WHOI-97-15
`
`.~~...
`
`Woods Hole
`Oceanographic
`Institution
`
`1930
`
`by
`
`Jason i. Gobat, Mark A. Grosenbaugh, and Michael S. Triantafyllou
`
`November, 1997
`
`. Technical Report
`
`Funding was provided
`
`by the Office of Naval Research under
`Contract Nos. N00014-92-J-1269. and N00014-95-1-01 06
`
`Copyright ~1997by Woods Hole Oceanographic Institution. All rights reserved.
`
`PGS v. WESTERNGECO (IPR2014-00689)
`WESTERNGECO Exhibit 2048, pg. 1
`
`

`
`WHOI-97-15
`
`WHOI Cable: Time Domain Numerical Simulation of
`Moored and Towed Oceanographic Systems
`
`Jason i. Gobat, Mark A. Grosenbaugh, and Michael S. Triantafyllou
`
`by
`
`Woods Hole Oceanographic Institution
`Woods Hole, MA 02543
`
`November, 1997
`
`Technical Report
`
`Funding was provided by the Offce of Naval Research through grants NOOOl4-92-J-1269
`and N00014-95-1-0106 and an Offce of Naval Research Graduate Fellowship.
`
`Reproduction in whole or in part is permitted for any purpose of
`the United States Government. This report should be cited as:
`Woods Hole Oceanographic Institution Technical Report WHOI-97-15.
`
`Copyright (£1997 by Woods Hole Oceanographic Institution. All rights reserved.
`
`Approved for Distribution:
`
`~ J¿ ~A:
`
`Timothy K. Stanton, Chairman
`Deparment of Applied Ocean Physics
`and Engineering
`
`~!!-!!~~lT== i.ir..Õ ..
`:i=ir3:_0::_ CJai-:: ,.. 0_lTo- o~-~
`
`PGS v. WESTERNGECO (IPR2014-00689)
`WESTERNGECO Exhibit 2048, pg. 2
`
`

`
`PGS V. WESTERNGECO (|PR2014-00689)
`WESTERNGECO Exhibit 2048, pg. 3
`
`PGS v. WESTERNGECO (IPR2014-00689)
`WESTERNGECO Exhibit 2048, pg. 3
`
`

`
`9 9 9
`
`10
`
`11
`
`11
`
`12
`
`12
`
`13
`
`13
`
`15
`
`15
`
`17
`
`18
`
`18
`
`18
`
`19
`
`19
`
`19
`
`20
`
`Contents
`
`Foreword
`
`About this Manual
`
`Acknow ledgements
`
`Typographical Conventions
`
`1 Introduction
`
`1.1 Overview of problem types .....
`
`1.2 WHOI Cable mathematical features
`
`1.3 WHOI Cable implementation features
`
`2 Mathematical and Numerical Theory
`
`2.1 General numerical approach .
`
`2.2 Numerical details of static problems
`
`2.2.1 Boundary conditions
`. . .
`
`Initialization
`
`2.2.2
`
`2.2.3 Coordinate integration .
`
`2.2.4 Bottom interaction .
`
`2.3 Numerical details of dynamic problems.
`2.3.1 Wave forcing . . . . . .
`
`2.3.1.1 Wave followers
`
`2.3.1.2 Morison's equation.
`
`2.3.1.3 Froude-Krylov forcing model
`
`1
`
`PGS v. WESTERNGECO (IPR2014-00689)
`WESTERNGECO Exhibit 2048, pg. 4
`
`

`
`2.3.2 Coordinate integration .
`
`2.3.3 Bottom interaction . . .
`
`2.3.4 Dynamic pay-in and pay-out of cable.
`2.4 Equations of motion . . . . . . . .
`2.4.1
`
`Two-dimensional problems
`
`2.4.1.1
`
`Static equations
`
`2.4.1.2 Dynamic equations
`
`2.4.2 Three-dimensional problems .
`. .
`
`Static equations
`
`2.4.2.1
`
`2.4.2.2 Dynamic equations
`
`2.5 Coordinate transformations
`
`2.5.1 Two-dimensional .
`
`2.5.2 Three-dimensional
`
`3 Structure of a cable Problem
`
`3.1
`
`Notation and coordinate systems
`
`3.2 Basic language features
`
`3.2.1 Expressions . . .
`
`3.2.1.1
`
`Continuous functions
`
`3.2.1.2 Discrete functions
`. . . . . . .
`3.3 Components of an input file
`
`3.2.2 Units
`
`3.3.1 Problem description
`
`3.3.2 Analysis parameters
`
`3.3.3 Environmental parameters .
`
`3.3.4 Cable, chain and rope materials .
`
`3.3.5 Connectors
`
`3.3.6 Buoys
`
`3.3.7 Anchors
`
`2
`
`20
`
`21
`
`21
`
`22
`
`22
`
`22
`
`23
`
`23
`
`23
`
`24
`
`26
`
`26
`
`27
`
`29
`
`29
`
`29
`
`30
`
`30
`
`31
`
`31
`
`31
`
`32
`
`32
`
`35
`
`37
`
`38
`
`39
`
`40
`
`PGS v. WESTERNGECO (IPR2014-00689)
`WESTERNGECO Exhibit 2048, pg. 5
`
`

