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MATERIALS SCIENCE AND TECHNOLOGIESES

FUNCTIONALLY GRADED

MATERIALS

NATHAN J REYNOLDS

EDITOR

Nova Science Publishers Inc New York

Copyright copy 2012 by Nova Science Publishers Inc

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DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS

Additional color graphics may be available in the e-book version of this book

Library of Congress Cataloging-in-Publication Data

Functionally graded materials editor Nathan J Reynolds p cm Includes index ISBN 97S-1-6 I 209-616-2 (hardcover) I Functionally gradient materials 1 Reynolds Nathan 1 TA4IS9FS5FS42011 620 J IS--dc23 2011027544

Published by Nova Science Publishers Inc t New York

L Llicignono A Gug lielllloli olld F QII _ltjrill i

319

CONTENTS

vii

A Linear Multi-Layered Model and Its Applications in Fracture and Contact Mechanics of Elastic Functionally G-aded Materials Liao-Liang Ke and Yue-Sheng Wang

Functionally Graded Materials Obtained by Combustion Synthesis Techniques A Review 93 Roberlo Rosa and Paolo Veronesi

The Method of Fundamental Solutions fOl- Thennoelastic Analysis of Functionally Graded Materials 123 Hili Wang and Qing-Hua Qin

Three-Dimensional The-mal Buckling Analysis of Functionally Graded Arbitrary Straight-Sided Quadrilateral Plates 157 p Malekzadeh

The Mechanical Response of Metal-Ceramic Functionally Graded Materials Models and Experiences 181 Gahriella Bolzon

Simulation of Quasi-Static Crack Propagation in Functionally Graded Materials 193 Marlin Sl eigemann

Cylind rically- or Spherically-Symmetric Problems of Functionally Graded Materials 249 Xian-Fang Li and Xu-Long Peng

Functionally Graded Foams for Filter Fabrication 305

tm or mical

~sed or - No out of

ial ~ of or

~ Icated

~~d in anlage

- nise

ni 10 the

F THE

Preface

Chapter 1

Chapter 2

Chapter 3

Chaptemiddot 4

Chapter 5

bapter 6

C pteI 7

pt ) 8

In Functionally Graded Materials ISBN 978-1-61209-616-2

Editor Nathan J Reynolds copy 2012 Nova Science Publishers Inc

Chapter 3

THE METHOD OF FUNDAMENTAL SOLUTIONS FOR

THERMOELASTIC ANALYSIS OF FUNCTIONALLY

GRADED MATERIALS

Hui Wang12

and Qing-Hua Qin3

1Institute of Scientific and Engineering Computation

Henan University of Technology

Zhengzhou 450052 China 2State Key Laboratory of Structural Analysis for

Industrial Equipment Dalian University of Technology

Dalian 116024 PRChina 3Research School of Engineering Australian National University

Canberra ACT 0200 Australia

ABSTRACT

Thermoelastic simulation of functionally graded materials is practically important for

engineers Here the extension and assembly of our two previous papers (Computational

Mechanics 2006 38 p51-60 Engineering Analysis with Boundary Elements 2008 32

p704-712) is presented to evaluate the transient temperature and stress distributions in

two-dimensional functionally graded solids In this chapter the analog equation method

is used to obtain an equivalent homogeneous system to the original nonhomogeneous

governing equation after which radial basis functions and fundamental solutions are used

to construct the related approximated solutions of particular part and complementary part

respectively Finally all unknowns are determined by satisfying the governing equations

at interior points and boundary conditions at boundary points Numerical experiments are

performed for different 2D functionally graded material problems and the meshless

method described in this chapter is validated by comparing available analytical and

numerical results

Corresponding author Email qinghuaqinanueduau Fax +61 2 61250506

Hui Wang and Qing-Hua Qin 124

Keywords Functionally graded materials Thermoelasticity Method of fundamental

solutions Radial basis functions Analog equation method

1 INTRODUCTION

Functionally graded materials (FGMs) can usually be viewed as special inhomogeneous

materials whose properties are dependent on spatial coordinates In FGMs due to the

continuous change of material properties in space the absence of interfaces between different

constituents or phases largely reduces the degree of material property mismatch and brings

appealing physical behaviors superior to homogeneous and conventional materials For

example for the classic ceramicmetal FGMs the ceramic phase offers thermal barrier effects

and protects the metal from corrosion and oxidation and the FGM is toughened and

strengthened by the metallic constituent A smooth transition between a pure metal and a pure

ceramic may result in a multifunctional material that combines the desirable high temperature

properties and thermal resistance of the ceramic with the fracture toughness and strength of

the metal Thus FGMs can be applied to many engineering structures subjected to severe

thermal loadings such as high temperature and thermal shocks to reduce thermal stresses and

suffer less thermal damage [1]

