Catenary mooring lines experience lifto from and grounding on the seabed when undergoing large dynamic motions. Numerical line mooring models account for this interaction using various seabed models and it is known that the action of lifto and grounding may lead to large dynamic tension uctuations. These uctuations may be spurious due to the inability of discretised mooring models to adequately account for the e ect of the seabed on the mooring line. In this work, the root cause and conditions that lead to the production of the large dynamic tension uctuations is determined. The e ect of line discretisation and seabed model on the tension uctuations is investigated using the widely used spring-mattress approach and a modi ed seabed reaction force model. An in-house mooring code was developed to perform these investigations. For code validation and benchmarking, and to illustrate the existence of the tension uctuations problem due to nodal grounding in existing mooring line simulation codes, comparisons are made to a commercial software. Keywords: Mooring, Seabed impact, Snap load, Tension uctuations, Lumped-mass 1. Introduction Catenary mooring lines provide the restoring force necessary for station- keeping of oating structures primarily by varying its suspended weight in Preprint submitted to Applied Ocean Research August 24, 2018 response to the tension applied at the fairlead connection to the oating plat- form. The seabed has a signi cant e ect on mooring line motions and loads and consequently the dynamics of the connected oating structure as well. One description of the seabed forces is the spring-mattress model, or variously referred to as the elastic seabed model, such as that proposed by Webster [1]. Similar variations of this approach are used in Refs. [2, 3, 4, 5] and is implemented in commercial codes such as Orca ex [6] and aNyMOOR [7]. The main advantage of this approach is that the entire line length re- mains active over a simulation. Hence, the e ect of the grounded section on the suspended part, and vice versa, is fully accounted for. In contrast to the spring-mattress model, in which the dynamics of the grounded sec- tion are calculated, Chatjigeorgiou and Mavrakos [8] proposed a model which calculates the touchdown point from a quasi-static solution, and truncating it from the touchdown point onwards. Consequently, the e ects of the line lifto and touchdown forces are neglected. Thomas [9] proposed a seabed model in which the mass of the rst two suspended nodes adjacent to the seabed are gradually reduced as they approach the sea-bed. The motiva- tion for the development of this model was to eradicate the uctuations in line tension associated with nodal grounding. However, a limitation of the method is that the mass modi er coecients used to perform the nodal mass reduction have to be determined, by trial and error, for individual grounding nodes and recalibrated for each speci c fairlead excitation time history. Wang et al. [10] used the lumped mass approach to model a mooring line in conjunction with a seabed model based on rigid body collision analysis. It was noted by the authors that, upon line impact with the seabed, there were uctuations in the fairlead tension which they attributed to the spa- tially discrete nature of the line structural model. Triantafyllou et al. [11] showed analytically that tension shocks may occur during both the loading and unloading phases of a dynamic mooring cable motion period and derived a condition for its occurrence. Gobat and Grosenbaugh [12] experimentally veri ed the condition, noting however that the occurrence of unloading shocks may not a ect the fairlead tension. In gist, there have been a variety of seabed modelling methods proposed. For time-domain analysis, the spring mattress method is the most widely used due to its exibility, simplicity, and completeness of analysis for the entire mooring line. However, previous studies have suggested that it is prone to numerical errors associated with line contact with the seabed. The occurrence of line tension uctuations due to nodal grounding in dis- 2 crete, lumped mass mooring models is studied using an in-house code and Orca ex [6] in this work. As noted by Yang et al. [13], such irregularities in the tension results have a direct impact on the assessment of the fatigue life of mooring lines. The spurious nature and cause of the uctuations is de- termined, and the e ectiveness of reducing the uctuations using a modi ed spring-mattress model is evaluated. 2. Lumped mass Mooring Line Model The applicability of the lumped mass approach to mooring line modelling has been proven in many implementations [2, 4, 14]. In a lumped mass model, a mooring line is discretised into nodes and elements, and a typical con gura- tion of a discretised catenary line is presented in Figure 1a. Figure 1b shows the connectivity between nodes and elements in the present model. The Or- ca ex theory manual [6] documents the force calculation procedures used in that software, while the methods in the current in-house code is presented in this section. (a) Discretised catenary mooring line. (b) Node and element connectivity. Figure 1: Typical lumped mass mooring line discretisation. 2.1. Distribution of Mass The equations of motion are solved for the nodes. The mass of each element is distributed equally to the adjacent nodes by way of a diagonal mass matrix. Mi=mj1Lj1 2I3+mjLj 2I3=X =j;j1mL 2I3 (1) wheremandL,=fj; j1gare the mass per unit length and lengths of the elements jandj1, and I3is a 3x3 identity matrix. 3 2.2. Line Tension The tension in a section of the line is represented as the tension, Tj, in the associated element jgiven by Tj=( Kjj; j>0 0; j0(2) wherejis the strain and Kjis the linear sti ness of the element. The element strain is calculated as j= 1jri+1rij Lj(3) where ri+1andriare the positions of the nodes bounding element j, andLj is the unstretched element length. With reference to Figure 1b, the resultant tension force vector acting on a node is the sum of the tension forces from its connected elements given by Ti=Tj=Kjj^ejKj1j1^ej1 (4) where the element unit direction vectors are given by ^ej= (ri+1ri)=Ljand ^ej1= (riri1)=Lj1. 2.3. Seabed Forces Figure 2 shows the seabed coordinate system and an element jin contact with the seabed. The seabed nominal elevation is zB;0, while the seabed force cuto elevation is zB;c; the elevation at which the element section is not in contact with the seabed. The unit vector normal to the seabed is ^nB. n^ B e^ jf,A e^ jf,NElement j Node iNode i+1z zB,0zB,cθ Figure 2: Seabed coordinate system de nitions. The unit vector ^ef;Ais a unit vector that is the projection of the element direction unit vector ^ejon the seabed tangent plane, while the unit vector ^ef;Nis the unit vector orthogonal to both ^ef;Aand^nB. 4 ^ef;A j=^ej(^ej^ej^nB) j^ej(^ej^ej^nB)j(5) ^ef;N j=^ef;A j^nB ^ef;A j^nB (6) 2.3.1. Spring Mattress Reaction Force Model In the usual seabed spring mattress model as described by Webster [1] and Gobat and Grosenbaugh [5], a seabed reaction force is directly applied on a node as a function of its own vertical elevation, FB;r i=kB i zB;c izi ^nB (7) wherekB iis the spring constant, zB;c iis the seabed force cuto elevation and ziis the nodal elevation. A damping force proportional to nodal velocity may also be included [5]. The nodal seabed sti ness coecient kB iis given by kB i=Wi NB;cDizB;0(8) whereWiis the nodal weight, zB;0is the nominal seabed elevation, NB;cis the seabed thickness coecient and Diis the line diameter at the s-coordinate of Nodei. 2.3.2. Modi ed Spring Mattress Reaction Force Model As will be discussed in Sections 4 and 5, the spring mattress model can We present a modi ed seabed spring mattress model in which an element can be in four states with respect to the seabed, as shown in Figure 3. Figure 3: Element embedment states. 5 States I and IV respectively describe the element in fully embedded and suspended states with respect to the seabed. State II refers to an element in the state of partial embedment with the center of the element below the seabed force cuto elevation, zB;c. State III describes a partially embedded element with the element center above zB;c. The four states determine how the seabed forces are calculated and distributed to the bounding nodes. The seabed forces acting on an element, FB, are comprised of forces that are tangential and normal to the seabed surface, FB; andFB;, FB=FB; +FB;(9) In the present model, the forces that are normal to the seabed surface are comprised of the seabed reaction, damping, and added mass components, respectively represented as FB;r,FB;dandFB;a, in Equation 10. FB;=FB;r+FB;d+FB;a(10) The seabed reaction force on element j,FB;r j, is given by, FB;r j= Zs2;j s1;jkB jh zB;c jzj(s)i ds! ^nB (11) where ^nBis the seabed normal direction unit vector, sis the line axial coor- dinate, as shown in Figure 1a, and zis the vertical coordinate, as shown in Figure 3. The quantity, zB;c jzj(s), in Equation 11 is the per unit length embedment depth. The element seabed spring sti ness is given by kB j=Wj=Lj zB;c jzB;0(12) whereWjandLjare the element weight and unstretched lengths respectively. The relationship between the seabed force cuto elevation zB;c jandzB;0is given by zB;c j=NB;c jDj+zB;0(13) whereNB;c jis the seabed thickness coecient and Djis the diameter of the element, or equivalently taken to be the physical outer diameter of the line section. The parameter NB;c jDjdenotes the distance over which the seabed force is increased linearly from zero to the element weight per unit length. 6 Equation 14 shows that when the entire element is resting on the seabed at the seabed nominal elevation, zB;0, the seabed reaction force on element jis equivalent to its weight. FB;r j= Zs2;j s1;jWj=Lj zB;c jzB;0h zB;c jzB;0i ds! ^nB = Zs2;j s1;jWj Ljds! ^nB=Wj^nB(14) Since each element is bounded by two nodes, with reference to Figure 3, the elevation of any point along an element is described with a linear inter- polation between the elevations of its bounding nodes, zj(s) =zl;j+zu;jzl;j Ljs=zl;j+zj Ljs (15) wherezuandzlrefer to the z-coordinates of the upper and lower nodes. Di erentiating Equation 15 with respect to zallows a change of integration variable from stozin Equation 11, FB;r j=kB jLj zjZz2;j z1;j zB;c jzj dz (16) Each element can be split into two half-elements of equal lengths, and the force on each half-element is assigned to its adjoining node. With reference to Figure 3, for an element in State I, fully embedded in the seabed, the seabed reaction force contribution of element jon the adjoining node of lower elevation can be calculated by integrating Equation 16 with the limits of integration zl;j=zlandz2;j=zmid;j = (zl;j+zu;j)=2, which gives FB;r l=kB jLj zj2 64 zB;c j(zmid;jzl;j) +z2 l;j zB;c j2 23 75 (17) For the node with the higher elevation, the limits of integration are z1;j= zmid;j andz2;j=zu. Integrating Equation 16 with these limits leads to the reaction force on the upper node being given by FB;r u=kB jLj zj zB;c j(zu;jzmid;j) +z2 mid;jz2 u;j 2 (18) 7 For a partially embedded element in State II, the seabed reaction force on the lower node is as shown in Equation 17. To obtain the seabed reaction force on the upper node, Equation 16 is integrated with the limits z1;j=zmid;j andz2;j=zB;c j, which leads to FB;r u=kB jLj zj2 64 zB;c j2 +z2 mid;j 2zB;c jzmid;j3 75 (19) For a partially embedded element in State III, the seabed reaction force on the upper node is taken to be zero as zmid;j> zB;c j. For the lower node, using the integration limits z1;j=zl;jandz2;j=zB;c jin Equation 16 leads to FB;r l=kB jLj zj2 64 zB;c j2 +z2 l;j 2zB;c jzl;j3 75 (20) An element in State IV is fully suspended and does not experience any seabed forces. For Node i, the total seabed reaction force, as with all other forces, are contributed by elements jandj1, FB;r i=FB;r j( j;zi;zi+1) +FB;r j1( j1;zi1;zi) (21) where the contributions from the elements are dependent on the elemental grounding state, =fI; II; III; IVg, and the relative elevations of nodes i+ 1,iandi1. In this modi ed spring mattress model, a node experiences a seabed reaction force when either of the half elements adjacent to it is in contact with the seabed, when zmid;jzB;c jorzmid;j1zB;c j1. 