`
`3.3.8 System layout. . .
`
`3.3.9 The end statement
`
`3.4 Tips and tricks . . . . .
`
`3.4.1 Static problems.
`
`3.4.2 Dynamic problems
`
`4 The cable Application
`4.1 Basic operation . . . . . . . . . . . .
`
`4.2 Using the run-time solution controls
`4.3 Using the C pre-processor . . . . . .
`
`4.4 Summary of command line parameters
`
`4.5 Interpreting the output from cable
`
`5 Post-processing cable Results
`
`5.1 Using cable results with Matlab
`
`5.1.1
`
`Format of the Matlab file
`
`5.1.2 Example Matlab manipulations .
`
`5.1.3
`
`res2mat command line parameters
`
`5.2 The animate post-processing application.
`
`5.2.1
`
`The main animation window
`
`5.2.2 Coordinates and zooming . .
`
`5.2.3 Animate command line parameters .
`5.3 ASCII output . . . . . . . . . . . . . . . .
`
`5.3.1 res2asc command line parameters.
`
`6 cable's Windows Interface
`6.1 Introduction......
`6.2 Building an input fie .
`6.3 Solving a problem ..
`
`6.4 Viewing and converting results
`
`3
`
`40
`
`45
`
`45
`
`45
`
`46
`
`47
`
`47
`
`48
`
`49
`
`50
`
`53
`
`55
`
`55
`
`55
`
`56
`
`57
`
`57
`
`58
`
`60
`
`60
`
`63
`
`63
`
`65
`
`65
`
`65
`
`67
`
`69
`
`PGS v. WESTERNGECO (IPR2014-00689)
`WESTERNGECO Exhibit 2048, pg. 6
`
`

`
`6.5 Working with files .
`
`6.6 Command reference
`
`6.7 Installng WHOI Cable for Windows
`
`6.7.1 System requirements. .
`
`6.7.2 Installation instructions
`
`6.7.3 Printing from animate under Windows.
`
`6.7.4 Modifying the installation . .
`
`6.7.4.1 File and pathnames
`6.7.4.2 Templates .
`
`A Subsurface Mooring Example
`
`B Shallow Water Surface Mooring
`
`C Deep Water S-tether Mooring
`
`D Horizontal Array Mooring
`
`E Towed Vehicle Example
`
`References
`
`70
`
`70
`
`71
`
`71
`
`72
`
`72
`
`73
`
`73
`
`73
`
`75
`
`79
`
`83
`
`89
`
`95
`
`99
`
`4
`
`PGS v. WESTERNGECO (IPR2014-00689)
`WESTERNGECO Exhibit 2048, pg. 7
`
`

`
`List of Figures
`
`2.1
`
`3.1
`
`Local and global two-dimensional coordinate systems.
`
`Geometric definitions for cable. . . . . . . . . . .
`
`4.1 cable's graphical information and control dialog. .
`
`4.2 The binary fie format for cable results files.
`
`5.1 The main window of animate. . . . . . . . .
`
`5.2 A time plot of the forces at two marked nodes.
`
`26
`
`30
`
`49
`
`54
`
`58
`
`59
`
`6.1 The relationships between the WHOI Cable component programs. 66
`
`6.2 The main window of the WHOI Cable Windows interface. . . . . . 66
`
`6.3 The solution control dialog in the WHOI Cable Windows interface. 68
`
`6.4 The results control dialog in the WHOI Cable Windows interface. . 69
`
`6.5 The setup dialog used to configure WHOI Cable pathnames. . 73
`
`A.l Example result from animate for a subsurface mooring. ......... 77
`
`A.2 A plot of the time history of forces for the subsurface mooring example. 77
`
`B.l The static configuration of the surface mooring example problem. . 81
`B.2 The time history of total tension for the surface mooring example. 82
`
`C.l Static and dynamic results for an S-tether mooring.
`
`D.l Animation result for the horizontal mooring example problem.
`
`D.2 Motion records for nodes on the horizontal mooring. . . . . . .
`
`87
`
`93
`
`93
`
`5
`
`PGS v. WESTERNGECO (IPR2014-00689)
`WESTERNGECO Exhibit 2048, pg. 8
`
`

