i n g program, FACGEN (Isaacson, to  l a r g e members  (or s u b t r a c t i n g ) one  i n the three  half  d i r e c t i o n s to  the the  edges. These are the c r o s s s e c t i o n s that are p l o t t e d . In  this  of  the elements  of  nearest to  way,  c e n t r e of  the  only the c o o r d i n a t e s of  the h u l l s  are r e q u i r e d .  the nodes on the twelve These nodes  would normally  needed i f elements were p l o t t e d i n d i v i d u a l l y . The f a c e t s need decreases  not  (and  be p l o t t e d . number of  c o o r d i n a t e s of nodes  As  the s i z e  be  individual the  facets  the number  not determined  T h i s c o n s i d e r a b l y reduces the computational area and p l o t t i n g  of  facets increase),  which have  edges  of  increases.  effort,  storage  time r e q u i r e d .  A three-dimensional  effect  o b j e c t s which are f u r t h e s t from  is  obtained  the observer  by first  plotting so  that  26  these o b j e c t s observer.  are  overwritten  T h i s i s done by  of the elements. If the may  break down and  l a r g e changes i n  this  by  determining  the d i s p l a y mechanism and  the nodes, and  the  the c e n t r o i d s of  all  in  of  number  it  be may  a  matrix.  the  There are  in t h i s case. The colour  change  in  recompilation,  and  the  components.  pattern  The  would  method that has vary  in  specified:  it  in  been  and  program. its' In  the  r e q u i r e d to p l o t  the  were  p o s s i b l e by  the a n a l y s t  the  in a t r i a l  assigned  pattern. This f l e x i b i l i t y  not have  B),  of  requires  each f i l e  been suggested. Since  on machine  pattern  program  from machine to machine (seven may  but green  of  the p l o t t i n g  four f i l e s were  fill  to the d i s c r e t i o n  time consuming p r o c e s s .  elements  or to  be a d j u s t e d  plotting  a c o s t l y and  d i f f e r e n t c o l o u r s and fill  ways  is  fill  adjustments to  p l o t t i n g of the r i g , two  be  two  of  advantage of t h i s procedure i s  combination  e r r o r manner without  the  as an e x t r a column  been l e f t  components of the s t r u c t u r e may  Every  heading,  be determined by the a n a l y s t when the p l o t t i n g c o l o u r has  to  importance.  descriptive  i n c l u d e d i n the l o c a t o r matrix  the a n a l y s t that  Hence,  of elements, the c o o r d i n a t e s  the l o c a t o r  be performed. The  with  described also applies  which the c o l o u r code f o r the elements may may  process  i t s ' four nodes.  takes on added  consists  number of nodes and  to  i s e s p e c i a l l y t r u e f o r elements  z coordinates  output  nearest  element i s too l a r g e , t h i s  the t h i r d r e s t r i c t i o n p r e v i o u s l y  The  those  in  the  the other  the c o l o u r code be  red on machine  has  the  option  may A of  2.7 changing  the p l o t t i n g . code f o r each  Instead of having  to t r a n s f e r  d e s i r a b l e to put a l l the e d i t i n g a f t e r transfer  four f i l e s ,  file.  present d u r i n g  may  D).  be  f i l e and  more to  the t r a n s f e r has been completed. When  i s t a k i n g p l a c e with  each  (Appendix  it  the elements i n t o one  one  performed but with four f i l e s , for  element  This  r e q u i r e s that  the t r a n s f e r i n g  the  f i l e , other t a s k s can  the commands must be  do  be  entered  an  operator  must  process  to check  on  be  its'  s t a t u s when he c o u l d be doing more p r o d u c t i v e t a s k s . The  coordinates  of  t r a n s f e r e d . To minimize the output  of the  the  entire  the time  structure  spent  subsequent f i l e s  are  first  transfering  need not  files,  include  the  c o o r d i n a t e s of the nodes. I f the number of nodes i s zero indicated  i n the  second l i n e of  the o u t p u t ) , the  plotting  program i n t e r p r e t s the l i n e s which f o l l o w to be the m a t r i x . T h i s c o n s i d e r a b l y reduces needed to p l o t the  r i g s i n c e the  need not be read i n f o r each time Two  different  transformed  processes are  r i g on the VAX  the displacement  each node must be determined  the storage space and c o o r d i n a t e s of the  translate  calls perform  to  be  nodes  r e q u i r e d to  display  the once  i s known, the c o o r d i n a t e s  of  before the p l o t t i n g program can  the b u i l t - i n  of the  subroutines  rotate  origin and  the t r a n s f o r m a t i o n for the e n t i r e data set  in a few m i l l i s e c o n d s . Rotate to  time  step.  be used. With the IRIS, once the displacement i s known,  locator  and the IRIS. For the VAX,  of the o r i g i n  (as  rotated (in  tenths of  has two  parameters: the  degrees) and  the a x i s  angle about  28 which  the  rotation  correspond to the three parameters  takes  last  place.  These  three degrees  for translate  angles  of motion.  v i z the three  would  There  are  displacements  (surge, sway and heave). One of the problems encountered  was  t r y i n g to decide the order i n which these commands should be executed. Rotate r o t a t e d and  then  t r a n s l a t e means that moved a c c o r d i n g  to  the body i s the  new  first  coordinate  system while with t r a n s l a t e - r o t a t e , the t r a n s f o r m a t i o n and r o t a t i o n are system.  both  performed  Since both these  practice, neither r e q u i r e d to  the  original  the  is  r i g on  c o r r e c t . The the  VAX  of the c o o r d i n a t e s produced  for  f o r the IRIS r e v e a l s  approximates  based  on  Comparison  the VAX with each of  that r o t a t e - t r a n s l a t e  the t r u e s i t u a t i o n .  in  coordinates  are  simultaneous a p p l i c a t i o n of these two p r o c e d u r e s .  produced  coordinate  p r o c e s s e s occur s i m u l t a n e o u s l y  combination  display  in  those best  5. RESULTS AND DISCUSSION  5. 