Finite Element Modeling of Net Panels Using a Net Element

Ocean Engineering 30 (2003) 251270www.elsevier.com/locate/oceaneng

Finite element modeling of net panels using aconsistent net element

Igor Tsukrov , Oleg Eroshkin, David Fredriksson,M. Robinson Swift, Barbaros Celikkol

Mechanical Engineering Department, University of New Hampshire, Durham, NH 03824, USAReceived 10 September 2001; accepted 3 January 2002

Abstract

A consistent finite element is proposed to model the hydrodynamic response of net panelsto environmental loading. This equivalent net element is constructed to reproduce the drag,buoyancy, inertial and elastic forces exerted on the netting by current and waves. To evaluatethe accuracy of the proposed finite element modeling, numerical predictions have been com-pared with the experimental observations and (simplified) analytical results of other authors.This new modeling technique has been applied to evaluate the performance of a tension legfish cage in the open ocean environment. 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Net modeling; Finite element analysis; Open ocean aquaculture

1. Introduction

Netting is a basic functional and structural component of fishing nets, offshoreaquaculture net pens and various fish enclosures. Advancements in the engineeringanalysis of nets are needed to improve design, performance and reliability of suchstructures. Research into the mechanical performance of nets has a long history (seeTerada et al. (1914); and Milne, 1972). It progresses along two major lines: experi-mental studies (sometimes combined with in-situ observations) andanalytical/numerical predictions. This paper contributes to the latter area of the analy-

Corresponding author. Tel.: +1 (603) 862-2086; fax: +1 (603) 862-1865..E-mail address: [email protected] (I. Tsukrov).

0029-8018/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved.PII: S 00 29 -8018( 02 )0 0021-5

252 I. Tsukrov et al. / Ocean Engineering 30 (2003) 251270

sis of nets. We propose a consistent net element to be used in a highly effectivecomputer-based numerical procedure the finite element method to model thedynamic response of net panels to open ocean environmental loading. In the follow-ing text, net panel refers to a section of netting between supports. Note that theanalysis of the modes of net failure, though important, is not included into this study.Instead, the focus is centered on the numerical modeling of deformation and overalldynamic behavior of nets subjected to wave and current loading.

Direct finite element modeling of all the strands comprising the net is impracticalbecause it requires the enormously large number of finite elements. For example,according to Bessonneau and Marichal (1998), the number of cells in a fishing netcan be more than 3 million. It is also impossible to exactly represent a net panel byan equivalent structural element, as shown in Section 2.2. That is why we developa special element that consistently models all hydrodynamic, hydrostatic and inertialforces acting on a portion of netting. Our approach is based on the observation thatdrag and inertia components of the hydrodynamic forces [eq. (1)] uncouple. Below,we discuss some previous work relevant to our studies.

Simple analytical formulae to evaluate the resistance of nets to currents were pro-posed by Kawakami (1959, 1964). Aarsnes et al. (1990) performed a series of testsand proposed an analytical method to calculate the forces produced on arrays of fishcages by a constant current. An algorithm based on the introduction of a mappingcoefficient was proposed and successfully applied by Gignoux and Messier (1999)to model aquaculture nets using Abaqus/Aqua with beam elements. This algorithmwas validated using experimental data of Mannuzza (1995). Our results are comparedagainst predictions of Kawakami and Aarsnes et al. in Section 4.

Bessonneau and Marichal (1998) used rigid cylindrical bar elements to investigatethe dynamics of submerged supple nets (trawls). The strong point of their approachwas the inclusion of all hydrostatic, hydrodynamic, gravitational and linking forcesinto the analysis. They proposed the mesh grouping method in which several actualmeshes are grouped into a fictitious equivalent mesh having the same specific mass,apparent weight and approximately the same drag resistance. The finite differencescheme was used for time discretization, and several examples of the trawl towingwere considered.