2.3.3. Vertical Seabed Damping Force The seabed normal damping force on node i,FB;d i, is given by FB;d i=FB;d j+FB;d j1=X =j;j1FB;d  (22) with the contributions from elements jandj1 as given by FB;d =( CB;d _zi^nB;_zi<0;jFB;r j>0 0; otherwise(23) 8 where _zis the vertical velocity of node i. The damping force is active only when a node has downward velocity and is experiencing a reaction force from the seabed contributed by the elements attached to it. The damping force coecient, CB;d , following the theory of vibrations [15], is given by CB;d = 2B;dq kB m, wheremare the masses per unit length of their respective elements, and B;dis the damping ratio [15]. 2.3.4. Seabed Added Mass Model Borrowing from the theory of hydrodynamic added mass, in addition to the reaction and damping forces, an added mass force is introduced as a vertical seabed force. To the best knowledge of the authors, existing time domain seabed models typically model the seabed normal forces on a mooring line as a spring and damper system. The seabed added mass force on node iis given by FB;a i=FB;a j+FB;a j1=X =j;j1FB;a  (24) where the force contribution from the adjacent elements are FB;a =( CB;a zi^nB;zi<0;jFB;r j>0 0; otherwise(25) and whereCB;a ,=fj; j1grepresent the added mass coecients of ele- mentsjandj1. The added mass force is active only when the acceleration of the node  ziis downwards, and the node experiences a seabed reaction force. The added mass coecients are prescribed as CB;a = (B;amL)=2, whereB;ais the added mass coecient, and mandL,=fj; j1g represent the structural masses per unit length and lengths of the elements jandj1. 2.3.5. Seabed Friction Force The friction force formulation is based on the Coulomb friction model. The friction force acts tangent to the seabed surface and is the sum of the components in the element axial and normal directions, FB;f;A,FB;f;N, FB; i=FB;f;A j +FB;f;N j +FB;f;A j1+FB;f;N j1=X =A;NX =j;j1FB;f;  (26) 9 where the axial and normal elemental contributions FB;f; ,=fA;Ngare given by FB;f;  =8 >>>< >>>:CB;f; _rf; ; CB;f;  _rf;  <B;f; FB;r  B;f; FB;r  _rf;  _rf;  ; CB;f;  _rf;  B;f; FB;r  (27) where friction coecients CB;f;  are de ned as CB;f;  =B;f;q kB m,= fA;Ng, andB;f;  are the static friction coecients in the _ef; ,=fA;Ng, directions. The vectors _rf; ,=fA;Ng, are given by _rf; = _ri^ef;  ^ef; . 2.4. Weight and Buoyancy The present mooring line model also takes into consideration the weight, buoyancy and hydrodynamic forces on the line. The weight of a line section submerged in water is given by W=(wA)L^k (28) wherewis the weight per unit length of the section and Ais the mean cross sectional area. 2.5. Hydrodynamic Forces The hydrodynamic forces are evaluated using Morison's equation, which consists of the added mass, Froude-Krylov and viscous drag components [16]. FH=FH; +FH; =FA; +FA;+FI; +FI;+FD; +FD; = 4D2[(C a+ 1)_u C ar ] +1 2DC dju relju rel + 4D2[(C a+ 1)_uC ar] +1 2DC dju relju rel(29) where FA; andFA;are the added mass, FI; andFI;are the Froude- Krylov and FD; andFD;are the drag forces in the line tangential and normal directions. C a,C aandC d, andC dare the sectional tangential and normal added mass and drag coecients, Dis the hydrodynamic diameter,  is the water density, _uis the acceleration of the water, and urelis the relative velocity between the water and the line structure. 10 2.6. Time Integration of Equations of Motion The equations of motion are assembled as Miri=Ti+FS i+FB i+Wi+FH i=Ftot i (30) A state vector, Y, containing the nodal positions and velocities, rand_r, is de ned and its time derivative can be written as dY dt=_r r =_r M1Ftot (31) Time integration is performed integrating the rst order ordinary di er- ential equation system in Equation 31 with the fourth-order Runge-Kutta algorithm [17]. 3. Validation of Mooring Line Model Without Seabed Contact In this section, the tension results from Orca ex and the current in-house code are compared without considering seabed contact so as to compare the results from the two codes involving all other forces including weight, buoyancy, hydrodynamic and tension forces. The two test con gurations consisting of fully suspended mooring chains, one with a xed end and the other a free end, are shown in Figure 4. X (m)Z (m) -750 -500 -250 0-800-600-400-2000 Configuration #1 Configuration #2Fairlead Fixed Free Figure 4: Suspended line con gurations for validation study. 11 The line is assumed to be fully submerged in water for the entire duration of the simulation. The water density is 1025 kg/m3. The structural parame- ters of the mooring line are listed in Table A.1. For static analysis, there are no external loads except for weight, hence in both con gurations the entire line is in the X-Z plane. The static line tensions for both con gurations are shown in Table 1. For Con guration 1, the line end tensions at both ends of the line are in agreement up to 6 signi cant gures. For Con guration 2, the discrepancy in the xed and fairlead end tensions are both 0.021%. Table 1: Initial position and tension at line ends. Con g Line End (X,Y,Z)-coordinates Tension- Orca ex (kN)Tension- Inhouse (kN) 1 Free (0.0, 0.0, 711.3) 0.00 0.00 Fairlead (0.0, 0.0, 0.0) 2275.58 2275.58 2 Fixed (-683.74, 0.0, -82.5) 2542.82 2542.28 Fairlead (0.0, 0.0, 0.0) 2806.61 2806.01 To validate the dynamic response of the mooring line, prescribed position and velocity are imposed at the fairlead position according to the functions, r(t) =R(t)3X i=1 rWF isin2t TWF i+ WF i +rLF isin2t TLF i+ LF i ^ei(32) _r(t) =R(t)3X i=12rWF i TWF icos2t TWF i+ WF i +2rLF i TLF icos2t TLF i+ LF i ^ei (33) where the index i=f1;2;3g, refers to the surge ( X), sway (Y) and heave ( Z) directions, and ( :)WFand (:)LFrefer to the wave and slow-drift frequency fairlead components for the quantity in ( :),riare the motion amplitudes, Tiare the motion periods, and iare the phase angles. R(t) is a ramp-up function implemented in Orca ex [6]. Three prescribed motions pro les, as presented in Table B.1, are used for validation of dynamic line response. Figure 5 presents the fairlead tensions time histories for Con gurations 1 and 2 with the imposed fairlead motions included in Table B.1. The results are shown for time windows that are slightly wider than one period of the tension uctuations, and when the 12 uctuations are fully developed; the ramp up time region is excluded. The results show that the time histories of the fairlead tensions from both codes are in good agreement, despite the di erence in the time integration methods used for the two codes. In particular, the tension peaks and troughs predicted are consistent. Time (s)Ttot fl (kN) 78 80 82 84 862250226022702280229023002310 (a) Con guration 1, motion pro le 1. Time (s)Ttot fl (kN) 78808284868890921000200030004000500060007000 (b) Con guration 2, motion pro le 1. Time (s)Ttot fl (kN) 78808284868890921000150020002500300035004000 (c) Con guration 1, motion pro le 2. Time (s)Ttot fl (kN) 7880828486889092200024002800320036004000 (d) Con guration 2, motion pro le 2. Time (s)Ttot fl (kN) 7880828486889092180020002200240026002800 (e) Con guration 1, motion pro le 3. Time (s)Ttot fl (kN) 7880828486889092200024002800320036004000 (f) Con guration 2, motion pro le 3. Figure 5: Fairlead tensions for fully suspended line from Orca ex and current in-house code. 13 4. E ects of Nodal Grounding on Line Tension Having validated the fairlead tension response for a fully suspended line without considering seabed model e ects, in this section we examine the e ects of nodal grounding on line tensions with di erent fairlead excitations and line spatial discretisations. As the line material speci ed is a studlink mooring chain, the axial length of a chain link is typically six times that of its diameter. For the line speci ed in Table A.1, the length of one of its chain links is thus 0.84 m. However, for numerical simulations element sizes are typically much larger than the size of a discrete chain link. For a catenary mooring line, the grounding and lifto of the individual chain links is the primary factor that determines the fairlead tension. When element sizes that are much larger than the physical chain links are used, a discretisation error is introduced. 4.1. Test Parameters Two single-harmonic fairlead motion pro les from Ref. [18], respectively surge motions in the wave frequency and slow-drift frequency ranges, are used as test cases. The water depth is 82.5 m. The fairlead excitation functions for position and velocity follow Equations 32 and 33. The values for the input parameters Cases 1 and 2 are shown in Table B.2. The initial line geometry is shown in Figure 6. X (m)Z (m) -720-640-560-480-400-320-240-160 -80 0-100-80-60-40-20020 Figure 6: Con guration 1 and fairlead motion 1. The seabed parameters used in Orca ex are presented in the shallow water coecient set shown in Table C.1.. The spring mattress (SM) seabed model described in Section 2.3.1, using the SM seabed force coecient set presented in Table C.2, was used to de ne the seabed forces. 14 4.2. In uence of Element Size Eight uniform line discretisations with element sizes of 11.86 m, 9.48 m, 8.0 m, 7.11 m, 4.74 m, 2.85 m, 2.03 m and 0.84 m are used for each test case. The nest discretisation using an element size of 0.84 m, equivalent to the size of a chain link for the present model, most closely represents physical reality. For all the test cases used in this study a uniform timestep size of 0.00025 s was applied to satisfy the stability limits of the time integration scheme applied to the 0.84 m-element discretization. 4.2.1. Case 1: Pure Surge Motion at Wave Frequency The following discussion focuses on a time window for one period of fair- lead tension oscillation in Case 1 during which a segment of the line under- goes grounding and lifto from the seabed. Figure 7 shows the location of the touchdown point, sTDP, as a function of the s-coordinate (see Figure 1a) and indicates that the line is experiencing grounding in the time period from 82.25 s to 87.0 s and the length of line in contact with the seabed increases from approximately 520 m to 560 m. Between 81.0 s to 82.25 s and 87.0 s to 91.0 s, the line is lifting o from the seabed. The instantaneous locations of the touchdown point tracked by all discretisations are in reasonably close agreement. Time (s)STDP 82 84 86 88 90500520540560580 L = 11.86 m L = 9.48 m L = 8.0 m L = 7.11 m L = 4.74 m L = 2.85 m L = 2.03 m L = 0.84 m Figure 7: Time history of touchdown point location, sTDP, for Case 1 from in-house code. Figures 8a and 9a present the fairlead tension time histories for Case 1, from the in-house code and Orca ex respectively, using the eight element lengths mentioned and show that the tension time series from both codes are 15 smoothly varying and in good mutual agreement from 81.0 s up to approxi- mately 83.5 s and from 87.0 s to 91.0 s. Between 83.5 s and 87.0 s, signi cant uctuations develop, coinciding with the line grounding time period as shown in Figure 7. 82 84 86 88 90 Time (s)200400600800100012001400Tension (kN)L=11.86 m L=9.48 m L=8.00 m L=7.11 m L=4.74 m L=2.85 m L=2.03 m L=0.84 m (a) Fairlead tension time histories for Case 1 (current in-house code). 