`
`E.l Steady-state configuration of the towing example. . . . . .
`
`E.2 Depth profile of the tow sled during a tow-yow maneuver.
`
`97
`
`98
`
`6
`
`PGS v. WESTERNGECO (IPR2014-00689)
`WESTERNGECO Exhibit 2048, pg. 9
`
`

`
`List of Tables
`
`5.1 The names that res2mat assigns to Matlab variables. . . . . . . . . . . . .. 56
`
`6.1 Complete command structure for the WHOI Cable for Windows encapsulator. 71
`
`7
`
`PGS v. WESTERNGECO (IPR2014-00689)
`WESTERNGECO Exhibit 2048, pg. 10
`
`

`
`8
`
`PGS V. WESTERNGECO (|PR2014-00689)
`WESTERNGECO Exhibit 2048, pg. 11
`
`PGS v. WESTERNGECO (IPR2014-00689)
`WESTERNGECO Exhibit 2048, pg. 11
`
`

`
`Foreword
`
`A bout this Manual
`
`This report documents version 1.0 of WHOI Cable. While it is our intention to pro-
`vide up-to-date, comprehensive, accurate documentaion, WHOI Cable is very much a work
`in progress and as such undergoes frequent change. If you find something that behaves
`diflerently than the way this document says it should behave then please let us know.
`
`This report is presented largely as a user's guide for WHOI Cable. We generally provide
`technical algorithmic details only to give the user a loose understanding of how problems are
`solved or in places where the information is not published elsewhere. User should consult
`the references listed in the bibliography for additional details (particularly ¡4, 121).
`
`Acknowledgements
`
`Though the current implementation of WHOI Cable is a relatively recent development, it
`does owe much to several pieces of work completed over the last few years. The time domain
`simulation algorithm is based on code originally developed by Christopher Howell for his
`Ph.D. thesis ¡4J and modified by Thanassis Tjavares for his Ph.D. thesis ¡12J.
`
`The funding for the development of WHOI Cable has been provided by the Offce of
`Naval Research through grant numbers NOOOI4-92-J-1269 at WHOI and N00014-95-1-0106
`(ONR. Code 321, Ocean Engineering and Marine Systems Program) and an Offce of Naval
`Research Graduate Fellowship.
`
`WHOI Cable is copyright (£1997 by Woods Hole Oceanographic Institution. WHOI
`Cable is proprietary software; free redistribution of WHOI Cable software is not permitted.
`
`Matlab is a trademark of The Mathworks, Inc. Pentium and Pentium Pro are trade-
`marks of Intel Corportation. Windows 95 and Windows NT are trademarks of Microsoft
`Corporation.
`
`9
`
`PGS v. WESTERNGECO (IPR2014-00689)
`WESTERNGECO Exhibit 2048, pg. 12
`
`

`
`Typographical Conventions
`
`This report employs a number of typographical conventions to mark buttons, command
`names, menu options, screen interaction, etc.
`
`Bold Font
`
`Used to mark buttons, and menu options in graphical environments.
`
`Italics Font
`
`Used to indicate an application program name, e.g. res2mat.
`
`Typewri ter Font
`Used to represent screen interaction at the shell prompt. Also used for
`example input files, and keywords that belong in input files.
`
`I Key I
`
`Represents a key (or key combination) to press, as in press I Return I to con-
`tinue.
`
`10
`
`PGS v. WESTERNGECO (IPR2014-00689)
`WESTERNGECO Exhibit 2048, pg. 13
`
`