So far two models have been used to characterize the material gradation One is the so-

called continuum model in which analytical functions such as exponent and power-law

functions are commonly used to describe the continuously varying material properties

Although the continuum model may not be physical in practice this model is convenient for

conducting mathematical analysis The other is the micromechanics model which takes into

account interactions between constituent phases and uses a certain representative volume

element (RVE) to estimate the average local stress and strain fields of the composite after

which the local average fields are used to evaluate the effective material properties The

Mori-Tanaka method [2] and the self-consistent method [3] are two representatives of these

models In this paper attention is focused on the continuum model only

From the view point of mathematics the thermoelastic analysis in FGMs is described by

partial differential equations with variable coefficients to which a closed-form analytical

solution is difficult to obtain and is available for limited problems with simple geometries

certain types of gradation of material properties specific types of boundary conditions and

special loading cases Therefore numerical methods have been developed for investigating

static or dynamic problems mainly involving the evaluation of temperature field and stress

fields to reduce dependency on costly and time consuming experimental analysis Among the

established numerical methods the finite element method (FEM) [4-6] or the graded finite

element method [7 8] the boundary element method (BEM) or boundary integral equation

method (BIEM) [9-11] are most versatile to deal with thermoelastic analysis More recently

as alternatives to the FEM and BEM meshless methods have been used for thermal analysis

of FGMs The method employs a set of scattered points instead of elements to approximate

solutions and exhibits advantages of avoiding mesh generation simple data preparation and

easy post-processing The corresponding developments in thermal and stress computation in

FGMs include Rao and Rahman [12] used element-free Galerkin method (EFGM) to

simulate stress fields near the crack tip in FGMs The same method was used by Dai et al

The Method of Functional Solutions hellip 125

[13] to study thermomechanical behavior of FGM plates Ching and Yen [14 15] analyzed

the static and transient responses of FGMs under mechanical and thermal loads by means of

the meshless local PetrovndashGalerkin (MLPG) method [16 17] Moreover Sladek et al solved

dynamic anti-plane shear crack problem and transient heat conduction in FGMs by a meshless

local boundary integral equation (LBIE) method [18 19]

As a Greenrsquos function-based meshless method the method of fundamental solution

(MFS) has been well established to determine the steady-state temperature distribution in

linear or nonlinear FGM with temperature-dependent thermal conductivity [20 21] by means

of the corresponding fundamental solutions or Greenrsquos functions [22] There are other similar

methods such as the virtual boundary collocation method [23] and charge simulation method

[24] F-Trefftz method [25] and the singularity method [26] These methods use essentially

fictitious source points outside the solution domain of interest and the corresponding

fundamental solutions to approximate the target function The unknown coefficients of the

fundamental solutions and the coordinates of the fictitious sources are found by forcing the

approximation to satisfy the boundary conditions Advantages of MFS include pure boundary

collocations good adaptivity and little data preparation This is because the Greenrsquos

functions used satisfy a priori the governing partial differential equation (PDE) for the

problem Moreover no any singular evaluations of fundamental solutions are encountered in

the MFS due to the distinctive locations of source points Although the conventional MFS

has been successfully applied to FGMs the application is yet very limited due to the fact that

the corresponding fundamental solutions or Greenrsquos functions for general FGMs are either not

available or mathematically too complex [22 27] The nonhomogeneous nature of FGMs

prohibits a simple construction and implementation of fundamental solutions for general

FGMs with various gradations Moreover when dealing with nonzero body forces or transient

problems the conventional MFS seems to be very inefficient

The objective of the chapter is to present a mixed meshless algorithm based on the MFS

and radial basis function (RBF) for analyzing two-dimensional thermomechanical problems

of FGMs with various graded behaviors In the present algorithm the analog equation method

(AEM) [28] or dual reciprocity method (DRM) [29] is used to obtain the equivalent

homogeneous system to the original nonhomogeneous equation and then RBF and MFS are

used to approximate the related particular part and complementary part respectively Finally

the enforcing satisfaction of governing equations at interpolation points and boundary

conditions at boundary nodes is used to determine all unknowns

The structure of the chapter is organized as follows Section 2 provides a full description

of the 2D thermomechanical system in FGMs In Section 3 the material properties of FGMs

used in this chapter are reviewed and the detailed solution procedure is presented in Sections

