A REAL VARIABLE BOUNDARY ELEMENT METHOD FOR TWO DIMENTIONAL THERMAL ANALYSIS IN ISOTROPIC AND HOMOGENEOUS

-~~ --

~---A RE~---AL V~---ARI~---ABLE B

O

UN

D

ARY ELEMENT METH

OD F

OR TWO

­

DIMENTIONAL THERMAL ANALYSIS IN ISOTROPIC AND

HOMOGENE

O

US

ONGTHMMENG

Universiti Malaysia Sarawak

2001

TA

418.24

058

A REAL VARIABLE BOUNDARY ELEMENT METHOD

FOR TWO-DIMENSIONAL THERMAL ANALYSIS IN

..

ISOTROPIC AND HOMOGENEOUS SOLIDS

By

Ong Thiam Eng

A report submitted

in

partial fulfillment of the requirements for the degree of

Bachelor of Engineering (Hons.)

Faculty of Engineering

UNIVERSITY MALAYSIA SARAW AK

March 2001

APPROVAL SHEET

This project report attached hereto, entitled "A Real Variable Boundary Element Method For Tw~Dimensional Thermal Analysis In Isotropic Solids' prepared and submitted by Ong Thiam Eng in partial fulfillment of the requirement of Bachelor's degree with Honours in Mechanical Engineering is hereby accepted.

s/p~

,

Date

,~, ~I

(Associate p r l : ' : . T. Ang) Project Supervisor

Universiti Malaysia Sarawak

Kota Samarahan

fk BORAN6 PENYERAHAN TESIS

ludul: A REAL VARIABLE BOUNDARY ELEMENT METHOD (RVBEM) FOR TWO-DIMENTIONAL THERMAL ANALYSIS IN

ISOTROPIC AND HOMOGENEOUS SOLIDS

SESI PENGAJIAN: 1998 - 2001

Saya ONG TIflAM ENG

(HURUF BESAR)

mengaku membeoarkan tesis ini disimpan di Pusat Khidmat Maklumat Akademik, Universiti Malaysia Sarawak dengan syarat-syarat kegunaan seperti berikut:

1. Hakmilik kertas projek adalah di bawah nama penulis melainkan penulisan sebagai projek bersama dan dibiayai oleb UNIMAS, bakmiliknya adalah kepunyaan UNIMAS.

2. Naskhah salinan di dalam bentuk kertas atau mikro hanya bolch dibuat dengan kebenaran bertulis daripada penulis.

3. Pusat Kbidmat Maklwnat Akademik, UNIMAS dibenarkan membuat saIinan untuk pengajian mereka.

4. Kertas projek banya boleh diterbitkan dengan kebenaran penulis. Bayaran royaIti adaIah mengikut kadar yang dipersetujui keJak.

5. • Saya membenarkanltidak memly:parlcan Perpustakaan membuat salinan kertas projek ini sebagai bahan pertukaran di antara institusi pengajian tinggi.

6. •• Sila tandakan ( 3 )

[:=JSULIT (Mengandungi maklumat yang berdaJjah keselamatan atau kepentingan Malaysia seperti yang termaktub di daIam AKTA RAHSIA RASMI 1972).

c=J

TERHAD (Mengandungi. maklumat TERHAD yang telah ditentukan oleh organisasil

badan di mana penyelidikan dijalankan). ~ TIDAK TERHAD

Disahkan oleh

(TANDATANGAN PENULIS) PENYEUA)

Alamat tetap: 2369, Jinjang North,

East Road, 52000, Associate Professor W. T. Ang (Nama Penyelia)

Kuala Lwnpur.

Tarikh: Tarikh:

b

A~

l

.lc'DI

I

CATATAN • Potoog yaog tidak berkenaan.

Jib Kerlas Projek ini SULIT atau TERHAD, sUa lampirkan sural daripada pihak berkuasa/ orgaaisasi berkenaan dengan menyertakan sekali lempob kertas projek. Ini perlu dikelaskan sebagai SUUT alau TERHAD.