Lee and Pei-Wen (2000) investigated the dynamic behavior of a tension leg plat-form with a netcage system and obtained an analytical solution (in an infinite series)for motion of the two-dimensional model of such a platform. They subdivided thearea occupied by the system into four regions and formulated the scattering andradiation problems for each region. To obtain the analytical solutions, assumptionsof small motion and of inviscid, incompressible and irrotational flow had to be used.Information on the net mesh geometry was taken into account through the solidityratio and the equivalent net-tether diameter. The Morison equation (Morison et al.,1950) was linearized using the Lorentz hypothesis of equivalent work (Solliitt andCross, 1972).

Tsukrov et al. (1999, 2000) analyzed open ocean aquaculture fish cages using theequivalent truss approach. In finite element simulations, a deformable truss structurewas used to represent each net panel. This truss had the same drag force, buoyancy,

253I. Tsukrov et al. / Ocean Engineering 30 (2003) 251270

gravity and stiffness as the original net panel, but the condition of the same inertiawas not satisfied. The necessity to improve the model was one of the motivationsof the current work.

In our paper, hydrodynamic forces on the structural elements are calculated usingthe Morison equation modified to account for relative motion between the structuralelement and the surrounding fluid. Following Haritos and He (1992), the fluid forceper unit length acting on a cylindrical element is represented as

f C1VRn C2VRt C3V n C4V Rn, (1)where VRn and VRt are the normal and tangential components of the fluid velocityrelative to the structural element, V n is the normal component of total fluid acceler-ation and V n is the normal component of fluid acceleration relative to the structuralelement (see Fig. 1). Bold is used to denote vectors and matrices. The coefficientsin the formula above are given by C1

12rwDCnVRn, C2 Ct, C3 rwA and

C4 rwACm, where D and A are the diameter and the cross-sectional area of theelement in the deformed configuration, rw is the water density, Cn and Ct are thenormal and tangential drag coefficients, Cm is the added mass coefficient. Note thatCn and Cm are dimensionless, while Ct has the dimension of viscosity. Equation (1)is known to adequately predict the hydrodynamic force on a submerged cylindricalelement whose diameter is small compared to the length of the wave (Haritos andHe, 1992; Webster, 1995; Tsukrov et al., 2000).

The normal and tangential drag coefficients can be either found from physicalexperiments or expressed in term of Reynolds number, Ren, as follows (Choo andCasarella, 1971):

Fig. 1. Tangent and normal component of fluid velocity vector.


Cn 8p

Rens(10.87s2) (0 Ren1),

1.45 8.55Re0.90n (1 Ren30),1.1 4Re0.50n (30 Ren105)

(2)

where Ren rwDVRn /m, s 0.077215665 ln(8 /Ren) and m is the water vis-cosity.

This paper is organized as follows. In Section 2, the hydrostatic and hydrodynamicforces acting on a net panel are quantified, and a consistent net element is developedto model the elastodynamic performance of nets. Section 3 describes the finiteelement computer program Aqua-FE used to model the dynamic response of par-tially or completely submerged structures in an ocean enviroment. In Section 4, theresults obtained using the consistent net element are compared with the predictionsmade by utilizing other approaches. Section 5 presents a practically important appli-cation of the proposed technique analysis of dynamic performance of the tensionleg fish cage Refa used in open ocean aquaculture.

2. Consistent finite element to model net panels

In order to accurately model the dynamic performance of a net panel, it is neces-sary to take into account the forces exerted on all strands of the netting by the fluidenvironment. It is not feasible to model the behavior of each strand comprising thenet separately, because of the large number of finite elements that are required forsuch direct modeling. That is why a special consistent net element is proposed inthis paper. The real net is substituted by the equivalent consistent net element(s)having the same hydrodynamic and elastic parameters.

2.1. Hydrostatic and hydrodynamic forces acting on a net panel

The action of the surrounding fluid on the net is manifested by buoyancy, dragand fluid acceleration forces, where drag and fluid acceleration forces are caused bycurrent and wave-related water motion. The weight and mass inertia of the net mustalso be taken into account in the modeling process. Thus, five parameters have tobe considered while modeling a net panel:

fluid drag and inertia forces; buoyancy; weight; mass inertia; elastic forces.