4 and 5 for transient thermal response and thermoelastic analysis respectively Some

conclusions are presented in Section 6

2 MATHEMATICAL FORMULATION

In this section basic formulations of thermoelasticity in FGMs are reviewed so that the

chapter is self-contained For the convenience of presentation the Cartesian tensor notation is

adopted The subscript comma in the following equations indicates a space derivative and


repeated subscripts in a variable represent summation Because FGMs can be viewed without

loss of generality as isotropic nonhomogeneous materials the following formulations and

processes are provided for general thermomechanical problems in 2D elastic solids

Furthermore it is well known that for a fully coupled thermomechanical problem such as

forging and casting it is not only the thermal field that influences the displacement and stress

fields but also the deformation itself that induces change in temperature distribution Here

for the sake of simplicity the thermomechanical deformation is considered to be sequentially

coupled in that sense that the temperature change influences the stress distributions only

21 Basic Equations of Heat Conduction in FGMs

(1) Heat Conduction Equation

Let us consider an isotropic and linear elastic domain bounded by the boundary

The Cartesian coordinates T

1 2( )x xx are used to describe temperature distribution and

infinitesimal static deformations The transient heat conduction in isotropic heterogeneous

media is then governed by the following relation

(1)

or

(2)

where T is the desired temperature field in the domain under consideration 1 2 i j

and represents the plane gradient and Laplace operators respectively

0t stands for spatial variable Parameters k c are the thermal conductivity density

and specific heat respectively which are assumed to depend on the space coordinate in our

analysis f denotes the internal heat source generated per unit volume

(2) Thermal boundary and initial conditions

To keep the system complete Eq (1) or (2) should be supplemented with the following

thermal boundary conditions

(3)

and the initial condition

(4)

0T t

k T t f t c tt

xx x x x x x

2

T tk T t k T t f t c

t

xx x x x x x x

2

11 22

1

2

3

T t T t

Tq t k q t

n

q t h T T

x x x

x x x

x x

00T Tx x


In Eq (3) T and q are specified values on the boundary 1 and

2 respectively h

and T stand for the coefficient of convection and the temperature of ambient fluid

respectively is the unit outward normal to the boundary 1

2 and 3 are

complementary parts of the boundary ie 1 2

2 3 1 3 and

1 2 3

22 Basic Equations of Thermoelasticity in FGMs

(1) Governing Equations

The governing equations for thermoelasticity involve the equilibrium equation

constitutive equation and strain-displacement relation For 2D continuously

nonhomogeneous isotropic and linear elastic FGMs the mechanical equilibrium requires

(5)

where ij denotes the components of Cauchy stress tensor and

ib the components of body

force per unit volume

The stress tensor ij and strain tensor

are related by the constitutive equation or the

generalized Hookersquos law which is given in the form

(6)

with

where E have different values for plane stress and plane strain states such that

and parameters ( ) ( )E x x and ( ) x are functions of space coordinates and represent

elastic modulus Poisson ratio and linear coefficient of thermal expansion respectively T

denotes the temperature change the material experiences with respect to the stress-free

reference configuration which can be determined by solving the heat conduction system If

the change in temperature is positive we have thermal expansion and if negative thermal

contraction

n

0 Ωij j ib x

ij

2ij ij kk ij ijm T

2

1 2

2 1

E

1 2

Em

2

for plane strain

1 2 1 for plane stress

1 1 21

E E

E E

x


If the displacement components are small enough that the square and product of its

derivatives are negligible then the relation of strain component and displacement

components iu can be written as

(7)

Substituting Eqs (6) and (7) into the equilibrium equation (5) yields the second-order

partial differential equation (PDE) in terms of displacement components as

(8)

(2) Mechanical Boundary Conditions

The boundary value problem (BVP) defined by Eqs (5) (6) and (7) is completed by

adding the following displacement and surface traction boundary conditions

(9)

where iu is the prescribed displacements on

u and it the given tractions on

t For a well-

posed problem we have nullu t and u t

3 MATERIAL PROPERTIES OF FGMS

Material properties of FGMs such as thermal conductivity density elastic modulus and

so on usually vary in space For illustrate this variation we take the ceramicmetal FGM as

an example The metalceramic FGM is often a mixture of two kinds of materials one is the

metal and the other is ceramic Without losing generality we assume that the left surface of

the FGM plate is ceramic rich and right is metal rich The region between the two surface

consists of material blended with both of them For convenience the x-axis is set along the

horizontal direction as illustrated in Figure 1 At any position x in the ceramicmetal FGM

the local volume fraction of metal is assumed to be ( )V x which can be used to characterize

the gradation Generally speaking ( )V x can be any non-singular non-negative function of x

