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2D Model For Steady State Temperature Distribution Finite Element Method Vinh Nguyen, Giuliano Basile, Christine Rohr University of Massachusetts Dartmouth September 23, 2010 Introduction Advisor Dr. Nima Rahbar: Civil Engineering Project

SLIDE 1

2D Model For Steady State Temperature Distribution

Finite Element Method Vinh Nguyen, Giuliano Basile, Christine Rohr

University of Massachusetts Dartmouth

September 23, 2010

SLIDE 2

Introduction

Advisor

Dr. Nima Rahbar: Civil Engineering

Project Objective To learn the fundamentals of matrices and how to analyze them. To learn how to use Matlab and finite element method to construct a 2D computer model for temperature distribution.

Nguyen, Basile, Rohr 2D Model For Temperature Distribution

SLIDE 3

Temperature Distribution in Materials

At steady state different materials have different temperature distributions;

This is due to different atomic structures

Metals – Crystalline = high thermal conductivity Ceramics – Amorphous = low thermal conductivity Polymers – Chains = low thermal conductivity

This knowledge can be used to choose the correct materials for engineering designs

Nguyen, Basile, Rohr 2D Model For Temperature Distribution

SLIDE 4

Thermal Distribution in Materials

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

Materials Values Wood 0.04-0.4 Rubber 0.16 Polypropylene 0.25 Cement 0.29 Glass 1.1 Soil 1.5 Steel 12.11-45.0 Lead 35.3 Aluminum 237.0 Gold 318.0 Silver 429.0 Diamond 90.0-2320.0

Nguyen, Basile, Rohr 2D Model For Temperature Distribution

SLIDE 5

Why Do We Study 2D Temperature 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

Nguyen, Basile, Rohr 2D Model For Temperature Distribution

SLIDE 6

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 < x < 5 (2) ϑ(y, 0) = 0 < y < 10 (3) ϑ(x, 10) = 100 sin(πx 10 ) 0 < x < 5 (4) δϑ δx (5, y) = 0 < y < 10 (5) The Exact Solution is given: ϑ(x, y) = 100 sinh(πy

10 ) sin(πx 10 )

sinh(π) (6)

Nguyen, Basile, Rohr 2D Model For Temperature Distribution

SLIDE 7

Description-Building The Mesh

The problem was first approached by creating 25 node- 32 element triangular mesh. The nodes are built from left to right and bottom up. An element is formed by connecting 3 nodal points. No heat is applied to the sides and the bottom. Heat is applied at the top

f the plate:

ϑ = 100 sin(πx 10 ) (7)

Nguyen, Basile, Rohr 2D Model For Temperature Distribution

SLIDE 8

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

1 2 3 4 5 1 2 3 4 5 6 7 8 9 10 Temperature Distribution Horizontal Side Vertical Side 45 50 55 60 65 70 75 80 85 90 Nguyen, Basile, Rohr 2D Model For Temperature Distribution

SLIDE 9

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

1 2 3 4 5 1 2 3 4 5 6 7 8 9 10 Temperature Distribution Horizontal side Vertical side 70 75 80 85 90 Nguyen, Basile, Rohr 2D Model For Temperature Distribution

SLIDE 10

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

1 2 3 4 5 1 2 3 4 5 6 7 8 9 10 Temperature Distribution Horizontal side Vertical side 82 84 86 88 90 92 94 96 Nguyen, Basile, Rohr 2D Model For Temperature Distribution

SLIDE 11

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

1 2 3 4 5 1 2 3 4 5 6 7 8 9 10 Temperature Distribution Horizontal Side Vertical Side 90 91 92 93 94 95 96 97 98 Nguyen, Basile, Rohr 2D Model For Temperature Distribution

SLIDE 12

Temperature Distribution (Right Side)

δ(x, y) = 100 sinh πy

10

πx

10

sinh(π)

1 2 3 4 5 6 7 8 9 10 10 20 30 40 50 60 70 80 90 100 YAxis Temperature

32 Elements

Nguyen, Basile, Rohr 2D Model For Temperature Distribution

SLIDE 13

Temperature Distribution (Right Side)

δ(x, y) = 100 sinh πy

10

πx

10

sinh(π)

1 2 3 4 5 6 7 8 9 10 10 20 30 40 50 60 70 80 90 100 YAxis Temperature

32 elements 128 elements

Nguyen, Basile, Rohr 2D Model For Temperature Distribution

SLIDE 14

Temperature Distribution (Right Side)

δ(x, y) = 100 sinh πy

10

πx

10

sinh(π)

1 2 3 4 5 6 7 8 9 10 10 20 30 40 50 60 70 80 90 100 YAxis Temperature

32 elements 128 elements 512 elements Nguyen, Basile, Rohr 2D Model For Temperature Distribution

SLIDE 15

Temperature Distribution (Right Side)

δ(x, y) = 100 sinh πy

10

πx

10

sinh(π)

1 2 3 4 5 6 7 8 9 10 10 20 30 40 50 60 70 80 90 100 YAxis Temperature 32 elements 128 elements 512 elements 1682 elements

Temperature at The Right Side of The Plate

The temperature lines converge to a smooth line as the number of elements increases Nguyen, Basile, Rohr 2D Model For Temperature Distribution

SLIDE 16

Error Computing

The exact solution is shown: ϑ(x, y) = 100 sinh(πy

10 ) sin(πx 10 )

sinh(π) (8) Error is calculated by: Error = Exact Solution − Nodal Point temperature Exact Solution .100 (9)

Nguyen, Basile, Rohr 2D Model For Temperature Distribution

SLIDE 17

Maximum Error Plot

1 2 3 4 5 1 2 3 4 5 6 7 32 elements Xaxis Percentage Error 1 2 3 4 5 0.5 1 1.5 2 128 elements Xaxis Percentage Error 1 2 3 4 5 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 512 elements Xaxis Percentage Error 1 2 3 4 5 0.02 0.04 0.06 0.08 0.1 0.12 0.14 1682 elements Xaxis Percentage Error

Nguyen, Basile, Rohr 2D Model For Temperature Distribution

SLIDE 18

Building The Mesh For Hole Defect Model

1 2 3 4 5 1 2 3 4 5 6 7 8 9 10 Mesh

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)

Xaxis Yaxis Nguyen, Basile, Rohr 2D Model For Temperature Distribution

SLIDE 19

Hole Defect Model vs. Original Model

1 2 3 4 5 1 2 3 4 5 6 7 8 9 10 Temperature Distribution Xaxis Yaxis 55 60 65 70 75 80 85 90 95 1 2 3 4 5 1 2 3 4 5 6 7 8 9 10 Temperature Distribution Xaxis Yaxis 45 50 55 60 65 70 75 80 85 90 95

Nguyen, Basile, Rohr 2D Model For Temperature Distribution

SLIDE 20

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

SLIDE 21

Questions?

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Nguyen, Basile, Rohr 2D Model For Temperature Distribution