Download presentation

Published byCarlos Ramos Modified over 4 years ago

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 = 0.251.5 d 0.5 d 0.5 d5 d4 = -0.250.5 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.

Similar presentations

OK

Lial/Hungerford/Holcomb/Mullins: Mathematics with Applications 11e Finite Mathematics with Applications 11e Copyright ©2015 Pearson Education, Inc. All.

Lial/Hungerford/Holcomb/Mullins: Mathematics with Applications 11e Finite Mathematics with Applications 11e Copyright ©2015 Pearson Education, Inc. All.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on digital advertising Ppt on history of indian mathematicians Free ppt on physical features of india Ppt on understanding secularism in india Ppt on electricity for class 10 download Ppt on viruses and bacteria worksheet Ppt on andhra pradesh Ppt on disk formatting pdf Ppt on credit policy in banks Ppt on earthquake resistant design of structures