0 1 2 3 4 5 6 Frequency (Hz)10−210−1100101102103104Tension Amplitude (kN)L=11.86 m L=9.48 m L=8.00 m L=7.11 m L=4.74 m L=2.85 m L=2.03 m L=0.84 m (b) Fairlead tension frequency spectra for Case 1 (current in-house code). Figure 8: Fairlead tension time history and frequency spectra for Case 1 from current in-house code during nodal grounding. Figures 8b and 9b present the frequency spectra plots for the fairlead tension time histories from the in-house code and Orca ex respectively, and show that at lower frequencies of approximately 1 Hz and below, the mag- nitude of the frequency components in the tension time histories generate 16 by both codes, for the varying element sizes, are in good mutual agreement. However, the frequency spectra begin to diverge at frequencies above 1 Hz. Figures 8b and 9b further illustrate that as the element size is reduced, the magnitudes of the higher frequency components are correspondingly reduced as well; the tension time histories for the 0.84 m-element discretization are smoothly varying. 82 84 86 88 90 Time (s)200400600800100012001400Tension (kN)L=11.86 m L=9.48 m L=8.00 m L=7.11 m L=4.74 m L=2.85 m L=2.03 m L=0.84 m (a) Fairlead tension time history for Case 1 (Orca ex). 0 1 2 3 4 5 6 Frequency (Hz)10−210−1100101102103104Tension Amplitude (kN)L=11.86 m L=9.48 m L=8.00 m L=7.11 m L=4.74 m L=2.85 m L=2.03 m L=0.84 m (b) Fairlead tension frequency spectra for Case 1 (Orca ex). Figure 9: Fairlead tension time history and frequency spectra for Case 1 from Orca ex during nodal grounding. 17 Using the results for the 0.84 m-element size as a baseline and de ning the deviation from this benchmark as the magnitude of the uctuations, the maximum peak-to-peak tension uctuation amplitude for each discretisation can be quanti ed and are plotted in Figure 10a, where it is illustrated again that the uctuation amplitudes are reduced along with a reduction in the element size. The peak tensions from both Orca ex and the in-house code with the various element sizes are presented in Figure 10a and show that as element size is reduced the peak tension converges towards to the benchmark values of the 0.84 m-element discretisation. 11.869.488.07.11 4.74 2.85 2.031298.01298.51299.01299.51300.01300.51301.01301.5Peak Tension (kN)In-house Orcaflex 0.84 Element Size (m) (a) Peak fairlead tension with varying discretisations. 11.869.488.07.11 4.74 2.85 2.030.0100.0200.0300.0400.0Tension Fluc. Amplitude (kN)In-house Orcaflex 0.84 Element Size (m) (b) Fairlead tension uctuation amplitude with varying discretisations. Figure 10: Peak fairlead tension and fairlead tension uctuation amplitudes for Case 1 with uniform line element sizes ranging from 11.86 m to 0.84 m. 18 The deviation of the peak tension values given by the various discreti- sations, from their respective benchmark peak tension values of the 0.84 m-element discretisation, are generally small. In contrast, the amplitudes of the tension uctuations, particularly for the coarser discretisations, are large in comparison to the smoothly varying tension benchmark results and degrade the quality of the solution. To investigate the cause of the tension uctuations, the discussion is now focused on the results for the 8.0 m and 0.84 m discretisations. The 0.84 m discretisation is chosen because the tension oscillations in this discretisation are the most subdued among all the discretisations and thus provides the conditions of the desired outcome to compare against. The 8.0 m discretisa- tion is chosen because the magnitude of the tension uctuations, as shown in Figure 10b, are signi cant and thus facilitates ease of identi cation of the onset of the tension uctuations while, as will be discussed, the nodal density is suciently high such that the case provides a mixture of nodal grounding occurrences that do and do not generate tension uctuations. s (m)ε 100 200 300 400 500 600 7000.000150.00020.000250.00030.000350.0004 t=83.12 s t=83.20 s t=83.28 s t=83.36s t=83.44 s t=83.52 s t=83.60 s (a) Strain during grounding of Nodes 66 and 67 (Case 1) with 8.0 m-element line. s (m)dε/ds 100 200 300 400 500 600 700-5E-0705E-071E-061.5E-06 t=83.12 s t=83.20 s t=83.28 s t=83.36s t=83.44 s t=83.52 s t=83.60 s (b) Strain spatial gradient during grounding of Nodes 66 and 67 (Case 1) with 8.0 m-element line. Figure 11: Strain and strain spatial gradient distribution with nodal grounding time window (83 :12st83:60 s) for Case 1 with 8.0 m-element line. 19 Figure 7 shows that line grounding starts at approximately 83.1 s. How- ever, the tension uctuations appear at approximately 84.0 s, coinciding with the grounding of Node 68. The grounding of Nodes 66 and 67, at 83.12 s and 83.64 s respectively, do not cause any tension uctuations. The strain, , and strain spatial gradient, d =ds, within the associated time window are shown in Figures 11a and 11b respectively. It is noted the temporal and spatial gradients of strain are smoothly varying. s (m)ε 100 200 300 400 500 600 700-0.0004-0.000200.00020.0004 t=84.80 s t=84.88 s t=85.20 s t=85.28 s t=85.36 s t=85.44 s t=85.52 s (a) Strain during grounding of Nodes 68, 69, 70 and 71 in Case 1 with 8.0 m-element line. s (m)dε/ds 100 200 300 400 500 600 700-2E-05-1E-0501E-052E-053E-05 t=84.80 s t=84.88 s t=85.20 s t=85.28 s t=85.36 s t=85.44 s t=85.52 s (b) Strain spatial gradient during grounding of Nodes 68, 69, 70 and 71 in Case 1 with 8.0 m-element line. Figure 12: Strain, strain spatial gradient during nodal grounding time window (84:80st85:60 s) for Case 1 with 8.0 m-element line. In contrast, as Nodes 68, 69 and 70 are grounded between 84.08 s and 84.97 s, Figure 12a illustrates that within the time window 84 :88t85:52 s the strain curves are highly irregular and large spatial gradients are ob- served in Figure 12b. Figure 12a also reveals the formation of localised low and negative strain zones in the strain curves of 84.20 s, 85.28 s, and 85.36 s at locations close to the touchdown point ( s500 m). As shown in Equa- tion 2, a mooring cable does not support compressive loading hence negative 20 strain indicates a slack region where line is slack. Equation 3 shows that element strain is determined from the distance between its two bounding nodes, hence, a low strain region is a compression zone where the separa- tion distances between nodes are low. Conversely, a high strain region is a rarefaction zone where the nodes are spaced further apart. s (m)ε 100 200 300 400 500 600 700-0.0002-0.000100.00010.00020.0003 t=84.80 s t=84.85 s t=84.88 s t=84.90 s t=84.92 s t=84.96 s t=85.02 s Initial formation of compression zoneReversal Development of compression zoneCollapse of compression zoneExpansion of rarefaction zone (a) Development and reversal of low strain zone after grounding of Node 69. s (m)ε 100 200 300 400 500 600 700-0.0004-0.000200.0002 t=85.20 s t=85.28 s t=85.36 s t=85.38 s t=85.40 s t=85.44 s t=85.48 sInitial formation of compression zoneDevelopment of compression zoneReversal Collapse of compression zoneExpansion of rarefaction zone (b) Development and reversal of low strain zone after grounding of Node 70. Figure 13: Development and reversal of low strain zones post-nodal grounding (Case 1). Figure 13 provides a closer look at the development and time evolution of the compression and rarefaction zones. Figure 13a presents the strain curves of the 8.0 m-element line after the grounding of Node 69 at 84.48 s, and illustrates the development of a slack region in the region of 400 s550 m. The spatial extent of the slack zone is time-varying. At 84.96 s, the slack zone has become a high-strain, rarefaction region. Similarly, Figure 13b illustrates that after the grounding of Node 70 at 84.97 s, at 85.20 s a slack region spanning 390 s590 m develops in the vicinity of the touchdown point (s560 m). At 85.28 s, the elements in the vicinity of the touchdown point have undergone further compression and the maximum negative strain 21 has increased further. At 85.44 s, the slack span has become a high tension zone. The high tension zone then expands, as shown by an enlarged high strain region in the strain curve at 85.48 s relative to 85.44 s. s (m)ε 100 200 300 400 500 600 70000.00010.00020.00030.0004 t=83.12 s t=83.28 s t=83.44 s t=83.60 s t=84.80 s t=84.96 s t=85.12 s t=85.28 s t=85.44 s t=85.60 s (a) Strain during nodal grounding time window (83 :12st85:60 s) in Case 1 with 0.84 m-element line. s (m)dε/ds 100 200 300 400 500 600 700-5E-0705E-071E-061.5E-062E-06 t=83.12 s t=83.28 s t=83.44 s t=83.60 s t=84.80 s t=84.96 s t=85.12 s t=85.28 s t=85.44 s t=85.60 s (b) Strain spatial gradient during nodal grounding time window (83:12st85:60 s) in Case 1 with 0.84 m-element line. Figure 14: Strain, strainrate and strain spatial gradient distribution during nodal grounding time window (83 :12st85:60 s) for Case 1 with 0.84 m-element line. Theand d=dscurves for the 0.84 m-element line are shown in Figures 14a and 14b respectively. The time period shown in these plots span the pe- riod prior to ( t<84:08 s) and during (84 :08t85:60 s) the development of the tension uctuations presented in Figure 12. Figure 14b illustrates that the d=dscurves during the period of occurrence of tension uctuations are smoothly varying with the exception of a kink that occurs consistently in the vicinity of the touchdown point. During the grounding of Nodes 66 and 67, the d =dscurves presented in Figure 11b have similar pro les compared to the d =dscurves for 0.84 m-element line presented in Figure 14b. Both situations were free of tension uctuations. In contrast, the d =dscurves during the grounding of Nodes 22 68 to 70 for the 8.0 m-element line, presented in Figure 12b, exhibit large gradients which are up to an order of magnitude larger than in Figures 11b and Figure 14b at s-coordinates that vary widely in time. The nodal vertical and horizontal velocities as well as the vertical coordi- nate for the 8.0 m and 0.84 m-element lines are shown in Figures 15 and 16 respectively. The times of grounding of each node are also shown to demar- cate the pre and post-grounding time regions for each node. The selected grounding nodes shown in Figure 16 for the 0.84 m-element line were chosen ats-coordinates similar to the grounding nodes for the 8.0 m-element line, as shown in Table 2, in order to compare the grounding velocities at the same s-coordinates with di erent discretisations. The fairlead forcing of Case 1 is a single-harmonic surge motion in the X(surge) direction, and a negative vx at the fairlead corresponds to the unloading phase of the motion. Table 2: Grounding nodal indices and s-coordinates (Case 1.) Node index8.0 m 66 67 68 69 70 71 0.84 m 619 629 639 648 658 667 TDP s- coord (m)8.0 m 519.48 527.48 535.47 543.47 551.46 559.45 0.84 m 519.60 528.00 536.41 543.98 552.39 559.95 As the fairlead velocity magnitude increased from 1.289 m/s to 3.088 m/s, the vertical velocities of the grounding nodes in the 8.0 m-element and 0.84 m-element discretisations respectively increased from 1.127 m/s to 3.348 m/s and 1.085 m/s to 3.227 m/s. Comparing the grounding velocities shown in Figures 15a and 16a with the corresponding fairlead velocities shown in Figures 15d and 16d respectively, it is observed that the magnitudes of the nodal vertical grounding velocities are positively correlated with the fairlead velocity. Comparing Figure 15a with 16a, Figure 15b with 16b, and Figure 15c and 16c, it is noted that the post-grounding velocity and positional uc- tuations for the 8.0 m-element line are more pronounced than that of the 0.84 m-element line. Figures 15a, 15b, 16a, and 16b illustrate that the post-grounding nodal velocities in Case 1 eventually settle at magnitudes close to zero. Com- paring Figures 15a with 16a, and 15b with 16b, and 15d with 16d, it is observed that the post-grounding velocities and vertical positions of the 8.0 m-element disretisation are underdamped and experience several periods of oscillations, while the post-grounding velocities and vertical positions are critically damped. 