1 / 82

ECIV 720 A Advanced Structural Mechanics and Analysis

ECIV 720 A Advanced Structural Mechanics and Analysis. Lecture 16 & 17: Higher Order Elements (review) 3-D Volume Elements Convergence Requirements Element Quality. Higher Order Elements. Complete Polynomial. 4 Boundary Conditions for admissible displacements. Quadrilateral Elements.

henry-noel
Download Presentation

ECIV 720 A Advanced Structural Mechanics and Analysis

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 16 & 17: Higher Order Elements (review) 3-D Volume Elements Convergence Requirements Element Quality

  2. Higher Order Elements Complete Polynomial 4 Boundary Conditions for admissible displacements Quadrilateral Elements Recall the 4-node 4 generalized displacements ai

  3. Higher Order Elements Quadrilateral Elements Assume Complete Quadratic Polynomial 9 generalized displacements ai 9 BC for admissible displacements

  4. 9-node quadrilateral BT18x3 D3x3 B3x18 ke 18x18 9-nodes x 2dof/node = 18 dof

  5. 9-node element Shape Functions 4 3 Corner Nodes h 7 6 9 8 Mid-Side Nodes x 5 1 2 Middle Node Following the standard procedure the shape functions are derived as

  6. N1,2,3,4 Graphical Representation

  7. N5,6,7,8 Graphical Representation

  8. N9 Graphical Representation

  9. Polynomials & the Pascal Triangle x y 1 0 x2 xy y2 2 x3 x2y xy2 y3 3 x4 x3y y4 xy3 4 x2y2 x5 x4y xy4 y5 x2y3 5 x3y2 Pascal Triangle Degree 1 …….

  10. Polynomials & the Pascal Triangle 1 4-node Quad Q1 Q2 x y x2 xy y2 x3 x2y xy2 y3 x4 x3y y4 xy3 x2y2 x5 x4y xy4 y5 x2y3 x3y2 9-node Quad ……. To construct a complete polynomial etc

  11. Incomplete Polynomials 1 x y x2 xy y2 x3 x2y xy2 y3 x4 x3y y4 xy3 x2y2 x5 x4y xy4 y5 x2y3 x3y2 ……. 3-node triangular

  12. Incomplete Polynomials 1 x y x2 xy y2 x3 x2y xy2 y3 x4 x3y y4 xy3 x2y2 x5 x4y xy4 y5 x2y3 x3y2 …….

  13. 8-node quadrilateral h 4 3 7 x 6 8 5 1 2 Assume interpolation 8 coefficients to determine for admissible displ.

  14. 8-node quadrilateral BT16x3 D3x3 B3x16 ke 16x16 8-nodes x 2dof/node = 16 dof

  15. 8-node element Shape Functions 4 3 Corner Nodes 7 6 8 5 Mid-Side Nodes 1 2 Following the standard procedure the shape functions are derived as h x

  16. N1,2,3,4 Graphical Representation

  17. N5,6,7,8 Graphical Representation

  18. Incomplete Polynomials 1 x y x2 xy y2 x3 x2y xy2 y3 x4 x3y y4 xy3 x2y2 x5 x4y xy4 y5 x2y3 x3y2 …….

  19. 6-node Triangular 3 6 5 1 2 4 Assume interpolation 6 coefficients to determine for admissible displ.

  20. 6-node triangular 3 BT12x3 D3x3 B3x12 6 5 ke 12x12 1 2 4 6-nodes x 2dof/node = 12 dof

  21. 6-node element Shape Functions 3 Corner Nodes 6 5 Mid-Side Nodes 1 2 4 Following the standard procedure the shape functions are derived as Li:Area coordinates

  22. Other Higher Order Elements 1 h 4 3 x y x2 xy y2 x3 x2y xy2 y3 x x4 x3y y4 xy3 x2y2 1 2 x5 x4y xy4 y5 x2y3 x3y2 ……. 12-node quad

  23. Other Higher Order Elements 1 h 4 3 x y x2 xy y2 x3 x2y xy2 y3 x x4 x3y y4 xy3 x2y2 1 2 x5 x4y xy4 y5 x2y3 x3y2 x3y2 ……. 16-node quad

  24. 3-D Stress state

  25. 3-D Stress State Assumption Small Deformations

  26. Strain Displacement Relationships Material Matrix

  27. 3-D Finite Element Analysis 12 11 10 9 3 8 7 2 6 1 5 4 Solution Domain is VOLUME Simplest Element (Lowest Order) Tetrahedral Element

  28. 3-D Tetrahedral Element 3 (0,0,1) z 4 (0,0,0) h 2 (0,1,0) 1 (1,0,0) Parent (Master) x Can be thought of an extension of the 2D CST

  29. 3-D Tetrahedral 4 3 z 1 2 h x Shape Functions Volume Coordinates

  30. Geometry – Isoparametric Formulation In view of shape functions

  31. Jacobian of Transformation

  32. Strain-Displacement Matrix B is CONSTANT

  33. Stiffness Matrix Element Strain Energy

  34. Force Terms Body Forces

  35. Element Forces 4 3 1 2 Surface Traction Applied on FACE of element eg on face 123

  36. Stress Calculations Constant Stress Tensor se= DB qe Stress Invariants

  37. Stress Calculations Principal Stresses

  38. Other Low Order Elements 4 5 6 7 8 5 3 6 1 1 2 3 2 18 dof 5-hedral 24 dof 6-hedral

  39. Degenerate Elements 5 5 6 7 8 ,7 6 4 8 1 3 2 1 2 ,3 Still has 24 dof

  40. Degenerate 5,6,7,8 4 5 6 7 8 1 1 2 3 2,3 Still has 24 dof

  41. Higher Order Elements 10-node 4-hedral 4 4 9 10 9 8 1 7 3 10 8 3 5 6 7 15 6 1 2 5 14 13 Z 2 Y X

  42. 15-node 5-hedral z 15 12 4 6 L 2 10 11 5 14 10 5 13 13 15 4 11 12 L 14 14 6 3 9 1 3 13 7 8 15 2 L 2 7 1 1 8 9 3 Z Y X

  43. 15-node 5-hedral Shape Functions

  44. 20-node 6-hedral z 24 23 8 16 15 22 5 20 h 7 x 13 14 6 17 4 19 12 11 18 1 3 Z 9 10 2 Y X

  45. 20-node 6-hedral Shape Functions

  46. Convergence Considerations For monotonic convergence of solution Requirements Elements (mesh) must be compatible Elements must be complete

  47. Monotonic Convergence FEM Solution Exact Solution No of Elements For monotonic convergence the elements must be complete and the mesh must be compatible

  48. Mixed Order Elements Consider the following Mesh 4-node 8-node Incompatible Elements…

  49. Mixed Order Elements We can derive a mixed order element for grading 4-node 8-node 7-node By blending shape functions appropriately

  50. Convergence Considerations For monotonic convergence of solution Requirements Elements (mesh) must be compatible Elements must be complete

More Related