The drag force acting on a net panel is assumed to be proportional to the projected


area of the panel, normal to the direction of the fluid motion (this assumption neglectsthe interaction between individual net strands). The projected area of the net panelcan be expressed in terms of the twine diameter d and the total strand length Ltotalas follows

Apr Ltotald. (3)For example, the total strand length and the projected area of a rectangular net panelof dimensions a b with half-mesh size l and twine diameter d (see Fig. 2), are

Ltotal 2ab

l , Apr 2abdl abS, (4)

where S 2d / l is the solidity ratio used, for example, by Aarsnes et al. (1990).Theweight and buoyancy of a net panel can be represented as

w pgrLtotald24 , (5)

fb pg(rwr)Ltotald24 , (6)

where r is the mass density of net material, rw is the fluid mass density, and g isthe acceleration of the free fall vector directed downward.

Note that even though eq. (4) assumes net cells are square shaped, the approachpresented in this section can also be applied to non-square (rectangular or rhombic)mesh geometries.

2.2. Comment on the equivalent truss/beam approach to net modeling

In most finite element simulations, the net panels are represented by some equival-ent structural elements, usually trusses or beams [see, for example, Gignoux and

Fig. 2. Net mesh geometry.


Messier (1999); or Tsukrov et al. (2000)]. Let us show that neither truss nor beamelements can be used to exactly reproduce the dynamics of a net panel subjected tocurrents and waves, even in the assumption of non-interacting net strands.

Consider an equivalent element used to model the rectangular section of nettingwith mesh parameters as shown in Fig. 2. The geometry of this truss or beam elementis characterized by its length Lel and cross-sectional area Ael. Assuming circular cross-section with diameter Del and material mass density rel, the weight, buoyancy andprojected area of the element are as follows

wel pgrelLelD2el4 (7)

fb el pg(rwrel)LelD2el4 (8)

Apr el DelLel. (9)To satisfy the condition of the same projected area, the diameter Del of the equivalenttruss/beam member must be found from equation DelLel Apr using eq. (4):

Del dLtotalLel

. (10)

To obtain the same total buoyancy for the equivalent truss/beam structure, the massdensity of its material, rel, must satisfy the following equation

pD2el4 Lelg(rwrel) fb. (11)

The expression for rel is obtained by substituting eq. (6) into eq. (11)

rel rwLel

Ltotal(rwr). (12)

But then the element weight (and thus, inertia) is greater than the weight of theactual net panel:

welw pgrwd24 L2totalL2elLel 0.

Thus, it is impossible for a truss/beam element to reproduce the net panel exactly,but in some cases the effect of additional weight on the overall dynamic performanceof the recticulated structure is insignificant and can be neglected (see Tsukrovet al., 2000).

2.3. Development of net element: Morison forces

In this paper, the consistent net element is proposed. It is constructed to adequatelyaccount for the hydrodynamic forces acting on any section of netting or an entire


net panel. We decided to use one-dimensional net elements: their development isstraightforward and they are easily compatible with the existing finite element codesused to analyze mooring systems and floating structures. Any two-dimensional netpanel can be modeled as a set of perpendicular or inclined one-dimensional netelements. The hydrodynamic behavior of the proposed net element is based on theMorison equation [eq. (1)]. Since drag force and inertia force in this equation areuncoupled, we can treat them separately. In the following analysis, we shall assumethat the water surface is not altered and there is no interaction between the individualnet strands [see discussion after eq. (17)].

Let us construct the net element of length L to model a section of a net panelhaving the following physical and geometrical properties:

r mass density of net material;A outline area of the modeled section of a net panel ( ab in Fig. 2);E Youngs modulus of net material;d twine diameter;l half-mesh size.