ACKNOWLEDGEMENT

I would like to express my appreciation to many individuals in bringing this thesis project

to a successful completion. First of all, I would like to thank my supervisor Dr. W. T.

Ang who gave guidance, support, encouragement, comments and suggestions in last few

months to complete this thesis project together. I would like to thank my dad, Ong Mam

Piang, my mom, Oh Guat Liang, my two sisters and my brother. Thanks for their support

and encouragement. For my friends, I would like to thank for their co-operation, support

and kind help. I would like to thank my friend especially Miss Tan Chan Boon who gave

TABLE OF CONTENTS

CONTENTS

PAGE

Acknowledgements Table of Contents 11 Abstract Vl Abstrak Vll CHAPTER 1 (PRELIMINARIES) 1.1 ) Introduction 1 1.2) Objective 1

1.3) Boundary Element Method (BEM) ]

1.4) Thermal Conductivity 3

1.5) Heat Flux 4

1.6) Heat Capacity 5

1.7) Fundamental Law 6

1.7.1) Law of Thermodynamics 6

l.7.2) Fourier' s Law of Conduction 7

1.8) Energy Equation 8

1.9) Governing Equation for Thermal Analysis 10

1.10) Problem in Project 11

l.] 0.]) Problem 1 ₁₁

l.1O.2) Problem 2 12

CHAPTER 2 (BEM FOR THE 2-D STEADY STATE THERMAL PROBLEMS) 14 2.2) Singular Solution 15 2.1) Introduction 2.3) Reciprocal Theorem 16 2.4) Integral Equation 17 2.5) BEM 25 2.6) Program 27 2.6.1) Subroutine: Data_Temperature 27 2.6.2) Subroutine: Integral_Gamma 28 2.6.3) Subroutine: Matrix 29

2.6.4) Subroutine: DLudcmp and DLubksb 30

2.6.5) Subroutine: Answer 30 2.7) Problems 32 2.7.1) Problem 1 32 2.7.2) Problem 2 35 2.7.3) Problem 3 37 2.7.4) Problem 4 39 2.7.5) Problem 5

...

41 2.7.6) Problem 6 46

Chapter 3 (BEM FOR THE 2-D TIME-DEPENDENT THERMAL PROBLEMS)

3.1) Introduction 49

3.2) Dual-Reciprocity Method (DRM) 49

3.3) Integral Equation 3.4) BEM

3.5) Program

3.5.1) Subroutine: Data_Time_Temperature 3.5.2) Subroutine: Integral_Time _Gamma 3.5.3) Subroutine: Integral_Time _Psi 3.5.4) Subroutine:: Integral_Time_Sigma 3.5.5) Subroutine: Matrix A 3.5.6) Subroutine: Matrix B 3.5.7) Subroutine: Stehfest 3.5.8) Function: V 3.5.9) Function: P 3.5.10) Function: LV 3.5.11) Function: LP 3.6) Problem

Chapter 4 (SUMMARY AND EXTENSION OF PROJECT)

References

Appendix 1 Main Program Code for steady state problem Appendix 2 Main Program Code for time dependent problem Appendix 3 Program Code for DLUDCMP and DLUBKSB Appendix 4 Program Code for Problem 1 (Steady state problem) Appendix 5 Program Code for Problem 2 (Steady state problem)

51 54 56 56 56 57 57 57 57 57 57 57 . 57 58 58 67 69 IV

Appendix 6 Program Code for Problem 3 (Steady state problem)

Appendix 7 Program Code for Problem 4 (Steady state problem)

Appendix 8 Program Code for Problem 5 (Steady state problem)

Appendix 9 Program Code for Problem 6 (Steady state problem)

Appendix 10 Program Code for Problem (Time dependent problem)

ABSTRACT

In the present project, the Boundary Element Method (BEM) will apply to solve heat conduction in two-dimensional body. It will be implemented on the computer through coding in FORTRAN.

BEM will used to solve thermally and homogeneous steady-state body in some heat

conduction problems.