`
`Chapter 1
`
`Introduction
`
`1.1 Overview of problem types
`
`The types of systems that we classify as oceanographic mooring systems include simple teth-
`ered buoys, towed and drifting systems, and complex strings of instrumentation suspended
`in deep water. From an engineering design perspective it is important that we can predict
`how these systems wil respond to a variety of environmental factors, particularly waves
`and current. We might want to know just how much current it wil take to pull a surface
`buoy under water or what the maximum tension wil be in a mooring line during a large
`storm. The scientific purposes of a system might require that the motion of a particular
`instrument not exceed a certain level in typical operating conditions. The unifying problem
`behind analyzing these kinds of systems is one of nonlinear cable mechanics.
`
`Typical oceanographic mooring systems consist of rope, wire, and chain connected to-
`gether by shackles, instruments, and buoys and terminated at the ends with buoys, ships,
`sinker weights, or anchors. WHOI Cable is a collection of computer programs for cable
`mechanics designed specifically to solve this nonlinear problem for systems which can be
`defined in these terms and which fit into one of several basic categories.
`
`For traditional single point moorings WHOI Cable (sometimes referred to by the name
`of the primary application program, cable) can solve subsurface and both taut and slack
`surface moorings. The system can consist of any combination of diflerent cable/chain/rope
`segments in series with one another. Instruments, floats, and connectors between segments
`are treated as lumped masses (i.e., the rigid body dynamics of an in-line instrument are
`not modeled, but the mass, weight and drag eflects of the instrument are considered). cable
`can also solve problems in which both ends of the mooring are anchored on the bottom
`and towing/drifter problems in which the subsurface end is unconstrained and the surface
`
`11
`
`PGS v. WESTERNGECO (IPR2014-00689)
`WESTERNGECO Exhibit 2048, pg. 14
`
`

`
`end is free to move either under the influence of current (drifters) or with a prescribed
`(possibly time varying) speed (towing). Towing problems can also include the eflects of
`time varying pay-in and pay-out rates. In all cases cable can produce solutions in either two
`or three dimensions and can solve either the the static (steady-state) problem given forcing
`by current or the dynamic problem given forcing by both current and waves.
`
`1.2 WHOI Cable mathematical features
`
`WHO
`
`I Cable is built around a mathematical model that includes the eflects of arbitrary
`geometric nonlinearities, material nonlinearities, material bending stiflness, and material
`torsion. Including geometric nonlinearities and bending stiflness means that WHOI Cable
`can accurately model systems in which cable segments go slack. The nonlinear, one-sided
`boundary condition at the seabed is modeled as an elastic foundation for systems with cable
`lying on the bottom. The numerical implementation includes an adaptive time stepping
`algorithm to speed the solution of problems with high nonlinearity.
`
`1.3 WHOI Cable implementation features
`
`WHO! Cable is a suite of applications, all of which are centered around the primary solver
`prograni. cable. cable is responsible for processing user input files and generating results for
`all of t.ll( various problem types. Input files are constructed using an intuitive, object based
`syiitax. This high-level syntax allows for the use of symbolic expressions in assignment
`st.at.~iients. the re-use of object descriptions that may be stored in central database files,
`aiid a largely free-form construction of input files. It also faciltates detailed error reporting.
`
`Results can be post-processed either by converting them to Matlab format with res2mat
`or by viewing them with the graphical application animate. animate provides a simple
`enviroiiment for viewing system configurations in both two- and three-dimensions and for
`generating graphs of result variables. Spectra of time domain results are also available.
`
`WHOI Cable for Windows includes an encapsulator application that allows for control
`of all of the component programs from within a familiar Windows style interface. This
`int.erface gives the user total push button control of the various options for solving problems
`and analyzing results.
`
`12
`
`PGS v. WESTERNGECO (IPR2014-00689)
`WESTERNGECO Exhibit 2048, pg. 15
`
`

`
`Chapter 2
`
`Mathematical and Numerical
`Theory
`
`2.1 General numerical approach
`
`For all combinations of boundary conditions, 2D or 3D and static or dynamic problems, the
`matlwmatical problem is posed as a system of coupled, nonlinear partial diflerential equa-
`tioiis. cable solves these systems numerically by discretizing the continuous (exact) forms
`of t.hese governing equations onto a grid of nodes on which it wil calculate an approxi-
`mate solution. As the grid becomes finer and finer the approximate solution wil approach
`solution. The cost of these finer discretizations which buy better solutions is an
`iiicrease in computation time.
`
`tli(' exact
`
`I3ot.h the static and the dynamic cable problems can be generalized as a system of N
`first-order nonlinear partial diflerential equations
`åY - ( -)
`ås + F s, Y = 0,
`the N dependent variables, s is the position variable along the cable
`(th(' Lagrangian coordinate), and F is a vector of functions that depends on the form of
`the governing equations. For example, in the 2D static problem (the simplest of all possible
`cases). equation 2.1 represents four equations in four unknowns: strain (from which we
`can always derive tension via a constitutive relationship), shear force, inclination angle, and
`curvature. This equation is discretized at the n nodal points using centered finite diflerences
`written on the half-grid points (which makes the diflerences second order accurate ¡12, 13)).
`At node k the discretized result is
`- - Sk - Sk-l ( - - )
`Yk - Yk-l + 2 Fk + Fk-1 = O.
`
`(2.1)
`
`(2.2)
`
`where 1"1 is the vector of
`
`13
`
`PGS v. WESTERNGECO (IPR2014-00689)
`WESTERNGECO Exhibit 2048, pg. 16
`
`

