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1 Finite Element Method MODELLING TECHNIQUES for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 11:

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Finite Element Method by G. R. Liu and S. S. Quek 2 CONTENTS INTRODUCTION CPU TIME ESTIMATION GEOMETRY MODELLING MESHING – Mesh density – Element distortion MESH COMPATIBILITY – Different order of elements – Straddling elements

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Finite Element Method by G. R. Liu and S. S. Quek 3 CONTENTS USE OF SYMMETRY – Mirror symmetry – Axial symmetry – Cyclic symmetry – Repetitive symmetry MODELLING OF OFFSETS – Creation of MPC equations for offsets MODELLING OF SUPPORTS MODELLING OF JOINTS

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Finite Element Method by G. R. Liu and S. S. Quek 4 CONTENTS OTHER APPLICATIONS OF MPC EQUATIONS – Modelling of symmetric boundary conditions – Enforcement of mesh compatibility – Modelling of constraints by rigid body attachment IMPLEMENTATION OF MPC EQUATIONS – Lagrange multiplier method – Penalty method

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Finite Element Method by G. R. Liu and S. S. Quek 5 INTRODUCTION Ensure reliability and accuracy of results. Improve efficiency and accuracy.

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Finite Element Method by G. R. Liu and S. S. Quek 6 INTRODUCTION Considerations: – Computational and manpower resources that limit the scale of the FEM model. – Requirement on results that defines the purpose and hence the methods of the analysis. – Mechanical characteristics of the geometry of the problem domain that determine the types of elements to use. – Boundary conditions. – Loading and initial conditions.

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Finite Element Method by G. R. Liu and S. S. Quek 7 CPU TIME ESTIMATION ( ranges from 2 – 3) Bandwidth, b, affects - minimize bandwidth Aim: – To create a FEM model with minimum DOFs by using elements of as low dimension as possible, and – To use as coarse a mesh as possible, and use fine meshes only for important areas.

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Finite Element Method by G. R. Liu and S. S. Quek 8 GEOMETRY MODELLING Reduction of a complex geometry to a manageable one. 3D? 2D? 1D? Combination? (Using 2D or 1D makes meshing much easier)

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Finite Element Method by G. R. Liu and S. S. Quek 9 GEOMETRY MODELLING Detailed modelling of areas where critical results are expected. Use of CAD software to aid modelling. Can be imported into FE software for meshing.

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Finite Element Method by G. R. Liu and S. S. Quek 10 MESHING To minimize the number of DOFs, have fine mesh at important areas. In FE packages, mesh density can be controlled by mesh seeds. Mesh density (Image courtesy of Institute of High Performance Computing and Sunstar Logistics(s) Pte Ltd (s))

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Finite Element Method by G. R. Liu and S. S. Quek 11 Element distortion Use of distorted elements in irregular and complex geometry is common but there are some limits to the distortion. The distortions are measured against the basic shape of the element – Square Quadrilateral elements – Isosceles triangle Triangle elements – Cube Hexahedron elements – Isosceles tetrahedron Tetrahedron elements

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Finite Element Method by G. R. Liu and S. S. Quek 12 Element distortion Aspect ratio distortion Rule of thumb:

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Finite Element Method by G. R. Liu and S. S. Quek 13 Element distortion Angular distortion

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Finite Element Method by G. R. Liu and S. S. Quek 14 Element distortion Curvature distortion

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Finite Element Method by G. R. Liu and S. S. Quek 15 Element distortion Volumetric distortion Area outside distorted element maps into an internal area – negative volume integration

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Finite Element Method by G. R. Liu and S. S. Quek 16 Element distortion Volumetric distortion (Contd)

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Finite Element Method by G. R. Liu and S. S. Quek 17 Element distortion Mid-node position distortion Shifting of nodes beyond limits can result in singular stress field (see crack tip elements)

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Finite Element Method by G. R. Liu and S. S. Quek 18 MESH COMPATIBILITY Requirement of Hamiltons principle – admissible displacement The displacement field is continuous along all the edges between elements