According to Morison equation, the drag force per unit length acting on a one-dimensional element is

fD C1VRn C2VRt. (13)The equivalency of drag force is provided by the correct choice of C1 and C2 coef-ficients. The following procedure is used to calculate C1 and C2 of the consistentnet element. First, the value of the Reynolds number Ren is calculated using thediameter of a single strand. Second, the normal and tangential drag coefficients Cnand Ct are found by substituting this value into eq. (2). Third, the Morison equationcoefficients C1 and C2 are calculated:

C1 rwAdlL CnVRn, C2 Ct. (14)

With such a choice of coefficients, the drag induced on the part of the actual netwill be modeled with good accuracy by the corresponding net element.

The inertia components of the Morison force are

fI C3V n C4V Rn, (15)where the coefficients C3 and C4 are calculated to provide the same inertia as theinertia of the corresponding part of the actual net panel:

C3 rwpd2A2lL , C4 rw

pd2A2lL Cm. (16)

Thus, the total hydrodynamic force acting on the consistent net element is

f rwAdlL CnVRnVRn CtVRt rw

pd2A2lL (V n CmV Rn). (17)


In the analysis to follow, we assume that the flow around the net element is notdisturbed by interaction with other net elements (approximation of non-interactingstrands). The assumption of non-interacting net strands was also used by Besson-neau and Marichal (1998) and Gignoux and Messier (1999). It is obvious, that suchan approximation becomes inaccurate at high solidity ratios caused, for example, bybiofouling. To improve the accuracy of modeling in such cases, one must eithersolve the three-dimensional problem of fluid dynamics for the flow around multiplenon-collinear cylinders, or introduce some kind of interaction or shadowing coef-ficients [see, for example, Aarsnes et al. (1990); Palczynski (2000) or Fredriksson(2001)].

Please note that our approach can be modified to approximately account for strandinteraction; this modification will not be presented in this paper since all the con-sidered examples deal with low solidity ratio nets.

2.4. Development of net element: discretization

To approximate the section of a net, we use a one-dimensional finite element withtwo nodes. The displacements of these nodes (three components at each node) consti-tute the six degrees of freedom of the element. Let us denote the vector of degreesof freedom by q and the shape function matrix by N. Below we derive the elementstiffness, damping and mass matrices and the nodal dynamic force vector for theconsistent net element.

Let us define the following directional unit vectors: nRV- in the direction of therelative velocity; nRa- in the direction of the relative acceleration; na- in the directionof the fluid acceleration; and t- parallel to the structural element (see Fig. 3). Thenthe fluid force per unit length can be expressed as

f C1nRVnTRVVC1nRVnTRVu C2ttTVC2ttTu C3V n C4nRanTRaV (18)C4nRanTRau,

where u and u are the velocity and acceleration of a point of the element. Using thefinite element shape functions to approximate u and u as functions of nodal displace-ments q, velocities q and accelerations q, we obtain the expression for the equivalentnodal force vector due to the wave and current loads:

F mqCq H (19)where m and C are the virtual mass and damping matrices given by

m L

0

C4NTnRVnTRVNds (20)

C L

0

C1NTnRVnTRVNds L

0

C2NTttTNds (21)


Fig. 3. Schematic representation of the possible directions of the fluid acceleration a, relative fluid accel-eration aR, relative fluid velocity vR and the corresponding unit vectors.

and the vector H is

H L

0

C1NTnRVnTRVVds L

0

C2NTttTVds L

0

C3NTnanTaV ds

L

0

C4NTnRanTRaV ds.

Note that a similar expression for H is presented in Gosz et al. (1997) for the equival-ent nodal force vector due to the wave and current loads on a truss element. But inthe case of a consistent net element, the formulae for coefficients C1, C2,C3 andC4 are different.

The buoyancy force per unit length is chosen to produce the same total buoyancyas the corresponding section of the net panel:

fb g(rwr)pd2A2lL . (22)

In general, the element may be partially submerged. Let part 0sLw of the element


be submerged and part LwsL be above the surface. Then, the elementbuoyancy/weight force vector can be calculated as follows

Fbw Lw

0

NTg(rwr)pd2A2lL ds

L

Lw

NTgrpd2A2lL ds. (23)

Note that Lw L for completely submerged element, and Lw 0 for a completelydry one.