The dual-Reciprocity Laplace Transform Boundary Element Method (DRL TBEM) will be used to solve time-dependent thermally and homogeneous body heat conduction problems. The Laplace transformation will be applied to this problem with respect to time, t. All domains integral in the formulation were converted into line integrals by used of dual-reciprocity method and radial basis function. A numerical inversion of Laplace transformation will be used in final stage to recover the solution in the physical space.

ABSTRAK

Pada projek ini, Boundary Element Method (BEM) digunakan untuk menyelesaikan

masalah-masalah yang berkaitan dengan konduksi haba dalam 2-dimension pepejal. Bahasa pengatucaraan FORTRAN akan digunakan untuk membentuk satu atucara yang berdasarkan kaedah ini.

BEM akan digunakan untuk menyelesaikan masalah-masalah thermally dan

homogeneous steady-state pepejal dalam konducksi haba.

Dual-Reciprocity Laplace Transform Boundary Element Method (DRL THEM) pula akan

digunakan untuk menyelesaikan masalah-masalah thermally dan homogeneous pepejal

dalam konduksi haba yang bergantung pada masa. Laplace transformation akan

digunakan untuk menyelesaikan masalah ini yang bertujuan menghapuskan anu masa,t.

Semua kamiran domain akan ditukarkan kepada kamiran garisan dengan dual-reciprocity

method dan radial basis function . Cara numerical untuk Laplace $ongsangan akan

digunakan untuk memulihkan ruang fizikal.

CHAPTER 1: PRELIMINARIES

1.1) Introduction

Heat transfer occurs in a material body between two points which have different temperature. One of the fundamental mechanisms of heat transfer is heat conduction, where heat energy is transported from region of high temperature within a medium to regions of lower temperature due to the random vibration of

molecules. For fluids and gases, heat energy can also be transported by

convection, a process where heat energy is carried by flowing particles.

1.2) Objective

The main objective of this project is to devise a numerical method for calculating the temperature in a finite two-dimensional body, given that either the temperature or heat flux (not both) is known at each and every point on the boundary of the body. The shape of the body and the boundary conditions are

arbitrary. The method, which will be used, is called the "Boundary Element

Method' (BEM). It will be implemented on the computer through coding in

FORTRAN.

1.3) Boundary Element Method

Boundary Element Method (BEM) is a numerical method for solving continuum mechanics problem such as fluid mechanics, solid mechanics, thermal behavior of object and etc. Those problems can be characterized mathematically by governing

differential equation and boundary and initial conditions. Usually it is difficult to

find analytical solution to these problems. Numerical solution techniques such as

the BEM will allow us to obtain quantitative answers. Generally, BEM involves

the following step;

i) Finding an integral expression for the solution of the partial differential

equation.

ii) Discretising the boundary of the solution domain and approximating

boundary values of unknown quantities to obtain a system of linear

algebraic equations.

iii) Solving the system of linear algebraic equation and post-processing the

solution.

The main advantages of BEM, compared to other numerical method such as Finite Element Method (FEM) are:

i) For the BEM. only the boundary has to be discretized. For the FEM, the

entire solution domain has to be discretized.

ii) The total number of unknown is less in BEM than FEM. The system of

linear algebraic equation in HEM is therefore smaner.

But the BEM also has some disadvantages. For further details, refer to James H.Kane [4].

1.4) Thermal Conductivity

The thermal conductivity k of a heat conducting medium can be defined as the heat flow per unit time when the temperature decreases by one degree in unit distance. The S.I unit of thermal conductivity is usually written as,

WImK or kWImK

Materials with high thermal conductivity are classified as good conductors of heat, whereas materials with low thermal conductivity are classified as good thermal insulators.

Thermal conductivity is location independent in a homogeneous material but it can be a function of location in an inhomogeneous materiaL That is, k in general

can be a function of x, y and z for an inhomogeneous material, i.e. k

=

k(x,y,z)

In this project, we consider only thermally isotropic and homogeneous materials. Thus, k is a positive constant here. For further details, refer to D.Paulikakos [5] and J.P. Holman [6].