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Finite Element Method by G. R. Liu and S. S. Quek 19 Different order of elements Crack like behaviour – incorrect results

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Finite Element Method by G. R. Liu and S. S. Quek 20 Different order of elements Solution: – Use same type of elements throughout – Use transition elements – Use MPC equations

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Finite Element Method by G. R. Liu and S. S. Quek 21 Straddling elements Avoid straddling of elements in mesh

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Finite Element Method by G. R. Liu and S. S. Quek 22 USE OF SYMMETRY Different types of symmetry: Mirror symmetry Axial symmetry Cyclic symmetry Repetitive symmetry Use of symmetry reduces number of DOFs and hence computational time. Also reduces numerical error.

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Finite Element Method by G. R. Liu and S. S. Quek 23 Mirror symmetry Symmetry about a particular plane

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Finite Element Method by G. R. Liu and S. S. Quek 24 Mirror symmetry Consider a 2D symmetric solid: u 1x = 0 u 2x = 0 u 3x = 0 Single point constraints (SPC)

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Finite Element Method by G. R. Liu and S. S. Quek 25 Mirror symmetry Deflection = Free Rotation = 0 Symmetric loading

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Finite Element Method by G. R. Liu and S. S. Quek 26 Mirror symmetry Anti-symmetric loading Deflection = 0 Rotation = Free

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Finite Element Method by G. R. Liu and S. S. Quek 27 Mirror symmetry Plane of symmetry uvw x y z xyFree Fix Free yzFixFree Fix zxFreeFixFreeFixFreeFix Symmetric No translational displacement normal to symmetry plane No rotational components w.r.t. axis parallel to symmetry plane

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Finite Element Method by G. R. Liu and S. S. Quek 28 Mirror symmetry Anti-symmetric No translational displacement parallel to symmetry plane No rotational components w.r.t. axis normal to symmetry plane Plane of symmetry uvw x y z xyFix Free Fix yzFreeFix Free zxFixFreeFixFreeFixFree

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Finite Element Method by G. R. Liu and S. S. Quek 29 Mirror symmetry Any load can be decomposed to a symmetric and an anti-symmetric load

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Finite Element Method by G. R. Liu and S. S. Quek 30 Mirror symmetry

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Finite Element Method by G. R. Liu and S. S. Quek 31 Mirror symmetry

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Finite Element Method by G. R. Liu and S. S. Quek 32 Mirror symmetry Dynamic problems (e.g. two half models to get full set of eigenmodes in eigenvalue analysis)

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Finite Element Method by G. R. Liu and S. S. Quek 33 Axial symmetry Use of 1D or 2D axisymmetric elements – Formulation similar to 1D and 2D elements except the use of polar coordinates Cylindrical shell using 1D axisymmetric elements 3D structure using 2D axisymmetric elements

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Finite Element Method by G. R. Liu and S. S. Quek 34 Cyclic symmetry u An = u Bn u At = u Bt Multipoint constraints (MPC)

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Finite Element Method by G. R. Liu and S. S. Quek 35 Repetitive symmetry u Ax = u Bx

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Finite Element Method by G. R. Liu and S. S. Quek 36 MODELLING OF OFFSETS, offset can be safely ignored, offset needs to be modelled, ordinary beam, plate and shell elements should not be used. Use 2D or 3D solid elements. Guidelines:

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Finite Element Method by G. R. Liu and S. S. Quek 37 MODELLING OF OFFSETS Three methods: – Very stiff element – Rigid element – MPC equations

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Finite Element Method by G. R. Liu and S. S. Quek 38 Creation of MPC equations for offsets Eliminate q 1, q 2, q 3

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Finite Element Method by G. R. Liu and S. S. Quek 39 Creation of MPC equations for offsets

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Finite Element Method by G. R. Liu and S. S. Quek 40 Creation of MPC equations for offfsets d 6 = d 1 + d 5 or d 1 + d 5 d 6 = 0 d 7 = d 2 d 4 or d 2 d 4 d 7 = 0 d 8 = d 3 or d 3 d 8 = 0 d 9 = d 5 or d 5 d 9 = 0