1 WAVE TRANSFORMATION AROUND AN ARTIFICIAL The use  of  a  finite  element  technique  ISLAND to  examine the  t r a n s f o r m a t i o n of waves around an a r t i f i c i a l i s l a n d suggests ( i n h e r e n t l y ) t h a t the s t r u c t u r e being a n a l y s e d can e a s i l y be d i s p l a y e d through the use of one of mediums p r e s e n t l y representation of  a v a i l a b l e . The  (numbers of  a number of  data  needed  graphical  for  nodes and elements,  such  a  coordinates  the nodes, and c o n n e c t i v i t y of the nodes) i s p a r t of the  f o r m u l a t i o n of the f i n i t e element problem. Since a graphics f a c i l i t y system are f u l l y  i s being  coloured  used, the c a p a b i l i t i e s of the  e x p l o i t e d through the use of d i f f e r e n t  and  c o n t r a s t i n g c o l o u r s t o represent the d i f f e r e n t components of the  system. The  island  i s 150.0m  i n diameter with  shallow  water depth of 3.0m, berm slope 1 i n 4 and deep water of  depth  12.0m. The waves a r e 7.5m high and have a p e r i o d of 10.0  seconds  (Fig.  14). Waves  are t r a v e l l i n g  i n the  positive  x - d i r e c t i o n . Though i t i s not used i n the a n a l y s i s , a is  included  continuity  to and  give to  the whole put  the i s l a n d  p e r s p e c t i v e . The w a l l i s b u i l t that s m a l l changes  structure  up  into  a  wall  sense  the  of  proper  of a s e r i e s of l e v e l s  so  i n runup around the i s l a n d can e a s i l y  be  seen. The l e v e l s a c t as contours and are 30 cm h i g h . The p l o t t i n g  program, developed  f o r use on  system, determines the c e n t r o i d of each those f u r t h e s t  the VAX  element and  plots  from the observer f i r s t . T h i s r e s u l t s i n the 29  30 c l o s e s t elements o v e r w r i t i n g those the problems encountered the i n a b i l i t y  of the  c o o r d i n a t e s of  i n the p l o t t i n g  program  the c e n t r o i d  plotting  of  f o r each element.  process  would  P r e s e n t l y , the c e n t r o i d of the arise  x, y  in  changes  and z  some of  in  wave  drawn. One  of  of the i s l a n d  is  to p r e c i s e l y  r e q u i r e the c a l c u l a t i o n of the fixed point  initially  determine  each element.  the  This  would  area and moment arm about With over 1100 elements,  take  considerable  more  a the  time.  i s determined by f i n d i n g the average  c o o r d i n a t e s of the s c a l e d  profile  each element.  views where  near the  Problems  there are  vertical  face  large  of  the  i s l a n d . Since the c e n t r o i d s of the elements were i n c o r r e c t l y determined, elements  f u r t h e r from  a f t e r those  to  closest  i n t r u s i o n of the the c u t t i n g  the viewer  the viewer.  yellow elements i n t o  of the  yellow elements  This  are  plotted  results  i n an  the r e d elements by the  red  Examples can be seen on the i n s i d e of the i s l a n d and on the effect  r i g h t hand s i d e  of the i s l a n d  or  elements.  in F i g .  i n F i g . 16.  15 The  i s much more n o t i c e a b l e when the g r a p h i c a l process i s  t a k i n g p l a c e on the screen. by elements nearer to the  Some of the e r r o r s are  covered  viewer. The problem becomes  apparent as the a m p l i f i c a t i o n  more  f a c t o r s are i n c r e a s e d .  Within the p l o t t i n g program, t h e r e a r e s e v e r a l  options  a v a i l a b l e t o the user (Appendix D ) . The i s l a n d may be  moved  ( r e l a t i v e to the viewer) so that the v a r i a t i o n  runup  i n the  or the s u r f a c e p r o f i l e around the i s l a n d may be more c l o s e l y examined. Each  view r e q u i r e s  the, entire structure  to  be  31 redrawn, a process which takes over c o s t l y and snapshots  four minutes. T h i s i s  time consuming p r o c e s s . The taken by a viewer  be more a p p r o p i a t e  miss some  of the  is a series  high above the i s l a n d . I t  i f the viewer  the i n f o r m a t i o n i n the  result  important  be  that  recorded s i n c e he  processes  or may  of  would  had a v i d e o camera so  scene may  a  may  respond  too  s l o w l y when r e c o r d i n g them. Sample r e s u l t s crude  of the  model of the i s l a n d  Of the i s l a n d  f o r the f i r s t  time  stepping procedure  on  are shown. There are three  views  time step ( F i g . 17 to F i g .  Then, the r e s u l t s of the f i r s t  two  time  19).  steps for the  first  view are i n c l u d e d ( F i g . 20 and F i g . 21). For each of the time s t e p s , there are three d i f f e r e n t views. The (-1.0, -1.0, the i s l a n d  -0.3) with  view (1.0,  1.0,  has  the a n a l y s t l o o k i n g  waves coming  -0.5), the a n a l y s t  hand s i d e of the i s l a n d ; the f r o n t the t h i r d view  (2.5, -1.5,  i n the  of the i s l a n d  Comparison  views but at d i f f e r e n t e v o l v e s . A movie was VAX.  be made  right  i s shown i n views,  transformed  between the  time steps to see how  the wave  same field  made using the r e s u l t s d i s p l a y e d on  U s i n g p o l y e t h y l e n e sheets,  of  second  -0.5). By combining these  can then  16 view  i s o b s e r v i n g the  the a n a l y s t has a good idea of the shape of the wave f i e l d .  first  at the back  towards him;  a  s u c c e s s i v e time steps  the were  recorded on video tape and e d i t e d so that the t r a n s f o r m a t i o n of  the s i m u l a t e d wave may  on which  a border,  subsequent sheets  title showing  be observed. and a x i s  With a master  are shown  the s t r u c t u r e  and  (Fig. border  sheet 22), were  32 o v e r l a y e d on the master sheet to ensure proper alignment the  of  i s l a n d at the end of each t i m e s t e p ( F i g . 