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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 MARTIN HODGKINSON B.A.Sc. University 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 Engineering We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA March 1987 © Derek Anthony Martin Hodgkinson, 1987 ^1 In presenting t h i s thesis in p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the The University of B r i t i s h Columbia, I agree that the Library s h a l l make i t freely available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his or her representatives. It is understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of C i v i l Engineering The University of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: March 1987 Abstract The r e s u l t s of hydrodynamic analyses of two problems involving offshore structures are displayed graphically. This form of presentation of the results and the l i b e r a l use of colour have been found to s i g n i f i c a n t l y help the ease in which the r e s u l t s are interpreted. For the transformation of waves around an a r t i f i c i a l island, a time history of the evolution of the regular, u n i d i r e c t i o n a l wave f i e l d around an a r t i f i c i a l island is obtained. Through the use of colour, regions in which wave breaking occurs have been c l e a r l y defined. The numerical technique used i s based on the f i n i t e element method using eight noded isoparametric elements. The determination of the transformed wave f i e l d takes wave breaking, wave re f r a c t i o n , d i f f r a c t i o n , r e f l e c t i o n and shoaling into account. The graphical display i s achieved by using a p l o t t i n g program developed for the output of f i n i t e element analyses. The motions of a semi-submersible r i g are computed from the RAO curves of the r i g , used to obtain i t s ' small response in a random sea. The numerical technique used in the analysis assumes that the v e r t i c a l members are slender and may be analysed using the Morison equation whereas the hul l s are treated as large members which are d i s c r e t i s e d and analysed using 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 cylinders and h u l l s together with the time history of the rig' s motions are displayed graphically. Once again, the graphical display i s plotted using a program developed for i i the output of f i n i t e element analyses for four noded elements. In thi s case, a f i n i t e element technique has not been employed but the res u l t s were ordered to act as though thi s i s the case. The slender members (cylinders) and large members (hulls) are c l e a r l y distinguishable by using d i f f e r e n t colours. The elements used in the analysis are also c l e a r l y shown. The VAX 11/730 system was used to obtain the results shown. A video tape, using the results of a time stepping procedure, was made by successively recording the hardcopies produced by the VAX p r i n t e r . The time stepping could also be seen, in real time, on the IRIS. Table of Contents A b s t r a c t . i i L i s t of F i g u r e s v i Acknowledgements v i i i 1 . I n t r o d u c t i o n 1 2. Review of Hydrodynamic Problems 5 2.1 Wave Transformation Around an A r t i f i c i a l Island ..5 2.2 Wave E f f e c t s on Hybrid Offshore Structures 9 2.2.1 Large Diameter Members 10 2.2.2 Slender Members 12 2.2.3 Total Structure 13 3. The D i s p l a y Systems 15 3.1 VAX/VMS " 15 3.2 Integrated Raster Imaging System 15 4. Computer G r a p h i c s C o n s i d e r a t i o n s 17 4.1 Wave Transformation Around an A r t i f i c i a l Island .17 4.2 Wave E f f e c t s on Hybrid Offshore Structures 23 5. R e s u l t s and D i s c u s s i o n 29 5.1 Wave Transformation Around an A r t i f i c i a l Island .29 5.2 Wave E f f e c t s on Hybrid Offshore Structures 33 6. C o n c l u s i o n s and Recommendations ...38 6.1 Wave Transformation Around an A r t i f i c i a l Island .38 6.2 Wave E f f e c t s on Hybrid Offshore Structures 40 7. R e f e r e n c e s 42 APPENDIX A : THE ISLAND 45 APPENDIX B : THE RIG 51 APPENDIX C : TRANSFER VAX/VMS FILE 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 Figures F i g . 1 Sketch of an a r t i f i c i a l island 65 Fi g . 2 Sketch of boundaries 66 Fi g . 3 Sketch of a semi-submersible r i g 67 Fi g . 4 An element for a slender member 68 Fi g . 5 Undistorted display of an a r t i f i c i a l i sland 69 Fi g . 6 Scaled island (NODES-* 1 . 0 , H=1.0, V=5.0) 70 Fi g . 7 Scaled island (NODES=1.0, H=1.5, V=10.0) 71 Fi g . 8 Scaled island (NODES=1.0, H=1.5, V=5.0) 72 Fi g . 9 Scaled island (NODES=1.5, H=1.5, V=5.0) 73 Fi g . 10 Mesh on exterior of h u l l s 74 Fi g . 1 1 Outline of hulls 75 Fi g . 1 2 Arrangement of elements on cylinder 76 Fi g . 1 3 Edges of elements on the cylinders 77 Fi g . 1 4 Dimensions of the island 78 Fi g . 1 5 Intrusion of yellow elements 79 Fi g . 1 6 Intrusion of red elements 80 Fi g . 1 7 Back of island (t=0) 81 Fi g . 18 Right side of island (t=0) 82 Fi g . 1 9 Front of island (t=0) 83 Fi g . 20 Back of island (t=l) 84 Fi g . 21 Back of island (t=2) 85 Fi g . 22 Master sheet for island 86 Fi g . 23 Overlay sheet for island 87 Fi g . 24 Plan view of island showing breaking 88 waves vi F i g . 25 Dimensions of the r i g 89 Fi g . 26 Rig showing the f l e x i b i l i t y of program 90 Fi g . 27 Plan view of semi-submersible r i g 91 Fi g . 28 . Side view of semi-submersible r i g 92 F i g . 29 Elevation of semi-submersible r i g 93 Fi g . 30 Rig at.end of time step #1 94 Fi g . 31 Rig at end of time step #2 95 v i i Acknowledgements This work would not have been possible without the encouragement, guidance and helpful suggestions of Prof. M. de St. Q. Isaacson. The constructive c r i t i c i s m s of Prof. W. F. Caselton are humbly acknowledged. Mr. A. Martin was instrumental in helping with the modifications of the C i v i l Engineering Graphics Package for use in these problems. Mr. R. Nelson and his staff at Media Services should be complemented for the work that they have done in preparing the video tape. The f i n a n c i a l support provided by the Natural Sciences and Engineering Research Council i s greatly appreciated. v i i i 1. INTRODUCTION Computer graphics i s the use of a computer to define, store, manipulate, interrogate and present p i c t o r i a l output (Mufti, 1983). Graphical displays play an important part in the interpretation of r e s u l t s from a f i n i t e element analysis. As the complexity of the problem increases, the interpretation of the large amounts of data becomes increasingly d i f f i c u l t . The time spent by the engineer in examining and interpreting the re s u l t s increases at a very high rate: the chance of making an error increases more rapidly. The use of a p i c t o r i a l description of the results may reveal a l o t about a system which would otherwise not be obvious from an inspection of 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 the engineer with the complex human processes involved in des ign. One picture i s worth more than several meters of numerical output i f the engineer must make decisions based on the numerical r e s u l t s . The computer can be used to present the results v i s u a l l y , in a form that is more natural to man. Graphic output devices s h i f t the burden of data interpretation from the human i n t e l l e c t u a l and vis u a l systems to the human v i s u a l system only. What we see i s a highly structured, synthesized and summarized form of the results without the presence of the massive amounts of computational data that was required. 1 2 The offshore industry i s a rapidly growing f i e l d and an increasing amount of r e s p o n s i b i l i t y i s being placed on upper management's shoulders. Decisions based on the results of numerical procedures must often be made quickly and can quite often have far reaching or costly consequences. Any advances which can help in the v i s u a l i z a t i o n of the r e s u l t s of analyses performed on offshore structures i s welcome. There are several numerical methods presently av a i l a b l e to determine parameters such as the p r o f i l e of the wave f i e l d in the v i c i n i t y of an offshore structure, the deflected shape of marine r i s e r s , the motion of a barge and i t s ' moorings, etc. but l i t t l e has been done to present these results in v i s u a l l y appealing and informative forms. This thesis attempts to make some progress in t h i s d i r e c t i o n . The results of hydrodrodynamic analyses performed on two d i s t i n c t offshore structures are displayed. In each case, the necessary numerical output can be obtained from various computer programs. These programs are then modified so that the necessary output i s in a form which i s compatible with the requirements of the p l o t t i n g program. Since the hydrodynamic analyses are based on d i s c r e t i s a t i o n s of the flow f i e l d or structure (either using the f i n i t e element method or the boundary element method), the results (runup, motions etc.) are usually known at s p e c i f i c points. In displaying the results of the analyses, i t i s useful to show how the d i s c r e t i z a t i o n was performed. The coordinates of the vertex points (nodes), are usually 3 known. If the r e s u l t s have been determined at points other than the vertex points, they must be transposed 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 building blocks of the picture and associated with each block i s an algorithm which explains how the nodes are connected. This algorithm i s sometimes c a l l e d the locator matrix. Perhaps one of the most commonly known applications of computer graphics i s the CAT (computer - a x i a l - tomography) scanner. It helps medical personnel to determine the necessity of needless and p o t e n t i a l l y harmful surgery. In the engineering f i e l d , graphics is used as a tool for gaining insight into concepts and analysing problems. In a l l of the displays, the l i b e r a l use of colour contributes to the ease of reading the output and speed at which the reader assimilates the information being presented. Animation for engineering i s quite d i f f e r e n t from the more well known animation for entertainment. The engineering industry has neither the time nor the money for the production procedures required to obtain r e a l i s t i c images; the only requirements on the graphical displays are that each separate part of the system must be unambigiously i d e n t i f i e d and the animation must be produced quickly, in real time, i f possible (Noma and Kunii, 1985). In the present work, a l l the programs are written in FORTRAN and the p l o t t i n g i s done using a VAX 11/730 under the EUNICE operating system i n s t a l l e d at The University of 4 B r i t i s h Columbia. The Integrated Raster Imaging System (IRIS) was used to display the time history of the systems in real time. A f i l m showing the time v a r i a t i o n of the results was made using hardcopies produced on the VAX printer. 