23 Time (s)vz (m/s) 81 82 83 84 85 86 87 88 89 90 91-4-2024 Node 66 Node 67 Node 68 Node 69 Node 70 Node 71(a) Nodal vertical velocities, vz, in Case 1 for 8.0 m-element line. Time (s)vx (m/s) 81 82 83 84 85 86 87 88 89 90 91-0.4-0.200.20.4 Node 66 Node 67 Node 68 Node 69 Node 70 Node 71 (b) Nodal horizontal velocities, vx, in Case 1 for 8.0 m-element line. Time (s)Z (m) 81 82 83 84 85 86 87 88 89 90 91-82.6-82.4-82.2-82-81.8-81.6 Node 66 Node 67 Node 68 Node 69 Node 70 Node 71 (c) Nodal vertical coordinate, Z, in Case 1 for 8.0 m-element line. Time (s)vx (m/s) 81 82 83 84 85 86 87 88 89 90 91-4-2024 vx (m/s) (d) Fairlead horizontal velocity vxin Case 1 for 8.0 m-element line. Figure 15: Nodal vertical coordinate and velocities in ZandXdirections of grounding nodes in Case 1 with element length of 8.0 m. 24 Time (s)vz (m/s) 81 82 83 84 85 86 87 88 89 90 91-4-2024 Node 619 Node 629 Node 639 Node 648 Node 658 Node 667(a) Nodal vertical velocities, vz, in Case 1 for 0.84 m-element line. Time (s)vx (m/s) 81 82 83 84 85 86 87 88 89 90 91-0.4-0.200.20.4 Node 619 Node 629 Node 639 Node 648 Node 658 Node 667 (b) Nodal horizontal velocities, vx, in Case 1 for 0.84 m-element line. Time (s)Z (m) 81 82 83 84 85 86 87 88 89 90 91-82.6-82.4-82.2-82-81.8-81.6 Node 619 Node 629 Node 639 Node 648 Node 658 Node 667 (c) Nodal vertical coordinates, Z, in Case 1 for 0.84 m-element line. Time (s)vx (m/s) 81 82 83 84 85 86 87 88 89 90 91-4-2024 vx (m/s) (d) Fairlead and horizontal velocity vxin Case 1 for 0.84 m-element line. Figure 16: Nodal vertical coordinate and velocities in ZandXdirections of grounding nodes in Case 1 with element length of 0.84 m. 25 From Figure 15b, an interesting observation of the oscillatory behavior of the horizontal nodal velocities vxin the vicinity of the touchdown zone, for the 8.0 m-element discretization, is that after a node is grounded, for a short period of time the slope of the horizontal velocity curves (d vx=dt), or equivalently the horizontal acceleration, for that node and the nearby nodes on the side of the line that is grounded is negative, while that of the nodes that are suspended are positive. For example, when Node 67 is grounded at 83.68 s, the slopes of the vx curves for itself and Node 66, which is on the grounded side of the line, are negative, while slopes of the vxcurves for Nodes 68 to 71 on the suspended side of the line are positive. When Node 68 grounded at 84.08 s, the slopes of thevxcurves for the grounded Nodes 66 and 68 are negative, while the slopes of the vxcurves for the suspended Nodes 69 to 71 are positive. Simi- larly, when Node 69 grounded at 84.48 s, the slopes of the vxcurves for the grounded Nodes 66 to 69 are negative, while the slopes of the vxcurves for the suspended Nodes 70 and 71 are positive. Finally, when Node 70 grounded at 84.97 s, the slopes of the vxcurves for grounded Nodes 66 to 70 are nega- tive, while slope of the vxcurve for the suspended Node 71 is positive. This behavior suggests that after a node is grounded, the nodes on the grounded side of the line as well as the recently grounded node itself, accelerate towards the anchor node while the nodes on the suspended side accelerate towards the fairlead node. However, as the horizontal velocities are oscillating, the accelerations of the grounded nodes are soon reversed until the grounding of the next node. 4.2.2. Case 2: Pure Surge Motion at Slow-Drift Frequency Figure 17a presents the peak tensions for Case 2 with the range of element sizes and shows that the variation in the peak tension associated with varying element size is small compared to the peak tension value. Figure 17b presents the peak-to-peak tension uctuation amplitude for each discretisation and reveals that the tension uctuations, in the range between 3 kN to 41.5 kN, are an order of magnitude smaller than in Case 1 as presented in Figure 10b. The grounding time window for Case 2 is 650 t740 s, as shown in Figure 18, during which the length of line in contact with the seabed increases from approximately 250 m to 600 m. The number of nodes on the seabed resting on the seabed from 32 to 77 for the 8.0 m-element line, and 291 to 720 for the 0.84 m-element line. 26 11.86 9.48 8.0 7.11 4.74 2.85 2.034562.04562.54563.04563.54564.0Peak Tension (kN)In-house Orcaflex 0.84 Element Size (m)(a) Peak fairlead tension with varying discretisations. 11.86 9.48 8.0 7.11 4.74 2.85 2.030.010.020.030.040.0Tension Fluc. Amplitude (kN)In-house Orcaflex 0.84 Element Size (m)(b) Fairlead tension uctuation amplitude with varying discretisations. Figure 17: Peak fairlead tension and fairlead tension uctuation amplitudes for Case 2 with uniform line element sizes ranging from 11.86 m to 0.84 m. Time (s)STDP 600 650 700 750 800200300400500600 L = 11.86 m L = 9.48 m L = 8.0 m L = 7.11 m L = 4.74 m L = 2.85 m L = 2.03 m L = 0.84 m Figure 18: Time history of touchdown point location for Case 2 from in-house code. Figures 19a and 19b respectively present the fairlead tension histories and frequency spectra for Case 2 from the current in-house code, while Figures 20a and 20b show the fairlead tension history and frequency spectra for Case 2 from Orca ex. Figures 19b and 20b con rm that the frequency contents of the tension time histories are dominated by the lower frequencies below approximately 0.1 Hz, which are in good agreement. 27 600 650 700 750 800 Time (s)10002000300040005000Tension (kN)L=11.86 m L=9.48 m L=8.00 m L=7.11 m L=4.74 m L=2.85 m L=2.03 m L=0.84 m(a) Fairlead tension time history for Case 2 (current in-house code). 0 1 2 3 4 5 6 Frequency (Hz)10−510−310−1101103105Tension Amplitude (kN)L=11.86 m L=9.48 m L=8.00 m L=7.11 m L=4.74 m L=2.85 m L=2.03 m L=0.84 m(b) Fairlead tension frequency spectra for Case 2 (current in-house code). Figure 19: Time history and frequency spectra of fairlead tension from current in-house code during nodal grounding for Case 2. 600 650 700 750 800 Time (s)10002000300040005000Tension (kN)L=11.86 m L=9.48 m L=8.00 m L=7.11 m L=4.74 m L=2.85 m L=2.03 m L=0.84 m (a) Fairlead tension time history for Case 2 (Orca ex). 0 1 2 3 4 5 6 Frequency (Hz)10−510−310−1101103105Tension Amplitude (kN)L=11.86 m L=9.48 m L=8.00 m L=7.11 m L=4.74 m L=2.85 m L=2.03 m L=0.84 m(b) Fairlead tension frequency spectra for Case 2 (Orca ex). Figure 20: Time history and frequency spectra of fairlead tension from Orca ex during nodal grounding for Case 2. The vertical and horizontal velocities of selected grounding nodes for Case 2 with the two discretisations are presented in Figures 21a and 21b respec- tively. The fairlead horizontal velocity vxis shown in Figure 21d. Figures 21a, 21b and 21c show that the velocities and vertical coordinates of the grounding nodes of the 8.0 m-element and 0.84 m-element discretisations, at comparable s-coordinates, are in close agreement. In contrast, the positional and velocity uctuations of the grounding nodes in Case 1 with the 8.0 m- element discretisation are higher than with the 0.84 m-element discretisation. 28 Time (s)vz (m/s) 660 680 700 720 740-1.5-1-0.500.511.5 L=8.0 m, Node 39 L=8.0 m, Node 45 L=8.0 m, Node 51 L=8.0 m, Node 57 L=8.0 m, Node 64 L=8.0 m, Node 70 L=0.84 m, Node 358 L=0.84 m, Node 417 L=0.84 m, Node 477 L=0.84 m, Node 536 L=0.84 m, Node 596 L=0.84 m, Node 655(a) Nodal vertical velocities, vzfor 8.0 m-element and 0.84 m-element line. Time (s)vx (m/s) 660 680 700 720 740-0.15-0.1-0.0500.050.10.15 L=8.0 m, Node 39 L=8.0 m, Node 45 L=8.0 m, Node 51 L=8.0 m, Node 57 L=8.0 m, Node 64 L=8.0 m, Node 70 L=0.84 m, Node 358 L=0.84 m, Node 417 L=0.84 m, Node 477 L=0.84 m, Node 536 L=0.84 m, Node 596 L=0.84 m, Node 655 (b) Nodal horizontal velocities, vxfor 8.0 m-element and 0.84 m-element line. Time (s)Z (m) 660 680 700 720 740-82.6-82.4-82.2-82-81.8-81.6 L=8.0 m, Node 39 L=8.0 m, Node 45 L=8.0 m, Node 51 L=8.0 m, Node 57 L=8.0 m, Node 64 L=8.0 m, Node 70 L=0.84 m, Node 358 L=0.84 m, Node 417 L=0.84 m, Node 477 L=0.84 m, Node 536 L=0.84 m, Node 596 L=0.84 m, Node 655 (c) Nodal vertical coordinate, Zfor 8.0 m-element and 0.84 m-element line. Time (s)vx (m/s) 660 680 700 720 740-1-0.500.51 vx (m/s) (d) Fairlead and horizontal velocity vx. Figure 21: Nodal vertical coordinate and velocities in ZandXdirections of grounding nodes in Case 2 with element lengths of 8.0 m and 0.84 m. 29 Table 3 shows the s-coordinates of the selected nodes whose velocities and vertical coordinates within the grounding time window are illustrated in Figure 21 and shows that the maximum vertical grounding velocity magni- tudes in Case 2 are 1.12 m/s and 1.11 m/s for the 0.84 m-element and 8.0 m-element discretisation respectively, which are lower than the grounding velocities in Case 1. The maximum fairlead velocity magnitude in Case 2 is 0.628 m/s compared to 3.392 m/s in Case 1. Table 3: Grounding nodal indices and s-coordinates (Case 2.) Node index8.0 m 39 45 51 57 64 70 0.84 m 358 417 477 536 596 655 TDP s- coord (m)8.0 m 303.70 351.65 399.61 447.56 503.50 551.46 0.84 m 300.15 349.76 400.21 449.81 500.26 549.87 s (m)ε 100 200 300 400 500 600 70000.00050.0010.00150.0020.0025L=8.0 m, t=660.0 s L=8.0 m, t=670.0 s L=8.0 m, t=680.0 s L=8.0 m, t=690.0 s L=8.0 m, t=700.0 s L=8.0 m, t=710.0 s L=8.0 m, t=720.0 s L=0.84 m, t=660.0 s L=0.84 m, t=670.0 s L=0.84 m, t=680.0 s L=0.84 m, t=690.0 s L=0.84 m, t=700.0 s L=0.84 m, t=710.0 s L=0.84 m, t=720.0 s (a) Strain for 8.0 m-element and 0.84 m-element lines with time window 660:0t720:0 s. s (m)dε/ds 100 200 300 400 500 600 700-5E-0705E-071E-061.5E-062E-06 L=8.0 m, t=660.0 s L=8.0 m, t=670.0 s L=8.0 m, t=680.0 s L=8.0 m, t=690.0 s L=8.0 m, t=700.0 s L=8.0 m, t=710.0 s L=8.0 m, t=720.0 s L=0.84 m, t=660.0 s L=0.84 m, t=670.0 s L=0.84 m, t=680.0 s L=0.84 m, t=690.0 s L=0.84 m, t=700.0 s L=0.84 m, t=710.0 s L=0.84 m, t=720.0 s (b) Strain spatial gradient for 8.0 m-element and 0.84 m-element lines with time window 660 :0t720:0 s. Figure 22: Strain, strain rate and strain gradient distribution for 8.0 m-element and 0.84 m-element lines for Case 2 between 660.0 s and 720.0 s. 30 Figures 22a and 22b respectively present the and d=dscurves of the 8.0 m-element and 0.84 m-element lines within the period of nodal grounding (660:0t720:0 s) for Case 2, and con rm that the strain distributions for the two discretisations are in good agreement at all the presented timesteps, and that there are no large strain spatial gradients. 