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

FFinite Element Methodinite Element Method

MODELLING TECHNIQUES

for readers of all backgroundsfor readers of all backgrounds

G. R. Liu and S. S. Quek

CHAPTER 11:

2Finite Element Method by G. R. Liu and S. S. Quek

CONTENTSCONTENTS

INTRODUCTIONCPU TIME ESTIMATIONGEOMETRY MODELLINGMESHING

– Mesh density– Element distortion

MESH COMPATIBILITY– Different order of elements– Straddling elements

CONTENTSCONTENTS

USE OF SYMMETRY– Mirror symmetry– Axial symmetry– Cyclic symmetry– Repetitive symmetry

MODELLING OF OFFSETS– Creation of MPC equations for offsets

MODELLING OF SUPPORTSMODELLING OF JOINTS

CONTENTSCONTENTS

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

INTRODUCTIONINTRODUCTION

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

0% 50% 100%

EFFORT

ACCURACY

(ANALYSIS)

(RESULTS)

INTRODUCTIONINTRODUCTION

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 ESTIMATIONCPU TIME ESTIMATION

CPU doft n ( ranges from 2 – 3)

b 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 MODELLINGGEOMETRY MODELLING Reduction of a complex geometry to a manageable one. 3D? 2D? 1D? Combination?

Shell Neutral surface

Neutral surface

Beam member

Bulky solids 3-D solid element mesh

2-D shell element mesh

1-D beam element mesh

Centroid

(Using 2D or 1D makes meshing much easier)

GEOMETRY MODELLINGGEOMETRY 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.

MESHINGMESHING

To minimize the number of DOFs, have fine mesh at important areas.

In FE packages, mesh density can be controlled by mesh seeds.

Mesh densityMesh density

(Image courtesy of Institute of High Performance Computing and Sunstar Logistics(s) Pte Ltd (s))

Element distortionElement 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

Aspect ratio distortion

3 Stress analysis

10 Displacement analysis

Rule of thumb:

Angular distortion

skew Taper b a

Curvature distortion

Volumetric distortion

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

Volumetric distortion (Cont’d)

Mid-node position distortion

>b/4 b

Shifting of nodes beyond limits can result in singular stress field (see crack tip elements)

MESH COMPATIBILITYMESH COMPATIBILITY

Requirement of Hamilton’s principle – admissible displacement

The displacement field is continuous along all the edges between elements

Different order of elementsDifferent order of elements

Linear

2 Quad

Linear

Crack like behaviour – incorrect results

Different order of elementsDifferent order of elements Solution:

– Use same type of elements throughout

– Use transition elements

– Use MPC equations

Quad Quad

Linear

Quad Linear Transition

Transition Element

Vary quadratically along this edge Vary linearly along this edge

Straddling elementsStraddling elements

Avoid straddling of elements in mesh

USE OF SYMMETRYUSE OF SYMMETRY

Different types of symmetry:

Mirror symmetryAxial symmetry

Cyclic symmetry Repetitive symmetry

Use of symmetry reduces number of DOFs and hence computational time. Also reduces numerical error.

Mirror symmetryMirror symmetry

Symmetry about a particular plane

Planes of symmetry

Modelling of quarter model is sufficient

Consider a 2D symmetric solid:

u1x = 0

u2x = 0

u3x = 0

Single point constraints (SPC)

Deflection = FreeRotation = 0

Symmetric loading

Anti-symmetric loading y

Deflection = 0Rotation = 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

Symmetric

•No translational displacement normal to symmetry plane

•No rotational components w.r.t. axis parallel to symmetry plane

Mirror symmetryMirror symmetryAnti-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 Fix Free Free Free Fix

yz Free Fix Fix Fix Free Free

zx Fix Free Fix Free Fix Free

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

Anti-Symmetric loading

Symmetric loading

Asymmetric loading

Full frame structure

P/2 + =

Anti-sym.

All nodes on this line fixed against vertical displacement.

All nodes on this line fixed against the horizontal displacement and rotation.