23). The IRIS i s used to o b t a i n  processes t a k i n g p l a c e . The U n i v e r s i t y  a r e a l time p i c t u r e of  The system  of B r i t i s h  presently  Columbia  is  the  installed  a small  one  at  (half  megabyte c a p a c i t y ) . T h i s put severe r e s t r i c t i o n s on the in which  the s t r u c t u r e  was drawn  steps which c o u l d be observed. are  The  perhaps best understood when  c l o s e l y examined. were too  large  The f i n i t e to g i v e  and the  number of  time  l i m i t a t i o n s of the  IRIS  the d i s c r e t i z e d domain  elements used i n the  any  way  is  analysis  reasonable r e s u l t s  on  which  engineers c o u l d make d e s i g n d e c i s i o n s . There were 429  nodes  and  126 elements i n t h i s numerical model.  In an a n a l y s i s  the  same i s l a n d c o n f i g u r a t i o n by Talukdar (1986), over  of 1400  nodes and 400 elements were used. With the generation of the other s i d e of the i s l a n d and the v e r t i c a l  face, the  g r a p h i c s model  1140 elements.  r e q u i r e s 2826  nodes and  computer  produce a p i c t u r e of T a l u k d a r ' s a n a l y t i c a l model, over nodes and  2000 elements  would have  been needed.  A  To 6700  large  p o r t i o n of the nodes and elements i s as a r e s u l t of b u i l d i n g the  vertical  wall  i n t e g r a l part  in  wave  and  crests  in a  number  of l e v e l s  the v i s u a l i z a t i o n troughs.  A  of  of  occuring in a  the  island  r e g i o n that  (Fig.  finer  24)  form  an  the movement of mesh,  r e a l i s t i c , would be pushing the machines plan view  which  to t h e i r  shows  i s concentric  though  the more  limits.  wave  with the  A  breaking island.  T h i s i s as a r e s u l t of the very c o a r s e mesh used to generate  33 the model. breaking The  With a  f i n e r mesh,  the boundary  and non-breaking regions  exact  shape of the wave  where runup i s g r e a t e s t or  defining  would be more  field  irregular.  i s not known but  where wave breaking  the  regions  takes  place  are c l e a r l y shown.  5.2 WAVE EFFECTS ON HYBRID OFFSHORE STRUCTURES The  program which generates the  the semi-submersible r i g (RIG)  computer g r a p h i c s model i s r e s t r i c t e d to r i g s  of  which  c o n s i s t of r e c t a n g u l a r h u l l s and v e r t i c a l c y l i n d e r s . The r i g c o n s i s t s of with 8  4 vertical  elements. The  20.0m h i g h , centred apart  c y l i n d e r s , each h u l l s are  at 40.0m  of diameter  100.0m long,  below the  30.0m  wide,  s u r f a c e and  70.0m  ( F i g . 25). The elements on the h u l l are 10.0m  These parameters (1986b). The 12.0sec and  d e f i n e the  disturbance has  DNV r i g  i s 5.3m  a principal  d i s c r e t i z a t i o n of the r e c t a n g u l a r the program FACGEN different  analysed  high,  wave  d i r e c t i o n of  which, i n t u r n ,  period  7 ) . RIG i s , t h e r e f o r e ,  60°. The using eight  (IS = 2, 4, 5  r e s t r i c t e d by the c a p a b i l i t y  i n FACGEN. As i t i s p r e s e n t l y w r i t t e n , one  f e a t u r e s of FACGEN i s  the d i s c r e t i z e d  form of  that i t i s capable a pair  of  configurations,  and  cylinders  Isaacson  i s r e s t r i c t e d to  i s s u i t a b l e f o r use on four of these cases  the  square.  h u l l s i s performed  s t r u c t u r a l shapes. Of these e i g h t  options  by  has a  RIG  these  12.0m  of  of h o r i z o n t a l  of of  generating rectangular  (IS = 7 ) . On c a r e f u l examination of the l i s t i n g of  the subroutine  which performs t h i s process  (BDBOX2), one can  34 see  that  with  the  statements, the  of  program may  d i s c r e t i z e d form h u l l s . A large  addition  of  an  be  five  or  six  extended to  unlimited  number  number of r e c t a n g u l a r  executable  generate  of  rectangular  hulls is unlikely  occur i n p r a c t i c e but three or four h u l l s are not These  changes  in  FACGEN  would  further  the  to  uncommon.  enhance  the  c a p a b i l i t i e s of RIG. RIG  can be  used with  an u n l i m i t e d  piercing vertical, circular cylinders the h u l l s .  There  are  several  a n a l y s t . For each c y l i n d e r , used to represent  the  number of  which extend down  options  available  c y l i n d e r or the  second of these o p t i o n s  to get  a  feature  is  accurate levels  computer  When combined with the  in  representation would  numerical  procedure. the h u l l s .  coordinates  of the c e n t r e  curved  that  members  the  elements  cylinders  Their position of  may  model  The  and  i s an of  in  the  used be  this  number  positioned  i s defined  by  the  the plane which occurs at  the  i n t e r s e c t i o n of the v e r t i c a l c y l i n d e r and the f r e e As an example of  hulls,  surface,  the  the prototype.  to  The  anywhere on  the  ensuring  of  correspond  in  i s not needed. However, i n order  d i s p l a y of  important  chords  number of l e v e l s  the c y l i n d e r s are u s u a l l y c l a s s i f i e d as slender  to  to the  the diameter, number of  each c y l i n d e r may be v a r i e d .  the  surface  the c a p a b i l i t i e s of  surface.  the program, F i g .  26  shows a r i g with three v e r t i c a l c y l i n d e r s , each of d i f f e r e n t - diameter and number  of l e v e l s . I t  program w i l l be extended t o i n c l u d e  i s a n t i c i p a t e d that i n c l i n e d slender  the  members  35 where the a b i l i t y to change the numerical  analysis  the number of elements used  on each c y l i n d e r  c r u c i a l . These i n c l i n e d members in wave kinematics  would have s m a l l e r  analysis.  more  changes  per u n i t l e n g t h than v e r t i c a l members  the same environment and hence, the  would become  in  WELSAS2  in  may use longer elements  (Isaacson,  1986a)  is  in  capable  of  a n a l y s i n g s t r u c t u r e s which c o n t a i n i n c l i n e d s l e n d e r members. However, the  data needed  for a graphical  description  of  these members i s not generated.  For each a n a l y t i c a l  element,  the c o o r d i n a t e s of p o i n t s along  i t s ' edges would be  needed.  