Maritime Transportation and Exploitation of Ocean and Coastal Resources – Guedes Soares, Garbatov & Fonseca (eds)

© 2005 Taylor & Francis Group, London, ISBN 0 415 39036 2

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1 INTRODUCTION

Users of flexible structures such as trawls or fish farm-ing cages, generally want to know the shape of suchstructures to assess, for example, the loss of volumeof the fish cage in the current or the possibility for smallfishes to escape from the trawl. Such assessment can becarried out by tests in tank, but numerical models aremore and more used.

Flexible structures such as trawls or fish farmingcages are made of netting, cables, chains, bars, floats,dead weights. Such components can be regarded as sur-face element (netting), linear element (cables, barsand chains) or point element (floats, doors and deadweights).

This paper focuses mainly on net modeling becauselinear and point elements have been largely studied(Zienkiewicz 1989, Desai 1972).

Few net models devoted to nets exist:

– Theret (1993), Bessonneau (1997), Niedzwiedz(2001), Pashen (2004), Lee (2004), Tsukrov (2003)have developed 3D numerical methods whichdescribe twines of the net by numerical bars. Thesemethods take into account a large number of twinesin each numerical bar. The forces considered are thedrag due to the water flow, but also the weight andthe buoyancy of the net. Some of these methodstake into account the twine elasticity. An iterativemethod gives the equilibrium shape of the net. Ferro(1988) uses nearly the same method but he uses thefinite element method, where elements are bars. Thedrawback of these models is that the numerical barsmust be parallel to the actual twines of the netting,

that means that the user of the modeling is not freefor creating the numerical bars.

– Tronstad and Larsen (1997) have developed a mem-brane element used by the finite element method in3D. The element has 4 corners and each side is par-allel to the diagonal of net meshes. The deforma-tion along each side of the membrane element canvary and the element stays plane. This element is atwo-dimensional element, contrary to the previousmethods which use one-dimensional elements: i.e.bars. The drawback is near the same as the previous:the sides of membrane element must be parallel tothe diagonals of the actual meshes, once again theuser is not free for creating the membrane elements.

– O’Neill (1997) has developed a model for axi-symmetrical structures, such as trawl cod-end. It isa 2D model. He takes into account the twine ten-sion, as previously, but also the mesh opening stiff-ness and the pressure of the fish catch on the net.The drawback of this modeling is that it is devotedto only axi-symmetrical structures.

To avoid the problem of constrained numerical ele-ments and of axi-symmetry hypothesis, and to take intoaccount further mechanical behaviors, a Finite ElementMethod (FEM) model of the net based on a triangularelement has been developed (Priour 1997, 1999, 2001,2003). The triangle is the simplest shape to describea surface element, it is why it has been chosen.

The FEM model takes into account the inner twinestension, the drag force on the net due to the current, thepressure created by the fish in the cod-end, the float-ability and weight of the net, the mesh opening stiff-ness and the bending stiffness of the net.

FEM modeling of flexible structures made of cables, bars and nets

D. PriourIFREMER, Plouzane, France

ABSTRACT: A Finite Element Method (FEM) modeling devoted to flexible structures made of nets, cablesand bars, such as fishing gears and fish farming cages is described. The paper focuses on a specific triangularelement devoted to net modeling. The main hypothesis is that the net twines remain straight in each triangularelement. The twines tension is described in mathematical form. The virtual work principle is used to calculateforces on the 3 vertices of the triangular element from the twines tension. The other forces taken into accountin this model are weight in water, mesh opening stiffness, twines contact, bending stiffness, catch pressure, andhydrodynamic loading. Equilibrium is found using the Newton-Raphson method. Results are given for trawlsmade of diamond and hexagonal meshes, for cod-end with grid and catch and for floating fish cage.

The FEM model is able to describe all the net andcables, which means that: – for a trawl, the cod-end, thewings, the headline and also the rigging up to the boatare taken into account, and – for a fish farming cage, thefloats, the netting, the dead weights and the mooringlines are taken into account. The net is modeled by tri-angular elements and cables, warps, bridles and barsare modeled by linear element (bars).