`
`Equation 2.2 represents an N x n system of coupled, nonlinear equations which cable solves
`using a Newton-Raphson like relaxation technique ¡8J.
`
`Equation 2.2 can only be satisfied by an exact solution for Yk. Given an inexact first
`guess at this solution, yko, we need to develop an iterative scheme to calculate successively
`better approximations, Y~, through a series of update vectors, ..Y~, such that
`
`yl+1 = Y~ + .. Y~
`where yl+l brings us closer to satisfying the equality in equation 2.2. In quantitative terms
`we want to iteratively minimize the error function
`
`(2.3)
`
`ek k, k-l - k k-l 2 -~ + -~-l .
`-¿ (y;-i y;-i ) _ y;-i _ y;-i + Sk - Sk-l (Fi pi )
`
`(2.4)
`
`Neglecting for clarity the dependence on the previous nodal point (k - 1), we can very
`loosely write
`
`ßëk~-
`
`~ ßYk'
`
`(2.5)
`
`(et+1 (Y~ + ..Y~) - et (Y~) J
`
`the known form of
`
`the discretized governing equations (equation 2.4). If
`
`Sy,ik
`The derivatives on the right hand side of equation 2.5 can be calculated analytically from
`we were to re-insert
`the dependence on Yk- 1, we would note that these derivatives actually constitute an N x 2N
`Jacobian matrix at each k (the matrix is composed of the derivatives of the N equations
`with respect to the 2N variables represented by Yk and Yk-l)' We can assemble the Jacobian
`matrices from each node into a single global Jacobian matrix (much like element stiflness
`matrices are assembled into global stiflness matrices in the finite element method), add
`boundary condition information and formulate a linear system that wil find ..Y~ to drive
`the updated error, et+1, to zero. If Ji is this global Jacobian matrix evaluated at yi then
`we see from equation 2.5 that
`
`Ji ..yi = -ë!.
`
`(2.6)
`
`Because only two nodes (k and k - 1) are coupled by each individual Jacobian matrix
`the assembled global Jacobian matrix in equation 2.6 wil be very sparse, with the only
`non-zero entries clustered near the main diagonaL. cable takes advantage of this sparsity
`in solving equation 2.6 by using a sparse Gaussian Elimination algorithm, NSPIV, due to
`Sherman ¡lOJ. Sparse algorithms such as NSPIV exploit sparsity to reduce both memory
`requirements and computation time (normal Gaussian elimination is an O(n3) operation,
`sparse algorithms can be as effcient as O(n)).
`
`The actual update to Y is scaled by a relaxation factor w
`yi+l = yi _ w..yi.
`
`(2.7)
`
`14
`
`PGS v. WESTERNGECO (IPR2014-00689)
`WESTERNGECO Exhibit 2048, pg. 17
`
`

`
`The purpose of this relaxation factor is to slow (under-relax) the update in cases where
`strong nonlinearities may mean that the update is not quite as robust as we would like.
`For highly nonlinear problems, where small changes in parameters can mean big changes
`in system configuration, the approximation of equation 2.5 becomes less valid and our
`update ..i\ if fully applied (w = 1), may actually increase the total system error. A
`small relaxation factor increases the accuracy of the linearized Taylor series expansion that
`equation 2.5 represents. By slowing the process down (w .c 1) the movement of the system
`from iteration to iteration towards equilbrium wil be smoother because the steps between
`iterations wil be smaller.
`
`The iterative updates of Y continue until the update vectors, ..Y, become smaller than
`a user defined convergence value. Given the update vector
`
`at each node k, the convergence condition is
`
`..Yk = ¡..Yi,k'" ..YN,kJ
`
`~ t ¡ n~ ~ IMi'kIJ "tolerance.
`
`(2.8)
`
`(2.9)
`
`Yi in the above are a set of canonical values that express the typical order of magnitude of
`the variable represented by Yi. A canonical value for strain, for example, is 0.01.
`
`2.2 Numerical details of static problems
`
`2.2.1 Boundary conditions
`
`Static boundary conditions for the various problem types can typically be described by an
`anchor restraint at the first end and an applied static force at the opposite, free end. The
`simplest case is a user prescribed force vector applied at the free end (general problems,
`see section 3.3.1). With a force that is known a priori, cable can generate a solution with one
`set of iterations directly. The problem is similar for subsurface moorings because cable can
`directly calculate the buoyancy and drag forces on the completely submerged buoy at the
`the system. Static (steady-state) towing problems can also be solved this way by
`fixing the position of the ship and applying weight and drag forces to the towed-body end of
`the system. The drag on the tow body and cable is generated both by the environmentally
`applied current and by an artificially superposed current that is equal in magnitude and
`opposite in direction to the steady-state tow speed.
`
`free end of
`
`In systems with a buoy on the free surface the problem is more complicated because we
`do not know the forces at the buoy end before solving the problem. Vertical and horizontal
`
`15
`
`PGS v. WESTERNGECO (IPR2014-00689)
`WESTERNGECO Exhibit 2048, pg. 18
`
`