These are determined Consequently,  RIG  for vertical may  be  slender  thought  members i n  of  as  RIG.  being  a  p o s t - p r o c e s s i n g program f o r FACGEN. Three views of (Fig.  27),  the r i g are shown. They  views  ( F i g . 29). They were obtained by changing the normal  vector  the  normal  c o l i n e a r ; the system p l a n view,  ( F i g . 28)  vector  and  aborts when  f o r example,  and  right  plan  side  so that  elevation  are the  viewing  vector  t h i s occurs.  the ( d e f a u l t )  a r e not  To get the  normal  vector  is  changed from (0.0, 0.0, 1.0) to (0.0, 1.0, 0.0). The r i g , as shown i n -2.0,  the r e s u l t s ,  -1.0),  i s from  the viewing  vector  the d e f a u l t v a l u e .  A time stepping procedure was performed u s i n g There are  (-3.0,  10  time steps  evenly  spaced through  p e r i o d . The motioms are a m p l i f i e d 100 times  WELSAS2. the  wave  so that they may  be more e a s i l y seen. The r i g a c t s as a r i g i d body. Once  the  coordinates  the  of  any  point  on  the  r i g are  known,  36 c o o r d i n a t e s of the The  other p o i n t s may  r e s u l t s of the f i r s t  and F i g . 31. A  was  made.  p r o d u c t i o n process was was  in  sheet's. I t  was  even more  aligned  the  i n t h i s case  The  major  same  than  time  and  title  and  to keep  order  i n which  the VAX. varying  border  also  steps were  different  true  overcome  one master sheet  the four  p l o t t i n g the r i g on the IRIS was  previous  the  subsequent sheets  showing for  each  only. executed  in  to the order  for  T h i s i s because the IRIS does not allow for f i l l i n t e n s i t y . Instead  elements on  the h u l l  h u l l s , the s o l i d  the e x t e r i o r  of p l o t t i n g the  before p l o t t i n g  fill  were p l o t t e d b e f o r e  o u t l i n e of  the o u t l i n e  of  r e p r e s e n t i n g the s u r f a c e s of the the elements which  of the  h u l l . No  change was  required in  the r i g ,  only the  initial,  sharp c o n t r a s t to the process used to d i s p l a y the  each time s t e p must be  the  on the  the  undisplaced  s t r u c t u r e has been s t o r e d i n the computer's memory. T h i s  on the IRIS where the new  the  cylinders.  Because of the ease with which the IRIS can perform t r a n s f o r m a t i o n of  of  hull  formed the g r i d  a l g o r i t h m f o r the p l o t t i n g of the v e r t i c a l  in  the that  so that the  step showing the d i s p l a c e d r i g and a border The  in  the  be shown. T h i s was  by u s i n g p o l y e t h y l e n e sheets with the a x i s , border  as  f o r the i s l a n d  movement of the r i g c o u l d  over  each sheet so  position  important  of the r i g  difficulty  t r y i n g to a l i g n  border  determined.  time steps are shown i n F i g . 30  v i d e o showing the movement  several periods  its'  two  be e a s i l y  is  island  c o n f i g u r a t i o n of the s t r u c t u r e f o r  s t o r e d . T h i s put severe  limitations  37 on the c a p a b i l i t y of the IRIS  to show more than a few  steps. While there are many elements which have been in d i s p l a y i n g the many more time each  time  r i g , the  steps because  step  need  not  s t r u c t u r e i s drawn, the steps may  be  system i s  displayed.  results.  the s t r u c t u r e at be  stored.  r e s u l t s of a  Once  filled handling  the end the  may  be  time  performed  any s i z e increment and may  u n t i l a l l the i n f o r m a t i o n  of  initial  l a r g e number of  Time s t e p p i n g  forwards or backwards and i n repeated  capable of  time  has been drawn from  be the  6. CONCLUSIONS AND  Two d i f f e r e n t  RECOMMENDATIONS  methods have  been used  to d i s p l a y  i s l a n d and the r i g . For s t r u c t u r e s where the f i l l the  same f o r a l l elements,  the program  the  pattern i s  developed f o r the  i s l a n d may be used. When a l l of the elements do not have the same f i l l  p a t t e r n , the program developed to d i s p l a y the r i g  should be used. In a d d i t i o n , the i s l a n d ' s program a l l o w s the c o l o u r code of each element  to be pre-determined whereas  t r i a l and e r r o r method may be used f o r the r i g . The r i g be used to  d i s p l a y the r e s u l t s  elements c o n s i s t i n g of valid for  up to  of numerical a n a l y s e s 4 nodes while  the r i g may  elements w h i l e  e a s i l y be extended the  island's  accomodate elements w i t h d i f f e r e n t  6.1 WAVE TRANSFORMATION The r e s u l t s  of an  an  of a  i n t o account have  be  noded  revised  to  patterns.  computer  artificial  r e f r a c t i o n , d i f f r a c t i o n and energy breaking  fill  AROUND AN ARTIFICIAL  analysis  t r a n s f o r m a t i o n around  may  is  program  to i n c l u d e e i g h t  program  may with  the i s l a n d  f o r elements w i t h up to 8 nodes. The g r a p h i c a l  a  ISLAND model  island,  of  wave  taking  wave  d i s s i p a t i o n due to  wave  been d i s p l a y e d g r a p h i c a l l y .  The  e v o l u t i o n of the wave f i e l d has been obtained by the use of a  time  stepping  procedure.  d i s c r e t e p o i n t s around the  By  positioning  the i s l a n d , the  himself  viewer may  at  observe  f r e e s u r f a c e . I f the viewer remains s t a t i o n a r y f o r some  time, he may observe The VAX  has  the changing face  been used  to  of the wave  simulate these  38  field.  conditions.  By  39 combining  these two  processes, the  idea of how the wave  viewer may  get a  f i e l d changes. For the more  good  realistic  case of both the observer and wave f i e l d changing,  the  IRIS  has been used. Some of the r e s o l u t i o n o b t a i n e d with the has been l o s t . Because of the s m a l l c a p a c i t y of the  VAX  system,  the p i c t u r e s were not l i k e those on the VAX. I n s t e a d of the elements  being f i l l e d with the a p p r o p i a t e c o l o u r , they  o u t l i n e d with the c o l o u r . Hidden performed.  