Finite Element Method for netting Daniel.Priour@ifremer.fr IFREMER - PowerPoint PPT Presentation

Fish cages and fishing gears Newton Raphson method Finite element method Finite element for netting Finite Element Method for netting Daniel.Priour@ifremer.fr IFREMER November 4, 2010 Daniel.Priour@ifremer.fr Finite Element Method for netting

Fish cages and fishing gears Newton Raphson method Fish cages Finite element method Fishing gears Finite element for netting Fish cages and fishing gears Fish cages Fishing gears Newton Raphson method Spring with 1DOF Spring with 2DOF Structure with several DOF Newmark an alternative to Newton Raphson method Finite element method A simple example Finite element for netting Triangular element Equations Tension in twines Drag on twines Twines flexion Catch pressure Daniel.Priour@ifremer.fr Finite Element Method for netting

Fish cages and fishing gears Newton Raphson method Fish cages Finite element method Fishing gears Finite element for netting Fish cages ◮ Salmon in Norway (1Mtons/y) ◮ Sea-bass in Greece ◮ Tuna in Japan (Ktons/y) Daniel.Priour@ifremer.fr Finite Element Method for netting

Fish cages and fishing gears Newton Raphson method Fish cages Finite element method Fishing gears Finite element for netting Fish cages ◮ What are the tension in cables? ◮ What is the volume inside the cage? Daniel.Priour@ifremer.fr Finite Element Method for netting

Fish cages and fishing gears Newton Raphson method Fish cages Finite element method Fishing gears Finite element for netting Fishing gears ◮ Bottom trawl common in Europe ◮ Purse seine ◮ Dredge ◮ Fish trap Daniel.Priour@ifremer.fr Finite Element Method for netting

Fish cages and fishing gears Newton Raphson method Fish cages Finite element method Fishing gears Finite element for netting Fishing gears ◮ What is the drag of the trawl? And the fuel consumption? ( 1 l Kg ) ◮ What are the mesh opening in the cod-end? And the fish escapment? Daniel.Priour@ifremer.fr Finite Element Method for netting

Fish cages and fishing gears Spring with 1DOF Newton Raphson method Spring with 2DOF Finite element method Structure with several DOF Finite element for netting Newmark an alternative to Newton Raphson method Newton Raphson method: Spring with 1DOF Question: What is its length? ◮ F ( x ) = Ax x − l 0 − Mg l 0 ◮ Stiffness not constant ◮ What is the length ( x ) at equilibrium? ◮ Equilibrium: F ( x ) = 0 Daniel.Priour@ifremer.fr Finite Element Method for netting

Fish cages and fishing gears Spring with 1DOF Newton Raphson method Spring with 2DOF Finite element method Structure with several DOF Finite element for netting Newmark an alternative to Newton Raphson method Spring with 1DOF ◮ Arbitrary length x = x 0 ◮ F ( x ) = Ax x − l 0 − Mg l 0 ◮ F ′ ( x ) = A l 0 (2 x − l 0 ) ◮ Newton Raphson leads to: F ( x k ) ◮ x k +1 = x k + − F ′ ( x k ) ◮ x 0 − > F ( x 0 ) − > F ′ ( x 0 ) − > x 1 F ( x + h ) − F ( x ) ◮ F ′ ( x ) = lim h → 0 h Daniel.Priour@ifremer.fr Finite Element Method for netting

Fish cages and fishing gears Spring with 1DOF Newton Raphson method Spring with 2DOF Finite element method Structure with several DOF Finite element for netting Newmark an alternative to Newton Raphson method Spring with 1DOF ◮ Convergence is pretty quick ◮ But sometimes F ′ ( x ) = 0 F ( x k ) ◮ x k +1 = x k + − F ′ ( x k ) ◮ Additional stiffness Daniel.Priour@ifremer.fr Finite Element Method for netting

Fish cages and fishing gears Spring with 1DOF Newton Raphson method Spring with 2DOF Finite element method Structure with several DOF Finite element for netting Newmark an alternative to Newton Raphson method Spring with 2DOF x 2 + y 2 � ◮ l = ◮ T = Al l − l 0 Stiffness not constant l 0 ◮ F x ( X ) = T x l ◮ F y ( X ) = T y l − Mg Daniel.Priour@ifremer.fr Finite Element Method for netting

Fish cages and fishing gears Spring with 1DOF Newton Raphson method Spring with 2DOF Finite element method Structure with several DOF Finite element for netting Newmark an alternative to Newton Raphson method Spring with 2DOF � x k X k = y k � F x ( X k ) F ( X k ) = F y ( X k ) � l 2 k − l 0 l k + y 2 x k y k F ′ ( X k ) = A k l 0 l k l 2 k − l 0 l k + y 2 x k y k k F ( X k ) X k +1 = X k + − F ′ ( X k ) Daniel.Priour@ifremer.fr Finite Element Method for netting

Fish cages and fishing gears Spring with 1DOF Newton Raphson method Spring with 2DOF Finite element method Structure with several DOF Finite element for netting Newmark an alternative to Newton Raphson method Spring with 2DOF � F 2 x + F 2 force norme: Residue of force norme y Daniel.Priour@ifremer.fr Finite Element Method for netting