To gain insight into the effect of material gradation on the thermoelastic behavior of the

FGM it is assumed that 1P and

2P are material parameters of ceramic and metal phases

respectively

ij

1( )

2ij i j j iu u

0k ki i kk i k k k i k k i i i iu u u u u mT m T b

i i u

i ij j i t

u u

t n t

x

x


Figure 1 Illustration of FGM structure

(1) Power-Law Type FGM (P-FGM)[30]

In this case the local volume fraction of metal ( )V x is assumed in the form of a simple

power-law distribution

(10)

where the power is the volume fraction exponent and L is the thickness of the FGM

layer It can be seen that the gradation given in Eq (10) implies that the FGM layer always

has 100 metal when ( ) 1V h and pure ceramic when (0) 0V which is of course

desirable

As a first order approximation the effective properties of a functionally graded material

can be obtained using the rule of mixtures for example

(11)

Figure 2 shows the variation of the effective material property versus non-dimensional

length with different power

(2) Exponential Type FGM (E-FGM)[31]

In this case the local volume fraction of metal ( )V x is assumed as

(12)

from which the effective properties of a functionally graded material can be given by

(13)

The gradient parameter in Eq (13) in fact can be determined by means of specified

material properties of the ceramic and metal phases

( ) V x x L

1 2( ) 1 ( ) ( )P x V x P V x P

( )x

LV x e

1 1( ) ( )x

LP x V x P Pe


(14)

and then the variation of the effective property along the graded direction is displayed in

Figure 2 for the purpose of comparison

Figure 2 Variation of the effective material property vs the non-dimensional thickness

It can be seen that the variation of graded parameter changes the material property of

FGMs Thus in the present work the effect of graded parameter is investigated to illustrate

the thermal and elastic behaviors of FGMs

4 THE METHOD OF FUNDAMENTAL SOLUTIONS FOR THERMAL

ANALYSIS

The boundary value problem (BVP) consisting of Eqs (1)-(4) can be converted into a

Poisson-type equation using the analog equation method (AEM) For this purpose suppose

2

1

lnP

P


( ) ( )tT T tx x is the sought solution to the BVP under consideration which is a continuously

differentiable function with up to two orders in If the Laplacian operator is applied to this

function namely

2 ( ) ( ) t tT b x x x (15)

then the solution of Eq (1) can be established by solving the linear equation (15) under the

same boundary conditions (3) and initial condition (4) if the fictitious source distribution

( )tb x is known

Itrsquos well known that the solution to the linear equation (15) can be written as a sum of the

complementary solution ( )t

hT x satisfying the following homogeneous equation

2 ( ) 0t

hT x (16)

and the particular solution satisfying the inhomogeneous equation

(17)

Then the total solutions for temperature field and heat flux at time instance t can be given

by

(18)

where ( )t

hq x and ( )t

pq x are the complementary and particular solutions for heat flux

respectively

41 Complementary Solutions

To obtain a weak solution of Laplace equation (16) the method of fundamental solution

is employed here In the MFS the desired solution can be expressed as a linear combination

of fundamental solutions or Greenrsquos functions associated with the governing equation under

consideration to guarantee prior the analytical satisfaction of the governing equation For this

purpose N fictitious source points ( 12 )si i Nx lying on the pseudo boundary the

virtual boundary similar to the physical boundary are selected as shown in Figure 3

Moreover it is assumed that at each source point there exists a virtual load t

i As a result

the potential ( )t

hT x and the boundary heat flux ( )t

hq x at any field point in the domain or on

the physical boundary can be written as [32-38]

( )t

pT x

2 ( ) ( )t t

pT b x x

( ) ( ) ( ) ( ) ( ) ( )t t t t t t

h p h pT T T q q q x x x x x x

x


1

1

( ) ( )

( ) ( )

Nt t

h i si si

i

Nt t

h i si si

i

T T

q Q

x x x x x

x x x x x

(19)

in which ( )siT x x and ( )siQ x x are the fundamental solution of the Laplace equation and its

normal derivative respectively

(20)

with

Figure 3 Illustration of the collocation points discretization on the physical and pseudo (virtual)

boundaries

42 Particular Solutions

RBFs are usually expressed in terms of Euclidian distance so they can work well in any

dimensional space Due to these advantages RBFs have been widely used in many practical

problems over the past decades In this section RBF approximation is presented for

evaluating the approximated particular solution at any given time t Firstly the right-hand

term ( )tb x in Eq (17) can be approximated by the standard dual reciprocity procedure

1 1 1 2 2 22

1( ) ln

2

( ) 1( )

