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# F inite Element Method - PowerPoint PPT Presentation

F inite Element Method. A Practical Course. CHAPTER 6:. FEM FOR 3D SOLIDS. CONTENTS. INTRODUCTION TETRAHEDRON ELEMENT Shape functions Strain matrix Element matrices HEXAHEDRON ELEMENT Shape functions Strain matrix Element matrices Using tetrahedrons to form hexahedrons

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### Finite Element Method

A Practical Course

CHAPTER 6:

FEM FOR 3D SOLIDS

• 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

• CASE STUDY

• For 3D solids, all the field variables are dependent of x, yandzcoordinates – most general element.

• The element is often known as a 3D solid elementor simply asolid element.

• A 3-D solid element can have a tetrahedron and hexahedron shape with flat or curved surfaces.

• At any node there are three components in x, y and z directions for the displacement as well as forces.

• 3D solid meshed with tetrahedron elements

Consider a 4 node tetrahedron element

where

Use volume coordinates (Recall Area coordinates for 2D triangular element)

Similarly,

Can also be viewed as ratio of distances

(Partition of unity)

since

(Delta function property)

i= 1,2

Therefore,

i

l = 4,1

j

l

j = 2,3

k

(Cofactors)

k = 3,4

where

(Volume of tetrahedron)

Therefore,

Since,

Therefore,

where

(Constant strain element)

where

Eisenberg and Malvern, 1973 :

Alternative method for evaluating me: special natural coordinate system

Jacobian:

• 3D solid meshed with hexahedron elements

8

fsz

6

4

7

1

0

z

fsy

fsx

2

0

y

3

x

Shape functions

(Tri-linear functions)

whereby

Note: Shape functions are expressed in natural coordinates – chain rule of differentiation

Chain rule of differentiation

where

Since,

or

Used to replace derivatives w.r.t. x, y, z with derivatives w.r.t. , , 

Gauss integration:

For rectangular hexahedron:

(Cont’d)

where

(Cont’d)

or

where

(Cont’d)

E.g.

(Cont’d)

Note: For x direction only

(Rectangular hexahedron)

8

fsz

6

4

7

1

0

z

fsy

fsx

2

0

y

3

x

Element matrices

• Hexahedrons can be made up of several tetrahedrons

Hexahedron made up of 5 tetrahedrons:

• Element matrices can be obtained by assembly of tetrahedron elements

Hexahedron made up of 6 tetrahedrons:

• Tetrahedron elements

• Tetrahedron elements (Cont’d)

20 nodes, cubic:

• Brick elements

nd=(n+1)(m+1)(p+1) nodes

Lagrange type:

where

• Brick elements (Cont’d)

Serendipity type elements:

• Brick elements (Cont’d)

32 nodes, tri-cubic:

E (Gpa)

GaAs

86.96

0.31

InAs

51.42

0.35

CASE STUDY

• Stress and strain analysis of a quantum dot heterostructure

GaAs cap layer

InAs wetting layer

InAs quantum dot

GaAs substrate

30 nm

30 nm

CASE STUDY