Since the mass per unit length is m r(pd2A) / (2lL), the element consistent massmatrix M is given by the following expression:

M rpd2A2lL

L

0

NTNds. (24)

The elasticity force in the net element is modeled as the resultant of tensions inall tethers deforming in the direction parallel to element (we neglect the transverseresistance). Thus, the element stiffness matrix is be calculated using the fiber materialYoungs modulus E and the effective cross sectional area given by

Aeff pd2A2lL . (25)

The elasticity force in the consistent net element is then the same as in the trusselement with the equal effective cross sectional area Aeff, and the expression for theelement stiffness matrix is

K EAeff

L L

0

BTBdetJds, (26)

where B J1N /s is the element strain-displacement transformation matrix thatmust be calculated at each time increment and J is the Jacobian. Formula (26)assumes small deformation and constant elastic stiffness of the net material through-out the entire range of deformation. To account for nonlinear material behavior, weuse the tangent elastic modulus calculated on the previous step of the time integrationprocedure. To account for large nonlinear strain, we implement the updated Lagrang-ian approach (Bathe, 1982). A total Lagrangian approach can also be used in thiscase.

3. Finite element program description

The Aqua-FE program, developed at the University of New Hampshire, is anadvanced computer design and analysis tool to model the dynamic response of par-tially or completely submerged structures in an ocean environment. It uses a commer-


cially available program, Msc. Mentat, as a graphical user interface. The core finiteelement code is written in Fortran, the numerical procedure implemented in thecode is described in detail by Gosz et al. (1997) and Swift et al. (1997). Note thatsome corrections to the procedure have been recently introduced they are reflectedin eqs. (31) and (32). Truss, buoy, cable and net elements are incorporated into theprogram to model various parts of marine structures and mooring systems. The non-linear Lagrangian formulation is employed to account for large displacements ofstructural elements. To solve the nonlinear equations of motion the unconditionallystable Newmark direct integration scheme is adopted as described next.

The standard finite element discretization of a structural system (including netelements) in a moving fluid environment results in the following system of differen-tial equations

Mq Kq R F (27)

where q is the (time dependent) vector of nodal displacements, M is the time inde-pendent consistent mass matrix, K is the global stiffness matrix, R is the equivalentnodal force vector due to gravity and buoyancy forces and F is the equivalent nodalforce vector due to relative fluid motion, wave and current loads. This equation ishighly nonlinear, because the right hand side part (vectors R and F) depends ontime, motion, deformation and whether the individual elements are submerged, par-tially submerged or dry.

The equivalent nodal force vector F is calculated using eq. (19). Substitutingexpression for F into eq. (27) yields

(M m)q Cq Kq R H. (28)

The solution technique of this equation is based upon Newmarks b method to inte-grate the equations in time and full NewtonRaphson iteration scheme to find nodaldisplacement at every time step. First, we discretize eq. (28) in time:

(M mt+t)qt+t Ct+tqt+t Ktq Rt+t Ht+tPtqt, (29)

where Kt is the tangent stiffness matrix, Pt is the internal force vector andq qt tqt. Following the Newmarks b method we then express velocity andacceleration at time instant t t as

qt+t qbt2

qtbt 12b1qt, qt+t qt (1g)tqt gtqt+t, (30)

where parameters b and g are chosen by the user in the intervals: 0b14,

0g12. Substituting these expressions into eq. (29) yields


(M mt+t Ct+tgt bt2Kt)qbt2

Rt+t Ht+tPtqt

(M mt+t Ct+t(gb)t)qtbt

M mt+t 1 g2b2gCt+ttqt.(31)

We use a NewtonRaphson approach (see Bathe, 1982) to solve the nonlineareq. (31) at each time step.