2

sj

sisj si si

T r

TQ k k x x n x x n

n r

x x

x xx x

2 2

1 1 2 2si sir x x x x


1

( ) ( ) M

t t

j j j

j

b

x x x x x (21)

where M is the number of interpolation points including interior and boundary points for the

domain of interest t

j are coefficients to be determined and are a set of global RBF

with different collocation points

The effectiveness and accuracy of the interpolation depends on the choice of the RBFs

Besides the adhoc function 1+r which is merely a special type of RBF that is used

almost exclusively and uncritically in the engineering literature [33 39 40] the three radial

basis functions polyharmonic splines 2 11 ( 1)nr n thin plate spline (TPS) 2 lnr r and

multiquadrics (MQ) are also commonly used in numerical formulations [36 41-44]

In the RBFs mentioned above the Euclidean distance related to the field and collocation

points is defined as

(22)

Similarly the particular solutions in the domain and defined on the

boundary can also be written as

(23)

with k n if the space interpolation functions are chosen so as to satisfy the

relationship

Table 1 Relation of radial basis function (RBF) and particular solution kernel (PSK)

for the case of Laplace operator

RBF PSK

( )

( )j x x

jx

( )j x x

2 2r c

r

2 2

1 1 2 2j jr x x x x

( )t

pT x ( )t

pq x

1

1

Mt t

p j j

j

Mt t

p j j

j

T

q

x x x

x x x

2 11 nr 1n

2 2 1

24 2 1

nr r

n

2 lnr r4 41 1

ln16 32

r r r

2 2r c 3 2 2 2ln 4

3 9

c c c r c


(24)

In Eq (23) usually refer to the particular solutions kernels (PSK) and the

corresponding expression of PSK for a given RBF is presented in Table 1

43 Complete Solutions

Based on the discussion above the complete solutions at a particular time t can be written

as

(25)

Moreover differentiating Eq (25) with respect to coordinate component yields

(26)

Next in order to obtain the temperature field and heat flux at any time a two-level finite

differencing in time is used For a typical time interval 1[ ] ( 0)k kt t k with time step

1k kt t t the relationship

(27)

leads to by the substitution of Eq (27) into Eq (2)

(28)

2 ( )j j x x x x

( )j x x

1 1

1 1

( ) ( ) s

N Mt t t

i si j j

i j

N Mt t t

i si j j

i j

T T

q Q

x x x x x

x x x x x

1 1

t N Mjsit t

i j

i jk k k

T T

x x x

x xx x x

1

1

1

1

1

k k

k k

k k

T t u u

f t f f

T TT

t t

x x x

x x x

x x

1 1

2 1

2

1

1

1 1

k k

k

k k

k

k k

k T c TT

k k t

k T c TT

k k t

f fk

x x x x xx

x x

x x x x xx

x x

x xx


In Eq (27) the time-step parameter usually assumes values between 1 (backward

differencing) and 0 (fully explicit scheme) Value 05 yields the Crank-Nicolson scheme

(central differences) known to be the most accurate two-level time stepping strategy

However for the first time step only backward differencing makes sense because other

schemes require that the initial values of the heat fluxes are known As these quantities are

not needed for the analytical solution they should also not arise in the numerical algorithm

On the other hand the backward scheme is unconditionally stable In the present work the

backward time stepping scheme is employed to perform the following analysis for simplicity

Let 1 then Eq (28) reduces to

(29)

At the same time the boundary conditions at 1kt time instance can be written as

(30)

Subsequently N points are chosen on the physical boundary to solve the system

consisting of Eqs (29) and (30) For this purpose substituting Eqs (25) and (26) into Eqs

(29) and (30) yields the following N M equations to determine all unknowns

(31)

where 1 1N

2 2N 3 3N and

1 2 3N N N N The operator L is defined for

convenience as fellows

(32)

1 1 1

2 1

k k k k

kk T c T c T f

Tk k t k t k

x x x x x x x x xx

x x x x

1 1

1

1 1

2

1 1

3

on

on

on

k k

k k

k k

T u t

q q t

h T q h T

x x

x x

x x

1

1

1 1

1 1

1

1

1

k kN Mm mk k

i m si j m j

i j m m

Nk

i n si

i

f c TT

k k t

m M

T

x x x xL x x L x x

x x

x x

1 1

2 2 2

3 3 3 3

1 1

1 1

1

1 1 1

2 2

1 1

1 1

1 1

1

1

Mk k

j n j n

j

N Mk k k

i n si j n j n

i j

N Mk k

i n si n i j n j n j

i j

u n N

Q q n N

h T Q h

x x x

x x x x x

x x x y x x x x

3 3 1

h u

n N

2

k c

k k t

x x xL I

x x


44 Numerical Examples

In order to demonstrate the efficiency and accuracy of the proposed meshless method and

the selected RBF and virtual boundary transient heat conduction in isotropic materials is first

considered since corresponding analytical results can be used for verification Then the

transient thermal response in FGMs is discussed Though the proposed meshless method has

no restrictions on the spatial variation of the material parameters of FGM the numerical

example presented here is restricted to an exponential variation of the material properties with