Fish cages and fishing gears Spring with 1DOF Newton Raphson method Spring with 2DOF Finite element method Structure with several DOF Finite element for netting Newmark an alternative to Newton Raphson method Structure with several DOF   F 1 k ( X k ) x 1 k         F 2 k ( X k ) x 2 k   X k = F ( X k ) = . .        x nk  F nk ( X k )    − dF 1 − dF 1 − dF 1 .  dx 1 dx 2 dx n   − dF 2 − dF 2 − dF 2  .  F ( X k ) − F ′ ( X k ) = dx 1 dx 2 dx n X k +1 = X k + − F ′ ( X k ) . . . .    − dF n − dF n − dF n  .  dx 1 dx 2 dx n Daniel.Priour@ifremer.fr Finite Element Method for netting

Fish cages and fishing gears Spring with 1DOF Newton Raphson method Spring with 2DOF Finite element method Structure with several DOF Finite element for netting Newmark an alternative to Newton Raphson method Structure with several DOF: Risk of singularity  − dF 1 − dF 1 − dF 1 dx 1 + K .  dx 2 dx n   − dF 2 − dF 2 − dF 2  dx 2 + K .  − F ′ ( X k ) = dx 1 dx n . . . + K .    − dF n − dF n − dF n  . dx n + K  dx 1 dx 2 The additional stiffness K is tuned Daniel.Priour@ifremer.fr Finite Element Method for netting

Fish cages and fishing gears Spring with 1DOF Newton Raphson method Spring with 2DOF Finite element method Structure with several DOF Finite element for netting Newmark an alternative to Newton Raphson method Spring with 1DOF: Newmark vs Newton ◮ F ( x ) = Ax x − l 0 − Mg l 0 ◮ M γ = F ( x ) ◮ ∆ t is choosen ◮ v k +1 = v k + γ k ∆ t ◮ x k +1 = x k + v k ∆ t + γ k ∆ t 2 2 Daniel.Priour@ifremer.fr Finite Element Method for netting

Fish cages and fishing gears Newton Raphson method A simple example Finite element method Finite element for netting Circle perimeter ◮ The perimeter is 2 π R ◮ The perimeter approximated by n times cord length ◮ Cord length 2 Rsin ( α 2 ) ◮ or 2 Rtan ( α 2 ) (outside) Daniel.Priour@ifremer.fr Finite Element Method for netting

Fish cages and fishing gears Newton Raphson method A simple example Finite element method Finite element for netting Circle perimeter Finite Element basic idea: ◮ Divided in finite elements ◮ Approximation by element ◮ Rebuilding all the elements Perimeter vs elements number Daniel.Priour@ifremer.fr Finite Element Method for netting

Triangular element Fish cages and fishing gears Equations Newton Raphson method Tension in twines Finite element method Drag on twines Finite element for netting Twines flexion Catch pressure Finite element for netting: How? Netting behaviour: ◮ Surface ◮ Elastic Model test in flume tank Daniel.Priour@ifremer.fr Finite Element Method for netting

Triangular element Fish cages and fishing gears Equations Newton Raphson method Tension in twines Finite element method Drag on twines Finite element for netting Twines flexion Catch pressure Finite element for netting: Triangle Triangular element for netting: ◮ twines parallel ◮ linear elasticity Triangular elements model all the netting Netting decomposed in triangular elements Daniel.Priour@ifremer.fr Finite Element Method for netting

Triangular element Fish cages and fishing gears Equations Newton Raphson method Tension in twines Finite element method Drag on twines Finite element for netting Twines flexion Catch pressure Finite element for netting: calculation of mesh sides ( U , V ) - Nodes are fixed to the netting - Cartesian coordinates: x 1 ... z 3 - Twine coordinates: U 1 ... V 3 - Triangle side 12 is linear combinaison of mesh sides U , V 12 = ( U 2 − U 1 ) U + ( V 2 − V 1 ) V 13 = ( U 3 − U 1 ) U + ( V 3 − V 1 ) V - 2 equations, 2 unknowns ( U , V ) Daniel.Priour@ifremer.fr Finite Element Method for netting

Triangular element Fish cages and fishing gears Equations Newton Raphson method Tension in twines Finite element method Drag on twines Finite element for netting Twines flexion Catch pressure Finite element for netting: calculation of mesh sides ( U , V ) Sides vectors: 2 previous equations with 2 X 2 − X 1 unknowns ( U and V ): 12 = | Y 2 − Y 1 Z 2 − Z 1 U = V 3 − V 1 12 − V 2 − V 1 13 d d X 3 − X 1 V = U 3 − U 1 12 − U 2 − U 1 13 13 = | Y 3 − Y 1 d d Z 3 − Z 1 and d = ( U 2 − U 1 )( V 1 − V 3 ) − ( U 3 − U 1 )( V 1 − V 2 ) Daniel.Priour@ifremer.fr Finite Element Method for netting