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

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

3. 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

4. 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

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

6. 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.

7. 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.

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

9. 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.

10. 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))

11. 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

12. Element distortion
Aspect ratio distortion
Rule of thumb:

13. Element distortion
Angular distortion

14. Element distortion
Curvature distortion

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

16. Element distortion
Volumetric distortion (Cont’d)

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

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

19. Different order of elements
Crack like behaviour – incorrect results

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

21. Straddling elements
Avoid straddling of elements in mesh

22. 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

23. Mirror symmetry
Symmetry about a particular plane

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

25. Mirror symmetry
Symmetric loading
Deflection = Free
Rotation = 0

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

27. 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

28. 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

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

30. Mirror symmetry

31. Mirror symmetry

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

33. 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

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

35. Repetitive symmetry
uAx = uBx

36. 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.

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

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

39. Creation of MPC equations for offsets

40. 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

41. MODELLING OF SUPPORTS

42. MODELLING OF SUPPORTS
(Prop support of beam)

43. MODELLING OF JOINTS
Perfect connection ensured here

44. 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)

45. MODELLING OF JOINTS
Using MPC equations

46. MODELLING OF JOINTS
Similar for plate connected to 3D solid

47. 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

48. 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

49. 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

50. 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

51. 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)

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

53. 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

54. 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

55. 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

56. 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

57. 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.