Cartesian coordinates for the purpose of comparison

Additionally itrsquos necessary to note that the location of the pseudo boundary is important

to the final numerical stability In the present work the source point is generated by [33-38]

(33)

where the nondimensional parameter 1 is named as similarity ratio and sx

bx and cx

are source point boundary point and central point of the domain respectively

Example 441 Thermal shock problem

To investigate the behavior of the algorithm in the presence of thermal shocks the

benchmark problem in [45] is considered and the solution obtained using the developed

technique is compared with an analytical solution The computing geometry is a unit square

[01]times[01] in which no internal heat source exists Zero initial temperature has been assumed

and homogeneous Neumann boundary conditions (insulation) are prescribed on the sides x =

0 y = 0 and y = 1 respectively The remaining side is subjected to a sudden unit temperature

jump Using the method of variable separation the analytic solution can be obtained as

2

0

4( ) 1 ( 1) cos( )exp( )

(2 1)

i

i i

i

T x t x ti

(34)

with (2 1) 2i i

In the computation thermal diffusivity a = 1 m2s and thermal conductivity of materials k

= 1W(m) is assumed The uniform interpolation scheme is used with the first order

interpolation function 1+r only A total of 20 fictitious source points are selected on the

virtual boundary and 121 uniform interpolation points are used unless there is a special

statement To study the effect of the location of the virtual boundary on the accuracy of the

proposed algorithm Figure 4 presents the relative error of temperature versus similarity ratio

at point (05 05) with time step ∆t= 001 s The results in Figure 4 show that good

computational accuracy and stability is achieved when the similarity ratio is greater than 2

and the optimal value of the similarity ratio is between 25ndash50 Although the virtual

boundary can theoretically be chosen arbitrarily outside of the domain either too small or too

great a distance between the virtual and physical boundaries will reduce accuracy due to the

singularity of the fundamental solution and the restriction of computer precision including

round-off error [46]

( )( 1) ( 1)s b b c b c x x x x x x


Figure 5 shows the percentage error of temperature for two different time steps It can be

seen that the smaller the time step the higher the accuracy of the results obtained However

more computational time will inevitably be required if a smaller time step is chosen

Additionally further reduction in the time step doesnrsquot reduce the relative error [47]

Figure 4 Effect of similarity ratio on temperature at point (05 05) with ∆t = 001 s

Figure 5 Effect of time step on relative error of temperature with γ = 30


Consider a functionally graded square [0 L]times[0 L] with a unidirectional variation of

thermal conductivity [48] In this example zero initial temperature is considered and the same

exponential spatial variation for thermal conductivity and diffusivity is assumed

1 15 2 25 3 35 4 45 5 0

1

2

3

4

5

6

7

Similarity ratio

Re

lative

err

or

in

te

mp

era

ture

t = 05s

t = 10s

0 01 02 03 04 05 06 07 08 09 1 0

1

2

3

4

5

6

7

8

9

x (m)

Re

lative

err

or

in

te

mp

era

ture

t = 05s t = 01s

t = 05s t = 001s

t = 10s t = 01s

t = 10s t = 001s


(35)

where k0=17W(moC) and a0 = 017 m

2s Two different exponential parameters η = 02 and

05 cm-1

are assumed in numerical calculation On the sides parallel to the y-axis two different

temperatures are prescribed The left side is kept at zero temperature and the right side has the

Heaviside function of time ie u = H(t) On the lateral sides of the square the heat flux

vanishes In the numerical calculation the side length L = 004 m is used The special case

with an exponential parameter η = 0 is considered first In this case the analytical solution is

given as

2 2

21

2 cos( ) sin exp

n

x T n n x an tT x t T

L n L L

(36)

which can be used to verify the accuracy of the present numerical method Numerical results

are obtained using 36 fictitious source points 169 interpolation points γ = 30 and time step

∆t = 1 s Figure 6 shows the temperature field at three points (x = 001 m 002 m and 003 m)

A good agreement between numerical and analytical results is observed from Figure 6

0 10 20 30 40 50 60

-01

0

01

02

03

04

05

06

07

08

Time t (second)