2. REVIEW OF HYDRODYNAMIC PROBLEMS 2. 1 WAVE TRANSFORMATION AROUND AN ARTIFICIAL ISLAND The f i r s t problem considered i s the prediction of the transformed wave f i e l d and an a r t i f i c i a l island (Fig. 1). The special case of a c i r c u l a r a r t i f i c i a l island with a berm of constant slope w i l l be examined. For thi s s i t u a t i o n , the incident wave f i e l d may be transformed by wave r e f r a c t i o n , d i f f r a c t i o n , r e f l e c t i o n , wave no n - l i n e a r i t i e s , wave-current interactions and energy d i s s i p a t i o n . However, in many sit u a t i o n s , some of these effects are unimportant and can be neglected. In the model considered here, wave r e f r a c t i o n , d i f f r a c t i o n and energy d i s s i p a t i o n due to wave breaking are taken into account. A f i n i t e element formulation of the extended mild slope equation is used. Berkhoff (1972) i n i t i a l l y presented the linear 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 the waves are of small amplitude (linear approximation) and that the seabed slope i s gentle and mildly varying. Consequently, lin e a r wave theory for constant depth may be applied over short distances; these assumptions give r i s e to the mild slope equation: (1 ) where c wave c e l e r i t y c 9 group c e l e r i t y angular frequency a two-dimensional surface potential at the 5 6 s t i l l water l e v e l V = horizontal gradient operator The v e l o c i t y p o t e n t i a l , of the three-dimensional flow i s : cosh k(z+d) $(x,y,z,t) = Re [A 4(x,y) e l w C ] (2) cosh kd where A -igH ( 2CJ H 0 = incident wave height g = acceleration due to gravity d = s t i l l water depth k = wave number = — c i = t = time In Eq. 1, both c and c^ are dependent on the seabed elevation. The v e l o c i t y p o t e n t i a l , <t>, i s made up of components due to the incident wave p o t e n t i a l , $j , and the unknown, scattered wave po t e n t i a l , <t>g: * = + * s (3) cosh k(z+d) i ( k x - u t ) where = A e cosh kd cosh k(z+d) -icot <t>e = A 0_(x,y) e a cosh kd 13 The mild slope equation i s subject to boundary conditions at a s o l i d boundary, S, and S 2, and a radiation condition in the far f i e l d , S 3 (Fig. 2). The s o l i d body boundary condition requires that there i s no flow normal to the body surface and i s applied along the island contour. The radiation condition would require that 7 0 g decays as the point of interest moves away from the o r i g i n . Eq. 1 may be solved by the f i n i t e element methods which involves a two-dimensional d i s c r e t i z a t i o n of the horizontal domain. The d i s s i p a t i o n of energy by wave breaking was f i r s t examained by Booij (1981) for linear r e f r a c t i o n and d i f f r a c t i o n by including an extra term in the mild slope equation: c_ V(cc V0) + ( -3- co2 + icoW)0 = 0 (4) " c where W i s a damping factor which may be due to various possible e f f e c t s . However, in the present formulation, only energy d i s s i p a t i o n due to wave breaking i s included. Dally, Dean and Dalrymple (1984) have developed a model for wave breaking and decay (based on an analogy to the hydraulic jump) for waves entering the surf zone for a plane beach. Isaacson (1985) has proposed an expression for W using t h i s model of wave breaking: c r 2d 2 W = K' -3 (1 - ) (5) d H 2 where K' = wave decay factor r = stable wave factor. These two factors are determined empirically: Dally et al have found that for beach slopes in the range 1/80 to 1/30, K' l i e s between 0.10 and 0.28 while T varies in the range 0.35 to 0.48. The application of Eq. 5 depends on the conditions for the onset and cessation of wave breaking. The wave height i s i n i t i a l l y unknown and in order to obtain wave 8 heights for use in the wave breaking c r i t e r i o n , the wave height over the berm i s assumed to be due to shoaling only. The expressions presented by Dally et al can be used for c a l c u l a t i n g the heights of the decaying waves once breaking has commenced. When waves are not breaking, W i s taken as zero since the other forms of energy d i s s i p a t i o n are generally small in comparison. In the f i n i t e element formulation of the problem, the Galerkin method i s used. The f i n a l f i n i t e element equations for each element may be written in the form:. [ k ] e { 0 1 } e = { q i } e (6) where <j>^ are the nodal values of the p o t e n t i a l function which are to be determined. The isoparametric l o c a l coordinate system for each element can then be transformed into the Cartesian global coordinate system of the entire domain. The system of equations can then be solved. Because of their a b i l i t y to accurately model curved boundaries, eight node two-dimensional isoparametric elements are used. Once the numerical values of the p o t e n t i a l function, <£(x,y), at d i f f e r e n t locations are known, the t o t a l potential can be obtained from Eq. 2. The wave elevation, 77 , can then be obtained from the l i n e a r i z e d dynamic free surface boundary condit ion. In previous work performed on t h i s topic (Talukdar 1986), the formulation and solution of the f i n i t e element equations have been examined. The sea state around the 9 a r t i f i c i a l island was obtained at a time when t i s equal to zero so that comparison could be made to other available r e s u l t s . In the present work, the transformation of the sea for an extended period of time around the a r t i f i c i a l i sland is represented graphically. This representation i s useful in determining the possible maneuvering of vessels in the neighbourhood of the island or with some further manipulation, the sediment movement in the v i c i n i t y of the isl a n d . 2.2 WAVE EFFECTS ON HYBRID OFFSHORE STRUCTURES It i s not uncommon for offshore structures to be made up of both slender and large' members. As an example, a semi-submersible r i g (Fig. 3) may consist of h u l l s which act as large members and braces and trusses which behave l i k e small members. In the analysis of wave-structure interaction problems, the two types of members respond d i f f e r e n t l y to the same wave conditions and, consequently, give r i s e to two d i f f e r e n t approaches by which wave force problems are treated. For slender structures which give r i s e to flow separation, wave loading and response calculations are generally based on the Morison equation which assumes that the body size i s small r e l a t i v e to the wave length so that the incident flow i s v i r t u a l l y unaffected by the presence of the structure i . e . the incident wave kinematics do not change s i g n i f i c a n t l y and flow separation dominates the loading behaviour. For large structures which span a 1 0 s i g n i f i c a n t f r a c t i o n of a wave length, the incident waves undergo s i g n i f i c a n t scattering or d i f f r a c t i o n and wave force c a l c u l a t i o n s should account for the scattering. In such cases, flow separation e f f e c t s can usually be neglected and the problem can be solved as one of potential flow. The complete solution using p o t e n t i a l flow theory requires the solution of non-linear equations. The problem i s usually l i n e a r i z e d by assuming that the wave heights are small. 2.2.1 LARGE DIAMETER MEMBERS The f l u i d (water) i s assumed incompressible and i n v i s c i d , and the flow i s assumed to be i r r o t a t i o n a l . The f l u i d motion may therefore be described by a velocity p o t e n t i a l , 0, which s a t i s f i e s the Laplace equation within the f l u i d region. The t o t a l flow pot e n t i a l of a structure o s c i l l a t i n g in waves i s made up of components associated with the incident waves (subscript 0), the scattered waves (subscript 7) and the forced waves (subscripts 1,...,6) due to each mode of motion, 0 may be expressed as: -iwH 6 - \ t s T -0 = [ U o + 0 7) + 2 -ito$.0, ] e w c (7) 2 k=1 k k where ^ ^ s the complex amplitude of each component motion. The unknown potentials, 0^, (k=1,...,7) may each be represented as due to a d i s t r i b u t i o n of point wave sources over the immersed equilibrium body surface, S^, and, at any general point x = (x,y,z) in the f l u i d , may be represented as 11 1 0 t ( x ) = — /„ f. (£) G(x,£) dS k = 1,...,7 (8) K 47T bb where = s o u r c e strength d i s t r i b u t i o n function I = ( f r S) on the body G (x,J_) = Green's function for the point x due to a source of unit strength located at i The source strength d i s t r i b u t i o n function is chosen so that the body surface boundary condition is s a t i s f i e d . The Green's function is chosen to s a t i s f y the Laplace equation, the l i n e a r i z e d free surface condition, the seabed condition and the radiation condition. Once the potentials, 0^, are known, the hydrodynamic pressure on the body surface may be determined from the li n e a r i z e d Bernoulli equation. The loads on the body may then be calculated: F. = -icop /<, 0 n. dS j = 1,...,6 (9) b where F 1 f F 2 i F 3 = force components in the x,y and z directions respectively ^ni^si^s ~ moment components about the x,y and z directions respectively Each component of the f l u i d force, Fy may be decomposed (e) into an exciting force component, F j , and a forced force component, F ^ f ^: F j - [ F ^ e ) + F J f ) ] e - i w t j = 1,..,6 (10) where F. ' = - - pHco2 J" (0 O + 0 7 ) n. dS 3 2 bb 3 1 2 and F<£> = . J / ^ j k + i " X j k > 5 k with = added-mass c o e f f i c i e n t Xj^ = 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 exciting forces may be obtained by a d i s c r e t i s a t i o n procedure in which the source strengths are f i r s t obtained at the centre of each facet on the submerged surface of the structure. The unknown potentials may then be evaluated. The hydrodynamic c o e f f i c i e n t s are determined by substituting Eq. 9 and Eq. 10 into Eq. 7 and c o l l e c t i n g corresponding terms. 