2D Model For Steady State Temperature Distribution ... · 2D Model For Steady State Temperature Distribution Finite Element Method ... plate with hole. 0.3.4 Matlab Code ... State

2D Model For Steady State TemperatureDistribution

Finite Element Method

Vinh Nguyen, Giuliano Basile, Christine Rohr

University of Massachusetts Dartmouth

September 23, 2010

Introduction

Advisor

Dr. Nima Rahbar: Civil Engineering

Project Objective

To learn the fundamentals of matrices and how to analyzethem.

To learn how to use Matlab and finite element method toconstruct a 2D computer model for temperature distribution.

Nguyen, Basile, Rohr 2D Model For Temperature Distribution

Temperature Distribution in Materials

At steady state different materials have different temperaturedistributions;

This is due to different atomic structures

Metals – Crystalline = high thermal conductivityCeramics – Amorphous = low thermal conductivityPolymers – Chains = low thermal conductivity

This knowledge can be used to choose the correct materialsfor engineering designs


Thermal Distribution in Materials

Table: Thermal Conductivity of Materials (W/m*K)

Materials Values

Wood 0.04-0.4Rubber 0.16

Polypropylene 0.25Cement 0.29

Glass 1.1Soil 1.5

Steel 12.11-45.0Lead 35.3

Aluminum 237.0Gold 318.0Silver 429.0

Diamond 90.0-2320.0


Why Do We Study 2D Temperature Distribution?

To generate new understanding and improve computermethods for calculating thermal distribution.

2D computer modeling is

cheapfast to processgives accurate numerical resultsparallel method can be used for higher efficiency


Description

This project is a PDE problem:

δ2ϑ

δx2+δ2ϑ

δy2= 0, Ω = 0 < x < 5; 0 < y < 10 (1)

With boundary conditions:

ϑ(x , 0) = 0 0 < x < 5 (2)

ϑ(y , 0) = 0 0 < y < 10 (3)

ϑ(x , 10) = 100 sin(πx

10) 0 < x < 5 (4)

δϑ

δx(5, y) = 0 0 < y < 10 (5)

The Exact Solution is given:

ϑ(x , y) =100 sinh(πy

10 ) sin(πx10 )

sinh(π)(6)


Description-Building The Mesh

!

!

!

!

The problem was firstapproached by creating 25node- 32 elementtriangular mesh.

The nodes are built fromleft to right and bottomup.

An element is formed byconnecting 3 nodal points.

No heat is applied to thesides and the bottom.

Heat is applied at the topof the plate:

ϑ = 100 sin(πx

10) (7)


25 Nodes (32 Elements) — Plate vs. MatLab Solution

0 1 2 3 4 50

1

2

3

4

5

6

7

8

9

10Temperature Distribution

Horizontal Side

Vert

ical S

ide

45

50

55

60

65

70

75

80

85

90

Student Version of MATLAB



0 1 2 3 4 50

1

2

3

4

5

6

7

8

9


Horizontal side

Ve

rtic

al sid

e

70

75

80

85

90




0 1 2 3 4 50

1

2

3

4

5

6

7

8

9


Horizontal side

Vert

ical sid

e

82

84

86

88

90

92

94

96




0 1 2 3 4 50

1

2

3

4

5

6

7

8

9


Horizontal Side

Ve

rtic

al S

ide

90

91

92

93

94

95

96

97

98



Temperature Distribution (Right Side)

δ(x , y) =100 sinh

(πy10

)sin

(πx10

)sinh(π)

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

60

70

80

90

100

Y!Axis

Temperature

32Elements




δ(x , y) =100 sinh

(πy10

)sin

(πx10

)sinh(π)

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

60

70

80

90

100

Y!Axis

Temperature

32elements

128elements




δ(x , y) =100 sinh

(πy10

)sin

(πx10

)sinh(π)

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

60

70

80

90

100

Y!Axis

Tem

pera

ture

32 elements

128 elements

512 elements




δ(x , y) =100 sinh

(πy10

)sin

(πx10

)sinh(π)