Te

mp

era

ture

(

)

Meshless x=001

x=002

x=003

Analytical x=001

x=002

x=003

Figure 6 Time variation of temperature in a finite square strip at three different positions with η = 0

The discussion above concerns heat conduction in homogeneous materials only since

analytical solutions can be used for verification To illustrate the application of the proposed

algorithm to the FGM consider now the FGM with η = 02 and 05 cm-1

respectively The

variation of temperature with time for three k-values and at position x = 002 m is presented

in Figure 7 As expected it is found from Figure 7 that the temperature increases along with

an increase in η-values (or equivalently in thermal conductivity) and the temperature

approaches a steady state when t gt20 s For final steady state an analytical solution can be

obtained as

0 0( ) ( )x xk x k e a x a e


(37)

Figure 7 Time variation of temperature at position x = 002m of functionally graded finite square strip

Analytical and numerical results computed at time t =70 s corresponding to stationary or

static loading conditions are presented in Figure 8 The numerical results are in good

agreement with the analytical results for the steady state case Simulateneously it is observed

from Figure 8 that the temperature increases along with an increase in η-values again This is

because the larger thermal conductivity results in smaller resistance to heat transfer from the

right to left

For comparison the results at some particular points obtained by both the proposed

method and the meshless local boundary integral equation method (LBIEM) [42] are listed in

Table 2 It can be seen from Table 2 that the results from the proposed method is slightly

larger than those obtained LBIEM and after t = 50 s the temperature reaches a relatively

steady state It should be mentioned here that the numerical solutions given in reference [42]

probably have certain error to practical computing results produced using LBIEM Moreover

different treatments of time domain may also be the main reason causing the discrepancy In

the derivation of LBIEM we noticed that Laplace transformation technology is used instead

of the time stepping scheme However to the steady-state temperature field at x = 001 m the

two methods provided almost same results as shown in Table 2

Table 2 Comparison of LBIEM and the proposed method at η =05cm-1

and x = 001 m

t=10s t=20s t=30s t=40s t=50s t=60s Stable

LBIEM 01871 03281 03800 03986 04019 04053 04581

MFS 03915 04497 04546 04550 04551 04551 04551

Exact 04551

1( ) ( with 0)

1

x

L

e xT x T

e L


Figure 8 Distribution of temperature along x-axis for a functionally graded finite square strip under

steady-state loading conditions

5 THE METHOD OF FUNDAMENTAL SOLUTIONS FOR

THERMOELASTIC ANALYSIS

For the thermoelastic equation (8) describing displacement responses in general

nonhomogeneous media the fundamental solutions are difficult to obtain in a closed form

However we can circumvent this obstacle by indirect ways From the viewpoint of

mathematics the displacement fields must be in terms of space coordinates regardless of the

particular forms of elastic properties and loading types So we can design an equivalent

elastic system as

(38)

to replace Eq (8) where and are elastic constants of a fictitious isotropic homogeneous

solid and ib the lsquofictitious body forcersquo induced by the sought displacement distributions and

the temperature change

For Eq (38) the solution variables iu can be divided into two parts ie the

complementary solutions h

iu and the particular solutions p

iu that is

(39)

in which the complementary solutions h

iu has to satisfy the homogeneous equation as

(40)

0k ki i kk iu u b

( ) ( ) ( )h p

i i iu u u x x x

0h h

k ki i kku u


while the particular solutions p

iu are required to satisfy the following inhomogeneous

equation

(41)

Obviously the particular solutions and homogeneous solutions satisfying Eqs (40) and

(41) respectively are not unique without considering the constraints of boundary conditions


To obtain an approximate solution of homogeneous equation (40) N fictitious source

points ( 12 )si i Nx locating on the pseudo boundary outside the domain under

consideration are selected Moreover assume that at each source point there is a pair of

fictitious point loads 1i and

2i along 1- and 2- directions respectively According to the

main construction of the MFS the approximate displacement fields at arbitrary points in

the domain or on the boundary can be expressed as a linear combination of fundamental

solutions in terms of assumed sources that is

1

sN

h

i nl li sn

n

u U

x x x (42)

in which the displacement fundamental solution ( )li snU x x denoting the induced displacement

distribution along the i-direction at the field point due to the unit concentrated load acting

in the l-direction at source point snx satisfies the following Navier equation

(43)

Such that is the Dirac delta function concentrated at the source point snx and

lie are the components of the 2 by 2 identity matrix For the case of plane strain the

displacement fundamental solution can be written as [49]

(44)