2.2.2 SLENDER MEMBERS The f l u i d force on a slender member inclined at an arb i t r a r y orientation acts in a di r e c t i o n perpendicular to the member axis and may be expressed by the r e l a t i v e v e l o c i t y formulation of the Morison equation (Fig. 4). Thus, the force per unit length, F^, (prime used for slender member c o e f f i c i e n t s ) , acting at a point x on the member axis and in a d i r e c t i o n n i s : 7TD2 7TD 2 F n = * — cm*n " p ~ ( C n f 1 ) % +• J * D Cd ( un- vn>l un- vnl where p = density of f l u i d u = f l u i d v e l o c i t y at x in the direction n n — v n = member v e l o c i t y component at x in the d i r e c t i o n n 1 3 = derivative with respect to time D = diameter of member C = i n e r t i a c o e f f i c i e n t m = drag c o e f f i c i e n t The drag term i s usually l i n e a r i z e d by taking (u -v )|u -v I = a (u -v ) (12) n n ' n n 1 n n n where a n = drag l i n e a r i z a t i o n factor. In addition, the flow v e l o c i t y and structure motions vary harmonically in time with angular frequency, co. The above formulation can be applied to two orthogonal d i r e c t i o n s for each element of a slender member. Suitable summations can then be used to obtain the t o t a l forces and moments on a l l the slender members. As with the large diameter members, the t o t a l force and moment components, F ^ , acting on a l l the slender members may be decomposed into an ex c i t i n g force, (e) . . . F^ , associated with the incident and scattered wave potentials and components associated with the forced potentials, F j ^ ^ , which themselves can be expressed in terms of added-mass c o e f f i c i e n t s , M--,, and damping T c o e f f i c i e n t s , X ., . jk 2.2.3 TOTAL STRUCTURE Once a l l the hydrodynamic c o e f f i c i e n t s are known, the equations of motion of the body may be solved to determine the six components of the body motion. The equations of motion may be written as follows: 1 4 U - « 2 < m j k + M j k + M j k > " i " ( * j k + V + ( c j k + C j k ) ] $ k = F ( e ) + p * ( e ) • = ] f m m t f 6 ( 1 3 ) where m.. = mass matrix Ilk C j k = hydrostatic s t i f f n e s s matrix C j k = ad d i t i o n a l s t i f f n e s s matrix due to moorings which may be present. The response amplitude operators, defined as 2^/H, may then be determined by solving Eq. 13 (these describe the body responses for incident waves of unit amplitude and d i f f e r e n t possible frequencies). The wave runup, R, around the waterline of the structure can also be obtained: R co2 6 Si, - = — 100 + <t>i + £ 0 k I „ _ n (14) H 2g k=i (H/2) k 2 - 0 For a fixed structure, the forced potential terms are excluded. 3. THE DISPLAY SYSTEMS 3.1 VAX/VMS The VAX/VMS system i s used to obtain 2D and 3D graphics using a display with 1024 x 780 p i x e l resolution. The plot t i n g i s performed using DI-3000, an integrated system of graphics software too l s . The application program defines computer graphics objects using 2D ( v i r t u a l coordinate system) or 3D (world coordinate system) graphics output primitives (moves, draws, e t c . ) . The application program defines the mapping from the world coordinate system (a window) into a v i r t u a l coordinate system (a viewport). This is c a l l e d the viewing transformation in which a "photograph" is taken of a world coordinate object, with sca l i n g , rotations, translations etc. already performed. This system allows the user to get hardcopies of the views and by recording successive hardcopies on f i l m , a r e a l i s t i c impression of the changes may be observed. 3.2 INTEGRATED RASTER IMAGING SYSTEM The .Integrated Raster Imaging System (IRIS) i s used to produce high resolution 2D and 3D computer graphics. The heart of the system i s the Geometry Engine which accepts points, vectors, polygons, characters and curves in a user defined coordinate system. These are transformed onto the screen in screen coordinates with a r b i t r a r y rotations, scaling and other transformations possible. In excess of 1 5 16 65000 coordinates per second can be processed. The IRIS screen i s 1024 x 768 p i x e l s . The orig i n i s located in the bottom l e f t corner of the display. This system is useful for a wide var i e t y of graphics applications, including simulation. One drawback of this system is that i t i s not portable. However, with the use of a video machine, the results may be stored in a compact and portable form. 4. COMPUTER GRAPHICS CONSIDERATIONS 4.1 WAVE TRANSFORMATION AROUND AN ARTIFICIAL ISLAND The size of the elements used in the analysis, to 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 large variations in the wave p r o f i l e over short distances, i . e . near the v e r t i c a l face of the a r t i f i c i a l island, smaller elements are needed. Neighbouring facet diameters should not d i f f e r by more than about 50% since any improvement made by the presence of the small facets w i l l be lost due to the computational i n e f f i c i e n c i e s associated with the larger diameter facets. However, there i s no e x p l i c i t mathematical rela t i o n s h i p between the size of an element and the accuracy of the solution. Consequently, the l i m i t on the size of an element i s dictated by the numerical accuracy required and the computational e f f o r t necessary to obtain convergence to the accurate solution. No ad d i t i o n a l elements are needed for the p l o t t i n g of the surface p r o f i l e . An eight noded isoparametric element i s used. The coordinates of each node are determined at the end of each time i n t e r v a l . Each node is fixed in the horizontal (x-y) plane. It i s free to move v e r t i c a l l y . Each node i s assigned a number and the outline of each element i s described by the numbers of the nodes touching the element. The system consists of four components: the breaking region, the non-breaking region, the berm and the v e r t i c a l wall. Each component i s uniquely coloured so that i t may e a s i l y be 1 7 18 distinguished from the other parts of the system. Dark blue is used for non-breaking regions and these are c l e a r l y distinguishable from the breaking (yellow) regions. The berm is coloured green while the v e r t i c a l wall i s red. These colours enable each of the components of the system to be readi l y i d e n t i f i e d . The output consists of a descriptive t i t l e , the numbers of nodes and elements in the picture and number of nodes required to display the island, coordinates of each node, the element number with i t s associated nodes and an assigned colour parameter. When breaking occurs, one normally expects to see whitecaps. This is c l e a r l y v i s i b l e on the VAX/VMS terminal screen but when .a hardcopy i s made, these areas show up as dark blue/black, barely distinguishable from the non-breaking regions. Consequently, yellow i s used for these areas. There are two ways in which the colour for the elements may be s p e c i f i e d : i t may be included in the locator matrix as an extra column or i t may be determined by the analyst when the p l o t t i n g i s being performed. In th i s case, the colour i s included in the locator matrix. Consequently, the locator matrix, for the eight noded element, i s dimensioned (number of nodes, 9). The colour i s assigned within the a n a l y t i c a l program. The advantage of th i s procedure i s that elements which s a t i s f y certain conditions (wave breaking or sloping surface, for example) may be assigned their colour d i r e c t l y . The number assigned for the colour need not be in a one-to-one correspondence with the d e f i n i t i o n of the 19 colour code in the p l o t t i n g program but the elements of each component should be assigned the same colour code. Wave breaking, for example, may be assigned a boolean variable: i f breaking occurs, the colour code for that element i s one and for non-breaking waves, the colour code i s zero. After the data has been read in by the p l o t t i n g program, there may be executable steps within the p l o t t i n g program which re-assign the colour: i f the colour code i s zero, then the program colour code may be set to f i v e ( f i v e may correspond to l i g h t blue) or i f the colour code i s three, the colour code may be reset to seven (this may be red). This enables the analyst to keep the two processes separate. The results may be generated on one system and when organized in a form such as has been suggested, they may be plotted on several other display systems, each of which may have i t s own colour code d e f i n i t i o n . To put the whole picture into perspective, the island i s included in the display. Since the structure i s symmetric about the x-axis, only one side of the i s l a n d was analysed. The coordinates of the nodes on t h i s side were determined prior to analysis. The other half of the island was generated by changing the sign of the y-coordinate of the nodes on the analysed side. A v e r t i c a l wall around the island enables the wave runup to be seen more c l e a r l y . The wall i s coloured (red)' to contrast the sloping berm (green) and the free surface (dark blue and yellow). 