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

60

70

80

90

100

Y!Axis

Tem

pera

ture

32 elements

128 elements

512 elements

1682 elements

Temperature at TheRight Side of ThePlate

The temperature linesconverge to a smooth lineasthe number of elementsincreases



Error Computing

The exact solution is shown:

ϑ(x , y) =100 sinh(πy

10 ) sin(πx10 )

sinh(π)(8)

Error is calculated by:

Error =Exact Solution − Nodal Point temperature

Exact Solution.100 (9)


Maximum Error Plot

0 1 2 3 4 50

1

2

3

4

5

6

732 elements

X!axis

Pe

rce

nta

ge

Err

or

0 1 2 3 4 50

0.5

1

1.5

2128 elements

X!axis

Pe

rce

nta

ge

Err

or

0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45512 elements

X!axis

Pe

rce

nta

ge

Err

or

0 1 2 3 4 50

0.02

0.04

0.06

0.08

0.1

0.12

0.141682 elements

X!axis

Pe

rce

nta

ge

Err

or



Building The Mesh For Hole Defect Model

0 1 2 3 4 50

1

2

3

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5

6

7

8

9

10Mesh

1 2 3 4 5

6 7 8 9 10

11 12 13 14

15 16 17 18 19

20 21 22 23 24

(1) (2) (3) (4)

(5) (6) (7) (8)

(9) (10)

(11) (12)

(13) (14)

(15) (16)

(17) (18) (19) (20)

(21) (22) (23) (24)

X!axis

Y!axis



Hole Defect Model vs. Original Model

0 1 2 3 4 50

1

2

3

4

5

6

7

8

9


X!axis

Y!

axis

55

60

65

70

75

80

85

90

95


(a) Hole model

0 1 2 3 4 50

1

2

3

4

5

6

7

8

9


X!axis

Y!

axis

45

50

55

60

65

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85

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95


(b) Original model

Figure 9: Matlab’s numerical results for the defected model and the original model from left to right (a), (b)

As in the figure we can see that heat is spreading further to the left at the top part and further down onthe right side of the plate with hole.

0.3.4 Matlab Code for Temperature Distribution of The Defected Model

The temperature distribution in this model is controlled by boundary conditions as below

%!!!!!!!!!!!!!!!!!!!!!!!!!!!!!%% Boundary cond i t i on s %%!!!!!!!!!!!!!!!!!!!!!!!!!!!!!%% pre s c r i b e d d i sp lacement ( e s s e n t i a l boundary cond i t i on )%% Idb ( i ,N)=1 i f the degree o f freedom i o f the node N i s p r e s c r i b e d% =0 otherw i s e%% 1) i n i t i a l i z e Idb to 0idb=zeros ( ndf , nnp ) ;% 2) enter the f l a g f o r p r e s c r i b e d d i sp lacement boundary cond i t i on sfor i = 1 : nxd

idb (1 , i )=1;end

for i = 1 : nxd : ( nyd"(nxd!1)+1)idb (1 , i )=1;

end

for i = nxd"(nyd!1)+1:nxd"( nyd )idb (1 , i )=1;

end

12


References

[Civil Engineer] Dr. Nima RahbarFundamental Matrix AlgebraUniversity of Massachusetts Dartmouth, Summer 2010.

[Thermal Conductivity of some common Materials]Thermal Conductivity of Materialswww. engineeringtoolbox. com , July 2010

Cu Atomic StructureCrystalline Atomic Structurehttp: // www. webelements. com , July 2010

Ceramic Atomic StructureAmorphous Atomic Structurehttp: // www. bccms. uni-bremen. de , July 2010

Polymer Atomic StructureChain Atomic Structurehttp: // www. themolecularuniverse. com , July 2010

www.engineeringtoolbox.com

http://www.webelements.com

http://www.bccms.uni-bremen.de

http://www.themolecularuniverse.com

Questions?

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Documents

2D Model For Steady State Temperature Distribution ... · 2D Model For Steady State Temperature Distribution Finite Element Method ... plate with hole. 0.3.4 Matlab Code ... State