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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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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

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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.

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

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

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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)

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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)

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

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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

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

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

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

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

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

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

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

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

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

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

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

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Finite Element Method by G. R. Liu and S. S. Quek 21 Shape functions z y x 3 0 f sz f sy f sx

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

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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

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

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

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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.,,

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

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

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

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

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

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

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Finite Element Method by G. R. Liu and S. S. Quek 33 Element matrices z y x 3 0 f sz f sy f sx For uniformly distributed load:

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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:

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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

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

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

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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

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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:

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

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

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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) GaAs InAs GaAs substrate GaAs cap layer InAs wetting layer InAs quantum dot

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

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

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

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

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