20 Once the analysis has been performed on the UBC mainframe, the results must be transfered to the VAX/VMS system (Appendix C). Using 3200 baud l i n e s , a f i l e containing the coordinates of over 2800 nodes and 1100 elements takes a l i t t l e more than half an hour to be transfered. If one period i s divided into sixteen equal time steps, the output produced from a numerial analysis w i l l take over eight hours to be transfered between the two systems. This process can be quickened i f the following observation i s made: the nodes needed to describe the structure may be divided into two categories, one for the is l a n d and one for the waves. The coordinates of the nodes which describe the island are independent of time. On the other hand, the coordinates of the nodes for the wave p r o f i l e change with each time step. The locator matrix i s the same for a l l time steps. Hence, i f only the coordinates of the nodes which describe the wave p r o f i l e at each subsequent time step are transfered, there w i l l be considerable time saved. More than half of the coordinates and a l l of the locator matrix w i l l have been discarded. With the information from the f i r s t time step already transfered, i t takes a s k i l l e d operator less than one minute to edit the f i l e s on the VAX to get each of them ready to be used as input for the p l o t t i n g program. This i s done by making one copy of the i n i t i a l time stepping f i l e for each subsequent time step. The lines which contain the coordinates for the wave p r o f i l e are replaced with those for the p a r t i c u l a r time 21 step. The time taken to transfer the f i l e s for three time steps i s cut in half and i s reduced l i n e a r l y as more time steps are added. Just over ten minutes are needed to transfer the results of each time step. Decisions on changes in the numerical program may be made more rapidly with savings in time and expense (computer time and personnel) being quite substantial. Because of the size of the elements and scale of the model being displayed, i f the ent i r e structure was shown unsealed, there would be l i t t l e ( i f any) knowledge gained of the p r o f i l e of the wave surface (Fig. 5). The wave elevation may be amplified to further emphasize any subtle variations which may be present. This i s achieved by having an extra parameter in the output: the number of nodes required to display the island. The nodes l i s t e d a f t e r these nodes in the output are those which describe the wave surface. Amplification of the l a s t group of nodes is performed immediately after they are read i n . The amplification factor must be changed within the p l o t t i n g program; each change requires i t s ' recompilation and reloading. This process is far less expensive and time consuming than having to re-run the numerical analysis program. It must be remembered that the v e r t i c a l wall's height i s fixed in the numerical program. If the waves are amplfied too much, they may overtop the structure. I n i t i a l l y , the waves were not amplified but the entire island and wave surface were scaled. A t r i a l and error 22 process has been used (Fig. 6 and F i g . 7) to achieve the best display (Fig. 8). The v e r t i c a l dimensions (z) are very small compared to the horizontal (x and y) dimensions. Consequently, the v e r t i c a l dimensions are amplified f i v e times while the horizontal dimensions are amplified by 50%. This stretches the island across the width of the screen while s t i l l keeping the slope within acceptable v i s u a l l i m i t s . The transformed wave f i e l d i s then amplified (Fig. 9): the v e r t i c a l coordinates of the nodes were increased by 50%. These two sets of amplifications are done as part of the p l o t t i n g process and involve a l l of the nodes. The resulting picture i s somewhat dis t o r t e d but the important parameter (variation of the wave p r o f i l e ) i s more e a s i l y seen. Crests and troughs are more c l e a r l y seen. In an attempt to obtain a more r e a l i s t i c idea of what i s r e a l l y happening, the time stepping procedure was performed on the IRIS. Several d i f f i c u l t i e s were encountered when using the IRIS. Because of the "large" number of nodes and elements, the system aborted on several attempts to load the data for the time stepping display. Hidden l i n e removal could not be performed. Though there are several packaged programs to perform this procedure, newer versions of the system contain t h i s feature and i t was f e l t that i t was not an e f f i c i e n t use of time to try to activate t h i s feature. The system could produce pictures l i k e those produced on the VAX only for the i n i t i a l condition. A large portion of the half megabyte memory was used to store the code needed to 23 produce the f i l l for the elements. In an attempt to harness some po s i t i v e r e s u l t s from the experience, a l l elements were made transparent; their outlines were lined with the same colours with which they would normally be f i l l e d . Using this process, i t was possible to show two time steps. A combination of these two processes (time stepping and element f i l l i n g ) would have produced the desired r e s u l t s . By separating them, the program was shown to be e f f e c t i v e within the l i m i t a t i o n s of the system. There would be no problem combining these processes on a more powerful system. The computer program has been c a r e f u l l y documented so that when a more powerful system i s ava i l a b l e , the proper images could be obtained. The extra steps have been included as comments in t h i s version of the program. 4.2 WAVE EFFECTS ON HYBRID OFFSHORE STRUCTURES Hybrid structures, as the name implies, are made up of two d i f f e r e n t types of members: small and large. To f u l l y exploit the c a p a b i l i t i e s of the system, the two c l a s s i f i c a t i o n s of members are highlighted with d i f f e r e n t colours. The large members (hulls) are coloured purple while the small members ( v e r t i c a l cylinders) are coloured yellow. The elements which have been used in the numerical procedure are shown on the members. The rectangular h u l l s are plotted in two steps. The analysis of large members involves a d i s c r e t i z a t i o n procedure in which the submerged body surface i s divided 24 into a number of small facets. These elements may be defined by the intersection of a series of horizontal and v e r t i c a l planes. Each crosssection which represents the surface p r o f i l e of the elements used for the wave source d i s t r i b u t i o n technique i s plotted. There i s no f i l l assigned to these elements (Fig. 1 0 ) . Next, the corners of the h u l l are connected, in an appropiate manner, to form the edges of the h u l l . These elements are f i l l e d with a transparent colour so that the crosssections which were previously drawn can be e a s i l y seen on the surface (Fig. 1 1 ) . The cylinders are also plotted in two steps. Slender members are analysed using Morison's equation applied to the centre of elements formed by taking transverse s l i c e s along the length of the member. I n i t i a l l y , the columns are plotted as long, slender rectangles with a rectangle going along the length of each element. By a t r i a l and error procedure, i t was found that in order to produce a c i r c u l a r p r o f i l e , sixteen short chords are s u f f i c i e n t for c y l i n d e r s . Arcs could not be used since the p l o t t i n g program i s only able to draw straight l i n e s . When p l o t t i n g elements, the program f i r s t draws the f i l l pattern for the i n t e r i o r of the element, including the boundary. Then, the boundary i s drawn on the edge of the f i l l area to overwrite the f i l l already present. In p l o t t i n g these elements, the outline colour and f i l l colour for each element are the same: yellow. The net result i s a s o l i d yellow mass in the shape of a c y l i n d e r . However, to show the arrangement of the computer graphics 25 elements of the c y l i n d e r , they are outlined blue (Fig. 12). Next, the top and bottom edges of each element of the slender member are outlined by p l o t t i n g the cross sections in another colour: black. This i s done by joining the sixteen chords in a head to t a i l manner (Fig. 13). In the d i f f r a c t i o n theory, the large members (hulls) are d i s c r e t i z e d and the wave source d i s t r i b u t i o n technique is applied to the centres of these elements. The facet generating program, FACGEN (Isaacson, 1986a), has been used to d i s c r e t i z e the h u l l s . In the graphical procedure, the coordinates of the nodes of each of these elements are not needed. Part of the output i s the coordinates of the centres of the elements. The coordinates of the four corners of the s l i c e s are determined by adding (or subtracting) one half of the width of the elements in the three directions to the coordinates of the centre of the elements nearest to the edges. These are the crosssections that are plotted. In t h i s way, only the coordinates of the nodes on the twelve edges of the h u l l s are required. These nodes would normally be needed i f elements were plo t t e d i n d i v i d u a l l y . The individual facets need not be plotted. As the size of the facets decreases (and number of facets increase), the number of coordinates of nodes which have not determined increases. This considerably reduces the computational e f f o r t , storage area and p l o t t i n g time required. A three-dimensional e f f e c t is obtained by p l o t t i n g objects which are furthest from the observer f i r s t so that 26 these objects are overwritten by those nearest to the observer. This i s done by