l i n e removal  Newer (and more powerful)  a l l e v i a t e these two problems. changes, can  be  used on  elements,  a  would be  produced.  was a l s o  not  systems can be used  The same programs, with  these  more r e a l i s t i c  were  newer systems.  computer model  Engineers  can  then  to  minor  With  more  and wave  field  confidently  base  design d e c i s i o n s on these models. Galvin H  0  and L  (1968) suggests a a r e the deep  0  r e s p e c t i v e l y and m used  to  classify  /3 = H / L m 0  water wave height  i s the slope different  ( s p i l l i n g , plunging, definition,  parameter,  types  collapsing  f o r a changing  and wave  of the berm  or  of  where  2  0  length  which can  breaking  surging).  be  waves  Using  this  s l o p e , the type of b r e a k i n g  wave  can be shown by a s s i g n i n g a c o l o u r code t o the b r e a k i n g wave w i t h i n the numerical program. There  is still  a  l o t more work to  be done before  d e s c r i p t i o n of the  transformed wave f i e l d  new techniques a r e  developed,  numerical  program.  If  the  they may format  unchanged, the p l o t t i n g programs  i s accurate.  the As  be i n c l u d e d i n the of  the  results  which have been  is  developed  40 may be used with  confidence.  6.2 WAVE EFFECTS ON HYBRID OFFSHORE STRUCTURES A computer program (RIG) has nodal c o o r d i n a t e s  been developed to arrange  of a semi-submersible  that the r i g can be  the  r i g i n a form  such  e f f i c i e n t l y p l o t t e d . P l o t t i n g has  been  c a r r i e d out on the VAX/VMS system. The program developed plot  the r i g on the VAX i s s u i t a b l e f o r use with  elements. The data i n a form s i m i l a r  four  needed to p l o t the r i g has been to the data of a f i n i t e element  to  noded  arranged analysis.  Three views ( p l a n , e l e v a t i o n and r i g h t  s i d e ) show the layout  of the r i g . The p o s i t i o n  the end of each  step of a s i m u l a t e d s u c c e s s i v e views design  of a r i g a t  storm  shows  i s d i s p l a y e d . Comparison how the  time  between  r i g would behave  i n the  storm. For a more i n f o r m a t i v e p r e s e n t a t i o n , the  time movement has been shown on the IRIS. There i s no  real limit  on the number of time steps which can be used. It  i s a n t i c i p a t e d that c r o s s b r a c i n g w i l l e v e n t u a l l y be  added to the r i g . The facet generating limited  to  vertical  cylinders.  c o o r d i n a t e s , as opposed to  program i s  The  use  presently  of  spherical  the p o l a r c o o r d i n a t e s  presently  employed, i s suggested f o r the c r o s s b r a c i n g to be i n c l u d e d . There w i l l be problems cylinders.  The  at the i n t e r s e c t i o n  development  d e s c r i p t i o n of the j o i n t e f f o r t . Since these displacements  of  of  an  accurate  requires considerable  d i s p l a y s are only  the  of two or  r i g , the j o i n t s  more  graphical thought  and  used to examine  the  need  not  be  well  41  defined. The  addition  of  d i f f i c u l t i e s . Their used to  draw the  where the  sixteen  description edges of chords  manner. The whipping of r i g may  lines  should  can be s i m i l a r  the elements were  joined  the c a b l e s  and  pose  no  to the  one  of the  in  a head  cylinders to  the bobbing of  tail the  then be observed s i m u l t a n e o u s l y .  The water s u r f a c e for  mooring  completeness s i n c e  water s u r f a c e  that  can a l s o i t is  i n the  display  the r i g ' s motions and not  i s examained  r i g ' s motions have been  be i n c l u d e d  i n t h i s a n a l y s i s . Since  calculated,  the v e l o c i t y  boundary  surface  profile.  equation  can then  be  used to  the  potential  of the surrounding f l u i d can be determined. The dynamic surface  the  give  free the  7. REFERENCES  1.  Berkhoff, J . Refraction Coastal  C.  W.,  (1972)  Diffraction", Engineering  "Computation Proc.  of  13th  Conference,  Combined  International  ASCE,  Vancouver,  pp471-490 2.  Booij,  N. ,  (1981)  Waves  on  Water  with  Non-Uniform Depth and Current", t h e s i s presented to  the  Technical  The  University  Netherlands, for 3.  "Gravity  in p a r t i a l  the degree  Byrne,  D.  of  Delft,  at  Delft,  f u l f i l m e n t of the  requirements  of Doctor of Philosophy  J.,  (1984)  "Pseudo  3-D  projections  from  i s o p a r a m e t r i c s u r f a c e s " , E n g i n e e r i n g i n Computers Voll, 4.  1984,  Sept pp219-226  De Jong, J . J . A. Construction  of  and Bruce, J . C , a  (1978) "Design  Caisson-Retained  Island  and  Drilling th  P l a t f o r m f o r the Beaufort Sea", O f f s h o r e Technology  Proceedings,  Conference, OTC  10  Annual  Paper 3294, Houston,  Texas 5.  DI-3000 User's Guide, March  6.  G a l v i n , C. J . ,  1984  (1968) "Breaker  Three Laboratory  Beaches", J .  Type C l a s s i f i c a t i o n Geophys. Res.,  Vol  on 73,  pp3651-3659 7.  IRIS User's Guide, V e r s i o n  8.  Isaacson, M. de S t . The  Coastal  Zone",  Q.,  1.0  (1985) "Wave T r a n s f o r m a t i o n  1985  Australasian  Conference  C o a s t a l and Ocean E n g i n e e r i n g , V o l I I , pp25-32 42  in on  43 9.  