4.3. E ects of Varying Seabed Force Coecients The choice of seabed force coecients is arbitrary and selected typically reduce the nodal penetration depth on the seabed, and post-grounding posi- tional uctuations [2, 4]. Gobat and Grosenbaugh [5] stated that the choice of coecients has little in uence on the gross response of the mooring line. This section examines the e ects of varying the coecients of the spring mat- tress model on line tension in Case 1. This section examines the e ects of applying di erent seabed force coecients sets presented in Table C.2 with the 4.74 m-element discretisation. The attention is focused on the coecients for the vertical reaction and damping forces in the spring mattress model, respectively NB;c(see Equation 8) and vertical damping B;d(see Section 2.3.3). The 4.74 m-element discretisation is chosen because no signi cant change in the peak tension result is achieved with further re nement of the nodal density; the di erence in the peak tension result between the 4.74 m and 2.85 m-element discretisations is 0.0015% and thus the solution for peak tension is considered to be converged. The applied time step, 0.00025 s, is signi cantly smaller than the maximum stable time step of 0.00328 s, and is sucient for time step convergence for Case 1. Figure 23a shows the fairlead tension time history for Case 1 with the SM coecient set, which produced the tension uctuations discussed in Section 4.2.1, compared with the tension time histories using a B;dvalue of 1.0 and NB;cvalues of 5 (SM 1), 10 (SM 2), 15 (SM 3), and 20 (SM 4). The e ect of increasingNB;cis that the seabed vertical spring sti ness, kB, is reduced and the seabed reaction force cuto elevation, zB;c(see Figure 2), is increased. It is evident from Figures 23a that the tension uctuations are progressively reduced as NB;cis increased. However, the peak tensions for the SM 1, SM 2, SM3and SM 4coecient sets, at 1285.18 kN, 1264.57 kN, 1250.44 kN, and 1232.86 kN respectively, are lower than the 1298.69 kN peak tension for the SM coecient set. 31 Time (s)Ttot fl (kN) 82 84 86 88 9040060080010001200 SM SM_1 SM_2 SM_3 SM_4(a) Fairlead tensions with SM, SM 1, SM2, SM 3, and SM 4coecients sets. Time (s)Ttot fl (kN) 82 84 86 88 9040060080010001200 SM SM_5 SM_6 SM_7 SM_8(b) Fairlead tensions with SM, SM 5, SM6, SM 7, and SM 8coecients sets. Time (s)Ttot fl (kN) 82 84 86 88 9040060080010001200 SM SM_1 SM_5 (c) Fairlead tensions with SM, SM 1, and SM 5coecients sets. Time (s)Ttot fl (kN) 82 84 86 88 9040060080010001200 SM SM_3 SM_7(d) Fairlead tensions with SM, SM 3, SM7coecients sets. Figure 23: Fairlead tensions for Case 1 with spring mattress seabed model and SM, SM 1, SM2, SM 3, SM 4, SM 5, SM 6, SM 7and SM 8coecient sets. Figure 23b shows the fairlead tension time history using the SM coecient set compared with the fairlead tension time histories using a B;dvalue of 0 and the same NB;cvalues of 5, 10, 15 and 20, corresponding to the SM 5, SM6, SM 7, and SM 8coecient sets respectively, and illustrates that the tension uctuations are progressively reduced as NB;cis increased. The peak tensions are 1283.81 kN, 1264.29 kN, 1251.66 kN and 1237.44 kN respectively for the SM 5, SM 6, SM 7, and SM 8coecient sets. With a NB;cvalue of 5, the mangnitude of the tension uctuations are reduced as the value of B;dwas changed from 1.0 to 0.0, however, the peak tension was also lowered slightly by 1.1%. With a value of 15 for NB;c, the tension uctuations are further reduced slightly as the value of B;dwas changed from 1.0 to 0.0, while the peak tension experienced a small increase of 0.1%. 32 Time (s)Rsb 82 84 86 88 9016001650170017501800 SM SM_1 SM_2 SM_3 SM_4(a) Seabed reaction force with SM, SM 1, SM 2, SM 3, and SM 4coecients sets. Time (s)Rsb 82 84 86 88 9016001650170017501800 SM SM_5 SM_6 SM_7 SM_8 (b) Seabed reaction force with SM, SM 5, SM 6, SM 7, and SM 8coecients sets. Figure 24: Seabed reaction force for Case 1 with spring mattress seabed model and SM, SM1, SM 2, SM 3, SM 4, SM 5, SM 6, SM 7and SM 8coecient sets. The cause for the reduction in fairlead tension is the increased seabed reaction force on the line as a result of increasing NB;c. Figures 24a and 24b respectively present, for B;dvalues of 1.0 and 0.0, the total seabed reaction force on the grounded line section for NB;cvalues of 5, 10, 15 and 20, as well as the seabed reaction force for the SM coecient set, and illustrate that the seabed reaction force on the line, which reduces the suspended line weight, generally increases together with NB;cover a period of oscillation. Figure 25b also illustrates that for di erent values of B;d, the vertical nodal coordinates begin to deviate at elevations closer to the seabed due to the increasing downward velocities of the grounding nodes as they approach the seabed (see Figure 15a) and leads to a larger disparity in the vertical damping 33 force arising from di ering values of B;d. Figure 25a shows the Z-coordinate of the grounding nodes when the SM and SM 4coecients are used, and it is evident that the penetration depths of the grounding nodes below zB;0increased with a larger NB;cvalue and, correspondingly, a lower seabed spring sti ness. Figure 25b presents the Z- coordinate of the grounding nodes when the SM 3and SM 7coecients are used, which respectively apply the values of 1.0 and 0.0 for B;dand the same value of 15 for NB;c, and illustrates that the nodal penetration depth increases with a reduction in the value of B;d. Time (s)Z (m) 82 84 86 88 90-82.5-82-81.5-81 Node 107, SM Node 109, SM Node 111, SM Node 112, SM Node 114, SM Node 116, SM Node 117, SM Node 119, SM Node 107, SM_4 Node 109, SM_4 Node 111, SM_4 Node 112, SM_4 Node 114, SM_4 Node 116, SM_4 Node 117, SM_4 Node 119, SM_4 (a) Nodal vertical positions, Z, with SM and SM 4coecients sets for Case 1. Time (s)Z (m) 82 84 86 88 90-82.5-82-81.5-81 Node 107, SM_3 Node 109, SM_3 Node 111, SM_3 Node 112, SM_3 Node 114, SM_3 Node 116, SM_3 Node 117, SM_3 Node 119, SM_3 Node 107, SM_7 Node 109, SM_7 Node 111, SM_7 Node 112, SM_7 Node 114, SM_7 Node 116, SM_7 Node 117, SM_7 Node 119, SM_7 (b) Nodal vertical positions, Z, with SM 3and SM 7coecients sets for Case 1. Figure 25: Nodal vertical positions, Z, with SM, SM 3, SM 4and SM 7coecients sets for Case 1. Figures 26a and 26b respectively present the strain and strain spatial gradient distributions with the 4.74 m-element discretisation and the SM 7 coecient set during the nodal grounding time window, 83 :12t85:60 s, and illustrates that the both the strain and strain gradient curves are free of 34 the large spatial gradients that are observed in Figures 12a and 12b. In rela- tion to the fairlead tension time histories, the smoothly varying strain curves and strain gradient curves, presented in Figures 26a and 26b respectively, correspond to the SM 7fairlead tension time history presented in Figure 23d, while the undulating strain and strain gradient curves in Figures 12a and 26b respectively correspond to the irregular fairlead tension time series for the 8.0 m-element line shown in Figure 8b. The key ndings from this series of test cases are, rstly, the seabed spring sti ness has a signi cant e ect on the fairlead peak tension and increasing it, while bene cial from the standpoint of reducing tension uctuations, leads to the underestimation of the peak tension results. Secondly, the e ect of the vertical damping coecient is mainly on the tension uctuations and its e ect on peak tension is small, and that a lower damping coecient has the e ect of reducing the uctuations. s (m)ε 100 200 300 400 500 600 7000.00010.00020.00030.0004 t = 83.12 s t = 83.60 s t = 83.84 s t = 84.32 s t = 84.96 s t = 85.60 s (a) Strain with 4.74 m-element line, SM 7coecient set. s (m)dε/ds 100 200 300 400 500 600 700-5E-0705E-071E-061.5E-062E-06 t = 83.12 s t = 83.60 s t = 83.84 s t = 84.32 s t = 84.96 s t = 85.60 s (b) Strain spatial gradient with 4.74 m-element line, SM 7coecient set. Figure 26: Strain and spatial gradient of strain with 4.74 m-element line, SM 7coecient set and time window (83 :12t85:60 s) for Case 1. 35 4.4. E ect of Grounding Element Orientation Figure 2 shows an element that is in the grounding state, with the vertical angle between the element axis and the seabed de ned as , the grounding angle of an element. For a catenary line, the of an element closer to the anchor is usually lower than for an element closer to the fairlead. Hence, to investigate the e ect of on the production of the tension uctuations, the 4.74 m-element line in a water depth of 82.5 m is speci ed together with three pretension values, respectively 706.9 kN, 1248.4 kN and 3136.5 kN; henceforth referred to as the low, mid-range and high pretension cases. The pretension value changes the sTDPrange, as shown in Figure 27. The fairlead excitation amplitude and frequency of 5.4 m and 10 s (Case 1) is uniformly applied for all three lines. 0.0 100.0 200.0 300.0 400.0 500.0 600.0 700.0 X-coord (m)-80.0-60.0-40.0-20.00.0Z-coord (m)T0= 706.9 kN T0= 1248.4 kN T0= 3136.5 kN Figure 27: Initial line geometries of 4.74 m-element discretisation with pretension values, T0, of 706.9 kN, 1248.4 kN and 3136.5 kN. Figure 28 presents the location of the touchdown point sTDPfor the lines with three pretension values in the time period for three oscillation cycles from 61 s to 91 s. As pretension is increased, sTDP is shifted towards the anchor as a longer segment of the line is suspended. For the low, mid- range and high pretension value cases, the sTDPrange for the same fairlead excitation speci ed here are 161.23 m to 396.53 m, 346.16 m to 497.91 m, and 507.39 m to 569.04 m respectively. The fairlead tension time histories are shown in Figure 29, where the disparities in the magnitudes of the tension uctuations between the tension time histories for the three pretension cases are clearly presented. The uctu- ation magnitude is largest for the low pretension case, which also experienced the highest sTDP range, and smallest for the high pretension case, with the 36 lowestsTDPrange. The magnitude of the tension uctuations for mid-range pretension case lies in between the high and low pretension cases. The velocity of the grounding node, which is the rst suspended node of a grounding element (Node i+1 in Figure 2), for the three pretension cases, are presented in Figure 30. The grounding node velocities for the two higher pretension cases are similar, and attained a maximum value of approximately 3.8 m/s, whereas that for the lowest pretension case achieved a maximum of 2.96 m/s. The grounding angle time histories are presented in Figure 31. Compar- ing Figures 31 and 29, it is evident that the onsets of the tension uctuations appear in tandem with higher grounding angles during the unloading phase of the fairlead excitation motion. The magnitude of the grounding angles achieved by the three pretension cases vary in the same order of the preten- sion values. The highest pretension case experienced the highest grounding angles, followed by the mid-range and low pretension cases. This variation in the grounding angles is due to the geometry of the catenary cable, and is also evident in the static line geometries, as seen in Figure 27. Time (s)STDP 65 70 75 80 85 90200300400500600 T0 = 706.9 kN T0 = 1248.4 kN T0 = 3136.5 kN Figure 28: Touchdown location sTDP of 4.74 m-element line with pretension values, T0, of 706.9 kN, 1248.4 kN and 3136.5 kN for Case 1 fairlead excitation parameters. Time (s)Ttot fl (kN) 65 70 75 80 85 900200400600 T0 = 706.9 kN T0 = 1248.4 kN T0 = 3136.5 kN Figure 29: Fairlead tension time histories of 4.74 m-element line with pretension values, T0, of 706.9 kN, 1248.4 kN and 3136.5 kN for Case 1 fairlead excitation parameters. 