Matrix Structural Analysis · 2D Steady State Temperature Distribution Matrix Structural Analysis Giuliano Basile Vinh Nguyen Christine Rohr University of Massachusetts Dartmouth

2D Steady State Temperature DistributionMatrix Structural Analysis

Giuliano BasileVinh Nguyen

Christine Rohr

University of Massachusetts Dartmouth

July 21, 2010

Introduction

Advisor

Dr. Nima Rahbar: Civil Engineer

Project Description

Learning the fundamentals for creating matrices. We will be workingwith 2 Dimensional frames. Constructing elements and nodes, whichwill be used to study temperature distribution through out ourspecimen.

Application of Research

Study the thermal distribution

Test different types of materials

Compare Numerical vs. Analytical results

Basile, Nguyen, Rohr (UMD) Matrix Structural Analysis July 21, 2010 2 / 24

Objectives

Use Matlab to calculate the 2D Steady State TemperatureDistribution

Consider the boundary conditions (will be discussed)

Use Triangular Elements

Compare your numerical solution with the exact analytical solution

Calculate number of nodes, elements needed for accurate results

Compute Errors


Thermal Distribution in Materials

We consider all materials to be at Steady State

Different materials have different temperature distributions;This is due to different atomic structures

Metals – CrystallineCeramics – AmorphousPolymers – Chains

Atomic structure leads to different Thermal Conductivity

(how heat travels throughout)

This knowledge can be to choose the correct material for engineeringdesigns


Metals

Figure: Crystalline Atomic Structure


Ceramics

Figure: Amorphous Atomic Structure


Polymers

Figure: Chain Atomic Structure


Thermal Distribution in Materials

Table: Thermal Conductivity of Materials (Watts/meter*Kelvin)

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 study 2D Thermal Distribution?

To generate new understanding and improve computer methods forcalculating thermal distribution.

2D computer modeling is

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


Short Description

!

!

!

!

Here we are modeling heatflux for a 2D plate

No heat is applied to the xand y axis (x-nodes =y-nodes = 0)

Flux is also consideredzero on the right side ofthe plate

Steady heat is beingapplied at the top of theplate:

θ = 100 sin(πx

10) (1)


What’s Included

Elements

We start with 32 triangular elements

Numbering left to right; bottom to top

Each element has 3 local and global nodes

Number of Elements and Global Nodes will change

Nodes

Local nodes are used to indicate Global nodes

Nodes are used to define elements

Independent Element Number ien (3,5) = 17

3 is the Local node number

5 is the Element number

17 is the Global node number


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



Maximum Error Computed

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



What’s Next???

Goals

Continue modeling temperature change

Add defect to material and relate it to original material

Add hole to the specimen

to be continued...


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

Thank You for Listening

We would like to take this time to thank some very special peopleduring this whole learning process.

Dr. Gottlieb

Dr. Davis

Dr. Kim

Dr. Rahbar

Dr. Hausknecht

CSUMS Staff

Daniel Higgs

Zachary Grant

Charels Poole

Sidafa Conde

CSUMS Students

Questions?

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Matrix Structural Analysis · 2D Steady State Temperature Distribution Matrix Structural Analysis Giuliano Basile Vinh Nguyen Christine Rohr University of Massachusetts Dartmouth

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