Exam3 Review

Exam3 Review

Date: April 22

Textbook: Chap 6 ~ 10

1

•

Chapter 6

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–

–

–

Outline

Jacobian of mapping (Q4)

Numerical integration

Q8, Q9

Incompatible modes, condensation

• Chapter 7

– Isoparametric triangles and tetrahedral

– Area coordinates

• Chapter 8

– Coordinate transformation (stress, stiffness)

– Transformation for BC, rigid body, different DOFs

– Singularity (fracture mechanics)

• Chapter 9

– Ill-conditioning (mathematically & physically)

– Convergence

– Mesh refinement, error estimate

• Chapter 10

– FE modeling issues

2

Incompatible Modes and Condensation

• Incompatible and nodeless modes (Q6)

4

u   Nui  (1  2 )a1  (1  2 )a2

i



x

a



y

b

i1

4

v   Ni v i  (1  2 )a3  (1  2 )a 4

i1

• Evaluate Jacobian at the center of element to represent

uniform strain

• Static condensation

– Eliminate internal nodes or nodeless DOF before assembly

 k rr

k

 cr

k rc   dr   rr 

  

k cc  dc  rc 



Boundary DOF

Internal DOF



k   k 

 k  k d





1

rr

rc

cc

cr

r

 rr   k rc k cc  rc 







condensed k 

1

condensed r 

3

Numerical integration and stress calculation

• Order of numerical integration

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–

–

–

Full integration

Reduce integration

Spurious modes

Hourglass control

• Stress calculation

– Nodal coordinates

r  3,

s  3

– Stress extrapolation (from Gauss point)

p   Nii

i  1 4

• Patch test

– Rigid-body test

– Constant strain test

4

Chap 7 Isoparametric Triangles & Tetrahedra

• Reference elements

J  x 21y 31  x 31y 21  2A

Need numerical integ.

• Area coordinates for analytical integration

A

A

A

1  1

 2  2 3  3

1  2  3  1

A

A

A

1  1  r  s 2  r 3  s

CST:

N1  1

N2  2

N3  3

• Triangular elements have different integration rules

5

Chap 8 Coordinate...