Isaacson, M. de program  S t . Q.,  WELSAS2",  (1986a) "User's  Department  of  Guide to the  Civil  Engineering,  U n i v e r s i t y of B r i t i s h Columbia, Vancouver, Canada 10. Isaacson, M. de S t . Q., (1986b) "Wave E f f e c t s on O f f s h o r e S t r u c t u r e s of E n g i n e e r i n g Report,  A r b i t r a r y Shape",  Department  of  Hybrid  Coastal/Ocean  Civil  Engineering,  U n i v e r s i t y of B r i t i s h Columbia, Vancouver, Canada 11. Isaacson, M. de  S t . Q. and  transformation  around  T a l u k d a r , K., (1986)  artificial  "Wave  islands",  Ocean  S t r u c t u r a l Dynamics Synposium 12. M u f t i , A f t a b A., (1983) "Elementary 13. Noma,  Tsukasa  "ANIMENGINE: Proceedings, 14. Rogers,  and An  Tosiyasu  Engineering  Graphics  David  Kunii,  Computer Graphics"  F.,  L.,  Animation  (1985) System"  I n t e r f a c e 1985 (1985)  "Procedural  Elements  for  Computer G r a p h i c s " , McGraw H i l l , Inc 15. Sarpkaya, T. and Isaacson, M. , (1981 ) "Mechanics of Wave Forces on O f f s h o r e S t r u c t u r e s " , Van Norstrand  Reinhold,  New York 16. Talukdar, Kushal, Artificial  (1986) "Transformation  Islands",  thesis  of Waves around  presented  in  partial  f u l f i l m e n t of the degree of Master of A p p l i e d Science to The U n i v e r s i t y of B r i t i s h Columbia, Vancouver, Canada i n April  1986  17. UBC Graphics Lab, "Computer Communications" 18. Walker,  J.  and  Headland,  J.,  (1982)  "Engineering  Approach to N o n - l i n e a r wave s h o a l i n g " , Proceedings,  18^  44 I n t e r n a t i o n a l C o a s t a l E n g i n e e r i n g Conference,  Ch 34,  pp  523-542 Weggel, Design",  J.  R.,  (1972)  Proceedings,  "Maximum 13^  Breaker  Height  International  E n g i n e e r i n g Conference, Ch 21, pp 419-431  for  Coastal  APPENDIX A : THE ISLAND A.1  USER'S MANUAL FOR DATGEN  The data g e n e r a t i n g program DATGEN generates the c o o r d i n a t e s of the nodes circular  and the  island  c o n n e c t i v i t y of these  with  linearly  sloping  nodes f o r  berm.  The  the data  r e q u i r e d to d e f i n e the system i s as f o l l o w s :  HD,  HS, AM,  DIA  (4F10.4)  HD = deep water depth HS = shallow water depth AM = berm slope DIA = i s l a n d diameter  NTT, NTT  SC  (I4,F8.4)  = p l o t t i n g code f o r the f i n i t e element mesh = 0 no p l o t = 1 plot  required  required  SC = s c a l e f a c t o r  ANINC  for plot  (F10.4)  ANINC = angular increment f o r each element  NR,  NA  (214)  NR = number of r a d i a l of the boundary  ( i n degrees)  island  increments from the v e r t i c a l to  position  i s loacted  45  where  the  face  radiation  46  NA = number of angular  increments (=  180 ANINC  )  DELTAT, TIME (2F10.4) DELTAT = time step  increment  TIME = d u r a t i o n of storm  DR(I) DR(I)  (consistent units)  (F10.4)  = r a d i a l increment from the l a s t element, from the v e r t i c a l face of the i s l a n d ; NR of these  values  J1 , J2, J3, J4 (414) J1  = p r i n t option  f o r data  = 0 no p r i n t  necessary  = 1 print J2 = breaking  required option  = 0 breaking  not c o n s i d e r e d  = 1 consider  breaking  J3 = g r i d p l o t t i n g  i n the a n a l y s i s  option  = 0 g r i d not needed = 1 generate g r i d J4 = g r a p h i c a l p l o t t i n g  option  = 0 f o r movie (IRIS) = 1 for s t i l l  frames (VAX)^  T, H, G (3F12.4) T = i n c i d e n t wave p e r i o d  starting  there  are  47 H = i n c i d e n t wave h e i g h t G = acceleration  due to g r a v i t y  With the data s u p p l i e d for  f o r DATGEN, the parameters  required  the wave t r a n s f o r m a t i o n program w i l l be determined.  A.2 EXECUTION OF DATGEN The program i s f i r s t  compiled.  R *FTN SCARDS = DATGEN The program i s then R -LOAD 5 = input  executed. file  If a p l o t of the f i n i t e required,  i t may  1 6 = output  element  file  mesh generated by DATGEN  be generated on  e i t h e r the p r i n t r o n i x  the QMS p r i n t e r , for  the p r i n t r o n i x p r i n t e r : CON *PRINT* RMPROUTE=CIVL R *PXPLOT COPY -PLOT# *PRINT* REL  for  *PRINT*  the QMS p r i n t e r : CON *PRINT* RMPROUTE=CNTR R *QMSPLOT COPY -PLOT# *PRINT* REL  *PRINT*  1 is or  48 A.3 EXECUTION OF ISLAND.TIME The  f i n i t e element  program ISLAND.TIME i s f i r s t  compiled.  R *FTN SCARDS = ISLAND.TIME With  input from DATGEN, the program i,s executed. R -LOAD+NICL:NEWSPARSPAK 1 = island 2 = z-coordi  geometry  for VAX  nat es for  5 = out put  file  6 = finite  element  I output  1 = wave height  around  8 = wave height  at  9 = wave elevation 12 = island  IRIS  file island  nodal at  nodal  points points  geometry  The output, i n a form compatible with the input requirements of the g r a p h i c s program, takes the f o l l o w i n g  form:  TITLE NNODES,NELEM,N X ( I ) , Y ( I ) , Z ( I ) , I = 1, NNODES I , LOCATRd ,J) , J = 1 , 9 The  l a s t column  assigned t o each  i n the l o c a t o r matrix i s the c o l o u r element.  code  49  A.4 EXECUTION OF GRAPHICS PROGRAM The commands r e q u i r e d a r e c o n t a i n e d i n a source  file,  Once the  to  output  files  have been  transfered  ROCK.  the VAX  system, the f o l l o w i n g command i s entered: $waveplot Follow the i n s t r u c t i o n s on the  screen ( i . e . h i t the  RETURN  key) and e n t e r : >sour rock In t h i s f i l e ,  the t y p i c a l sequence of commands i s :  read geom t i m e l . defn dsca  1.5,5.0,  defn view -1.0,-1.0,-0.4, p l o t elem 0813 plot axis p l o t head 1, plot  fram  Changes to the a m p l i f i c a t i o n recompilation waveplot.  relinking  of  the  graphics  a  program,  T h i s i s done as f o l l o w s :  $fort  waveplot  $di31oad For a r e a l field,  and  f a c t o r f o r the waves r e q u i r e  waveplot  share ex  time examination  of the e v o l u t i o n  of the wave  the user s i g n s on t o the IRIS and i s s u e s the command:  $run  histisle  Using the mouse and the legend d i s p l a y e d on the screen,  the  user may r o t a t e the i s l a n d , move i n or away, move the i s l a n d to the  side  or c a l l  any  one  of a  number  of  available  50 options. Any  changes  recompiltion $fort  to  this  would  also  require  its'  and r e l i n k i n g . The commands to be used a r e :  histisle  $irload  program  histisle  APPENDIX B : THE RIG  B.1 User's Manual f o r FAGGEN (Isaacson, 1986a) The  data  generating  conjunction GENeration for the  be  in  Program f o r WELSAS2) generates the f a c e t  geometry  f o r FACGEN  program  FACGEN.  