determining the centroids of a l l of the elements. If the element i s too large, t h i s process may break down and t h i s i s e s p e c i a l l y true for elements with large changes in z coordinates in i t s ' four nodes. Hence, the t h i r d r e s t r i c t i o n previously described also applies to the display mechanism and takes on added importance. The output consists of a descriptive heading, the number of nodes and number of elements, the coordinates of the nodes, and the locator matrix. There are two ways in which the colour code for the elements may be s p e c i f i e d : i t may be included in the locator matrix as an extra column or i t may be determined by the analyst when the p l o t t i n g i s to be performed. The colour has been l e f t to the d i s c r e t i o n of the analyst in t h i s case. The advantage of t h i s procedure i s that the colour combination and f i l l pattern of the components of the structure may be adjusted in a t r i a l and error manner without adjustments to the p l o t t i n g program. Every change in the p l o t t i n g program requires i t s ' recompilation, a costly and time consuming process. In the p l o t t i n g of the r i g , four f i l e s were required to plot the two components. The elements in each f i l e were assigned d i f f e r e n t colours and f i l l pattern. This f l e x i b i l i t y in the f i l l pattern would not have been possible by the other method that has been suggested. Since the colour code may vary from machine to machine (seven may be red on machine A but green on machine B), the analyst has the option of 2.7 changing the p l o t t i n g . code for each element (Appendix D). Instead of having to transfer four f i l e s , i t may be more desirable to put a l l the elements into one f i l e and to do the e d i t i n g after the transfer has been completed. When the transfer i s taking place with one f i l e , other tasks can be performed but with four f i l e s , the commands must be entered for each f i l e . This requires that an operator must be present during the transfering process to check on i t s ' status when he could be doing more productive tasks. The coordinates of the entire structure are f i r s t transfered. To minimize the time spent transfering f i l e s , the output of the subsequent f i l e s need not include the coordinates of the nodes. If the number of nodes i s zero (as indicated in the second l i n e of the output), the p l o t t i n g program interprets the l i n e s which follow to be the locator matrix. This considerably reduces the storage space and time needed to plot the r i g since the coordinates of the nodes need not be read in for each time step. Two d i f f e r e n t processes are required to display the transformed r i g on the VAX and the IRIS. For the VAX, once the displacement of the o r i g i n i s known, the coordinates of each node must be determined before the p l o t t i n g program can be used. With the IRIS, once the displacement of the o r i g i n i s known, c a l l s to the b u i l t - i n subroutines rotate and translate perform the transformation for the entire data set in a few milliseconds. Rotate has two parameters: the angle to be rotated (in tenths of degrees) and the axis about 28 which the rotation takes place. These angles would correspond to the l a s t three degrees of motion. There are three parameters for translate v i z the three displacements (surge, sway and heave). One of the problems encountered was trying to decide the order in which these commands should be executed. Rotate - translate means that the body i s f i r s t rotated and then moved according to the new coordinate system while with translate - rotate, the transformation and rotation are both performed in the o r i g i n a l coordinate system. Since both these processes occur simultaneously in practice, neither combination i s correct. The coordinates required to display the r i g on the VAX are based on simultaneous application of these two procedures. Comparison of the coordinates produced for the VAX with each of those produced for the IRIS reveals that rotate - translate best approximates the true s i t u a t i o n . 5. RESULTS AND DISCUSSION 5. 1 WAVE TRANSFORMATION AROUND AN ARTIFICIAL ISLAND The use of a f i n i t e element technique to examine the transformation of waves around an a r t i f i c i a l island suggests (inherently) that the structure being analysed can e a s i l y be displayed through the use of one of a number of graphical mediums presently available. The data needed for such a representation (numbers of nodes and elements, coordinates of the nodes, and connectivity of the nodes) i s part of the formulation of the f i n i t e element problem. Since a coloured graphics f a c i l i t y i s being used, the c a p a b i l i t i e s of the system are f u l l y exploited through the use of d i f f e r e n t and contrasting colours to represent the di f f e r e n t components of the system. The island i s 150.0m in diameter with shallow water depth of 3.0m, berm slope 1 in 4 and deep water depth of 12.0m. The waves are 7.5m high and have a period of 10.0 seconds (Fig. 14). Waves are t r a v e l l i n g in the positi v e x - d i r e c t i o n . Though i t is not used in the analysis, a wall i s included to give the whole structure a sense of continuity and to put the island into the proper perspective. The wall i s b u i l t up of a series of level s so that small changes in runup around the island can e a s i l y be seen. The le v e l s act as contours and are 30 cm high. The p l o t t i n g program, developed for use on the VAX system, determines the centroid of each element and plots those furthest from the observer f i r s t . This results in the 29 30 closest elements overwriting those i n i t i a l l y drawn. One of the problems encountered in the p l o t t i n g of the island is the i n a b i l i t y of the program to p r e c i s e l y determine the coordinates of the centroid of each element. This would require the ca l c u l a t i o n of the area and moment arm about a fixed point for each element. With over 1100 elements, the pl o t t i n g process would take considerable more time. Presently, the centroid is determined by finding the average of the x, y and z coordinates of each element. Problems arise in some of the scaled views where there are large changes in wave p r o f i l e near the v e r t i c a l face of the island. Since the centroids of the elements were incorrectly determined, elements further from the viewer are plotted after those closest to the viewer. This results in an intrusion of the yellow elements into the red elements or the cutting of the yellow elements by the red elements. Examples can be seen on the inside of the island in Fi g . 15 and on the right hand side of the island in F i g . 16. The effe c t i s much more noticeable when the graphical process is taking place on the screen. Some of the errors are covered by elements nearer to the viewer. The problem becomes more apparent as the amplification factors are increased. Within the p l o t t i n g program, there are several options available to the user (Appendix D). The is 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 in the runup or the surface p r o f i l e around the island may be more closely examined. Each view requires the, entire structure to be 31 redrawn, a process which takes over four minutes. This i s a co s t l y and time consuming process. The result i s a series of snapshots taken by a viewer high above the island. It would be more appropiate i f the viewer had a video camera so that the information in the scene may be recorded since he may miss some of the important processes or may respond too slowly when recording them. Sample results of the time stepping procedure on a crude model of the island are shown. There are three views Of the islan d for the f i r s t time step (Fig. 17 to F i g . 19). Then, the results of the f i r s t two time steps for the f i r s t view are included (Fig. 20 and F i g . 21). For each of the 16 time steps, there are three d i f f e r e n t views. The f i r s t view (-1.0, -1.0, -0.3) has the analyst looking at the back of the islan d with waves coming towards him; in the second view (1.0, 1.0, -0.5), the analyst i s observing the right hand side of the island; the front of the island is shown in the t h i r d view (2.5, -1.5, -0.5). By combining these views, the analyst has a good idea of the shape of the transformed wave f i e l d . Comparison can then be made between the same views but at d i f f e r e n t time steps to see how the wave f i e l d evolves. A movie was made using the results displayed on the VAX. Using polyethylene sheets, successive time steps were recorded on video tape and edited so that the transformation of the simulated wave may be observed. With a master sheet on which a border, t i t l e and axis are shown (Fig. 22), subsequent sheets showing the structure and border were 32 overlayed on the master sheet to ensure proper alignment of the island at the end of each timestep (Fig. 23). The IRIS i s used to obtain a real time picture of the processes taking place. The system presently i n s t a l l e d at The University of B r i t i s h Columbia i s a small one (half megabyte capacity). This put severe r e s t r i c t i o n s on the way in which the structure was drawn and the number of time steps which could be observed. The l i m i t a t i o n s of the IRIS are perhaps best understood when the d i s c r e t i z e d domain is cl o s e l y examined. The f i n i t e elements used in the analysis were too large to give any reasonable results on which engineers could make design decisions. There were 429 nodes and 126 elements in t h i s numerical model. In an analysis of the same island configuration by Talukdar (1986), over 1400 nodes and 400 elements were used. With the generation of the other side of the island and the v e r t i c a l face, the computer graphics model requires 2826 nodes and 1140 elements. To produce a picture of Talukdar's a n a l y t i c a l model, over 6700 nodes and 2000 elements would have been needed. A large portion of the nodes and elements i s as a result of building the v e r t i c a l wall in a number of leve l s which form an integral part in the v i s u a l i z a t i o n of the movement of the wave crests and troughs. A fi n e r mesh, though more r e a l i s t i c , would be pushing the machines to their l i m i t s . A plan view of the island (Fig. 24) shows wave breaking occuring in a region that i s concentric with the island. This i s as a result of the very coarse mesh used to generate 33 the model. With a finer mesh, the boundary defining the breaking and non-breaking regions would be more i r r e g u l a r . The exact shape of the wave f i e l d i s not known but regions where runup i s greatest or where wave breaking takes place are c l e a r l y shown. 