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## PowerPoint Slideshow about ' F inite Element Method' - blair-benson

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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

(Delta function property)

Shape functions

(Adjoint matrix)

i= 1,2

Therefore,

i

l = 4,1

j

l

j = 2,3

k

(Cofactors)

k = 3,4

where

Element matrices

where

Element matrices

Eisenberg and Malvern, 1973 :

Element matrices

Alternative method for evaluating me: special natural coordinate system

Element matrices

Jacobian:

Element matrices

For uniformly distributed load:

HEXAHEDRON ELEMENT

- 3D solid meshed with hexahedron elements

Shape functions

(Tri-linear functions)

Strain matrix

whereby

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

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:

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:

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

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