1.5) Heat Flux

Uj

Figure 1.1

Heat flux, q defined as the heat transfer rate per unit area normal to the direction

of heat flow (for heat flow in the x-direction),

au

where U lS temperature, IS temperature gradient, constant k IS thermal

ax

conductivity and the minus sign show that heat energy loss from the material to

surrounding.

For heat flux in other direction, we may have,

au

q =-k­

y Oy

for 1-D heat flow in the y and z direction respectively.

For general 3-D heat flow, we can define the heat flux

which is related to u by,

q

=

_k[OU

,

au ,

au] ..

.

.

.

.

..

.

.

(1.1)

- AX By OZ

and it is valid for a thennally isotropic material. For further details, refer to D.Paulikakos [5] and J.P. Holman [6].

1.6) Beat Capacity

Heat capacity of an object is the heat energy required to produce unit temperature change. A material with a high heat capacity, like water required a lot of heat energy to change its temperature. While, a material with a low heat capacity like silver required little heat energy to change its temperature. Two object made of

same material will have heat capacities proportional to their masses. Therefore, specific heat capacity is defined where it refers not to an object but a unit mass of the material of which the object is made.

Specific heat capacity, c

=

l

mdu

For further details, refer to D.Paulikakos [5] and JP. Holman [6].

1.7) Fundamental Law

Figure 1.2

Let us consider an arbitrary region bounded a closed surface, S in a body where the heat energy is loss to surrounding as in figure (1.]). The vector n is unit

magnitude and normal to S pointing away from body and q is heat flux. The

amount of heat energy flowing through the infinitesimal surface of area, ds from

body to surrounding per unit time is equal to q. n ds. So, total amount of heat

"

energy leaving the region V through S per unit time is

H~· ~

ds . • s

1.7.1) Law o(Thermodynamics First Law ofThermodynamic

The First Law of Thermodynamics (conservation of energy) is the cornerstone of the science of the heat transfer. According to the First Law of Thermodynamics,

the rate of change of the total energy in a volume with respect to time must be

equal the rate of the change of heat energy going into the region volume through

the surface. If S is the bounding surface of the region R then A

-

aQ

=

-ffn.qds

at

s

-

­

where

Q

is heat energy in the body,

aQ

is rate of change heat energy inside the

at

body, n is unit normal to the surface and q is heat flux. For further details, refer

to J.P. Holman [6].

Second Law ofThermodynamics

The Second Law of Thermodynamics is a supplementary fundamental principle in analysis of the heat transfer process. This law states that heat is transferred from high temperature region to low temperature region where the temperature gradient is negative. For further details, refer to J.P. Holman (6].

1.7.2) Fourier's Law o(Conduction

In 1882, the French mathematical physicist named J.Fourier proposed the Fourier

Law

of which is the first theoretical modeling of heat conduction. The Fourier

Law

of Conduction states that the rate of heat flow in the x-direction through a thermally isotropic material is directly proportional to the area

A

(perpendicular to the heat flow) and to the change of the temperature with respect to the length of path ofthe heat flow (temperature gradient), i.e.

OU Q oc A x

ax

au

or

Qx

=

-kA-... .... (1.2)

ax

Where

au

and Qx are temperature gradient and heat transfer in the x-direction.

ax

The positive constant k is thennal conductivity of the material and the minus sign inserted so that the Second Law of Thermodynamics is satisfied. For further details, refer to D.Paulikakos [5].

1.8) Energy Equation ./

\

II d Figure 1.3

Let us consider an arbitrary region V bounded by a closed surface, S in a body such as figure 1.2. The density of the body is p~ its specific heat capacity is c. The

energy equation states the amount of heat energy per unit time flowing through the surface, S is expressed as:

aQ 1\

-=-ffn.qds (from 1 st Law of Thermodynamics)

at

s

- ­

where Q is the total amount of heat energy in V. The vector n is unit magnitude

and nonnal to S pointing away from V. For an infinitesimal surface of area, ds and

from the definition of c as in section 1.5, we obtain

dQ

c = - ­

dmdu dQ=cdudm dQ=pcdudV

Total amount energy in the body, Q=

HI

pc(u - uJdV v

where U_o = 0 ifT is in Kelvin and U_o = -273.13° C if u is in Celsius. Thus,

aQ

=

~

fIfpc(u - u }tV

at at

v 0

=

HI

~

(p:u}iv

v

at

assuming p and c does not change with time

Where,

Q

represent the heat energy per unit time and

u represents the temperature which is a function of x, y, z and t (in

general)