5.2 WAVE EFFECTS ON HYBRID OFFSHORE STRUCTURES The program which generates the computer graphics model of the semi-submersible r i g (RIG) is r e s t r i c t e d to r i g s which consist of rectangular hulls and v e r t i c a l c y l i n d e r s . The r i g consists of 4 v e r t i c a l cylinders, each of diameter 12.0m with 8 elements. The hull s are 100.0m long, 30.0m wide, 20.0m high, centred at 40.0m below the surface and 70.0m apart (Fig. 25). The elements on the h u l l are 10.0m square. These parameters define the DNV r i g analysed by Isaacson (1986b). The disturbance is 5.3m high, has a period of 12.0sec and has a p r i n c i p a l wave d i r e c t i o n of 60°. The d i s c r e t i z a t i o n of the rectangular h u l l s i s performed using the program FACGEN which, in turn, i s r e s t r i c t e d to eight d i f f e r e n t structural shapes. Of these eight configurations, RIG i s suitable for use on four of these cases (IS = 2, 4, 5 and 7). RIG i s , therefore, r e s t r i c t e d by the c a p a b i l i t y of these options in FACGEN. As i t i s presently written, one of the features of FACGEN is that i t i s capable of generating the d i s c r e t i z e d form of a pair of horizontal rectangular cylinders (IS = 7). On careful 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 addition of fi v e or six executable statements, the program may be extended to generate the di s c r e t i z e d form of an unlimited number of rectangular h u l l s . A large number of rectangular h u l l s is unlikely to occur in practice but three or four hulls are not uncommon. These changes in FACGEN would further enhance the c a p a b i l i t i e s of RIG. RIG can be used with an unlimited number of surface piercing v e r t i c a l , c i r c u l a r c y l inders which extend down to the h u l l s . There are several options available to the analyst. For each cylinder, the diameter, number of chords used to represent the cylinder or the number of levels in each cylinder may be varied. When combined with the h u l l s , the cylinders are usually c l a s s i f i e d as slender members and the second of these options i s not needed. However, in order to get a computer display of the curved surface, t h i s feature i s important in ensuring that the model i s an accurate representation of the prototype. The number of levels would correspond to the elements used in the numerical procedure. The cylinders may be positioned anywhere on the h u l l s . Their position i s defined by the coordinates of the centre of the plane which occurs at the intersection of the v e r t i c a l cylinder and the free surface. As an example of the c a p a b i l i t i e s of the program, F i g . 26 shows a r i g with three v e r t i c a l cylinders, each of dif f e r e n t - diameter and number of l e v e l s . It is anticipated that the program w i l l be extended to include inclined slender members 35 where the a b i l i t y to change the number of elements used in the numerical analysis on each cylinder would become more c r u c i a l . These inclined members would have smaller changes in wave kinematics per unit length than v e r t i c a l members in the same environment and hence, may use longer elements in the analysis. WELSAS2 (Isaacson, 1986a) i s capable of analysing structures which contain in c l i n e d slender 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 coordinates of points along i t s ' edges would be needed. These are determined for v e r t i c a l slender members in RIG. Consequently, RIG may be thought of as being a post-processing program for FACGEN. Three views of the r i g are shown. They are the plan (Fig. 27), elevation (Fig. 28) and right side views (Fig. 29). They were obtained by changing the normal vector so that the normal vector and viewing vector are not colinear; the system aborts when this occurs. To get the plan view, for example, the (default) normal vector i s changed from (0.0, 0.0, 1.0) to (0.0, 1.0, 0.0). The r i g , as shown in the results, i s from the viewing vector (-3.0, -2.0, -1.0), the default value. A time stepping procedure was performed using WELSAS2. There are 10 time steps evenly spaced through the wave period. The motioms are amplified 100 times so that they may be more eas i l y seen. The r i g acts as a r i g i d body. Once the coordinates of any point on the r i g are known, the 36 coordinates of the other points may be e a s i l y determined. The results of the f i r s t two time steps are shown in F i g . 30 and F i g . 31. A video showing the movement of the r i g over several periods was made. The major d i f f i c u l t y in the production process was t r y i n g to align each sheet so that i t s ' border was in the same position as the previous sheet's. It was even more important to keep the border aligned in t h i s case than for the island so that the true movement of the r i g could be shown. This was also overcome by using polyethylene sheets with one master sheet showing the axis, border and t i t l e and subsequent sheets for each time step showing the displaced r i g and a border only. The order in which the four steps were executed in p l o t t i n g the r i g on the IRIS was d i f f e r e n t to the order for the VAX. This i s because the IRIS does not allow for f i l l of varying i n t e n s i t y . Instead of p l o t t i n g the outline of the elements on the h u l l before p l o t t i n g the outline of the h u l l s , the s o l i d f i l l representing the surfaces of the h u l l were plotted before the elements which formed the gri d on the exterior of the h u l l . No change was required in the algorithm for the p l o t t i n g of the v e r t i c a l cylinders. Because of the ease with which the IRIS can perform the transformation of the r i g , only the i n i t i a l , undisplaced structure has been stored in the computer's memory. This i s in sharp contrast to the process used to display the island on the IRIS where the new configuration of the structure for each time step must be stored. This put severe l i m i t a t i o n s 37 on the c a p a b i l i t y of the IRIS to show more than a few time steps. While there are many elements which have been f i l l e d in displaying the r i g , the system i s capable of handling many more time steps because the structure at the end of each time step need not be stored. Once the i n i t i a l structure i s drawn, the results of a large number of time steps may be displayed. Time stepping may be performed forwards or backwards and in any size increment and may be repeated u n t i l a l l the information has been drawn from the res u l t s . 6. CONCLUSIONS AND RECOMMENDATIONS Two d i f f e r e n t methods have been used to display the island and the r i g . For structures where the f i l l pattern i s the same for a l l elements, the program developed for the island may be used. When a l l of the elements do not have the same f i l l pattern, the program developed to display the r i g should be used. In addition, the island's program allows the colour code of each element to be pre-determined whereas a t r i a l and error method may be used for the r i g . The r i g may be used to display the results of numerical analyses with elements co n s i s t i n g of up to 4 nodes while the island i s v a l i d for elements with up to 8 nodes. The graphical program for the r i g may e a s i l y be extended to include eight noded elements while the island's program may be revised to accomodate elements with d i f f e r e n t f i l l patterns. 6.1 WAVE TRANSFORMATION AROUND AN ARTIFICIAL ISLAND The re s u l t s of an analysis of a computer model of wave transformation around an a r t i f i c i a l island, taking wave re f r a c t i o n , d i f f r a c t i o n and energy dissipation due to wave breaking into account have been displayed graphically. The evolution of the wave f i e l d has been obtained by the use of a time stepping procedure. By positioning himself at discrete points around the island, the viewer may observe the free surface. If the viewer remains stationary for some time, he may observe the changing face of the wave f i e l d . The VAX has been used to simulate these conditions. By 38 39 combining these two processes, the viewer may get a good idea of how the wave f i e l d changes. For the more r e a l i s t i c case of both the observer and wave f i e l d changing, the IRIS has been used. Some of the resolution obtained with the VAX has been l o s t . Because of the small capacity of the system, the pictures were not l i k e those on the VAX. Instead of the elements being f i l l e d with the appropiate colour, they were outlined with the colour. Hidden l i n e removal was also not performed. Newer (and more powerful) systems can be used to a l l e v i a t e these two problems. The same programs, with minor changes, can be used on these newer systems. With more elements, a more r e a l i s t i c computer model and wave f i e l d would be produced. Engineers can then confidently base design decisions on these models. Galvin (1968) suggests a parameter, /3 = H 0/L 0m 2 where H 0 and L 0 are the deep water wave height and wave length respectively and m i s the slope of the berm which can be used to c l a s s i f y d i f f e r e n t types of breaking waves ( s p i l l i n g , plunging, collapsing or surging). Using this d e f i n i t i o n , for a changing slope, the type of breaking wave can be shown by assigning a colour code to the breaking wave within the numerical program. There i s s t i l l a l o t more work to be done before the description of the transformed wave f i e l d i s accurate. As new techniques are developed, they may be included in the numerical program. If the format of the res u l t s is unchanged, the p l o t t i n g programs which have been developed 40 may be used with confidence. 