Properties are halved for this member

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

motion symmetric about this node

motion antisymmetric about this node

Rotation dof = 0 at this node

translational dof v = 0 at this node

Axial symmetryAxial symmetry Use of 1D or 2D axisymmetric elements

– Formulation similar to 1D and 2D elements except the use of polar coordinates

z w2 w1

w = W sin

Cylindrical shell using 1D axisymmetric elements

3D structure using 2D axisymmetric elements

Cyclic symmetryCyclic symmetry

Representative cell

Side B

F Side A

uAn = uBn

uAt = uBt

Multipoint constraints (MPC)

Repetitive symmetryRepetitive symmetry

Representative cell

uAx = uBx

MODELLING OF OFFSETSMODELLING OF OFFSETS Offset Length of beam l

Offset

Length of beam l

Corner Nodes

Joint point

, offset can be safely ignored l

100 5 , offset needs to be modelled

5, ordinary beam, plate and shell elements should not be used.

Use 2D or 3D solid elements.

Guidelines:

MODELLING OF OFFSETSMODELLING OF OFFSETS

Three methods:– Very stiff element– Rigid element– MPC equations

Very stiff element/ Rigid element

Corner Nodes

Rigid body connecting two corner nodes

Corner Nodes

Creation of MPC equations for offsetsCreation of MPC equations for offsets

1 1 3d q q

2 2d q

3 3d q

4 1d q

5 2 3d q q

6 3d q1 3 4

Eliminate q1, q2, q3

DOFs at all nodes to be DOFs of the Equation of MPC connected by the rigid body rigid bodyN N N

Creation of MPC equations for offsetsCreation of MPC equations for offsets

Rigid body

Node on neutral-surface of the plate

Node of the beam element

Neutral surface of the plate

Creation of MPC equations for offfsetsCreation of MPC equations for offfsets

d6 d9 d8 d7

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 SUPPORTSMODELLING OF SUPPORTS

Beam with “built-in end”

a) Full constraint only in the horizontal direction

c) Fully clamped support

b) Support provides full constraint only on the lower surface

MODELLING OF SUPPORTSMODELLING OF SUPPORTS

(Prop support of beam)

MODELLING OF JOINTSMODELLING OF JOINTS

turbine blade

turbine disc nodes at interface

(a) (b)

Perfect connection ensured here

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)

Using MPC equations

2a disc model

blade model

1 5d d

2 6 7d d ad

3 5d d

4 6 7d d ad

Mesh for plate

Mesh for Solid

Similar for plate connected to 3D solid

OTHER APPLICATIONS OF OTHER APPLICATIONS OF MPC EQUATIONSMPC EQUATIONS

Modelling of symmetric boundary Modelling of symmetric boundary conditionsconditions

Axis of symmetry

Axis of symmetry x

dn = 0

ui cos + vi sin=0

or ui+vi tan =0

for i=1, 2, 3

Enforcement of mesh compatibilityEnforcement of mesh compatibility

Quad Linear

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

Use lower order shape function to interpolate

Quad Quad

d5 d10

Use shape function of longer element to interpolate

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

Substituting the values of for the two additional nodes

d2 = 0.251.5 d1 + 1.50.5 d3 0.250.5 d5

d4 = 0.250.5 d1 + 0.51.5 d3

+ 0.251.5 d5

Quad Quad

d5 d10

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 Modelling of constraints by rigid body attachmentattachment

d1 d2 d3 d4

Rigid slab

d1 = q1

d2 = q1+q2 l1

d3=q1+q2 l2

d4=q1+q2 l3

(l2 /l1-1) d1 - ( l2 /l1) d2 + d3 = 0

(l3 /l1-1) d1 - ( l3 /l1) d2 + d4 = 0

Eliminate q1 and q2

(DOF in x direction not considered)

IMPLEMENTATION OF MPC IMPLEMENTATION OF MPC EQUATIONSEQUATIONS

CD Q 0 (Matrix form of MPC equations)

(Global system equation)

Constant matrices

Lagrange multiplier methodLagrange multiplier method

{ } 0T CD Q

2D KD D F CD QT T T { }

D FK C

λ QC 0

(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 methodLagrange 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 termsEfficiency of solving the system equations is

Penalty methodPenalty method

t CD Q (Constrain equations)

2D KD D F t tT T T

=1 2 ... m is a diagonal

matrix of ‘penalty numbers’

[ ]T T K C C D F C Q

Stationary condition of the modified functional requires the derivatives of p with respect to the Di to vanish

Penalty matrix

[Zienkiewicz et al., 2000] :

= constant (1/h)p+1

Characteristic size of element

P is the order of element used

4 61.0 10 max (diagonal elements in the stiffness matrix)

5 81.0 10 or

Young’s modulus

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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