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Examples Cavok V3 R9 Part 1 : RefEle construction Part 2 : Modeling Part 3 : Mesh import O. Tanneau & P. Lamary 2006 Copyright © Volage Limited 2001 - 2006

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Part 1 : RefEle Construction Cavok includes a constructor of reference element Main Entries : Topology Formal definitions of : U : DOF N : shape functions L : gradients and combinations of DOF D : matrix of physical coefficients

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General examples RefEleTest2.py : Q4 for 2D Poisson equation RefEleTest3.py : Q4 for 2D Piezzo-electricity RefEleTest4.py : Q8 and H8 for Poisson equation (2D and 3D respectively) RefEleTest5.py : H8 for 3D Acoustic RefEleTransfo_FM2UY.py and RefEleTransfo_FM2UYCT.py : introduction of rotations in 2D plates elements Bimorph. Beam – harmonic response – Voltage field and shape. Acoustic. Cavity – pressure field – harmonic response

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Porous Materials PoroGen2D.pyPoroGen3D.py PoroWeakVariational2D.pyPoroWeakVariational3D.py T3RefEle.py T6RefEle.py Q4RefEle.py Q8RefEle.py T4RefEle.py T10RefEle.py H8RefEle.py P6RefEle.py Main Script File Variational Form Definitions Topology Definitions Construction of libraries of various u-p Biot-Allard elements Poro2DSet.cao Poro3DSet.caoCavok Library File

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3D Plates RefElePlaq3DMidlin_harm_rot.py RefElePlaq3DMidlin_harm_rot_Q8.py Reissner-Mindlin formulation Selective integration (Batoz&Dhatt, Tome2, pp 303) Q4 or Q8 Generalized rotations, third rotation fictious Harmonic response of an assembly of 2 thin plates

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Others RefEleAcou3DH8.py RefEleCoupUPS_Z.py StrucRaid.py T Beam harmonic response with a plate model Torsional load H8 Acoustics element and a corresponding fluid-structure coupling element See section models for examples of use Example of use of plates elements From *.cao library

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Part 2 : mesh importation 3 mesh strategies are proposed : 1)Internal use of structured mesh cavok functions (see examples in others sections ) 2)Gmsh tool : usefull for unstructured mesh 2D and 3D (triangles and tetrahedrons) 3)From Ansys importation

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From GMSH gmshtest2.pygmshtest3.py gmshtest4.py Tube_kundt_gmsh.py 2D Poisson problem with an in-plane triangular mesh 3D Poisson problem with a tetrahedron mesh 2D Poisson problem over a 3D triangular mesh 3D Poroelastic primatic and hexahedric elements Frequency analysis

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From ANSYS Car_* 1 : mesh importation and saving in a *.cao (cavok) file 2 : analysis definition (CL, etc …) 3 : calculus and post-processing Conceptual Car-model inluding air (green) and porous (red) domains Pressure modulus at 1000 Hz

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Part 3 : Modeling 4 examples : - 3D Plate BenchMark - Active Control of a double-wall (with porous lining) coupled to an acoustical cavity - Use of the external Acoustical Boundary Element solver - Simple Box with one elastic side and a porous layer

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3D Plate BenchMark 2 plates : 0.5x0.5x0.002 Steel Free conditions, load on a corner in verical direction CasimirTest.py Comparison with ANSYS (FE) and ETAPE (continuous elements)

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Active Control of a double-wall (with porous lining) coupled to an acoustical cavity Script chain programming in Control_Flat_Panel_Cavity.py : Loop : For all chosen frequencies : - Calculus without control - Introduce transfer function between sensor and actuator - Calculus with control activated Actuator Sensor

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Use of the external Acoustical Boundary Element solver This examples illustrate how to create BEM analysis from Cavok A transfer between Cavok FE results and BEM boundary conditions is also performed BEM_Cyl_1.py Radiation from a cylindrical with Un = 1 BEM_Plaq_1.py Chain FEM for the plate and BEM for radiation

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Simple Box with one elastic side and a porous layer ShoesBox1.py Plate 0.5x0.4m2 + coupling elements Air 0.3 m Porous 0.1 m Response at 500 Hz Real displacement for plate Modulus of pressure in Air and Porous

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Finite Element Method in Geotechnical Engineering

Finite Element Method in Geotechnical Engineering

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