37 Time (s)VTDP (m/s) 65 70 75 80 85 90-4-3-2-1 T0 = 706.9 kN T0 = 1248.4 kN T0 = 3136.5 kNFigure 30: Vertical velocity of touchdown node of 4.74 m-element line with pretension values,T0, of 706.9 kN, 1248.4 kN and 3136.5 kN for Case 1 fairlead excitation parameters. Time (s)ThetaTDP (rad) 65 70 75 80 85 9000.050.10.150.2 T0 = 706.9 kN T0 = 1248.4 kN T0 = 3136.5 kN Figure 31: Grounding element angle of 4.74 m-element line with pretension values of 706.9 kN, 1248.4 kN and 3136.5 kN for Case 1 fairlead excitation parameters. With the same dynamic excitation applied to the three pretension cases, the di erence between them are in the pretension values and the sTDPranges. AsTDPrange close to the anchor, in general, leads to lower angles, due to the natural geometry of the catenary. Comparing the grounding velocities andangle pro les of the mid-range and high pretension cases, it is noted that the tension uctuations are more signi cant in the mid-range case, even though the grounding velocity pro les in the two cases are similar. Further- more, the grounding velocities in the low-pretension case is the lowest among the three cases, while also experiencing the highest angles, and it is notable that the tension uctuation magnitude experienced in this case is larger than in the mid-range and high pretension cases. Therefore, the magnitudes of the tension uctuations are positively correlated with . Consider a grounding element with the asymptotic values of at 0 and /2, corresponding to the perfectly vertical and horizontal orientations, as shown in Figure 32. The seabed reaction force acting in the axial direction of the element is thus FB;r j^ej ^ej=FB;r jsin^ej. Hence the maximum value of the seabed reaction force acting in the axial direction of the grounding 38 element isFB;r jwhenis/2 and the element is vertical, and acts to compress the element. Conversely, when the grounding element is horizontal and is 0, an isolated element will not experience compression due to seabed contact. Figure 32: Grounding element with angle of 0 and/2 rad. It was evident in Section 4.2.1 that the e ect of compression on a par- ticular segment of the line gives rise to large strain gradients which in turn lead to the production and propagation of stress waves. High values leads to a larger e ective seabed force component in the grounding element axial direction, which compresses the grounding element and leads to the rapid creation of a low strain region and large strain gradients in the line, and is therefore associated with the creation of the propagating stress waves. 4.5. Discussion of Results from Nodal Grounding Study Reducing the element size leads to a reduction of the magnitude of the tension uctuations. Taking Case 1 as an example, Figure 10b shows that for the in-house code an element size of 2.03 m is required to reduce the amplitude of the spurious tension uctuations to approximately 20 kN or 1.5% of peak tension. However, the peak tension value achieved convergence with a coarser discretisation comprising of 4.74 m length elements. The use of a ner discretisation necessarily entails longer computing times, and for the in-house code, the increase was approximately 1.93 times comparing the 4.74 m (150 elements) to the 2.03 m (350 elements) discretisations. Presently, the same time step was used for both discretisations, and considering that a larger time step may be applied for the coarser discretisation, the disparity in computing times can potentially be wider. Higher grounding velocities are associated with the development of ten- sion uctuations. In Case 2, the maximum fairlead velocity attained was 39 lower than Case 1. When considering only the grounding velocities in Case 1, Figures 15a and 16a show that the grounding velocities of Nodes 66 and 67 for the 8.0 m line and Nodes 619 and 629 for the 0.84 m-element line were lower than that for Nodes 68 to 71 and Nodes 639 to 667 for the two discretisations respectively. The same is observed in Case 2, where a com- parison of Figure 21a and Figure 21d shows that the grounding velocities of the nodes increase in tandem with fairlead velocity. In Case 1, the grounding of Nodes 66 and 67 in Case 1 with the 8.0 m-element line did not lead to the development of tension oscillations, while the grounding of Nodes 68 to 71 did, due to comparatively higher grounding velocities. The relatively low grounding velocities in Case 2 also did not lead to tension uctuations. The cause of the tension uctuations is the formation of localised low strain or slack regions in the vicinity of the touchdown point. The process is initiated by high speed nodal grounding situations and the formation of a localised low tension or compression zone, followed by the development and expansion of the compression zone. The compression zones are transient and eventually collapse, to be replaced by a local rarefaction, high strain zone, which then expands. High strain gradients are generated due to the presence of the compres- sion and rarefaction zones. The large strain gradients represent unbalanced tension forces and are responsible for driving the propagation of stress waves which travel along the length of the line. The transition from a compression to a rarefaction zone is due solely to the nodal motions. When the ground- ing of Node 70 produced a low tension zone at the vicinity of the touchdown point, this resulted in a net acceleration of the nodes outwards relative to the low strain region. The contraction and collapse of the compression zone is a result of the outward motion of the nodes relative to it; as the nodes at the high strain gradient interfaces experience an increase in the resultant tension force due to the high strain gradient (see Equation 4), they begin to accelerate in the direction of the resultant tension forces. As the nodes move outwards from the compression zone, the separation between the nodes within it and consequently the strain, increases again. At the boundaries, namely the anchor and fairlead nodes, the stress waves are re ected back into the line, as shown by Yang et. al [19]. The inertial and tension forces on the nodes give rise to oscillatory motions. Successive nodal groundings may occur and contribute to the creation of multiple stress waves that propagate back and forth along the length of the line, as was observed in Case 1. To accurately resolve high strain gradients close to the touchdown zone, 40 the discretisation has to be suciently high. If the number of elements is equivalent to that would be present in a physical line, the strain spatial gra- dient can be expected to follow that shown in Figure 14b. The kinks in the strain and strain gradient curves close to the touchdown point is char- acteristic of static catenary lines. In contrast, when the discretisation is insuciently re ned to capture a smooth spatial strain variation, localised strain discontinuities and low strain regions may form due to rapid nodal grounding. Of particular importance is the d =dscurve, because large spa- tial strain gradients deviating from the smoothly varying pro les as shown in Figures 14b and 11b indicate that there are large resultant nodal tension forces. This leads to the creation and propagation of stress waves following large nodal accelerations, velocities and motions; manifesting at the fairlead as tension uctuations. In addition to the grounding velocity, the grounding element angle has an e ect on the production of the tension uctuations and higher angles increase the likelihood of the creation of low strain regions and large strain gradients, particularly if the grounding velocities are signi cant. Locally re- ning the element size in such regions is expected to be helpful in reducing the e ects of the propagating stress waves created due to large strain gra- dients. Larger elements can be used in regions where is expected to be lower. Reducing the seabed spring sti ness and damping coecients, as dis- cussed in Section 4.3, have the e ect of reducing the magnitude of the fair- lead tension uctuations. The mechanisms that produce this e ect is the increased seabed penetration depths and the higher seabed force cuto el- evation,zB;c. Consider the motion of two successive grounding nodes, as the rst node is allowed to penetrate deeper into the seabed and the down- wards vertical speed of the second node is reduced due to a higher zB;c, the separation between the two nodes, which determines the element strain (see Equation 3) is less likely transition rapidly into a slack state, causing the large strain gradients that subsequently produce the propagating stress waves. However, reducing the linear spring sti ness and damping tends to lower the peak tensions in an oscillation period. 5. Application of Modi ed Spring Mattress Model In this section the e ectiveness of the modi ed spring mattress (MSM) model in mitigating the fairlead tension uctuations is evaluated for the 41 shallow and deep water depths of 82.5 m and 914.0 m respectively. The features of the MSM model that di er from the spring mattress (SM) model are the modi ed reaction force and added mass force models described in Sections 2.3.2 and 2.3.4. 5.1. Shallow water environment A single segment mooring line with structural parameters enumerated in Table A.1 is used for the shallow water environment. Test cases 1, 3 and 4, as shown in Table B.2, provide the fairlead excitation pro les. Compar- isons are made between the results from the SM, SM 7and modi ed spring mattress (MSM) seabed models using the relevant coecient sets shown in Table C.2. Comparisons are made between the results from the SM, SM 7 and modi ed spring mattress (MSM) seabed models using the relevant coef- cient sets shown in Table C.2. The SM 7coecient set is chosen because it was shown to be e ective at reducing the tension uctuations in Case 1 (see Section 4.3). The maximum tension results from the present code are compared to that from Ref. [18], shown in Table 4 as T[18], as a means of benchmarking. A second qualitative indicator for the quality of the results is the existence of spurious oscillations, which can be evaluated from the tension time series plots for Cases 1, 3 and 4 as shown in Figures 33, 36 and 43 respectively. For Case 1, the peak tension results, presented in Table 4, using the SM, SM7and MSM seabed models with the two line discretisations agree well withT[18] in general. The SM model with the 4.74 m and 0.84 m-element discretisations (SM4:74and SM0:84) respectively underpredict T[18] by 0.08% and 0.09%, while the MSM (MSM4:74) and SM 7(SM4:74 7) coecient sets, both using the 4.74 m-element discretisation, respectively underpredict T[18] by 0.43% and 3.72%. Time (s)Ttot fl (kN) 82 84 86 88 9020040060080010001200 SM, L=0.84 m SM, L=4.74 m SM_7, L=4.74 m MSM, L=4.74 m Figure 33: Fairlead tension time history for Case 1 with 0.84 m and 8.0 m-element lines and SM, SM 7and MSM models. 