4. Comparison with available results

There are certain difficulties involved with comparison/validation of differentapproaches to net modeling. Most of the numerical simulation results available inthe literature are provided for net structures of a particular design (Bessonneau andMarichal, 1998; Lee and Pei-Wen, 2000) with no exact geometry and materialproperties given. Experimental and in situ observations are usually made for thestructures containing several net panels or a net with some additional members whosecontribution to drag/inertia forces is not easily evaluated [see, for example, Palczyn-ski (2000); Kim et al. (2001); Fredriksson (2001)]. We provide here a comparisonof our predictions with the easy-to-use formulae of Kawakami (1959, 1964) andAarsnes et al. (1990).

The semi-empirical formula of Kawakami (1964) gives the drag force acting ona net having outline area A as a quadratic function of current velocity V

fD 12ACdrwV

2 (32)

where Cd is the mesh drag coefficient. For the knotless net, this coefficient is givenby the following empirical expression (see Milne (1972)

Cd 1 2.73dl 3.12dl2 (33)where d is the twine diameter and l is the half-mesh size (Fig. 2). This formulaprovides the resistance of a net to current acting in the direction perpendicular tothe net panel only. It does not cover the oblique currents and the resulting lift forces.

The formula proposed by Aarsnes et al. (1990) based on their experimental obser-vations, includes both drag and lift forces:

f 12ACDrwVV

12CLV

2nL, (34)

where the drag and lift coefficients are given by


Fig. 4. Orientation of the net panel is characterized by angle a between current V andunit normal vector n.

CD 0.04 0.04 2dl4.96dl2 109.6dl3cosa, (35)CL 1.14dl14.16dl2 80.8dl3sin2a. (36)

The direction of the lift force is defined by unit vector nL that can be expressed interms of current velocity vector V and unit vector n normal to the net panel anddirected leeward as follows

nL (V n) V|V n V| , (37)

where a is the angle between the current direction and the normal vector of netpanel (Fig. 4).

Figs. 5 and 6 present the comparison of the finite element simulations using the

Fig. 5. 1 m1 m net panel: variation of drag with current.


Fig. 6. 1m1m net panel: variation of total drag force with angle (current=1 m/s).

consistent net element with the results of Kawakami (1964) and Aarsnes et al. (1990).They are obtained for a square 1 m1 m panel of net having l 0.0155 m,d 0.0016 m, r 1150 kg/m3 and E 2 109 Pa. In Fig. 5, the drag is plottedas function of current which is assumed to be perpendicular to the panel. The calcu-lations have been performed using 4, 24 and 544 net elements to model the samepanel. The dependence of the total drag force (including both drag and liftcomponents) on the orientation angle a is depicted in Fig. 6. The results show thatthe increase in number of net elements per panel from 24 to 544 (Fig. 7) does notproduce a significant change in the predicted hydrodynamic forces for the consideredcase of current-only loading.

As can be seen in the figures, the drag forces obtained using our approach are

Fig. 7. Net panel in constant current V. (a) 24 elements, (b) 544 elements.


lower than the ones given by eqs. (32) and (34). It can be explained by the followingconsiderations. First, our model accounts for deformation of the net panel producingmore accurate distribution of tension: in-plane components of drag cancel each otherresulting in lower normal drag force (see Fig. 7a). Second, we use the drag coef-ficients that depend on Reynolds number while formulae (33), (35) and (36) containgeometrical parameters only. For example, when the velocity changes from 0.1 to1 m/s, the Reynolds number changes from 137 to 1375 resulting in Cd decreasingfrom 1.44 to 1.21 (compare with Kawakamis constant parameter Cd 1.31).

Note that formulae of Kawakami (1964) and Aarsnes et al. (1990) do not includethe fluid acceleration term (the inertia component of Morison equation). That is whyonly constant current loading (no wave) could be used to compare our approach withtheir analytical formulae.

Recently, a comprehensive experimental study of the hydrodynamic characteristicsof netting and the net shadowing effects has been initiated at the University of NewHampshire Ocean Engineering Program. This study will include physical testing ofnet panels at the UNH wave/tow basin facility and in situ monitoring of fish cagesat the UNH open ocean aquaculture demonstration site. The results will be reportedwhen enough data is accumulated and processed.