2 HYPOTHESIS AND METHOD

Due to the high number of knots in structures made ofnetting, the time required for the determination of theequilibrium position of knots is generally too long.Thus, in the FEM model described in this paper, thenet is split into triangular elements, each of them beingable to cover a large number of meshes (Figure 1). Theycan be also smaller than a mesh (Figure 7): there is noconstraint. The triangles are contiguous, which meansthat a vertex can belong to several triangles, and triangle

boundaries are not necessary parallel to the twines orto the meshes diagonal. So, the user is free to createthe triangular elements. The refinement is easy: a trian-gular element can be divided in 3 triangles without anymodification on the remaining. Each vertex is linkedto the net, so when the equilibrium position of each ver-tex is found, the equilibrium position of the net is found.

In this FEM model the diamond and square meshescan be modeled, as well as hexagonal meshes. Such netshave 2 directions of twines (called here u and v) fordiamond and square netting or 3 directions for hexag-onal one.

For diamond and square mesh netting (2 directionsof twines), each direction of twine is kept parallel ineach triangular element: in other words, all the u twinesare kept parallel, just as the v twines (Figure 2). Thehypothesis is the same for 3 directions of twines. Thisleads to a constant deformation for each direction oftwines, that means that all the u twines have the samedeformation, just as the v twines. Obviously there is avariation of orientation and deformation from one tri-angular element to an adjacent one. This hypothesis isreasonable if all the elements are small.

With these hypotheses the force on each vertex canbe calculated. These forces depend on the position ofvertices. From these forces the Newton-Raphson iter-ative method gives the equilibrium position of the net.

3 TWINES TENSION

Here the forces due to twines tension are describedfor diamond and square mesh netting, the method issimilar for hexagonal meshes.

All the twines of a triangular element are modeledas a bar made of isotropic and elastic material. Thematerial can have 2 elastic modulus, one in traction andone in compression (very low or zero), to take intoaccount the un-stretched twine.

In each triangular element the elongation of twinesof a same direction is constant. Thus, mesh sides alongu twine have the same length (|u|) and all mesh sidesalong v twine have the same length (|v|). The tensionis Tu (N) along the u direction and Tv along the vdirection:

(1)

(2)

Where E is the twine modulus of elasticity (Pa), A isthe twine section (m2), n0 is the un-stretched length ofmesh side (m) and |u| (|v|) is the stretched length ofmesh side along u (v) direction (m).

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Figure 1. The net is split into contiguous triangles. Thetwines are kept parallel in each triangular element.

Figure 2. Triangular element for nets with two directionsof twines (u, v). The vertices are 1, 2 and 3. The co-ordinatesin number of meshes are marked.

In order to calculate the tensions, the stretched lengthof mesh sides must be calculated. These two lengthsare calculated with the following equations (Figure 2):

(3)

(4)

Where U1, U2, U3, V1, V2 and V3 are the mesh numberof the three nodes (as marked on Figure 2), S12 (S13)is the vector from node 1 to node 2 (1 to 3) and u & vare the mesh side vector of u and v twines.

With these two previous equations the mesh sidevectors can be deduced:

(5)

(6)

Where n is the number of mesh sides in the trian-gular element:

(7)

The forces on each vertex are calculated with theprinciple of virtual work. For example, fx1 the X com-ponent of the force on vertex 1 is assessed by a dis-placement (dx1) along X axis of the vertex 1. Thisdisplacement leads to an external work for the trian-gular element:

(8)

This displacement (dx1) induces a variation of twineslength (d|u| & d|v|) and consequently an internal workfor the triangular element:

(9)

Due to the principle of virtual work the internalwork must be equal to the external one. Which gives:

(10)

That gives the forces due to twines tension on the 3vertices:

(11)

(12)

(13)

4 SUM OF THE FORCES

Up to now, the forces applied to each triangular ele-ment are calculated. These forces have been spread oneach node (vertices of each triangle and extremities ofbars for cables and chains). The force components ofall triangles and bars are added to get the whole forcesapplied to each node. The complete loading is the sumof forces which depend on tension, drag, buoyancy,weight, twines contact and catch:

(14)

Where F is the total loading on net and bars, Fe isdue to the twine and cable tension (described previ-ously), Fd is due to the drag force on twines and bars,Ff is due to the buoyancy of net and bars, Fw is due tothe weight of net and bars, Ft is due to twines contact,Fc is due to the catch, Fo is due to the mesh openingstiffness, and Fb is due to the bending stiffness (Ft, Foand Fb have been implemented in the FEM model onlyfor diamond and square mesh nets).