42 Table 4: Fairlead peak tensions with SM, SM 7and MSM models compared to benchmark tension T[18] for Cases 1 to 4. Model Case 1 Case 3 Case 4 Fairlead Tension (kN) T[18] 1300 6777 7488 TSM 4:74m 1298.98 6876.02 7384.98 TSM 0:84m 1298.87 6860.27 7414.92 TSM7 4:74m 1251.68 6752.64 7337.61 TMSM 4:74m 1294.38 6858.60 7355.22 Percentage deviation of tension from T[18] TSM 4:74m -0.079% 1.461% -0.975% TSM 0:84m -0.087% 1.229% -1.376% TSM7 4:74m -3.717% -0.359% -2.008% TMSM 4:74m -0.432% 1.204% -1.773% s (m)ε 100 200 300 400 500 600 70000.00010.00020.00030.0004 t = 83.12 s t = 83.44 s t = 83.76 s t = 84.08 s t = 84.40 s t = 84.72 s t = 85.04 s t = 85.36 s t = 85.60 s (a) Strain distributions for 8.0 m-element line using MSM model. s (m)dε/ds 100 200 300 400 500 600 700-5E-0705E-071E-061.5E-062E-06 t = 83.12 s t = 83.44 s t = 83.76 s t = 84.08 s t = 84.40 s t = 84.72 s t = 85.04 s t = 85.36 s t = 85.60 s (b) Strain spatial gradient distributions for 8.0 m-element line using MSM model. Figure 34: Strain and strain gradient distribution for Case 1 during nodal grounding time window (8 :12t85:52 s). The tension time series in Figure 33 show that the tension uctuations 43 are signi cantly reduced in the results for SM0:84, MSM4:74and SM4:74 7. The and d=dsdistributions in a line with 4.74 m elements and applying the MSM model are presented in Figures 34a and 34b, and illustrate that the MSM model is e ective in preventing severe strain gradients from occurring during nodal grounding. Time (s)Z (m) 82 84 86 88 90-82.5-82-81.5-81 Node 107, MSM Node 109, MSM Node 111, MSM Node 112, MSM Node 114, MSM Node 116, MSM Node 117, MSM Node 119, MSM Node 107, SM Node 109, SM Node 111, SM Node 112, SM Node 114, SM Node 116, SM Node 117, SM Node 119, SM Figure 35: Nodal vertical coordinate, Z, results for 8.0 m-element line using SM and MSM models during nodal grounding time window (8 :12t85:52 s). Figure 35 illustrates that the penetration depth of the nodes has increased when the MSM model is applied, compared to the nodal penetration depths of the SM model, due to a more gradual application of the vertical seabed force by the MSM model as well as the addition of the arti cial grounding added mass force which increases the mass of the nodes when they make contact with the seabed. The maximum penetration elevation in Case 1 with the MSM model is -82.84 m compared to that of -82.74 m with the SM 7 coecient set. Case 3 is a biharmonic fairlead excitation in the surge and heave directions with a maximum displacement of 25.4 m of the fairlead node from the initial position. From the tension peak values for Case 3 presented in Table 4, the results for SM4:74and SM0:84overpredicted T[18] by 1.46% and 1.23% respectively, while the MSM4:74results overpredicted T[18] by 1.20%. The SM4:74 7results underpredicted T[18] by 0.36%. Figure 36 presents the fairlead tension time history for one oscillation period (600 st800 s) in Case 3. The tension uctuations do not occur between 600 s to 665 s for all models and discretisations, as presented in Figure 37a, due to the low nodal grounding velocities and grounding angles shown respectively in Figures 38a and 39a. Between 665 s and 800 s, the uctuations appear at various points in time, as presented in Figure 37b, 44 with some variability between the models. From 665 s to 750 s, uctuations appear at the troughs as well as loading cycle (positive slope) of the tension results for all models, whereas between 750 s and 800 s, some uctuations occur at the troughs of the SM4:74fairlead tension results and are subdued in the results for the SM4:74 7, SM0:84, MSM4:74models. Time (s)Ttot fl (kN) 600 650 700 750 800200040006000SM, L=0.84 m SM, L=4.74 m SM_7, L=4.74 m MSM, L=4.74 m Figure 36: Fairlead tension time histories for Case 3 with time window (600 t800 s). Time (s)Ttot fl (kN) 600 610 620 630 640 650 660200040006000 SM, L=0.84 m SM, L=4.74 m SM_7, L=4.74 m MSM, L=4.74 m (a) Fairlead tension time histories for Case 3 with time window (600 t665 s). Time (s)Ttot fl (kN) 6706806907007107207307407507607707807908000200400600800 SM, L=0.84 m SM, L=4.74 m SM_7, L=4.74 m MSM, L=4.74 m (b) Fairlead tension time histories for Case 3 with time window (665 t800 s). Figure 37: Fairlead tension time histories for Case 3 with SM, SM 7, MSM seabed models and 4.74 m and 0.84 m-element discretisations. 45 Time (s)VTDP (m/s) 600 610 620 630 640 650 660-4-2024 SM, L=0.84 m SM, L=4.74 m SM_7, L=4.74 m MSM, L=4.74 m(a) Touchdown node vertical velocities for Case 3 with time window (600t665 s). Time (s)VTDP (m/s) 670680690700710720730740750760770780790800-4-2024 SM, L=0.84 m SM, L=4.74 m SM_7, L=4.74 m MSM, L=4.74 m (b) Touchdown node vertical velocities for Case 3 with time window (665t800 s). Figure 38: Touchdown node vertical velocities for Case 3 with SM, SM 7, MSM seabed models and 4.74 m and 0.84 m-element discretisations. Time (s)ThetaTDP (rad) 600 610 620 630 640 650 66000.20.40.60.8 SM, L=0.84 m SM, L=4.74 m SM_7, L=4.74 m MSM, L=4.74 m (a) Grounding element angles for Case 3 with time window (600 t665 s). Time (s)ThetaTDP (rad) 67068069070071072073074075076077078079080000.20.40.60.8 SM, L=0.84 m SM, L=4.74 m SM_7, L=4.74 m MSM, L=4.74 m (b) Grounding element angles for Case 3 with time window (665 t800 s). Figure 39: Grounding element angles for Case 3 with SM, SM 7, MSM seabed models and 4.74 m and 0.84 m-element discretisations. The characteristics for the some of the tension uctuations in Case 3 di er from Case 1 in that these uctuations are observed during the loading 46 cycle, and are preceded by signi cant uctuations at the troughs of the time series. The tension at the trough is close to zero, indicating that the fairlead tension is close to being slack. The strains in the line during the development of the tension uctuations at one of the troughs at t= 726:0 s is presented in Figure 40, and illustrates that a large segment of the line is under signi cant compression for all the models and discretisations used. s (m)ε 100 200 300 400 500 600 700-0.02-0.015-0.01-0.0050 SM_4.74 SM_0.84 SM7_4.74 MSM_4.74 Figure 40: Strain distributions at t= 726:0 s for Case 3 during development of tension uctuations. At this point, a distinction is made between the tension uctuations that occur in all the models between 665 s and 750 s, and those that occur in the SM4:74model from 750 s to 800 s. The former occurs due to the occurrence of snap loading, and is characterised by two waves of tensions uctuations or shocks; the rst occurs during rapid line unloading when line tension falls precipitously and parts of the line goes slack in the vicinity of the touchdown point, followed by a second wave of uctuations that occur when the line is loaded again and the part of the line under compression returns to a tensioned state, which leads to the propagation of shock waves. The occurrence of snap events has been demonstrated experimentally by Hsu et al.[20] and Azcona et al.[2]. The analytical shock condition pro- posed by Triantafyllou and Bliek [11] and experimentally veri ed by Gobat and Grosenbaugh[12] states that cable unloading and loading shocks develop when the condition in Equation 34 is met, dx0 dt r T0 m(34) wherex0is the position of the touchdown point, T0is the touchdown point tension, and mis the mass per unit length of the line, and the left and right hand sides respectively refer to the touchdown point and transverse wave speeds. In the present study, the touchdown point speed is approximated by the time rate of change of sTDP, whileT0is approximated with the horizontal component of the fairlead tension following [12]. 47 600 625 650 675 700 725 750 775 800 Time (s)050100Wave speed (m/s)Trans. wave speed TDP speedFigure 41: Touchdown point and transverse wave speeds for Case 3 during time window 600t800 s. 82 84 86 88 90 Time (s)02040Wave speed (m/s)Trans. wave speed TDP speed Figure 42: Touchdown point and transverse wave speeds for Case 1 during time window 81t91 s. The touchdown point and transverse wave speeds in Cases 3 and 1 are shown in Figures 41 and 42 respectively. Figure 41 shows the occurrence of the touchdown point speed exceeding the transverse wave speed between 665 s and 750 s, which corresponds to the occurrence of snap load tension uctuations that comprise of line unloading and loading shocks. Between 750 s and 800 s, the touchdown point speed is lower than the transverse wave speed in Case 3. Hence, the tension uctuations that occur in the SM4:74 tension results (see Figure 33) are not due to snap loading shocks. This is further illustrated in Figure 42 which shows that the touchdown point speed never exceeds the transverse wave speeds and hence not ful lling the shock condition, even though there are uctuations occurring in the tension. The tension uctuations occurring in the SM4:74model in Case 1, and be- tween 750 s and 800 s in Case 3, are caused by nodal grounding. Unlike the 48 snap loading shocks, which may occur when touchdown point velocities and angles are relatively low, as shown in Figures 38a and 39a respectively, and the fairlead tension approaches zero, the nodal grounding tension uctua- tions occur when grounding velocities and angles are comparatively higher, as presented in Figures 38b and 39b respectively, and the line does not be- come slack. Parts of the line in the vicinity of the touchdown point may become slack during nodal grounding, which generates strain discontinuities, however, the magnitude of the negative strains are lower than what occurs in snap loading situations. The MSM model is e ective in mitigating the nodal grounding tension uctuations, but not the snap load tension shocks. Case 4 is a biharmonic fairlead excitation in the surge direction with a maximum displacement of 18.0 m of the fairlead node from the initial position. For Case 4, Table 4 shows that the peak tensions with SM4:74 and SM0:84, and MSM models underpredict T[18] by under 1.8%, while the SM4:74 7model underpredicts Tby approximately 2%. Time (s)Ttot fl (kN) 700 720 740 760 780 800200040006000 SM, L=0.84 m SM, L=4.74 m SM_7, L=4.74 m MSM, L=4.74 m Figure 43: Fairlead tension time histories in Case 4 during (700 t800 s). Figure 44 presents the touchdown point and transverse wave speeds in Case 4 and illustrates that the snap loading shock condition [11] is met between 700 s and 760 s. From 760 s to 800 s, the touchdown point speed is lower than the transverse wave speed. Figures 43 and 45a respectively show the tension time histories for Case 4 for one period of fairlead motion and a close up view during the time window (700 t760 s) when snap load shocks occur. During 700 t760 s, even though the shock condition is not met, tension uctuations appear in the results for the SM4:74model, but are signi cantly reduced in the SM0:84and SM4:74 7and MSM4:74models. As was in Case 3, the tension uctuations occurring during 760 <t800 s are due to the occurrence of nodal grounding. As shown in the tension histories 49 700 720 740 760 780 800 Time (s)050100Wave speed (m/s)Trans. wave speed TDP speedFigure 44: Touchdown point and transverse wave speeds for Case 4 during (700 t800 s). for the SM0:84, SM4:74 7and MSM4:74models, this type of uctuations can be mitigated by re ning the nodal density and judiciously modifying the seabed force coecients. Time (s)Ttot fl (kN) 700 710 720 730 740 750 76002004006008001000 SM, L=0.84 m SM, L=4.74 m SM_7, L=4.74 m MSM, L=4.74 m (a) Fairlead tension time histories for Case 4 with time window (700 t750 s). Time (s)Ttot fl (kN) 760 770 780 790 800200300400500600 SM, L=0.84 m SM, L=4.74 m SM_7, L=4.74 m MSM, L=4.74 m (b) Fairlead tension time histories for Case 4 with time window (750 t800 s). Figure 45: Fairlead tension time histories for Case 4 with SM, SM 7, MSM seabed models and 4.74 m and 0.84 m-element discretisations. Figures 46 and 47 respectively show the touchdown node vertical velocity andangles during the two aforementioned time windows in Case 4. It is observed from Figure 47a that in the time window (700 t760s) during which snap loading uctuations occur, the angles were under 0.3 rad, and in general lower than the angles in the time window (760 < t800 s) during which nodal grounding uctuations occur, presented in Figure 47b. 50 An exception for this trend occurred at 756 s, when the angle exceeds 0.3 rad. Comparing Figures 46a and 46b, it is observed that the maximum grounding velocities in the time windows (700 t760) and (760 < t 800) are of comparable magnitudes approaching 4 m/s. Time (s)VTDP (m/s) 700 710 720 730 740 750 760-4-2024 SM, L=0.84 m SM, L=4.74 m SM_7, L=4.74 m MSM, L=4.74 m (a) Touchdown node vertical velocities for Case 4 during (700 t750 s). Time (s)VTDP (m/s) 760 770 780 790 800-4-2024 SM, L=0.84 m SM, L=4.74 m SM_7, L=4.74 m MSM, L=4.74 m (b) Touchdown node vertical velocities for Case 4 during (750 t800 s). Figure 46: Touchdown node vertical velocities for Case 4 with SM, SM 7, MSM seabed models and 4.74 m and 0.84 m-element discretisations. Time (s)ThetaTDP (rad) 700 710 720 730 740 750 76000.10.20.30.4 SM, L=0.84 m SM, L=4.74 m SM_7, L=4.74 m MSM, L=4.74 m (a) Grounding element angles for Case 4 during (700 t750 s). Time (s)ThetaTDP (rad) 760 770 780 790 80000.10.20.30.4 SM, L=0.84 m SM, L=4.74 m SM_7, L=4.74 m MSM, L=4.74 m (b) Grounding element angles for Case 4 during (750 t800 s). Figure 47: Grounding element angles for Case 4 with SM, SM 7, MSM seabed models and 4.74 m and 0.84 m-element discretisations. 