`
`forces applied by the buoy on the cable segment under the buoy are a function of the buoy
`draft and the known buoyancy and drag properties of the buoy. cable begins the solution
`with forces calculated from an initial guess of the draft (equal to the maximum available
`draft), H2. We then calculate the actual draft, H2, for these forces. The absolute error is
`H s 9
`If this error is positive then we can tell immediately that the buoy does not have suffcient
`buoyancy to coiie to the surface. If the error is negative then we know that we need less
`buoyancy and we proceed to make a series of guesses
`
`eO = HO _ HO
`
`(2.10)
`
`H~+I = ßH~.
`
`(2.11)
`
`uiit.il we get a solution such that
`
`ei = Hi _ Hi
`
`(2.12)
`
`. . .
`H s 9
`is positive. With the actual solution now bracketed between H2 and H~, we proceed to use
`a \'-giiia falsi root finding technique ¡2J to home-in on a final solution. This root finding
`11I'ol'l'd\le fÒrins a second, outer loop of iterations. At each new guessed draft we must
`go t lirougli a iiew series of iterations to solve the problem. This inner loop of iterations
`is IIll process of finding the equilibrium position for a given applied static force based on
`till nirnit best guess at the draft. Note that ß in equation 2.11 is an outer iteration
`"1''laxat.io!l" factor that controls how fast we search for the minimum draft that brackets
`till rl'l draft. It should always be smaller than unity (it defaults to 0.95). Making it too
`siiall caii result in singularities because very small drafts equate to forces which may not
`1)( largi' euougli to support the weight of the system.
`
`TIll idea of outer loop iterations is also used for problems with both ends anchored on
`till Iiotloiu (see example, Appendix D). In this case, we do not know a priori the reaction
`lill"(I'S at. t.he second anchor. Given the position of the second anchor, however, we can
`iii-rfillu out.er loop iterations by changing the applied force at the second end with each
`oiill'l it.eration, until that second end is brought to its actual known position. The adjusted
`applil'd f()l:e at. each outer iteration is calculated from
`pHI = pk _ ß ( Xk - X)
`
`(2.13)
`
`wlie\'' pI. is the applied force vector at outer iteration k, Xk is the calculated position of
`11)( second anchor at outer iteration k and X is the desired position of the second anchor.
`Ii is a "stiflness" factor defaults to 5.
`The final type of problem that requires outer loop iterations is drifting systems. In
`principle, solving a drifter problem requires the same boundary conditions as a towing
`
`16
`
`PGS v. WESTERNGECO (IPR2014-00689)
`WESTERNGECO Exhibit 2048, pg. 19
`
`