5 METHOD OF NUMERICAL CALCULATION

Forces (F) are calculated on each node. These forcesdepend on the node positions (X). Thus the relationF(X) is established. Obviously, in most of the net shapecalculations, the initial position (Xinitial) of the nodesis not at equilibrium so F(Xinitial) � 0. To achieve theequilibrium position (Xfinal), where F(Xfinal) � 0,the Newton-Raphson iterative method is used:

(15)

Where Xk�1 is the nodes position vector at itera-tion k � 1 (m), F(Xk) is the force vector on nodes atiteration k (N), and F�(Xk) is the matrix of derivativeof force (or stiffness) on vertices at iteration k (N/m).

A large displacement of the structure (net and cable)can be obtained by this usual method (Desai and all1972, Zienkiewicz and all 1989, Muttin 1991). It con-sists in applying total force (F(X)) and stiffness (F�(X))to revise the co-ordinates (X) of nodal points. At everystage total force and stiffness depend on the new posi-tion of nodes. The process is repeated until the totalforce is very small.

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The numerical method used here is currently usedin finite element method.

6 VALIDATION

6.1 Hydrostatic pressure

Comparisons are carried out between cod-end shapemeasurement under well-known condition and thecorresponding FEM model. O’Neill et al. (1997) des-cribed a test, which consists of a cod-end hanging undergravity. The catch force (Fc in equ. 14) is due to thepressure of the catch on the net. The pressure imple-mented in the FEM model is:

(16)

Where P is the pressure of the catch on the net (Pa),� is the mass density of the catch (Kg/m3), g is thegravity (9.81 m/s), and h is the height of the catch (m).

The parameters of the tested case are:Mesh size � 37.2 mm,Number of meshes around � 50,Number of meshes long � 50,Volume of catch � 0.0265 m3,Mass density of the catch � 1000 Kg/m3,Radius of the top ring � 25 cm.

In the FEM model the cod-end is made of 742 nodes,1360 triangular element, 1 rope to close the net and2 planes of symmetry.

Figure 3 shows that the FEM model gives a satisfac-tory description of the cod-end. This description isabout the formulation of forces due to the twine tensionand to the hydrostatic pressure (Fe and Fc in equ. 14).

6.2 Net tightened by its weight

The FEM model described in this paper is verified bycomparing the results with those obtained from a ref-erence model.

The net panel calculated by the FEM model and bythe reference model is square and is constituted by40 � 40 meshes and 3281 knots. The twine stiffnessis 10 000 N, its diameter is 0.01 m and the mesh size is1.2 m. The length of the top boundary is 32 m and thedensity of the net material is 2 000 Kg/m3. The net isheld by its top boundary and it is tightened by its ownweight.

The FEM model uses 1050 triangular elements and512 nodes with one symmetry plane. The criterion ofconvergence is that the mean of residual of force oneach node is less than 0.01 N. The calculated shape ison the middle and bottom of the Figure 4.

The reference model uses numerical bar for eachmesh side, as described by Ferro (1988). The reference

model uses 3136 bars and 1625 nodes with one sym-metry plane (Figure 4, top). The criterion of conver-gence is the same as previously.

The calculated shapes obtained by the FEM andthe reference models are similar.

6.3 Pressure of the caught fish

Results carried out by the FEM model are comparedwith flume tank tests. The flume tank tests were con-ducted on cod-end partly filled with bags of water(Premecs-I 2000). The implemented pressure used inthe catch force (Fc in equ. 14) is:

(17)

Where P is the pressure of catch on the netting (Pa),� is the mass density of water (kg/m3), Cd is the dragcoefficient, and V is the flow velocity (m/s).

For this purpose the distance between the front ofthe catch and the extremity of the cod-end was intro-duced in the FEM model for each test. This distancehas been measured during flume tank tests.

Figure 5 shows the results of the model (net) and ofthe flume tank tests (cross). This comparison showsthat the numerical model provides a good descriptionof the cod-end.

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Figure 3. Cod-end hanging with 26.5 kg of water. Compari-son between tested cod – end (left) and the FEM model (right).