used  (FACet  the  l a r g e members  file  should  FACGEN  with  ( h u l l s ) and  handle up t o 8 d i f f e r e n t input  RIG  program  has the  s t r u c t u r a l shapes. (unit 3)  i s of  capability  In g e n e r a l ,  the f o l l o w i n g  to the form  (Isaacson,1986):  IR,  IS  (215)  IR  = runup o p t i o n c o n t r o l tag  IS  = s t r u c t u r e shape o p t i o n c o n t r o l tag  ( I P ( I ) , 1=1,5)  (515)  IP  = integer structure  IP(1)  = number of l e n g t h d i v i s i o n s  IP(2)  = number of width  IP(3)  = number of height d i v i s i o n s  IP(4),IP(5)  are  not  parameters  divisions  defined for  conf i g u r a t i o n  (RP(I), 1=1,5) RP  = real structure  RP(1)  = length  RP(2)  = width  RP(3)  = height  (5G20.5) parameters  51  this  structure  52 RP(4)  .= z - c o o r d i n a t e of h o r i z o n t a l  RP(5)  = horizontal  distance  axis  between the two  cylinder  axes  The output f i l e  to FACGEN  ( u n i t 4) c o n s i s t s of the f o l l o w i n g  parameters:  IR,  (215)  ( I P ( I ) , 1=1,5)  (515)  (RP(I),  (5G20.5)  N N  IS  1=1,5)  (15)  = number  of f a c e t s  ( X ( I ) , Y ( I ) , 1(1),  1=1,N)  X, Y, Z = x, y, z c o o r d i n a t e s  The r e s t  of  the  output  interaction analysis  and i s  particular configuration, follows: 1,7, 10,3,2,0,0, 100.,30.,20.,-40.,70.,  is  (3G20.5)  of the f a c e t  used  in  the  not p e r t i n e n t  the input  file  centres  wave-structure here. For  to  FACGEN  this is  as  53  B.2 USER'S MANUAL FOR RIG The  input  to RIG takes the f o l l o w i n g  TITLE TITLE  (A80)  = d e s c r i p t i v e heading  NUMCYL NUMCYL  form:  (16)  = number of v e r t i c a l c y l i n d e r s  i n the r i g  DIA,  (4F7.2)  X, Y, ZED, NCIRCL, NLONG  DIA  = diameter of v e r t i c a l  cylinders  X  = x-coordinate of c e n t r e of c y l i n d e r  Y  = y - c o o r d i n a t e of c e n t r e of c y l i n d e r  ZED  = length  of c y l i n d e r  NCIRCL  = number  of p o i n t s  in z-direction to be used to represent  the  circle NLONG  = number of storeys the  which are used to  cylinder  HS, TP, A, NINCR  (3F11.2,  16)  HS  = s i g n i f i c a n t wave height  TP  = peak wave p e r i o d  A  = charcteristic  length  of barge  NINCR  = number  time  intervals  the  represent  of  displaced  shape  for  i s to be p l o t t e d  which  54  (RAO(I), RAO  1=1,6)  (6F11.3)  = Response Amplitude origin  Operator amplitudes of the  of the barge  RAO(I,1) => surge RAO(I,2)  => sway  RAO(I,3)  => heave  RAO(I,4)  =>  RAO(I,5)  => p i t c h  RAO(I,6)  => yaw  roll  1=1,6)  (PHASE(I), PHASE  The  = RAO  relevant  the input  (6F11.3)  phases  part of the output  from FACGEN i s a l s o part  f o r RIG.  B.3 EXECUTION OF FACGEN FACGEN must f i r s t  be compiled.  R *FTN SCARDS = FACGEN  The  f a c e t g e n e r a t i n g program i s then R -LOAD 3 = input  file  executed.  I 4 = output  file  I  of  55  B.4 EXECUTION OF RIG RIG  must f i r s t  be compiled.  R *FTN SCARDS = RIG  The  program i s then executed. 1 = cylinder  R -LOAD  2 = cross  outline section  3 = outline  of  4 = out put  file  5 = i nput  of  hull  element  description  file output  8 = echo  print  input  = cross  file  and  print  10  file  2  7 = echo  9 = test  output  output  geometry  out put  hull  file  I  file  6 = grid  output  file file  I 2  file section  of  cylinder  output  file  B.5 EXECUTION OF GRAPHICS PROGRAM All  of the commands needed t o p l o t the semi-submersible r i g  are c o n t a i n e d i n a source f i l e s have been t r a n s f e r e d system  (Appendix  file,  SUB. A f t e r the four  from the MTS  system  output  t o the VAX  C ) , the f o l l o w i n g command i s e n t e r e d :  $rigplot Follow the i n s t r u c t i o n s on the screen ( i . e . h i t the RETURN key) and e n t e r : >sour sub  56  In t h i s f i l e ,  the sequence of commands i s :  read geom ele22. p l o t elem 0404 p l o t head 1, plot axis plot  frame  read geom h u l l 3 3 . p l o t elem 0226 read geom c y l 1 1 . p l o t elem 0555 read geom c y l 2 2 . p l o t elem 0909  For a  structure  p l o t t i n g time To see the  with  496  nodes  i s approximately  and  704  elements,  the  2min 30sec.  r e a l time p r o c e s s e s ,  the user signs  on to the  IRIS and i s s u e s the command: $run  histrig  Using the mouse and the legend d i s p l a y e d on the screen, user may r o t a t e the r i g , move  i n or away, move to the  or c a l l any of a number of other o p t i o n s .  the side  APPENDIX C : TRANSFER VAX/VMS F I L E  The  c o l o u r e d g r a p h i c s f a c i l i t i e s used t o d i s p l a y the f i g u r e s  are contained  i n the VAX/VMS whereas  programs were compiled The  TO THE UBC/MTS SYSTEM  and  executed  the data  generating  on the UBC/MTS  system.  f o l l o w i n g sequence of commands i s d e s c r i b e d i n d e t a i l i n  the p u b l i c a t i o n "Computer Communications" a v a i l a b l e from the UBC  Graphics  Laboratory.  Commands  are  in  italics  while  comments are i n normal type. I t i s assumed t h a t the user f a m i l i a r with  the signon  procedures  f o r the VAX/VMS  is and  UBC/MTS systems.  signon on the VAX t e r m i n a l SD ALLOC TXAO: KERMIT  D i a l up UBC (228-1401) on modem SET  LINE TXAO:  CONN G "ccid"  SIG  "password"  RUN  *KERMIT  SERVER Ctrl  ] C  GET  "filename"  Repeat the GET command rec ieved 57  until a l l  f i l e s have  been  58  BYE  Perform (Appendix LO  the D)  appropiate  plotting  sequence  APPENDIX D : PLOTTING THE OUTPUT ON THE VAX/VMS SYSTEM In the  p l o t t i n g the elements, user.  They  commonly used  may as  be  s e v e r a l o p t i o n s are a v a i l a b l e used  individually  a combination  options. I n i t i a l l y ,  a trial  deide which f e a t u r e i n each  of  any or  and e r r o r  but  are  a l l of  to more  these  method i s used  to  o p t i o n and what combination  of  such f e a t u r e i s best s u i t e d f o r the p a r t i c u l a r  situation.  