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**Finite Element Method CHAPTER 11: MODELLING TECHNIQUES**

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

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**CONTENTS INTRODUCTION CPU TIME ESTIMATION GEOMETRY MODELLING MESHING**

Mesh density Element distortion MESH COMPATIBILITY Different order of elements Straddling elements

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**CONTENTS USE OF SYMMETRY MODELLING OF OFFSETS MODELLING OF SUPPORTS**

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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**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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**INTRODUCTION Ensure reliability and accuracy of results.**

Improve efficiency and accuracy.

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**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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**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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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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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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MESHING Mesh density To minimize the number of DOFs, have fine mesh at important areas. In FE packages, mesh density can be controlled by mesh seeds. (Image courtesy of Institute of High Performance Computing and Sunstar Logistics(s) Pte Ltd (s))

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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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Element distortion Aspect ratio distortion Rule of thumb:

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Element distortion Angular distortion

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Element distortion Curvature distortion

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**Element distortion Volumetric distortion**

Area outside distorted element maps into an internal area – negative volume integration

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Element distortion Volumetric distortion (Cont’d)

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**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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MESH COMPATIBILITY Requirement of Hamilton’s principle – admissible displacement The displacement field is continuous along all the edges between elements

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**Different order of elements**

Crack like behaviour – incorrect results

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**Different order of elements**

Solution: Use same type of elements throughout Use transition elements Use MPC equations

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Straddling elements Avoid straddling of elements in mesh

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**USE OF SYMMETRY Different types of symmetry:**

Use of symmetry reduces number of DOFs and hence computational time. Also reduces numerical error. Mirror symmetry Axial symmetry Cyclic symmetry Repetitive symmetry

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Mirror symmetry Symmetry about a particular plane

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**Mirror symmetry Consider a 2D symmetric solid: u1x = 0 u2x = 0 u3x = 0**

Single point constraints (SPC)

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Mirror symmetry Symmetric loading Deflection = Free Rotation = 0

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Mirror symmetry Anti-symmetric loading Deflection = 0 Rotation = Free

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**Mirror symmetry Symmetric**

No translational displacement normal to symmetry plane No rotational components w.r.t. axis parallel to symmetry plane Plane of symmetry u v w x y z xy Free Fix yz zx

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**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 u v w x y z xy Fix Free yz zx

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Mirror symmetry Any load can be decomposed to a symmetric and an anti-symmetric load

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Mirror symmetry

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Mirror symmetry

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Mirror symmetry Dynamic problems (e.g. two half models to get full set of eigenmodes in eigenvalue analysis)

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**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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Cyclic symmetry uAn = uBn uAt = uBt Multipoint constraints (MPC)

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Repetitive symmetry uAx = uBx

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**MODELLING OF OFFSETS Guidelines: , 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.

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**MODELLING OF OFFSETS Three methods: Very stiff element Rigid element**

MPC equations

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**Creation of MPC equations for offsets**

Eliminate q1, q2, q3

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**Creation of MPC equations for offsets**

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**Creation of MPC equations for offfsets**

d6 = d1 + d5 or d1 + d5 - d6 = 0 d7 = d2 - d4 or d2 - d4 - d7 = 0 d8 = d or d3 - d8 = 0 d9 = d or d5 - d9 = 0

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MODELLING OF SUPPORTS

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MODELLING OF SUPPORTS (Prop support of beam)

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MODELLING OF JOINTS Perfect connection ensured here

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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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MODELLING OF JOINTS Using MPC equations

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MODELLING OF JOINTS Similar for plate connected to 3D solid

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**OTHER APPLICATIONS OF MPC EQUATIONS**

Modelling of symmetric boundary conditions dn = 0 ui cos + vi sin=0 or ui+vi tan =0 for i=1, 2, 3

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**Enforcement of mesh compatibility**

Use lower order shape function to interpolate dx = 0.5(1-) d (1+) d3 dy = 0.5(1-) d (1+) d6 Substitute value of at node 3 0.5 d1 - d d3 =0 0.5 d4 - d d6 =0

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**Enforcement of mesh compatibility**

Use shape function of longer element to interpolate dx = -0.5 (1-) d1 + (1+)(1-) d (1+) d5 Substituting the values of for the two additional nodes d2 = 0.251.5 d 0.5 d 0.5 d5 d4 = -0.250.5 d 1.5 d 1.5 d5

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**Enforcement of mesh compatibility**

In x direction, 0.375 d1 - d d d5 = 0 d d3 - d d5 = 0 In y direction, 0.375 d6- d d d10 = 0 d d8 - d d10 = 0

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**Modelling of constraints by rigid body attachment**

d1 = q1 d2 = q1+q2 l1 d3=q1+q2 l2 d4=q1+q2 l3 Eliminate q1 and q2 (l2 /l1-1) d1 - ( l2 /l1) d2 + d3 = 0 (l3 /l1-1) d1 - ( l3 /l1) d2 + d4 = 0 (DOF in x direction not considered)

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**IMPLEMENTATION OF MPC EQUATIONS**

(Global system equation) (Matrix form of MPC equations) Constant matrices

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**Lagrange multiplier method**

(Lagrange multipliers) Multiplied to MPC equations Added to functional The stationary condition requires the derivatives of p with respect to the Di and i to vanish. Matrix equation is solved

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**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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**Penalty method (Constrain equations)**

=1 m is a diagonal matrix of ‘penalty numbers’ Stationary condition of the modified functional requires the derivatives of p with respect to the Di to vanish Penalty matrix

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**Penalty method [Zienkiewicz et al., 2000] : = constant (1/h)p+1**

P is the order of element used Characteristic size of element max (diagonal elements in the stiffness matrix) or Young’s modulus

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**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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