1.9) Governing Equation for Thermal Analysis

Since energy is conserved, we know that,

au

1\

IIf

fX-dV

=

-If

q. nds ...... .... (1.3)

v

at

s ­

To convert the right hand side of (1 .3) into volume integral, we shall now apply

Gauss-Ostrogradskii theorem, i.e.

~

H~·~ds= HIV.~dV

s v Letting F

=

q , we obtain ~

Hq·~ds=

HIV.qdV ............ (1.4) s - v

-Combining (1.3) and (1.4) together, we obtain

~

a

H

q·~ds=

fffv

·qdV

=

-HJ

fX~dV

s - v - v

at

So, JJJV.qdV=-JIJ fXoU dV v - v

at

au

v

·q=-fX-... .... (1.5)

-

at

According to Fourier Law for thermally isotropic and homogeneous materials

(section 1.5),

where k is a positive constant. Equation (1.5) becomes

The temperature u(x,y,z,t) in a thermally isotropic and homogeneous material must satisfy (1.6).

1.10) Problems in Project

In this project, we consider two-dimensional problems where the temperature depends on x, y, and t only.

1.10.1) Problem I:

In Chapter 2, a 2-D steady-state problem involving a thermally isotropic and homogeneous material is considered. The problem is to solve

for

(X,y)E

R, ... .... .. ... (1.7)

subject to,

u(x,y)

=

U(x,y) for

(x, y)E

C, , ... ... (1.8)

~[u(x,y)]=p(x,y)

for

(X,Y)EC

₂ ... .. ... (1.9)

an

where x and yare Cartesian co-ordinates, U(x, y) and p(x, y) are suitably given

functions, R is two-dimensional region bounded by the simple dosed curve C, C]

and C2 are non-intersecting curves which are such that C1 U C2

=

C and

Ou

=~.

(Vu)

with

~

being the unit norma1 vector to C2 at point

(x, y)

pointing

an -

­

awayfromR.

Equation (1.7) is a special case of Equation (1.6) above for the case where u does not change with time, t. The condition (1.8) implies that the temperature is known

on Cl while (1.9) tells us that the heat flux is known on C2.

A Boundary Element Method (BEM) will be used for solving this problem.

1. 10.2) Problem 2

In Chapter 2, we consider a time-dependent two-dimensional problem. The

problem is to solve

for (x,y)ER and t~O,.........(1.10)

subject to

u(x,y,O)=f(x,y) for (x,y)ER, ... (l.ll)

u{x,y,t)

=

k(x,y) for (X,y)ECI and t~O,... ....(1.12)

a

-[u(x,y,t)]= h(x,y)

on

for (X,y)EC2 and t~O...(l.13)

where x and y are Cartesian co-ordinates, p is density of the material, k is thennal

conductivity and c is specific heat capacity of the material, f(x,y) , k(x,y) and

h(x,y) are suitably given functions, R is two-dimensional region bounded by the

simple closed curve C, C1 and C] are non-intersecting curves which are such that

c.

uC2 = C and

au

=~.

(Vu)

with

~

being the unit normal vector to C2 at point

On - ­

(x,y)

pointing away from R.

Acrording to Equation (1. 11), the temperature is known at all point in the body at time 1=0. The boundary condition denotes that temperature is known at boundary CI while heat flux is known at boundary C] at any time.

A Dual-reciprocity Laplace Transform Boundary Element Method (DRL TBEM) will be used for solving this problem. The Laplace Transformation will be applied 10 this problem with respect to time, t. The radial basis function and a reciprocal theorem will be used to convert all domain integrals to line integrals. A numerical inversion of the Laplace transformation will be used in final stage to recover the solution in the physical space.