F inite Element Method

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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

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
• CASE STUDY
INTRODUCTION
• 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.
TETRAHEDRON ELEMENT
• 3D solid meshed with tetrahedron elements
TETRAHEDRON ELEMENT

Consider a 4 node tetrahedron element

Shape functions

where

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

Shape functions

Similarly,

Can also be viewed as ratio of distances

(Partition of unity)

since

Shape functions

(Delta function property)

Shape functions

i= 1,2

Therefore,

i

l = 4,1

j

l

j = 2,3

k

(Cofactors)

k = 3,4

where

Shape functions

(Volume of tetrahedron)

Therefore,

Strain matrix

Since,

Therefore,

where

(Constant strain element)

Element matrices

Eisenberg and Malvern, 1973 :

Element matrices

Alternative method for evaluating me: special natural coordinate system

Element matrices

HEXAHEDRON ELEMENT
• 3D solid meshed with hexahedron elements

5

8

fsz

6

4

7

1

0

z

fsy

fsx

2

0

y

3

x

Shape functions
Shape functions

(Tri-linear functions)

Strain matrix

whereby

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

Strain matrix

Chain rule of differentiation

where

Strain matrix

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

Element matrices

Gauss integration:

Element matrices

For rectangular hexahedron:

Element matrices

(Cont’d)

where

Element matrices

(Cont’d)

or

where

Element matrices

(Cont’d)

E.g.

Element matrices

(Cont’d)

Note: For x direction only

(Rectangular hexahedron)

5

8

fsz

6

4

7

1

0

z

fsy

fsx

2

0

y

3

x

Element matrices

Using tetrahedrons to form hexahedrons
• Hexahedrons can be made up of several tetrahedrons

Hexahedron made up of 5 tetrahedrons:

Using tetrahedrons to form hexahedrons
• Element matrices can be obtained by assembly of tetrahedron elements

Hexahedron made up of 6 tetrahedrons:

HIGHER ORDER ELEMENTS
• Tetrahedron elements

HIGHER ORDER ELEMENTS
• Tetrahedron elements (Cont’d)

20 nodes, cubic:

HIGHER ORDER ELEMENTS
• Brick elements

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

Lagrange type:

where

HIGHER ORDER ELEMENTS
• Brick elements (Cont’d)

Serendipity type elements:

HIGHER ORDER ELEMENTS
• Brick elements (Cont’d)

32 nodes, tri-cubic:

Material

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