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1 Finite Element Method FEM FOR 3D SOLIDS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 9:

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Presentation on theme: "1 Finite Element Method FEM FOR 3D SOLIDS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 9:"— Presentation transcript:

1 1 Finite Element Method FEM FOR 3D SOLIDS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 9:

2 Finite Element Method by G. R. Liu and S. S. Quek 2 CONTENTS INTRODUCTION TETRAHEDRON ELEMENT – Shape functions – Strain matrix – Element matrices HEXAHEDRON ELEMENT – Shape functions – Strain matrix – Element matrices – Using tetrahedrons to form hexahedrons HIGHER ORDER ELEMENTS ELEMENTS WITH CURVED SURFACES

3 Finite Element Method by G. R. Liu and S. S. Quek 3 INTRODUCTION For 3D solids, all the field variables are dependent of x, y and z coordinates – most general element. The element is often known as a 3D solid element or simply a solid element. A 3D solid element can have a tetrahedron and hexahedron shape with flat or curved surfaces. At any node there are three components in the x, y and z directions for the displacement as well as forces.

4 Finite Element Method by G. R. Liu and S. S. Quek 4 TETRAHEDRON ELEMENT 3D solid meshed with tetrahedron elements

5 Finite Element Method by G. R. Liu and S. S. Quek 5 TETRAHEDRON ELEMENT Consider a four node tetrahedron element

6 Finite Element Method by G. R. Liu and S. S. Quek 6 Shape functions where Use volume coordinates (Recall Area coordinates for 2D triangular element)

7 Finite Element Method by G. R. Liu and S. S. Quek 7 Shape functions Similarly, Can also be viewed as ratio of distances since (Partition of unity)

8 Finite Element Method by G. R. Liu and S. S. Quek 8 Shape functions (Delta function property)

9 Finite Element Method by G. R. Liu and S. S. Quek 9 Shape functions Therefore, where (Adjoint matrix) (Cofactors) i j k l i= 1,2 j = 2,3 k = 3,4 l = 4,1

10 Finite Element Method by G. R. Liu and S. S. Quek 10 Shape functions (Volume of tetrahedron) Therefore,

11 Finite Element Method by G. R. Liu and S. S. Quek 11 Strain matrix Since, Therefore,where (Constant strain element)

12 Finite Element Method by G. R. Liu and S. S. Quek 12 Element matrices where

13 Finite Element Method by G. R. Liu and S. S. Quek 13 Element matrices Eisenberg and Malvern [1973] :

14 Finite Element Method by G. R. Liu and S. S. Quek 14 Element matrices Alternative method for evaluating m e : special natural coordinate system

15 Finite Element Method by G. R. Liu and S. S. Quek 15 Element matrices

16 Finite Element Method by G. R. Liu and S. S. Quek 16 Element matrices

17 Finite Element Method by G. R. Liu and S. S. Quek 17 Element matrices

18 Finite Element Method by G. R. Liu and S. S. Quek 18 Element matrices Jacobian:

19 Finite Element Method by G. R. Liu and S. S. Quek 19 Element matrices For uniformly distributed load:

20 Finite Element Method by G. R. Liu and S. S. Quek 20 HEXAHEDRON ELEMENT 3D solid meshed with hexahedron elements

21 Finite Element Method by G. R. Liu and S. S. Quek 21 Shape functions 1 7 5 8 6 4 2 0 z y x 3 0 f sz f sy f sx

22 Finite Element Method by G. R. Liu and S. S. Quek 22 Shape functions (Tri-linear functions)

23 Finite Element Method by G. R. Liu and S. S. Quek 23 Strain matrix whereby Note: Shape functions are expressed in natural coordinates – chain rule of differentiation

24 Finite Element Method by G. R. Liu and S. S. Quek 24 Strain matrix Chain rule of differentiation where

25 Finite Element Method by G. R. Liu and S. S. Quek 25 Strain matrix Since, or

26 Finite Element Method by G. R. Liu and S. S. Quek 26 Strain matrix Used to replace derivatives w.r.t. x, y, z with derivatives w.r.t.,,

27 Finite Element Method by G. R. Liu and S. S. Quek 27 Element matrices Gauss integration:

28 Finite Element Method by G. R. Liu and S. S. Quek 28 Element matrices For rectangular hexahedron:

29 Finite Element Method by G. R. Liu and S. S. Quek 29 Element matrices (Contd) where

30 Finite Element Method by G. R. Liu and S. S. Quek 30 Element matrices (Contd) or where

31 Finite Element Method by G. R. Liu and S. S. Quek 31 Element matrices (Contd) E.g.

32 Finite Element Method by G. R. Liu and S. S. Quek 32 Element matrices (Contd) Note: For x direction only (Rectangular hexahedron)

33 Finite Element Method by G. R. Liu and S. S. Quek 33 Element matrices 1 7 5 8 6 4 2 0 z y x 3 0 f sz f sy f sx For uniformly distributed load:

34 Finite Element Method by G. R. Liu and S. S. Quek 34 Using tetrahedrons to form hexahedrons Hexahedrons can be made up of several tetrahedrons Hexahedron made up of 5 tetrahedrons:

35 Finite Element Method by G. R. Liu and S. S. Quek 35 Using tetrahedrons to form hexahedrons Hexahedron made up of six tetrahedrons: Element matrices can be obtained by assembly of tetrahedron elements

36 Finite Element Method by G. R. Liu and S. S. Quek 36 HIGHER ORDER ELEMENTS Tetrahedron elements 10 nodes, quadratic:

37 Finite Element Method by G. R. Liu and S. S. Quek 37 HIGHER ORDER ELEMENTS Tetrahedron elements (Contd) 20 nodes, cubic:

38 Finite Element Method by G. R. Liu and S. S. Quek 38 HIGHER ORDER ELEMENTS Brick elements Lagrange type: (n d =(n+1)(m+1)(p+1) nodes) where

39 Finite Element Method by G. R. Liu and S. S. Quek 39 HIGHER ORDER ELEMENTS Brick elements (Contd) Serendipity type elements: 20 nodes, tri-quadratic:

40 Finite Element Method by G. R. Liu and S. S. Quek 40 HIGHER ORDER ELEMENTS Brick elements (Contd) 32 nodes, tri-cubic:

41 Finite Element Method by G. R. Liu and S. S. Quek 41 ELEMENTS WITH CURVED SURFACES

42 Finite Element Method by G. R. Liu and S. S. Quek 42 CASE STUDY Stress and strain analysis of a quantum dot heterostructure MaterialE (Gpa) GaAs86.960.31 InAs51.420.35 GaAs substrate GaAs cap layer InAs wetting layer InAs quantum dot

43 Finite Element Method by G. R. Liu and S. S. Quek 43 CASE STUDY

44 Finite Element Method by G. R. Liu and S. S. Quek 44 CASE STUDY 30 nm

45 Finite Element Method by G. R. Liu and S. S. Quek 45 CASE STUDY

46 Finite Element Method by G. R. Liu and S. S. Quek 46 CASE STUDY


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