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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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F inite element method

Finite Element Method

A Practical Course

CHAPTER 6:

FEM FOR 3D SOLIDS


Contents
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
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
TETRAHEDRON ELEMENT

  • 3D solid meshed with tetrahedron elements


Tetrahedron element1
TETRAHEDRON ELEMENT

Consider a 4 node tetrahedron element


Shape functions
Shape functions

where

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


Shape functions1
Shape functions

Similarly,

Can also be viewed as ratio of distances

(Partition of unity)

since


Shape functions2
Shape functions

(Delta function property)


Shape functions3
Shape functions

(Adjoint matrix)

i= 1,2

Therefore,

i

l = 4,1

j

l

j = 2,3

k

(Cofactors)

k = 3,4

where


Shape functions4
Shape functions

(Volume of tetrahedron)

Therefore,


Strain matrix
Strain matrix

Since,

Therefore,

where

(Constant strain element)



Element matrices1
Element matrices

Eisenberg and Malvern, 1973 :


Element matrices2
Element matrices

Alternative method for evaluating me: special natural coordinate system





Element matrices6
Element matrices

Jacobian:


Element matrices7
Element matrices

For uniformly distributed load:


Hexahedron element
HEXAHEDRON ELEMENT

  • 3D solid meshed with hexahedron elements


Shape functions5

5

8

fsz

6

4

7

1

0

z

fsy

fsx

2

0

y

3

x

Shape functions


Shape functions6
Shape functions

(Tri-linear functions)


Strain matrix1
Strain matrix

whereby

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


Strain matrix2
Strain matrix

Chain rule of differentiation

where


Strain matrix3
Strain matrix

Since,

or


Strain matrix4
Strain matrix

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


Element matrices8
Element matrices

Gauss integration:


Element matrices9
Element matrices

For rectangular hexahedron:


Element matrices10
Element matrices

(Cont’d)

where


Element matrices11
Element matrices

(Cont’d)

or

where


Element matrices12
Element matrices

(Cont’d)

E.g.


Element matrices13
Element matrices

(Cont’d)

Note: For x direction only

(Rectangular hexahedron)


Element matrices14

5

8

fsz

6

4

7

1

0

z

fsy

fsx

2

0

y

3

x

Element matrices

For uniformly distributed load:


Using tetrahedrons to form hexahedrons
Using tetrahedrons to form hexahedrons

  • Hexahedrons can be made up of several tetrahedrons

Hexahedron made up of 5 tetrahedrons:


Using tetrahedrons to form hexahedrons1
Using tetrahedrons to form hexahedrons

  • Element matrices can be obtained by assembly of tetrahedron elements

Hexahedron made up of 6 tetrahedrons:


Higher order elements
HIGHER ORDER ELEMENTS

  • Tetrahedron elements

10 nodes, quadratic:


Higher order elements1
HIGHER ORDER ELEMENTS

  • Tetrahedron elements (Cont’d)

20 nodes, cubic:


Higher order elements2
HIGHER ORDER ELEMENTS

  • Brick elements

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

Lagrange type:

where


Higher order elements3
HIGHER ORDER ELEMENTS

  • Brick elements (Cont’d)

Serendipity type elements:

20 nodes, tri-quadratic:


Higher order elements4
HIGHER ORDER ELEMENTS

  • Brick elements (Cont’d)

32 nodes, tri-cubic:



Case study

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



Case study2

30 nm

30 nm

CASE STUDY