2D Steady State Temperature Distribution Matrix Structural Analysis

Giuliano Basile Vinh 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 working with 2 Dimensional frames. Constructing elements and nodes, which will be used to study temperature distribution through out our specimen.

Application of Research Study the thermal distribution Test different types of materials Compare Numerical vs. Analytical results Basile, Nguyen, Rohr (UMD)

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Objectives

Use Matlab to calculate the 2D Steady State Temperature Distribution 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

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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 – Crystalline Ceramics – Amorphous Polymers – Chains

Atomic structure leads to different Thermal Conductivity (how heat travels throughout)

This knowledge can be to choose the correct material for engineering designs

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Metals

Figure: Crystalline Atomic Structure

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Ceramics

Figure: Amorphous Atomic Structure Basile, Nguyen, Rohr (UMD)

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Polymers

Figure: Chain Atomic Structure

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

Basile, Nguyen, Rohr (UMD)

Materials

Values

Wood Rubber Polypropylene Cement Glass Soil Steel Lead Aluminum Gold Silver Diamond

0.04-0.4 0.16 0.25 0.29 1.1 1.5 12.11-45.0 35.3 237.0 318.0 429.0 90.0-2320.0

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Why study 2D Thermal Distribution?

To generate new understanding and improve computer methods for calculating thermal distribution. 2D computer modeling is cheap fast to process gives accurate numerical results parallel method can be used for higher efficiency

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Short Description Here we are modeling heat flux for a 2D plate No heat is applied to the x and y axis (x-nodes = y-nodes = 0) Flux is also considered zero on the right side of the plate Steady heat is being applied at the top of the plate: θ = 100 sin(

πx ) 10

(1)

! Basile, Nguyen, Rohr (UMD)

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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 Basile, Nguyen, Rohr (UMD)

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

10

90

9

85

8

80

Vertical Side

7

75

6

70

5

65

4 60 3 55 2 50 1 45 0

0

1

2 3 4 Horizontal Side

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5

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81 Nodes (128 Elements) — Plate vs. MatLab Solution Temperature Distribution

10 9

90 8

Vertical side

7

85

6 5

80

4 75

3 2

70

1 0

0

1

2 3 4 Horizontal side

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5

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324 Nodes (512 Elements) — Plate vs. MatLab Solution Temperature Distribution

10

96 9 94

8

Vertical side

7

92

6 90 5 88

4 3

86

2 84 1 0

82 0

1

2 3 4 Horizontal side

5

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900 Nodes (1682 Elements) — Plate vs. MatLab Solution Temperature Distribution

10

98

9 97 8 96

Vertical Side

7 95

6 5

94

4

93

3

92

2 91 1 90 0

0

1

2 3 4 Horizontal Side

5

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Temperature Distribution (Right Side)
100 sinh πy 10 sin δ(x, y ) = sinh(π)

πx 10




100 90 80

Temperature

70 60

32 Elements

50 40 30 20 10 0

0

1

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2

3

4

5 Y!Axis

6

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7

8

9

10

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Temperature Distribution (Right Side)
100 sinh πy 10 sin δ(x, y ) = sinh(π)

πx 10




100 90 80

Temperature

70 60 50

128 elements

40 30

32 elements

20 10 0

0

1

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3

4

5 Y!Axis

6

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8

9

10

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Temperature Distribution (Right Side)
100 sinh πy 10 sin δ(x, y ) = sinh(π)

πx 10




100 90 80

Temperature

70

128 elements

60 50

32 elements

40 30

512 elements

20 10 0

0

1

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3

4

5 Y!Axis

6

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7

8

9

10

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Temperature Distribution (Right Side)
100 sinh πy 10 sin δ(x, y ) = sinh(π)

πx 10




100

Temperature at The Right Side of The Plate

90 80

Temperature

70 60 50 The temperature lines converge to a smooth line as the number of elements increases

40 30

32 elements 128 elements 512 elements 1682 elements

20 10 0

0

1

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3

4

5 Y!Axis

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9

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Maximum Error Computed 32 elements

7

128 elements

2

5

Percentage Error

Percentage Error

6

4 3 2

1.5

1

0.5

1 0

0

1

2

3

4

0

5

0

1

2

X!axis

512 elements

0.45

3

4

5

4

5

X!axis

1682 elements

0.14

0.4

0.12 Percentage Error

Percentage Error

0.35 0.3 0.25 0.2 0.15

0.1 0.08 0.06 0.04

0.1 0.02

0.05 0

0

1

2

3

4

5

0

0

X!axis

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2

3 X!axis

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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...

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References [Civil Engineer] Dr. Nima Rahbar Fundamental Matrix Algebra University of Massachusetts Dartmouth, Summer 2010. [Thermal Conductivity of some common Materials] Thermal Conductivity of Materials www. engineeringtoolbox. com , July 2010 Cu Atomic Structure Crystalline Atomic Structure http: // www. webelements. com , July 2010 Ceramic Atomic Structure Amorphous Atomic Structure http: // www. bccms. uni-bremen. de , July 2010 Polymer Atomic Structure Chain Atomic Structure http: // www. themolecularuniverse. com , July 2010

Thank You for Listening We would like to take this time to thank some very special people during 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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