Once these f e a t u r e s have been determined, they can be s t o r e d in a  source f i l e  so that  on one  call  to  this f i l e ,  options can be executed s e q u e n t i a l l y . The o p t i o n s a r e :  defn view  x,y,z]  norm  x,y,z]  move  x,y,z]  axis  x,y,z]  alab  u,v,w]  asca  x]  zoom  x]  dsca  x,y,z]  proj  i]  port  i]  head  new heading]  lege  i ] [legend]  p l o t node  i]  elem  abed]  axis  i] 59  the  60 lege [ i ] head [ i ] f ram  save  [file]  logf  [file]  sour  [file]  paus wipe [ i ] stop exit  In the  call  "plot  element", a  4-digit  a c c o r d i n g to the f o l l o w i n g scheme: plot where  elem abed  numbering, a = 0 no numbering of nodes = 1 numbering of nodes outline colour, b = 0 white = 1 red = 2 green = 3 dark blue = 4 light  blue  = 5 yellow = 6 scarlet  number  is  entered  61 = 7 white = 8 black = 9 green interior  fill  pattern, c  = 0 no =  fill  1 solid  fill  = 2 transparent interior If  parameter b  parameter for  fill  colour,d i s zero,  i t is,  by d e f a u l t ,  s e t equal  to  d.  example  0002 => g r e e n 0022 =>  outline  transparent outline  green  fill  and  green  APPENDIX  The  video  tape  applications propogation  shows two  E  :  THE  MOVIE  examples  of  around an a r t i f i c i a l  first  artificial  example  i s of  exploration  Beaufort  Sea,  in  at the  approach the i s l a n d  i s l a n d and a second to wave  an  rig.  a n a l y t i c a l model  1 0 times  the  Mackenzie  northern  Bay  area  boundary of  the three views,  If we  past the  s i d e and  focus  island,  where waves which  we can  see  t h i s time  the viewer  f r o n t of  the  left  waves almost  that the  a region at the  have t r a v e l l e d around  second view  from t h i s  shows the same has been  island.  the  be  by a  relatively  of the  is  the moves  becoming  back of the  island  the other s i d e  of  side.  i s l a n d and waves,  but  moved so that we can see  the  Waves are  hand corner  wave as i t wave  is  simulated.  back of the i s l a n d are f a c i n g  the i s l a n d c o l l i d e with those  lower  Waves  s c a l e i s exaggerated  our a t t e n t i o n on one  s m a l l e r . There i s a l s o  The  an  the  the i n c i d e n t wave  f a c t o r of about 3 so that the berm appears to be left  of  Canada.  so that storm c o n d i t i o n s may  In t h i s p r e s e n t a t i o n , the v e r t i c a l  viewer.  of  from the upper r i g h t hand corner of  s c r e e n . For each of  steep. The  wave  i s l a n d . I t i s not u n l i k e the islands, used f o r o i l  and gas  repeated  graphics  i n o f f s h o r e hydrodynamics: one d e a l s with  induced motions of a semi-submersible The  computer  now  approaching  screen. At  from  the  the f r o n t ,  the  overtop the v e r t i c a l w a l l . Towards the back  of  the i s l a n d , the waves are c l e a r l y not as h i g h . Wave breaking has occured.  The  yellow  area 62  defines  the  region  where  63  breaking  takes  place.  When  waves  d i s s i p a t e d . S i n c e wave energy of the wave h e i g h t , the  break,  energy  i s p r o p o r t i o n a l to the  energy l o s s r e s u l t s  in a  is  square  reduction  in wave h e i g h t as the waves t r a v e l past the i s l a n d . In the l o o k i n g at  final  view of  the f r o n t  Waves are now  the  i s l a n d and  and r i g h t  wave,  hand s i d e  approaching the i s l a n d  we  are  of t h e ' i s l a n d .  from the lower  right  hand s i d e of the s c r e e n . The decrease i n wave height as the wave moves around the r e s u l t s p r e s e n t e d as  island  i s once  again seen. With  shown i n these  t h r e e views, an  engineer can c o n f i d e n t l y determine the p o s s i b l e of v e s s e l s  the ocean  maneuvering  i n the v i c i n i t y of the i s l a n d or the movement  of  any of the sediment over the berm of the i s l a n d . The second  example  of  the  application  of  computer  g r a p h i c s i n ocean e n g i n e e r i n g i s a semi-submersible r i g . The numerical model of the r i g of a  r i g which  motions a r e  i s very  shown  i s shown: t h i s  much l i k e  for a  wave  meters and have been a m p l i f i e d  the Ocean  h e i g h t of  Storm c o n d i t i o n s  having the  to 15  water s u r f a c e  (at the top  been shown. The r i g ' s motions  amplitude motions  that  waves. The  would  any i r r e g u l a r  can be  Ranger.  The  approximately  5  are simulated  by  l o c a t i o n of the  of the v e r t i c a l columns) has not  ocean engineer so  model  100 times i n r e l a t i o n to the  s t r u c t u r e ' s dimensions. r i g respond  i s a crude  predicted  like  to examine  movements i n the  or  design  the large  office  before c o n s t r u c t i o n begins. The motions of p e r s o n n e l who are at the  working  deck  can  then  be  calculated.  A  worker  64 s t a n d i n g a t t h i s l o c a t i o n may  experience unacceptably  large  motions which would preclude the c o n t i n u a t i o n of h i s d u t i e s .  j  65  Fig.  1  Sketch of an a r t i f i c i a l  (Adapted from De Jong and Bruce,  island 1978)  66  y  Fig. 2  Sketch of boundaries  (Adapted from Talukdar,  1986)  Fig. 3  Sketch of semi-submersible r i g  (Adapted from Sarpkaya and Isaacson,  1981)  68  Fig. 4  An element  f o r a s l e n d e r member  (Adapted from Isaacson, 1986b)  Fig. 5  U n d i s t o r t e d d i s p l a y o f an a r t i f i c i a l  island  F i g . 10  Mesh on e x t e r i o r of  hulls  Fig.  11  Outline of h u l l s  F i g . 12  Arrangement o f elements on c y l i n d e r  ig.  13  Edges o f elements on the c y l i n d e r  78  Fig.  14  Dimensions of the  island  i g . 15  I n t r u s i o n o f y e l l o w elements  81  83  84  85  WAVE F i g . 22  HEIGHT  Master sheet  WITH  BREAKING  for island  SHOWN  89  30m  E  O lO  Side  view  20m lODm t  4 columns, 12 m diameter  35 m  30m •»  40m  Elevation  30 m  100m F i g . 25  Dimensions  of the r i g  (Adapted from Isaacson, 1986b)  -B»Y  F i g . 29  SEMI—SUBMERSIBLE E l e v a t i o n of semi-submersible r i g  RIG vO  

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