6.2 WAVE EFFECTS ON HYBRID OFFSHORE STRUCTURES A computer program (RIG) has been developed to arrange the nodal coordinates of a semi-submersible r i g in a form such that the r i g can be e f f i c i e n t l y plotted. 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 to plot the r i g on the VAX i s suitable for use with four noded elements. The data needed to plot the r i g has been arranged in a form s i m i l a r to the data of a f i n i t e element analysis. Three views (plan, elevation and right side) show the layout of the r i g . The position of a r i g at the end of each time step of a simulated storm is displayed. Comparison between successive views shows how the r i g would behave in the design storm. For a more informative presentation, the real time movement has been shown on the IRIS. There is no l i m i t on the number of time steps which can be used. It i s anticipated that cross bracing w i l l eventually be added to the r i g . The facet generating program i s presently l i m i t e d to v e r t i c a l cylinders. The use of spherical coordinates, as opposed to the polar coordinates presently employed, i s suggested for the cross bracing to be included. There w i l l be problems at the intersection of two or more cyl i n d e r s . The development of an accurate graphical description of the jo i n t requires considerable thought and e f f o r t . Since these displays are only used to examine the displacements of the r i g , the joints need not be well 41 defined. The addition of mooring l i n e s should pose no d i f f i c u l t i e s . Their description can be s i m i l a r to the one used to draw the edges of the elements of the cylinders where the sixteen chords were joined in a head to t a i l manner. The whipping of the cables and the bobbing of the r i g may then be observed simultaneously. The water surface can also be included in the display for completeness since i t i s the r i g ' s motions and not the water surface that i s examained in t h i s a n a l y s i s . Since the rig ' s motions have been calculated, the v e l o c i t y potential of the surrounding f l u i d can be determined. The dynamic free surface boundary equation can then be used to give the surface p r o f i l e . 7. REFERENCES 1. Berkhoff, J. C. W., (1972) "Computation of Combined Refraction D i f f r a c t i o n " , Proc. 13th International Coastal Engineering Conference, ASCE, Vancouver, pp471-490 2. Booij, N. , (1981) "Gravity Waves on Water with Non-Uniform Depth and Current", thesis presented to the Technical University of Delf t , at D e l f t , The Netherlands, in p a r t i a l fulfilment of the requirements for the degree of Doctor of Philosophy 3. Byrne, D. J., (1984) "Pseudo 3-D projections from isoparametric surfaces", Engineering in Computers 1984, V o l l , Sept pp219-226 4. De Jong, J . J. A. and Bruce, J. C , (1978) "Design and Construction of a Caisson-Retained Island D r i l l i n g t h Platform for the Beaufort Sea", Proceedings, 10 Annual Offshore Technology Conference, OTC Paper 3294, Houston, Texas 5. DI-3000 User's Guide, March 1984 6. Galvin, C. J., (1968) "Breaker Type C l a s s i f i c a t i o n on Three Laboratory Beaches", J. Geophys. Res., Vol 73, pp3651-3659 7. IRIS User's Guide, Version 1.0 8. Isaacson, M. de St. Q., (1985) "Wave Transformation in The Coastal Zone", 1985 Australasian Conference on Coastal and Ocean Engineering, Vol II, pp25-32 42 43 9. Isaacson, M. de St. Q., (1986a) "User's Guide to the program WELSAS2", Department of C i v i l Engineering, University of B r i t i s h Columbia, Vancouver, Canada 10. Isaacson, M. de St. Q., (1986b) "Wave Effects on Hybrid Offshore Structures of A r b i t r a r y Shape", Coastal/Ocean Engineering Report, Department of C i v i l Engineering, University of B r i t i s h Columbia, Vancouver, Canada 11. Isaacson, M. de St. Q. and Talukdar, K., (1986) "Wave transformation around a r t i f i c i a l islands", Ocean Structural Dynamics Synposium 12. Mufti, Aftab A., (1983) "Elementary Computer Graphics" 13. Noma, Tsukasa and Kunii, Tosiyasu L., (1985) "ANIMENGINE: An Engineering Animation System" Proceedings, Graphics Interface 1985 14. Rogers, David F., (1985) "Procedural Elements for Computer Graphics", McGraw H i l l , Inc 15. Sarpkaya, T. and Isaacson, M. , (1981 ) "Mechanics of Wave Forces on Offshore Structures", Van Norstrand Reinhold, New York 16. Talukdar, Kushal, (1986) "Transformation of Waves around A r t i f i c i a l Islands", thesis presented in p a r t i a l fulfilment of the degree of Master of Applied Science to The University of B r i t i s h Columbia, Vancouver, Canada in A p r i l 1986 17. UBC Graphics Lab, "Computer Communications" 18. Walker, J. and Headland, J . , (1982) "Engineering Approach to Non-linear wave shoaling", Proceedings, 1 8 ^ 44 International Coastal Engineering Conference, Ch 34, pp 523-542 Weggel, J. R., (1972) "Maximum Breaker Height for Design", Proceedings, 1 3 ^ International Coastal Engineering Conference, Ch 21, pp 419-431 APPENDIX A : THE ISLAND A.1 USER'S MANUAL FOR DATGEN The data generating program DATGEN generates the coordinates of the nodes and the connectivity of these nodes for the c i r c u l a r island with l i n e a r l y sloping berm. The data required to define the system i s as follows: HD, HS, AM, DIA (4F10.4) HD = deep water depth HS = shallow water depth AM = berm slope DIA = island diameter NTT, SC (I4,F8.4) NTT = p l o t t i n g code for the f i n i t e element mesh = 0 no plot required = 1 plot required SC = scale factor for plot ANINC (F10.4) ANINC = angular increment for each element (in degrees) NR, NA (214) NR = number of r a d i a l increments from the v e r t i c a l face of the island to po s i t i o n where the radiation boundary i s loacted 45 46 180 NA = number of angular increments (= ) ANINC DELTAT, TIME (2F10.4) DELTAT = time step increment TIME = duration of storm (consistent units) DR(I) (F10.4) DR(I) = r a d i a l increment from the l a s t element, s t a r t i n g from the v e r t i c a l face of the island; there are NR of these values J1 , J2, J3, J4 (414) J1 = print option for data = 0 no print necessary = 1 print required J2 = breaking option = 0 breaking not considered = 1 consider breaking in the analysis J3 = grid p l o t t i n g option = 0 grid not needed = 1 generate gri d J4 = graphical p l o t t i n g option = 0 for movie (IRIS) = 1 for s t i l l frames (VAX)^ T, H, G (3F12.4) T = incident wave period 47 H = incident wave height G = acceleration due to gravity With the data supplied for DATGEN, the parameters required for the wave transformation 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 executed. R -LOAD 5 = input file 1 6 = output file 1 If a plot of the f i n i t e element mesh generated by DATGEN i s required, i t may be generated on either the printronix or the QMS p r i n t e r , for the printronix p r i n t e r : CON *PRINT* RMPROUTE=CIVL R *PXPLOT COPY -PLOT# *PRINT* REL *PRINT* for the QMS p r i n t e r : CON *PRINT* RMPROUTE=CNTR R *QMSPLOT COPY -PLOT# *PRINT* REL *PRINT* 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 geometry for VAX 2 = z-coordi nat es for IRIS 5 = out put file I 6 = finite element output file 1 = wave height around island 8 = wave height at nodal points 9 = wave elevation at nodal points 12 = island geometry The output, in a form compatible with the input requirements of the graphics program, takes the following form: TITLE NNODES,NELEM,N X(I), Y(I), Z(I), I = 1, NNODES I , LOCATRd ,J) , J = 1 ,9 The last column in the locator matrix i s the colour code assigned to each element. 49 A.4 EXECUTION OF GRAPHICS PROGRAM The commands required are contained in a source f i l e , ROCK. Once the output f i l e s have been transfered to the VAX system, the following command is entered: $waveplot Follow the instructions on the screen ( i . e . h i t the RETURN key) and enter: >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 timel. defn dsca 1.5,5.0, defn view -1.0,-1.0,-0.4, plot elem 0813 plot axis plot head 1, plot fram Changes to the ampli f i c a t i o n factor for the waves require a recompilation and r e l i n k i n g of the graphics program, waveplot. This i s done as follows: $fort waveplot $di31oad waveplot share ex For a real time examination of the evolution of the wave f i e l d , the user signs on to the IRIS and issues the command: $run h i s t i s l e Using the mouse and the legend displayed on the screen, the user may rotate the isl a n d , move in or away, move the island to the side or c a l l any one of a number of available 50 options. Any changes to th i s program would also require i t s ' recompiltion and r e l i n k i n g . The commands to be used are: $fort h i s t i s l e $irload h i s t i s l e APPENDIX B : THE RIG B.1 User's Manual for FAGGEN (Isaacson, 1986a) The data generating program RIG should be used in conjunction with the program FACGEN. FACGEN (FACet GENeration Program for WELSAS2) generates the facet geometry for the large members (hulls) and has the c a p a b i l i t y to handle up to 8 d i f f e r e n t structural shapes. In general, the input f i l e for FACGEN (unit 3) is of the following form (Isaacson,1986): IR IS IR, IS (215) = runup option control tag = structure shape option control tag (IP(I), 1=1,5) (515) IP = integer structure parameters IP(1) = number of length d i v i s i o n s IP(2) = number of width di v i s i o n s IP(3) = number of height d i v i s i o n s IP(4),IP(5) are not defined for conf iguration t h i s structure (RP(I), 1=1,5) (5G20.5) RP = real structure parameters RP(1) = length RP(2) = width RP(3) = height 51 52 RP(4) .