5. Application to offshore aquaculture fishcages

The proposed approach to net modeling was applied to the analysis of a tensionleg fishcage (TLC Refa) considered for deployment at the University of NewHampshire open ocean aquaculture demonstration site (see Tsukrov et al., 2001). Thesite is located in 52 m of water 2.6 km south of the Isles of Shoals, New Hampshire.

The finite element model of the TLC Refa is presented in Fig. 8. The fish cageconsists of a net bag (height=16 m, diameter=16 m) with a floating collar(height=1 m, diameter=6 m). The cage is moored with six tension legs, and thestructural stability is provided by a hexagonal reinforcement ring. The mechanicaland geometric properties of the cage structural components are provided in Table 1.Note that for some of them (reinforcement ring, collar and buoy) the effective mech-anical properties are given to take into account the complicated cross-sectionalgeometry. In this cage, netting is not only used as a fish enclosure, but also as animportant structural component needed for overall structural integrity. The impor-tance of the net contribution into the dynamic performance of the fishcage was amajor practical motivation of the present research. The netting parameters arel 0.0155 m, d 0.0016 m, r 1150 kg/m3 and E 2 109 Pa.

The dynamic performance of the TLC Refa fish cage was investigated for bothtypical and extreme environmental loading conditions at the UNH aquaculture dem-onstration site. The regular loading condition for the site consists of a 1.2 m waveand the tidal component of the coastal current which is estimated to be 0.25 m/s.The extreme loading condition consists of a nominal 9 m wave (with a period of8.8 s) and a current which is set to change linearly from 1 m/s on the surface to0.25 m/s near the bottom. The wave data sets and the locations used to obtain these


Fig. 8. Finite element model of TLC Refa1800.

Table 1Effective mechanical and geometric properties of TLC Refa

Element Material Mass Cross-sectional area Youngsdensity(kg/m3) (m2) modulus(Pa)

Leg chain Steel 7850 2.726104 21011Reinforcement ring Glass fiber armed 1000 4.42103 101011

plasticVertical rope Polypropylene 950 3.14104 1.2109Horizontal rope Polypropylene 950 2104 1.2109Net twine Polyamide 1150 1.976106 2109Buoy Polyurethane 116 0.283Collar HDPE 409 1108

characteristics are described in detail by Fredriksson et al. (1999) and Fredriksson(2001).

Figs. 9 and 10 present the comparison of the special net element approach withthe previously used equivalent truss approach described in Tsukrov et al. (2000).The results are provided for the mooring leg and upper vertical rope tension underextreme environmental loading condition. The variations of tension in the upper ver-tical rope are given as functions of time for 10, 21 and 36 consistent net elementsper panel. Note that the initial low tensions in Fig. 9 correspond to the state whenthe cage is released from the undeformed configuration before the periodic pattern


Fig. 9. Comparison of the mooring line A tensions for different net modeling approaches.

Fig. 10. Tensions in the vertical rope B obtained using different net modeling approaches.

of motion develops. As can be seen from the figures, the equivalent truss elementapproach noticeably overestimates the tension in the ropes and time required todevelop the periodic pattern of motion. The reason for this is that the equivalenttruss approach overestimates the inertia of netting, as discussed in Section 2. It canbe also seen from Fig. 10 that the increase in number of net elements per panelfrom 21 to 36 does not cause significant tension difference in the fully developedperiodic motion.

Fig. 11 reports the results of computer modeling of the TLC Refa under bothtypical and extreme environmental loading conditions using 21 net elements perpanel (this corresponds to 342 net elements in the entire structure). The characteristicdeformed configurations and maximum tensions at the key locations are given. Under


Fig. 11. TLC Refa deformed shape and highest tension values under: (a) typical loading conditions;and (b) extreme loading conditions.

a typical loading condition, the fish cage keeps its shape, and both deformation andtensions are small. The extreme environmental loading results in large deformationsbut moderate tensions the safety factor k Tfail /Tcalculated of at least 3.4 wasobserved for all analyzed fish cage components, see Tsukrov et al. (2001). Our simul-ations (not provided here) show no big difference in the results obtained using differ-ent net modeling techniques for typical loading conditions it is explained by thelow levels of acceleration of the fish cage motion, so the inaccuracy in the inertiamodeling becomes insignificant.