7 APPLICATION

7.1 Trawl

The Figure 6 represents the calculated shape of a trawlfrom the doors to the cod-end. The trawl is made of net-ting panels of diamond meshes and hexagonal meshes(Van Marlen 1980).

The trawl consists of 4 panels of hexagonal mesh netin the front part, 36 panels of diamond mesh net and36 cables. More than 1 000 000 meshes and 2 000 000

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Figure 4. Comparison between a reference model (top)and the modeling described here (twines on the middle andtriangular elements on the bottom): shapes are very similar.

Figure 5. Comparison between the flume tank test (cross)and the numerical results of the FEM model (net). Lowsolidity cod-end with a 300 Kg catch on the top. PA cod-endwith a 40 Kg catch on the bottom.

Figure 6. Rear and side views of the calculated trawl.Cables are on the right, hexagonal meshes in the centre, dia-mond meshes on the left (only few twines are drawn) and the10 m3 catch on the left.

knots are used in the netting. The shape is calculatedfor a trawler speed of 2.08 m/s.

The structure is constituted by 1129 nodes and 91bars for modeling the cables, 1954 triangular elementsfor the diamond mesh, and 81 triangular elements forthe hexagonal mesh. The trawl holds 10 m3 of fish catch.The structure has one vertical plane of symmetry.

7.2 Bending stiffness

The FEM model is used to assess the bending stiff-ness (EI) of a netting.

The panel tested has only 9 meshes. The mesh sideis 0.069 m long. The twine diameter is 4 mm. The meshopening stiffness is 0.009 N.m/Rad. The neutral angle

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Figure 7. On the top, photo of a netting bent on its ownweight. Calculation of the bent panel: the twines are drawnon the middle and the triangular elements on the bottom.

Figure 8. Calculation of a cod-end with a grid: completeview on the bottom, few netting panels have been hidden onthe top and on the middle. The bold lines are the frame of thegrid.

Figure 9. Design plan of the circular cage. The bottom net-ting panel is on the middle surrounded by side panels.

between twines is 0.62 Rad (36°). These two parame-ters are used by Fo in equ. 14. The panel is hold partlyhorizontally, the other part is bent under its own weight(Figure 7).

The panel in the FEM model is made of 273 nodesand 491 triangular elements. The criterion of conver-gence is when the mean force on nodes is less than1.10�6N. The EI coefficient is adjusted to find thesame deformation as during the test. That gives450.10�6Nm2. This value is close to the one assessedby Sala (2004) who found values between 470.10�6

and 690.10�6Nm2.

7.3 Grid in a cod-end

The cod-end is made of 92 meshes around and 97meshes along. The mesh side is 72.5 mm. The grid is1.75 m by 1.23 m. The rectangular frame and the driv-ing netting panel behind the grid is visible on the topof the Figure 8. This panel drives out the small fish.Its mesh side is 30 mm. The towing speed is 1.5 m/sand the catch is 2 m3.

The cod-end in the FEM model is made of 1235nodes, 77 bars for the grid and cables, and 2442 trian-gular elements for the netting.

The centre view of Figure 8 shows the gap betweenthe grid and the netting which allow the larger fish topass through.

7.4 Circular cage

The cage is made of 12 netting panels around andone panel for the bottom (Figure 9). The diameter ofthe cage is 20 m, there is 1 200 meshes around and300 meshes high. The mesh side is 0.035 m long and1.2 mm thick. 1 200 Kg of dead weights are spread atthe bottom of the cage. The mooring is made of 3 chainsof 150 m long of 10 Kg/m. The 3 buoys are 3 m highand 2.3 m3 of volume. The current is 0.5 m/s.

The cage in the FEM model is made of 165 barsand 1400 triangular elements. The calculation of thecircular cage is shown on Figure 10.

8 CONCLUSION

For the calculation of the shape of flexible structuresmade of netting, cables and bars, a numerical modelbased on the finite element method has been developed.This FEM model is a 3 dimensional one, based on asurface triangular element that represents the behaviorof elastic net. These triangular elements have no con-straint relatively to the netting: the sides are not nec-essary parallel to the actual twines or to the diagonalof meshes. That leads the user free to model the struc-ture. The FEM method gives pretty good comparisonswith tests.

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Figure 10. Two zoom views of a circular cage moored by 3chains and 3 buoys.

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