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F inite Element Method

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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F inite Element Method

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  1. Finite Element Method 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 • 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 • 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:

  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. 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

  28. 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

  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 = d3 or d3-d8 = 0 d9 = d5 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-) 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

  49. 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

  50. 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

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