# Finite Element Method CHAPTER 11: MODELLING TECHNIQUES

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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

CONTENTS INTRODUCTION CPU TIME ESTIMATION GEOMETRY MODELLING MESHING
Mesh density Element distortion MESH COMPATIBILITY Different order of elements Straddling elements

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

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

INTRODUCTION Ensure reliability and accuracy of results.
Improve efficiency and accuracy.

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.

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.

GEOMETRY MODELLING Reduction of a complex geometry to a manageable one. 3D? 2D? 1D? Combination? (Using 2D or 1D makes meshing much easier)

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.

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))

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

Element distortion Aspect ratio distortion Rule of thumb:

Element distortion Angular distortion

Element distortion Curvature distortion

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

Element distortion Volumetric distortion (Cont’d)

Element distortion Mid-node position distortion
Shifting of nodes beyond limits can result in singular stress field (see crack tip elements)

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

Different order of elements
Crack like behaviour – incorrect results

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

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

Mirror symmetry Symmetry about a particular plane

Mirror symmetry Consider a 2D symmetric solid: u1x = 0 u2x = 0 u3x = 0
Single point constraints (SPC)

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

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

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

Mirror symmetry

Mirror symmetry

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

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

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

Repetitive symmetry uAx = uBx

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.

MODELLING OF OFFSETS Three methods: Very stiff element Rigid element
MPC equations

Creation of MPC equations for offsets
Eliminate q1, q2, q3

Creation of MPC equations for offsets

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

MODELLING OF SUPPORTS

MODELLING OF SUPPORTS (Prop support of beam)

MODELLING OF JOINTS Perfect connection ensured here

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)

MODELLING OF JOINTS Using MPC equations

MODELLING OF JOINTS Similar for plate connected to 3D solid

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

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

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 = 0.251.5 d 0.5 d 0.5 d5 d4 = -0.250.5 d 1.5 d 1.5 d5

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

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)

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

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

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

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

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

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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