= z-coordinate of horizontal axis RP(5) = horizontal distance between the two cylinder axes The output f i l e to FACGEN (unit 4) consists of the following parameters: IR, IS (215) (IP(I), 1=1,5) (515) (RP(I), 1=1,5) (5G20.5) N (15) N = number of facets (X(I), Y(I), 1(1), 1=1,N) (3G20.5) X, Y, Z = x, y, z coordinates of the facet centres The rest of the output i s used in the wave-structure interaction analysis and i s not pertinent here. For this p a r t i c u l a r configuration, the input f i l e to FACGEN is as follows: 1,7, 10,3,2,0,0, 100.,30.,20.,-40.,70., 53 B.2 USER'S MANUAL FOR RIG The input to RIG takes the following form: TITLE (A80) TITLE = descriptive heading NUMCYL (16) NUMCYL = number of v e r t i c a l cylinders in the r i g DIA, X, Y, ZED, NCIRCL, NLONG (4F7.2) DIA = diameter of v e r t i c a l cylinders X = x-coordinate of centre of cylinder Y = y-coordinate of centre of cylinder ZED = length of cylinder in z-direction NCIRCL = number of points to be used to represent the c i r c l e NLONG = number of storeys which are used to represent the 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 period A = c h a r c t e r i s t i c length of barge NINCR = number of time intervals for which the displaced shape is to be plotted 54 (RAO(I), 1=1,6) (6F11.3) RAO = Response Amplitude Operator amplitudes of the ori g i n of the barge RAO(I,1) => surge RAO(I,2) => sway RAO(I,3) => heave RAO(I,4) => r o l l RAO(I ,5) => pit c h RAO(I,6) => yaw (PHASE(I), 1=1,6) (6F11.3) PHASE = RAO phases The relevant part of the output from FACGEN i s also part of the input for RIG. B.3 EXECUTION OF FACGEN FACGEN must f i r s t be compiled. R *FTN SCARDS = FACGEN The facet generating program i s then executed. R -LOAD 3 = input file I 4 = output file I 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. R -LOAD 1 = cylinder outline output file 2 = cross section of hull output file 3 = outline of hull output file 4 = out put file I 5 = i nput file 2 6 = grid geometry and element description out put file 7 = echo print output file I 8 = echo print input file 2 9 = test file 10 = cross section of cylinder output file B.5 EXECUTION OF GRAPHICS PROGRAM A l l of the commands needed to plot the semi-submersible r i g are contained in a source f i l e , SUB. After the four output f i l e s have been transfered from the MTS system to the VAX system (Appendix C), the following command i s entered: $rigplot Follow the instructions on the screen ( i . e . h i t the RETURN key) and enter: >sour sub 56 In t h i s f i l e , the sequence of commands i s : read geom ele22. plot elem 0404 plot head 1, plot axis plot frame read geom hull33. plot elem 0226 read geom cyl11. plot elem 0555 read geom cyl22. plot elem 0909 For a structure with 496 nodes and 704 elements, the pl o t t i n g time i s approximately 2min 30sec. To see the real time processes, the user signs on to the IRIS and issues the command: $run h i s t r i g Using the mouse and the legend displayed on the screen, the user may rotate the r i g , move in or away, move to the side or c a l l any of a number of other options. APPENDIX C : TRANSFER VAX/VMS FILE TO THE UBC/MTS SYSTEM The coloured graphics f a c i l i t i e s used to display the figures are contained in the VAX/VMS whereas the data generating programs were compiled and executed on the UBC/MTS system. The following sequence of commands i s described in d e t a i l in the publication "Computer Communications" avail a b l e from the UBC Graphics Laboratory. Commands are in i t a l i c s while comments are in normal type. It i s assumed that the user i s familiar with the signon procedures for the VAX/VMS and UBC/MTS systems. signon on the VAX terminal SD ALLOC TXAO: KERMIT Dial up UBC (228-1401) on modem SET LINE TXAO: CONN G SIG "ccid" "password" RUN *KERMIT SERVER Ctrl ] C GET "filename" Repeat the GET command u n t i l a l l f i l e s have been rec ieved 57 58 BYE Perform the appropiate p l o t t i n g sequence (Appendix D) LO APPENDIX D : PLOTTING THE OUTPUT ON THE VAX/VMS SYSTEM In p l o t t i n g the elements, several options are ava i l a b l e to the user. They may be used i n d i v i d u a l l y but are more commonly used as a combination of any or a l l of these options. I n i t i a l l y , a t r i a l and error method i s used to deide which feature in each option and what combination of such feature is best suited for the p a r t i c u l a r s i t u a t i o n . Once these features have been determined, they can be stored in a source f i l e so that on one c a l l to t h i s f i l e , the options can be executed sequentially. The options are: defn view norm move axis alab asca zoom dsca proj port head lege plot node elem axis x,y,z] x,y,z] x,y,z] x,y,z] u,v,w] x] x] x,y,z] i ] i ] new heading] i ] [legend] i ] abed] i] 59 60 lege [ i ] head [ i ] f ram save [ f i l e ] logf [ f i l e ] sour [ f i l e ] paus wipe [ i ] stop exit In the c a l l "plot element", a 4-digit number i s entered according to the following scheme: plot elem abed where numbering, a = 0 no numbering of nodes = 1 numbering of nodes outline colour, b = 0 white = 1 red = 2 green = 3 dark blue = 4 l i g h t blue = 5 yellow = 6 s c a r l e t 61 = 7 white = 8 black = 9 green i n t e r i o r f i l l p a t t e r n , c = 0 no f i l l = 1 s o l i d f i l l = 2 t r a n s p a r e n t f i l l i n t e r i o r c o l o u r , d If parameter b i s zero, i t i s , by d e f a u l t , s e t equal to parameter d. fo r example 0002 => green o u t l i n e 0022 => tra n s p a r e n t green f i l l and green o u t l i n e A P P E N D I X E : T H E M O V I E The video tape shows two examples of computer graphics applications in offshore hydrodynamics: one deals with wave propogation around an a r t i f i c i a l island and a second to wave induced motions of a semi-submersible r i g . The f i r s t example is of an a n a l y t i c a l model of an a r t i f i c i a l i s l a n d . It i s not unlike the islands, used for o i l and gas exploration in the Mackenzie Bay area of the Beaufort Sea, at the northern boundary of Canada. Waves approach the islan d from the upper right hand corner of the screen. For each of the three views, the incident wave is repeated 10 times so that storm conditions may be simulated. In t h i s presentation, the v e r t i c a l scale is exaggerated by a factor of about 3 so that the berm appears to be r e l a t i v e l y steep. The l e f t side and back of the island are facing the viewer. If we focus our attention on one wave as i t moves past the i s l a n d , we can see that the wave i s becoming smaller. There i s also a region at the back of the island where waves which have travelled around the other side of the island c o l l i d e with those from th i s side. The second view shows the same island and waves, but t h i s time the viewer has been moved so that we can see the front of the i s l a n d . Waves are now approaching from the lower l e f t hand corner of the screen. At the front, the waves almost overtop the v e r t i c a l wall. Towards the back of the island, the waves are c l e a r l y not as high. Wave breaking has occured. The yellow area defines the region where 62 63 breaking takes place. When waves break, energy is dissipated. Since wave energy i s proportional to the square of the wave height, the energy loss results in a reduction in wave height as the waves t r a v e l past the isla n d . In the f i n a l view of the island and wave, we are looking at the front and right hand side of the' island. Waves are now approaching the island from the lower right hand side of the screen. The decrease in wave height as the wave moves around the island i s once again seen. With the results presented as shown in these three views, an ocean engineer can confidently determine the possible maneuvering of vessels in the v i c i n i t y of the island or the movement of any of the sediment over the berm of the islan d . The second example of the app l i c a t i o n of computer graphics in ocean engineering i s a semi-submersible r i g . The numerical model of the r i g i s shown: t h i s i s a crude model of a r i g which i s very much l i k e the Ocean Ranger. The motions are shown for a wave height of approximately 5 meters and have been amplified 100 times in r e l a t i o n to the structure's dimensions. Storm conditions are simulated by having the r i g respond to 15 waves. The location of the water surface (at the top of the v e r t i c a l columns) has not been shown. The ocean engineer would l i k e to examine the rig ' s motions so that any irregular movements or large amplitude motions can be predicted in the design o f f i c e before construction begins. The motions of personnel who are at the working deck can then be calc u l a t e d . A worker 64 standing at t h i s location may experience unacceptably large motions which would preclude the continuation of his duties. j 6 5 F i g . 1 Sketch of an a r t i f i c i a l island (Adapted from De Jong and Bruce, 1978) 6 6 y F i g . 2 Sketch of boundaries (Adapted from Talukdar, 1986) F i g . 3 Sketch of semi-submersible r i g (Adapted from Sarpkaya and Isaacson, 1981) 68 F i g . 4 An element for a slender member (Adapted from Isaacson, 1986b) F i g . 5 U n d i s t o r t e d d i s p l a y of an a r t i f i c i a l i s l a n d F i g . 10 Mesh on e x t e r i o r of h u l l s F i g . 11 Outline of h u l l s F i g . 12 Arrangement of elements on c y l i n d e r i g . 13 Edges of elements on the cy l i n d e r 78 F i g . 14 Dimensions of the island i g . 15 I n t r u s i o n of yellow elements 81 83 84 85 WAVE H E I G H T WITH B R E A K I N G SHOWN F i g . 22 Master sheet for island 89 30m E O lO Side view 20m t lODm 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 S E M I — S U B M E R S I B L E R I G F i g . 29 E l e v a t i o n of semi-submersible r i g vO 

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