51 The grounding velocity and angle att= 786 s, which are approximately 3 m/s and 0.3 rad respectively, are compared with the grounding velocities andangles att= 756 s, respectively 3.75 m/s and 0.35 rad, and are there- fore higher than necessary to induce nodal grounding uctuations. However, due to the snap load shock condition being met at 756 s, the more severe uctuations arising from the alternating slack and taut tension states were preeminent. 5.2. Deep water environment A multi-segment, chain-wire rope-chain line with structural parameters enumerated in Figure A.2 is used in the present deep water case, with a water depth of 914.0 m. The pretension is 1200 kN, and the initial line geometry is shown in Figure 48. The fairlead excitation for Case 4 with parameters listed in Figure B.2 is used in this case. Comparisons are made between the SM, SM7and MSM models for the in-house code and Orca ex (using the deep water seabed coecient set presented in Figure C.1), with element sizes of 8.0 m and 4.0 m. -1750.0 -1500.0 -1250.0 -1000.0 -750.0 -500.0 -250.0 0.0 X (m)-914.0-600.0 -750.0-400.0-200.00.0Z (m) zB, 0 Segment (1)Segment (2)Segment (3) Chain Wire rope Figure 48: Initial static line geometry for multisegment mooring line. The fairlead tension time series for one low frequency oscillation period of 100 s is included in Figure 49, while the touchdown point and transverse wave speeds are plotted in Figure 50. The results from Orca ex and the in-house code are in good agreement, as evident in Figure 49, with a max- imum di erence in the peak tension of 30 kN, or 1.2%, with no noticeable phase di erence in the time histories. The peak tensions from the in-house code using the SM, SM 7and MSM models for the 8.0 m and 4.0 m element discretisations di er by a maximum of 24 kN or 0.95%. 52 Figure 50 shows that the tension shock conditions are met in the time window 700t800 s. However, in contrast to the shallow water cases presented in Section 5.1, the tension uctuations in the deep water case are less pronounced. There are spikes in the SM 7tension results at the local tension peaks at 730.5 s, 740.5 s, 750.5 s, 760.5 s and 770.5 s but these are small in magnitude. 700 720 740 760 780 800 Time (s)5001000150020002500Tension (kN)Orcaflex, L=8 m Orcaflex, L=4 m SM, L=8 m SM, L=4 m SM7, L=8 m SM7, L=4 m MSM, L=8 m MSM, L=4 m Figure 49: Fairlead tension time histories for Orca ex and in-house code using SM, SM 7and MSM seabed models for multisegmented line in deep water with Case 4 fairlead excitation during time window (700 t800 s). 700 720 740 760 780 800 Time (s)020406080100Wave speed (m/s)Trans. wave speed TDP speed Figure 50: Touchdown point and transverse wave speeds for deep water case during time window (700t800 s). Figure 51 shows the strain distribution in the mooring line over one such tension shock cycle (744 :67t748:38 s) during which a section of the line goes slack and subsequently returns to a tensioned state. The location 53 of line section in compression is in the vicinity of the touchdown point. As is typical of slack line tension shocks, the magnitude of the compressive strain during the snap load cycle in the present case is signi cantly more severe than the negative strains associated with tension uctuations arising from nodal grounding (see Figure 12a). The large strain gradients present thus generate propagating stress waves. The absence of large uctuations in the fairlead tension is likely due to the signi cant physical separation between the source of the stress waves at the touchdown point and the fairlead. In the present deep water case, the distance between the fairlead and the line section under compression is approximately 1550 m, whereas in the shallow water Cases 1 and 3, the distances is considerable shorter at 160 m and 100 m, as shown in Figures 12a and 40 respectively. It is likely due to this larger separation distance that the travelling stress waves have been attenuated signi cantly due to uid and line structural damping when they arrive at the fairlead. s (m)ε 500 1000 1500 2000-0.02-0.015-0.01-0.0050 t=744.67 s t=747.08 s t=748.08 s t=748.38 s Figure 51: Strain distribution during slack and snap loading with time window (700t800 s). 5.3. E ect of Line Discretisation on Modi ed Spring Mattress (MSM) Seabed Model Coecients In Section 5.1, the proposed Modi ed Spring Mattress (MSM) model, with coecients for NB;candB;aboth set to 1.0 for the 4.74 m element line, was shown to be e ective at reducing the severity of the tension uctuations caused by nodal grounding. The fairlead tension results for Cases 1, 3, and 4 of the shallow water cases, using the same coecient set of the MSM model with element sizes of 8.0 m, 4.74 m, 2.85 m and 2.03 m, are shown in Figures 52, 53, and 54 respectively. For the time periods during which tension uctuations have occurred, 84 :0t87:0 s in Case 1, 750 :0t800:0 s in Case 3 and 760 :0t800:0 s in Case 4, disparities in the severity of the 54 reduced tension uctuations are small between the di erent discretisations. Hence, the same coecient set is shown to be applicable and e ective for a range of discretisations, which may be required for nodal density convergence studies. Time (s)Ttot fl (kN) 82 84 86 88 9040060080010001200 L=8.0 m L=4.74 m L=2.85 m L=2.03 m Figure 52: Fairlead tension time histories with MSM model ( NB;c=B;a=1.0) and element lengths of 8.0 m, 4.74 m, 2.85 m and 2.03 m for Case 1. Time (s)Ttot fl (kN) 700 750 8002004006008001000 L=8.0 m L=4.74 m L=2.85 m L=2.03 m Figure 53: Fairlead tension time histories with MSM model ( NB;c=B;a=1.0) and element lengths of 8.0 m, 4.74 m, 2.85 m and 2.03 m for Case 3. Time (s)Ttot fl (kN) 700 720 740 760 780 80005001000 L=8.0 m L=4.74 m L=2.85 m L=2.03 m Figure 54: Fairlead tension time histories with MSM model ( NB;c=B;a=1.0) and element lengths of 8.0 m, 4.74 m, 2.85 m and 2.03 m for Case 4. 5.4. Summary of Findings from Shallow and Deepwater Test Cases Tension uctuations arise from either line impact with the seabed and slack-snap load cycles. In both situations, negative strain zones may form 55 in the vicinity of the touchdown point, and is the source of the propagating stress waves that appear as uctuations in the fairlead tension time history. The tension uctuations are more pronounced in shallow water environments due to the close proximity of the touchdown zone to the fairlead in comparison with deep water cases. For deeper water environments, where distances from the touchdown point to the fairlead are further, the stress waves have to travel longer distances and thus undergo more damping before they arrive at the fairlead. The tension time histories of the three test cases presented in Figures 33, 36, and 43 suggest that reducing the element size is an e ective way to reduce the spurious tension uctuations due to line-seabed interactions. The element size required depends on the environmental excitation and the grounding velocities of the nodes. Increasing the seabed thickness coecient NB;cand thereby reducing the vertical reaction spring sti ness, as well as reducing the vertical damping co- ecientB;din the SM model are also e ective methods to mitigate the nodal grounding tension uctuations. However, the peak tension results shown in Table 4 suggest that using required NB;candB;dvalues tend to reduce peak tensions, leading to less conservative results for line tensions. This has an e ect of line overload failure predictions. The deviations of peak tension from Tfor Cases 1, 3 and 4, with the SM7coecient set, are between -3.73% and -0.36% (underprediction), while the peak tension deviation from Twith the SM coecient set is between - 0.98% (underprediction) and 1.46% (overprediction). With the MSM model, the peak tension deviation from Tis between -1.77% and 1.2%. Hence, the SM coecient set provides the most conservative line tension predictions, followed by the MSM model and SM 7coecient set. The MSM model, while providing slightly less conservative maximum tension predictions compared to the SM model, is able to signi cantly reduce the tension uctuations caused by line-seabed impact. The peak tensions calculated by the MSM model is also more conservative compared to the SM7coecient set, while providing similar e ectiveness in the mitigation of the tension uctuations. The MSM model achieves this primarily through a higher-order formulation of the vertical reaction force that calculates the reaction force based on the embedment depth of the half elements bounding a node. The application of an added mass force also lowers the natural frequencies of line section that is in contact with the seabed and further reduces the tension uctuations. However, a drawback is that the embedment 56 depth of the grounding nodes may be overestimated. The proposed Modi ed Spring Mattress (MSM) model uses the coe- cient set given in Table C.2 and its e ectiveness in reducing the severity of the tension uctuations arising from nodal grounding for a range of fairlead excitation frequencies and magnitudes, particularly in shallow water condi- tions, was shown in Section 5.1. Finally, the recommended values for the arti cial added mass, B;a, seabed thicknessNB;ccoecients in the MSM model, are both 1.0, and are generally applicable for di erent line materials and discretisations. 6. Conclusion This work presents an investigation into the cause of the spurious tension uctuations that a ect lumped mass mooring line models due to line-seabed interactions. It was shown that both the commercial code Orca ex as well as an in-house code may experience the tension uctuations under moderately severe line-seabed impact conditions. The typical strain and spatial strain gradient pro les of situations which would lead to or preclude the development of tension uctuations were pre- sented in Section 4. The tension uctuations develop due to the formation of low strain regions close to the touchdown point when nodal grounding velocities are relatively high. The rapid formation of the localised low strain regions after nodal grounding leads to the creation of large strain gradients close to the touchdown point. The strain gradients represent unbalanced tension forces and lead to the propagation of the elastic stress waves along the length of the line. A modi ed spring mattress model that calculates the vertical reaction force based on the orientation and elevation of the elements bounding a node was presented. This allows a more gradual application of the seabed vertical force without de ning a large seabed force cuto elevation as required in the conventional spring mattress model. The de nition of an arti cial added mass force also acts to reduce the natural frequencies of the line section in contact with the seabed. The model was able to avoid the fairlead tension uctuations for the severe excitation test cases when used together with a relatively coarse line discretisation, while also providing more conservative estimates of the peak line tensions compared with the spring mattress model applied with the force coecients required to adequately mitigate the tension uctuations. 57 The recommended values for the force coecients in the proposed model were also given and are generally applicable for di erent line materials, dis- cretisations and environments. Studies are underway on applying the ap- proach used in the modi ed spring mattress model to account for line bending and torsional e ects so that it can be applied to pipeline and riser models. Acknowledgements The authors would like to acknowledge the support of Lloyd's Register Group Limited, Energy Research Institute @ NTU (ERI@N) and Singapore Economic Development Board (EDB) under the EDP-IPP programme for their support in undertaking this work.