`
`problem. The situation is more complicated for drifters, however, because for a complex
`current profile we do not know the steady-state drift speed of the system. At steady-state,
`the sum of the integrated drag force in the horizontal directions must be zero,
`
`L
`
`! ~PCdX(S)S(s) IUc(s) - ul (Uc(s) - U) ds = 0,
`o
`
`(2.14)
`
`where the integrand represents the drag force in a horizontal direction as a function of
`position due to a relative velocity that is the value of the current at that position minus
`the steady-state drift speed, U. For Uc(s) constant, U = Uc satisfies this constraint and we
`know that the system wil drift with the current speed. For Uc(s) not constant, however, it
`is clear that at some points on the system U must be less than Uc (s) and at other points
`it must be greater than Uc(s). For drifting systems, the outer loop of iterations determines
`U such that equation 2.14 is satisfied.
`
`The initial value of U is set to 105% of the maximum current speed (setting it to 100% of
`the current speed leads to numerical problems because there may be no resultant horizontal
`drag for cases of constant current). The outer loop iteration procedure then uses the same
`regula falsi root finding scheme as discussed above to find the actual drift speed which must
`be bracketed between the maximum current value and zero. The calculated speed at the end
`of each iteration is determined from the drag force that is required to balance the tension
`and shear forces at the top of the system. The convergence of the procedure is based on the
`absolute relative diflerence between this calculated speed and the guessed speed that was
`used to begin the iteration.
`
`2.2.2 Initialization
`
`With the system discretized according to user inputs, the first step in solving a problem is
`to calculate a zero order approximate solution based on an inextensible catenary with no
`bending stiflness. This solution provides the initialization for the iterative scheme outlined
`in section 2.1 For multi-segment problems the catenary solution is based on a single equiv-
`alent stiflness and weight. The equivalent unit weight is calculated by summing the total
`wet weight of all components in the system (cable segments, connectors, and attachments)
`length of all cable segments. The equivalent stiflness is found by
`adding all cable segments together as simple linear springs in series and then dividing by
`total length.
`
`and dividing by the total
`
`17
`
`PGS v. WESTERNGECO (IPR2014-00689)
`WESTERNGECO Exhibit 2048, pg. 20
`
`

`
`2.2.3 Coordinate integration
`
`Because the global coordinate variables x, y, z do not appear in any of the governing equa-
`tions, they are simply integrated based on cable coordinates and cable orientation after
`each iteration. While the coordinates do not figure directly into the governing equations
`it is important that they be updated because they are used in evaluating the current at a
`node, determining if a node is lying on the bottom, and fixing the position of the top node
`to determine the draft of a buoy.
`
`The first node is always located at the origin. In 2D the position of any subsequent
`node, k, is then calculated from
`
`Zk
`
`Xk
`
`Zk-l + .6sk-1 cos cPk (1 + Ek) ,
`
`Yk-l + .6sk-1 sincPk (1 + Ek)'
`
`(2.15)
`
`(2.16)
`
`.6Sk-1 is the spacing between nodes k and k - 1. Ek and cPk are the strain and tilt angle at
`node k. A similar form applies in 3D, with the sin arid cos terms replaced by appropriate
`functions of the four Euler parameters (see section 2.5).
`
`2.2.4 Bottom interaction
`
`For problems with cable segments that may be lying on the bottom, cable models the sea
`bed as an elastic mattress with a linear spring. If Z is the vertical coordinate of a node
`along the cable and Z , 0 (where 0 is the vertical position of the sea floor) then the vertical
`reaction force applied by the bottom at that point is
`
`J?s = k I Z I , J?s ~ WQ
`
`(2.17)
`
`where WQ is the wet weight per unit length of the cable at that node and k is a parameter
`describing the stiflness of the bottom. This representation is reasonably smooth and has
`the advantage that it enforces a limit on the force exerted by the bottom (it cannot exceed
`the weight of the cable). The smoothness is important to avoid the abrupt changes in
`configuration that can occur between iterations in systems with high nonlinearity. The
`bottom boundary condition, being one-sided, is necessarily very nonlinear.
`
`2.3 Numerical details of dynamic problems
`
`The solution of dynamic problems proceeds by applying the same iterative scheme at each
`time step. With the continuous form of the governing equations written as
`
`åY ( -) åY - ( -)
`ås + M Y åt + F s, Y = 0
`
`18
`
`(2.18)
`
`PGS v. WESTERNGECO (IPR2014-00689)
`WESTERNGECO Exhibit 2048, pg. 21
`
`