6. Conclusions

In the prediction of overall dynamic behavior of net-containing structures (fishingnets, aquaculture net pens, etc.), it is important to accurately model the contributionfrom forces exerted on the nets by environmental loading. A reasonable estimate ofthe total force acting on a net panel must include weight, buoyancy, inertia and dragforces experienced by nets. In numerical modeling using finite elements it is notfeasible to model the behavior of each strand comprising the net separately, becauseof the large number of finite elements that are required for such direct modeling. Ithas been shown in this paper that hydrostatic and hydrodynamic forces exerted on themoving netting, including wave and current-related water motion, cannot be exactlyrepresented by equivalent structural elements. Both buoyancy and inertia of structuralelements are proportional to the volume, drag is approximately proportional to thearea, and it is impossible to choose the equivalent geometry to reproduce these threeforces simultaneously. That is why we have developed a special consistent netelement to model net panels or their parts.


The development of the consistent net element is based on the modified Morisonequation (1). It makes use of the fact that drag and inertia related terms in thisequation are uncoupled, so they can be treated separately. Thus, it is possible tochoose such a set of coefficients in eq. (1) that both the action of fluid and inertiaof a net panel are reflected. These coefficients are presented in eqs. (14) and (16).The stiffness, mass and damping matrices and the equivalent nodal force vector ofthe consistent net element are then given by eqs. (20)(26).

Note that the Morison equation is valid for cylindrical structural elements withthe diameter that is small compared to the wavelength. We also neglect the watersurface diffraction effects and the effect of interaction between individual strands ofnetting. The interaction effect refers to the disturbance in the fluid flow around a netstrand caused by the presence of other strands. In our approach, the effect of netinteraction can be approximately taken into account by introduction of so-calledshadowing coefficients into the Morison equation.

The consistent net element has been implemented in the finite element programAqua-FE used in the University of New Hampshire to analyze the dynamic perform-ance of various structures subjected to mechanical and current/wave-related environ-mental loading. The comparison with semi-empirical formulae of Kawakami (1964)and Aarsnes et al. (1990) shows that our simulations predict lower drag forces. Thereason for this is that our model accounts for deformation of the net panel reflectingthe fact that drag forces on net strands partially cancel each other. We also use thedrag coefficients that depend on the Reynolds number, while formulae of Kawakami(1964) and Aarsnes et al. (1990) have drag coefficients dependent upon geometricalparameters only.

The proposed approach to net modeling has been applied to the analysis of atension leg fish cage considered for deployment at the University of New Hampshireopen ocean aquaculture demonstration site. Various environmental loading con-ditions have been considered. The results have been compared with the predictionsobtained using the equivalent truss elements to model nets. It has been shown thatthe latter approach overestimates the stresses in the fish cage because it over predictsthe inertia of the system. The finite element simulations also show that 21 specialnet elements per net panel is enough to adequately represent the influence of the neton the overall dynamic characteristics of the considered fish cage system.

7. Acknowledgments

This work was supported by the Open Ocean Aquaculture Demonstration Project,NOAA/Sea Grant contract No. NA86GR60016. The authors would like to thank K.Baldwin and M. Grosenbaugh for useful discussions. The authors also acknowledgecontribution of M. Gosz into the original development of Aqua-FE computer pro-gram. The information on physical and geometrical properties of TLC Refa waspresented by Darco Lisac, Refa Mediterranean.


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Finite element modeling of net panels using a consistent net elementIntroductionConsistent finite element to model net panelsHydrostatic and hydrodynamic forces acting on a net panelComment on the equivalent truss/beam approach to net modelingDevelopment of net element: Morison forcesDevelopment of net element: discretization

Finite element program descriptionComparison with available resultsApplication to offshore aquaculture fishcagesConclusionsAcknowledgments

References

Documents

Finite Element Modeling of Net Panels Using a Net Element