1. 1 Copyright by PZ Bar-Yoseph © Finite Element Methods in Engineering-1 036015 Winter Semester Lecture 7

2. 2 FE Model for 2-D Boundary Value Problems 1. Weak formulation of the governing differential equations and boundary conditions. 2. Discretization (mesh generation) of the solution domain into a set of finite elements & data interpolation (preprocessing) 3. Derivation of the element basis functions. 4. Calculation of the element matrices. 5. Assembly. 6. Imposition of b.c.’s. 7. Solution of the algebraic set of equations. 8. Postprocesing computation of the FE solution and secondary variables. 1. Weak formulation of the governing differential equations and boundary conditions. 2. Discretization (mesh generation) of the solution domain into a set of finite elements & data interpolation (preprocessing) 3. Derivation of the element basis functions. 4. Calculation of the element matrices. 5. Assembly. 6. Imposition of b.c.’s. 7. Solution of the algebraic set of equations. 8. Postprocesing computation of the FE solution and secondary variables. The FE procedure includes the following steps:

3. 3 Poisson eq. in a complex domain (e.g. platelet) Mathematical model

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5. 5 FE mesh

6. 6 Bilinear (4 node) quadrilateral element Linear (3 node) triangular element Straight-sided elements

7. 7 A quadrilateral zone is discretized into triangular elements A triangular zone is discretized into quadrilateral elements Triangular vs. quadrilateral elements

8. 8 Figure 1: Mesh after 10 adaptive for geometry with 4 stent struts across aneurysm throat

9. 9 A combination of triangular and rectangular elements

10. 10 FE triangulation

11. 11 incompatible mesh refinement with 4-node elements

12. 12 - Transition element A combination of 4-node rectangular and 5-node transition elements Mesh refinement- (a)

13. 13 A combination of triangular and rectangular elements Mesh refinement- (b)

14. 14 Mesh refinement- (c) A combination of 4-node quadrilateral and 3- node triangular elements

15. 15 FE mesh can violate the symmetry of the problem

16. 16 Biquadratic (9 node) quadrilateral element quadratic (6 node) triangular element Curved-sided elements

17. 17 Linear interpolation on the eth triangular element Linear triangular elements

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20. 20 Length coordinates

21. 21 Area coordinates for a triangle element

22. 22 Linear map from triangle to master element Master element Real element Linear triangular element

23. 23 Linear interpolation on the eth triangular element A finite element approximation over a 3-node triangular element is a linear function of x and y:

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25. 25 where have the values 1,2,3 for i=1 and are permuted cyclically for i=2, and i=3. Area of the triangle

26. 26 Element Shape Quality

27. 27 Shape distortions that should be avoided in FE triangulations

28. 28 Local and global basis functions for the 3-node triangular element

29. 29 The coordinate map

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32. 32 Triangle integration formula:

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35. 35 On edge 3-1:

36. 36 Similar results are obtained by cyclic permutation when sides 1-2 and 2-3 are boundary sides

37. 37 Pascal’s triangle for various triangular elements having complete polynomial basis functions

38. 38 Gauss points and weights for Gauss quadrature rules on triangles

39. 39