`
`we can use backward finite diflerences in time for the time derivatives and write a discrete
`form of the governing equations that is of the same basic form as 2.2 because we know the
`solution at the previous time step. M in equation 2.18 is an N x N matrix that depends
`on the form of the governing equations. The initial guess for the solution at each new time
`step is simply the solution from the previous time step.
`
`There are limits to the maximum allowable time step that can be used to propagate the
`solution in time without giving rise to numerical instabilities. cable does have an adaptive
`time stepping algorithm whereby if an instability arises the time step wil be automatically
`reduced to try to get through that portion of the simulation. At each time step where the
`baseline time increment is not small enough to accurately propagate the solution, cable wil
`reduce the increment by a factor of ten and take ten steps at the smaller increment. It wil
`descend as low as five orders of magnitude from the baseline increment before giving up.
`
`Adaptive time stepping is only of limited usefulness, however, without some care being
`taken in the choice of a baseline time increment. If the program is deciding that it needs a
`smaller time increment at every step then it would be faster to have set a smaller time step
`in the first place (rather than wasting computational resources at each time step deciding
`t.hat. a smaller increment is necessary).
`
`2.3.1 Wave forcing
`
`2.3.1.1 Wave followers
`
`The dynamic excitation for wave following surface buoys is the simplest of the forcing
`models. For wave following buoys the model is forced by matching the vertical velocity at
`the free (buoy) end to the vertical velocity of the incident wave. In 2D
`
`Un
`
`Vn
`
`Us cos(cPn)
`
`Us sin(cPn)
`
`(2.19)
`
`(2.20)
`
`nation from vertical at the topmost node, and Us is the vertical
`
`where Un, Vn are the tangential and normal velocities at the topmost node, cPn is the incli-
`surface velocity. There is
`no imposed horizontal component of motion with this representation. Prescribed motion
`with both horizontal and vertical components can be imposed using the velocity forcing
`method (see section 3.3.3).
`
`2.3.1.2 Morison's equation
`
`Morison's equation provides a convenient way to model the hydrodynamic, wave-induced
`loads on ocean structures by linearly superposing the solutions to a viscid and an invis-
`
`19
`
`PGS v. WESTERNGECO (IPR2014-00689)
`WESTERNGECO Exhibit 2048, pg. 22
`
`

`
`cid problem ¡6, IJ. For cable problems it is most appropriate for modeling the forces on
`subsurface buoys.
`
`In inviscid theory we can derive a force term that is due to the wave induced inertia of
`the fluid surrounding the buoy. This force can be written as
`
`~ aOw
`
`Fi = p\l (1 + CM) at
`
`(2.21 )
`
`where ClIl is an added-mass coeffcient (= 0.5 for a sphere) and Ow is the velocity of the
`water particles under the waves. The viscous portion of the force is a drag term based on an
`buoy through
`
`experimentally derived drag coeffcient, CD, and the relative velocity of the
`
`the water, U;',
`
`~ 1 ~ I ~ i
`Fv = 2PCDSUR UR .
`
`(2.22)
`
`S is the projected area of the buoy.
`
`2.3.1.3 Froude-Krylov forcing model
`
`The forcing model that cable uses for surface buoys that are not wave followers is described in
`¡3J and derived in detail in ¡7J It combines a Froude-Krylov force (calculated by integrating
`the dynamic pressure of the wave field over the surface of the buoy) with the Haskind
`relations to calculate the wave exciting and damping forces. The Haskind relations calculate
`a wave damping coeffcient that is proportional to the square of the wave exciting force.
`
`2.3.2 Coordinate integration
`
`In a dynamic problem the coordinate integration given in equations 2.15 and 2.16 must be
`modified slightly in towing problems to account for the possibility that the first node may
`have moved and/or that cable may be paying out at the top node. At time step i, the
`integration begins by fixing the position of the topmost node
`
`Zin
`xin
`
`Zi-i + (ui cos d,i - vi sin d,i ) dt
`
`n n '+n n '+n ,
`n n '+n n '+n ,
`
`xi-i + (ui sin d,i + vi cos d,i ) dt
`
`(2.23)
`
`(2.24)
`
`where u~, v~ are the tangential and normal velocities of the topmost node at time step i.
`Integrating from the top down then, the coordinates for subsequent nodes, k, are calculated
`from
`
`Zk
`
`Xk
`
`ZkH - ßSk cos q;k (1 + Ek) ,
`
`Yk+i - ßSk sin q;k (1 + Ek) .
`
`(2.25)
`
`(2.26)
`
`20
`
`PGS v. WESTERNGECO (IPR2014-00689)
`WESTERNGECO Exhibit 2048, pg. 23
`
`

`
`For other types of problems in which the first node always remains fixed at the anchor
`(and thus at the origin), equations 2.15 and 2.16 stil hold.
`
`2.3.3 Bottom interaction
`
`The bottom boundary condition for the dynamic problem is modeled slightly differently
`than for the static problem. Because impact forces can cause the sea floor to exert a
`reaction force greater than the weight of the cable, the dynamic vertical reaction force of
`the elastic foundation is modeled as a simple linear spring/ dashpot combination,
`
`(2.27)
`for nodes with z -( O. If the