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Finite Element Method by G. R. Liu and S. S. Quek 41 MODELLING OF SUPPORTS

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Finite Element Method by G. R. Liu and S. S. Quek 42 MODELLING OF SUPPORTS (Prop support of beam)

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Finite Element Method by G. R. Liu and S. S. Quek 43 MODELLING OF JOINTS Perfect connection ensured here

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Finite Element Method by G. R. Liu and S. S. Quek 44 MODELLING OF JOINTS Mismatch between DOFs of beams and 2D solid – beam is free to rotate (rotation not transmitted to 2D solid) Perfect connection by artificially extending beam into 2D solid (Additional mass)

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Finite Element Method by G. R. Liu and S. S. Quek 45 MODELLING OF JOINTS Using MPC equations

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Finite Element Method by G. R. Liu and S. S. Quek 46 MODELLING OF JOINTS Similar for plate connected to 3D solid

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Finite Element Method by G. R. Liu and S. S. Quek 47 OTHER APPLICATIONS OF MPC EQUATIONS Modelling of symmetric boundary conditions d n = 0 u i cos + v i sin =0 or u i +v i tan =0 for i=1, 2, 3

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Finite Element Method by G. R. Liu and S. S. Quek 48 Enforcement of mesh compatibility d x = 0.5(1 ) d (1+ ) d 3 d y = 0.5(1 ) d (1+ ) d 6 Substitute value of at node d 1 d d 3 =0 0.5 d 4 d d 6 =0 Use lower order shape function to interpolate

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Finite Element Method by G. R. Liu and S. S. Quek 49 Enforcement of mesh compatibility Use shape function of longer element to interpolate d x = (1 ) d 1 + (1+ )(1 ) d (1+ ) d 5 Substituting the values of for the two additional nodes d 2 = d d d 5 d 4 = d d d 5

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Finite Element Method by G. R. Liu and S. S. Quek 50 Enforcement of mesh compatibility In x direction, d 1 d d d 5 = d d 3 d d 5 = 0 In y direction, d 6 d d d 10 = d d 8 d d 10 = 0

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Finite Element Method by G. R. Liu and S. S. Quek 51 Modelling of constraints by rigid body attachment d 1 = q 1 d 2 = q 1 +q 2 l 1 d 3 =q 1 +q 2 l 2 d 4 =q 1 +q 2 l 3 (l 2 /l 1 -1) d 1 - ( l 2 /l 1 ) d 2 + d 3 = 0 (l 3 /l 1 -1) d 1 - ( l 3 /l 1 ) d 2 + d 4 = 0 Eliminate q 1 and q 2 (DOF in x direction not considered)

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Finite Element Method by G. R. Liu and S. S. Quek 52 IMPLEMENTATION OF MPC EQUATIONS (Matrix form of MPC equations) (Global system equation) Constant matrices

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Finite Element Method by G. R. Liu and S. S. Quek 53 Lagrange multiplier method (Lagrange multipliers) Multiplied to MPC equations Added to functional The stationary condition requires the derivatives of p with respect to the D i and i to vanish. Matrix equation is solved

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Finite Element Method by G. R. Liu and S. S. Quek 54 Lagrange multiplier method Constraint equations are satisfied exactly Total number of unknowns is increased Expanded stiffness matrix is non-positive definite due to the presence of zero diagonal terms Efficiency of solving the system equations is lower

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Finite Element Method by G. R. Liu and S. S. Quek 55 Penalty method (Constrain equations) = m is a diagonal matrix of penalty numbers Stationary condition of the modified functional requires the derivatives of p with respect to the D i to vanish Penalty matrix

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Finite Element Method by G. R. Liu and S. S. Quek 56 Penalty method [Zienkiewicz et al., 2000] : = constant (1/h) p+1 Characteristic size of element P is the order of element used max (diagonal elements in the stiffness matrix) or Youngs modulus

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Finite Element Method by G. R. Liu and S. S. Quek 57 Penalty method The total number of unknowns is not changed. System equations generally behave well. The constraint equations can only be satisfied approximately. Right choice of may be ambiguous.

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