F inite Element Method

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F inite Element Method. 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

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### Finite Element Method

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
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
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.
CPU TIME ESTIMATION
• 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.

( ranges from 2 – 3)

Bandwidth, b, affects 

- minimize bandwidth

Aim:

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
• 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

Avoid straddling of elements in mesh

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

Deflection = Free

Rotation = 0

Mirror symmetry

Deflection = 0

Rotation = Free

Plane of symmetry

u

v

w

x

y

z

xy

Free

Free

Fix

Fix

Fix

Free

yz

Fix

Free

Free

Free

Fix

Fix

zx

Free

Fix

Free

Fix

Free

Fix

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

Fix

Fix

Free

Free

Free

Fix

yz

Free

Fix

Fix

Fix

Free

Free

zx

Fix

Free

Fix

Free

Fix

Free

Mirror symmetry
• Anti-symmetric
• No translational displacement parallel to symmetry plane
• No rotational components w.r.t. axis normal to symmetry plane
Mirror symmetry
• Any load can be decomposed to a symmetric and an anti-symmetric load
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)

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 offfsets

d6 = d1 + d5 or d1 + d5-d6 = 0

d7 = d2-d4 or d2-d4-d7 = 0

d8 = d3 or d3-d8 = 0

d9 = d5 or d5-d9 = 0

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-) d1 + 0.5(1+) d3

dy= 0.5(1-) d4 + 0.5(1+) d6

Substitute value of  at node 3

0.5 d1-d2 + 0.5 d3 =0

0.5 d4-d5 + 0.5 d6 =0

Enforcement of mesh compatibility

Use shape function of longer element to interpolate

dx = -0.5 (1-) d1 + (1+)(1-) d3 + 0.5 (1+) d5

Substituting the values of  for the two additional nodes

d2 = 0.251.5 d1 + 1.50.5 d3 -0.250.5 d5

d4 = -0.250.5 d1 + 0.51.5 d3 + 0.251.5 d5

Enforcement of mesh compatibility

In x direction,

0.375 d1 -d2 + 0.75 d3 - 0.125 d5 = 0

-0.125 d1 + 0.75 d3 -d4 + 0.375 d5 = 0

In y direction,

0.375 d6-d7+0.75 d8- 0.125 d10 = 0

-0.125 d6+0.75 d8 -d9 + 0.375 d10 = 0

Modelling of constraints by rigid body attachment

d1 = q1

d2 = q1+q2l1

d3=q1+q2l2

d4=q1+q2l3

Eliminate q1and 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

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 matrixis non-positive definite due to the presence of zero diagonal terms
• Efficiency of solving the system equations is lower
Penalty method

(Constrain equations)

=12 ... 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.