2D Model For Steady State Temperature Distribution

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1 2D Model For Steady State Temperature Distribution Finite Element Method Vinh University of Massachusetts Dartmouth December 14, 21

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.

3 What is finite element method? Finite Element Method is: A numerical method. A very popular technique used in engineering over the last 1 years. Can be used to find the approximate solutions for complicated problems such as partial differential equations.

4 What is finite element method?-example Calculate the area A of the given geometry. A can be Area,Temperature, Stress etc.

5 What is finite element method?-example Divide into smaller pieces (triangular, rectangular etc). Assemble the pieces together.

6 Project Description This project is a PDE problem (Laplace s 2nd order equation): δ 2 ϑ δx 2 + δ2 ϑ =, Ω = { < x < 5; < y < 1} (1) δy 2 With boundary conditions: ϑ(x, ) = < x < 5 (2) ϑ(y, ) = < y < 1 () ϑ(x, 1) = 1 sin( πx 1 ) < x < 5 (4) δϑ (5, y) δx = < y < 1 (5) The Exact Solution is found to be: πy πx 1 sinh( 1 ) sin( 1 ϑ(x, y) = ) sinh(π) (6)

7 Description-Using Finite Element Method Divide the the plate into small pieces. Assemble the pieces together.!

8 25 Nodes (2 Elements) Finite Element vs. Real Solution Temperature Distribution Vertical Side Horizontal Side

9 81 Nodes (128 Elements) Finite Element vs. Real Solution 1 Temperature Distribution Vertical side Horizontal side

10 24 Nodes (512 Elements) Plate vs. MatLab Solution Temperature Distribution Vertical side Horizontal side

11 Maximum Error Plot 7 2 elements elements 6 Percentage Error Percentage Error X!axis X!axis Percentage Error elements X!axis Percentage Error elements X!axis

12 Temperature Distribution (Right Side) 1 ϑ(x, y) = 1 sinh ( πy ) ( 1 sin πx ) 1 sinh(π) Temperature Vertical Axis

13 Building The Mesh For Hole Defect Model Mesh (21) (22) (2) (24) (17) (18) (19) (2) (15) (16) 6 (1) (14) Y!axis (11) (12) (9) (1) (5) (6) (7) (8) 1 (1) (2) () (4) X!axis

14 Defect Model vs. Original Model 1 9 Temperature Distribution Temperature Distribution Y!axis 5 75 Y!axis X!axis X!axis 45 (a) Hole model (b) Original model

15 Right Side Temperature Distribution Temperature Vertical Axis

16 Right Side Temperature Distribution Temperature Square Plate Original Vertical Axis

Current Project: Importing Mesh Meshing with Matlab is very difficult for complex geometries. Use ABAQUS R to sketch and mesh desired geometries.

17 Current Project: Importing Mesh Meshing with Matlab is very difficult for complex geometries. Use ABAQUS R to sketch and mesh desired geometries. ABAQUS R is a commercial engineering software for finite element analysis. After meshing with ABAQUS R, the mesh is imported to the heat code to do analysis.

18 Current Project: Eclipse Hole Model Sketch the plate Create the plate Mesh the plate

19 Importing The Mesh To Matlab 1 Eclipse Hole Mesh 9 8 Vertical Axis Horizontal Axis ABAQUS R exports the nodes and connectivity of elements as ipn.file. The ipn file need be to translated to.mat file to be readable by Matlab. After importing the elements and nodes to the heat code, the mesh is then generated as shown.

20 Project Description PDE problem (Laplace s 2nd order equation): δ 2 ϑ δx 2 + δ2 ϑ =, Ω = { < x < 5; < y < 1} (7) δy 2 With boundary conditions: Ellipse : (x 2.5)2 1.5 ϑ(x, ) = (8) ϑ(y, ) = (9) ϑ(x, 1) = 1 sin( πx 1 )(1) δϑ (5, y) δx = (11) δϑ (Ellipse) = δx (12) (y 5)2 + = (1)

21 1 Temperature Distribution Y!Axis X!Axis

22 Right Side Temperature Distribution Temperature Vertical Axis

23 Right Side Temperature Trend Temperature square hole Eclipse hole 2 1 Original Vertical Axis

My Code vs. ABAQUS 1 Temperature Distribution 9 5 8 Y!

24 My Code vs. ABAQUS 1 Temperature Distribution Y!Axis Abaqus X!Axis My Code

25 My Code vs. ABAQUS Temperature Vertical Axis Abaqus Result My Code s Result Student Version of MATLAB

26 Error Between Abaqua and Matlab!"##$% *#+,#-'$.-#)/)*"#)12("$)!2#) 4/# HI'J.8)1#8.9$ 56)7/#)1#8.9$ :--/-);<= % > >? >A>? B 9.4 CA@B %DA%% >A>E F 24.51?@AF >A>B D 4.2 B@A%G >A>F E >A>@ G 5.58 FBAFG >A>% C 6.96 D@A>B >A%% %> 78.6 EGA@B >A>C %% 1 %>> >

27 Result with heat source or heat sink at the hole 1 Temperature Distribution 1 Temperature Distribution Y!Axis 5 4 Y!Axis X!Axis Temperature at hole U= X!Axis Temperature at hole U=1

28 Future Research Goals Continue to learn finite element method. Do analysis with different materials of different thermal conductivities.

29 References [Civil Engineering Dept] Dr. Nima Rahbar Fundamental Matrix Algebra University of Massachusetts Dartmouth, Summer 21. [Mechanical Engineering Dept] Dr.R. Krishnakumar Introduction to Finite element Method http: // www. youtube. com/ watch? v= djd9-f-onls, June 28 Indian Institute of Technology, Madras

30 Questions?????????????????????????????????????????????????????????????

A numerical grid and grid less (Mesh less) techniques for the solution of 2D Laplace equation

Available online at www.pelagiaresearchlibrary.com Advances in Applied Science Research, 2014, 5(1):150-155 ISSN: 0976-8610 CODEN (USA): AASRFC A numerical grid and grid less (Mesh less) techniques for