It is apparent that Eq (42) completely satisfies Eq (40) in the domain based on the

definition of the fundamental solutions and the fact that source point and field point canrsquot

overlap in the MFS

0p p

k ki i kk iu u b

x

x

( ) ( ) lk ki sn li kk sn sn liU U e x x x x x x

sn x x

1 1 (3 4 ) ln

8 (1 )li li l iU v r r

v r

x y

snx x



In this section RBFs are used to derive the displacement particular solutions Firstly the

generalized fictitious body forces are approximated as

(45)

where M is the number of interpolating points in the domain m

l are coefficients to be

determined and ( )m x x is a set of RBFs

Similarly the particular solution ( )p

iu x is also approximated by means of the same

coefficient set

(46)

where ( )li m x x is a corresponding kernel of approximate particular solutions Because the

particular solution ( )p

iu x satisfies Eq (41) the precondition to this process is that such

relations

(47)

holds true

Generally the particular solution kernel li can be expressed by the second order

differential of Galerkin-Papkovich function liF as [50]

(48)

Substituting Eq (48) into the left hand term of Eq (47) yields

(49)

where 4 denotes the biharmonic operator As a result we have

(50)

Due to the property of RBFs associated with the Euclidian distance only itrsquos convenient

to write the biharmonic operator in polar coordinate for an assumed function in terms of r

only that is

1 1

( ) ( ) ( )M M

m m

i m i li m l

m m

b

x x x x x

1

( ) ( )M

p m

i li m l

m

u

x x x

( ) ( ) ( )lk ki m li kk m li m x x x x x x

1 1

2li li mm mi mlF F

4

1 = 11 2

kl ki li kk li mmkk liF F

4 1

1li liF


(51)

with Thus integrating Eq (50) yields the expression of liF and then the

required particular solution kernel can be derived using Eq (48)

For the typical conical spline 2 1 ( 1)nr n and thin plate spline (TPS)

2 ln ( 1)nr r n the corresponding set of particular solutions are given as [51]

(1) Conical spline

(52)

with

(2) Thin plate spline

(53)

with


According to Eq (39) the complete solutions of displacement components are written as

the sum of the particular and homogeneous solutions thus we have

1 1

( ) ( ) ( )N M

n m m

i li n l li l

n m

u U

x x y x (54)

Consequently the stress components can be expressed by substituting Eq (54) into Eqs

(7) and (6) as

4 2 2 1 d d 1 d d

d d d dr r r r

r r r r r r

mr x x

2 1

1 2 2

1 1

2 1 2 1 2 3

n

li li l ir A A r rn n

1

2

4 5 2 2 3

2 1

A n n

A n

2 2

1 2 3 2

1

32 1 1 2

n

li il i l

rA A r r

n n

22

1

2

8 29 27 8 2 2 1 2 4 7 4 2 ln

2 1 2 3 2 1 2 ln

A n n n n n n n r

A n n n n r


1 1

( ) ( ) ( )N M

n m

ij lij n l lij m l ij

n m

S m T

x x y x x (55)

where

(56)

Finally the satisfaction of Eq (54) to the governing Eq (8) at M interpolation points

and substitution of Eq (54) and (55) into boundary conditions (9) at N boundary nodes

produce a set of linear algebraic equations which can be written in matrix form as

(57)

where the unknown coefficient vector is


For simplicity the temperature distribution is taken in a close form in the following

computation rather than numerical solution of temperature boundary value problems (BVP)

consisting of heat conduction theory Furthermore in this section three examples of FGM

subjected to mechanical or thermal loads are considered to assess the proposed algorithm In

all three examples except for Poissonrsquos ratio the material properties vary exponentially or

according to a power law This is a reasonable assumption since variation on the Poissonrsquos

ratio is usually small compared with that of other properties

To assess the accuracy and convergence of the approximation the average relative error

( )Arerr defined by

(58)

is introduced where j and j are respectively the analytical and numerical results of

variable at the sample points of interest and L is the total number of these points

lij ij lk k li j lj i


S U U U

H A B

T

1 1 1 1

1 2 1 2 1 2 1 2

N N M M A

2

1

2

1

L

j j

j

L

j

j

Arerr


Example 541 Hollow circular plate under radial internal pressure

Consider a hollow circular plate as shown in Figure 9 with inner radius 5 mma and

outer radius 10 mmb under internal radial pressure Suppose that the plate is graded along

the radial direction so that elastic modulus For 0 the Youngrsquos

modulus increases as the radius increases When 0 the problem is reduced to the

analysis of homogeneous media Analytical solutions of stress components for the case of

plane stress state are given in closed form as [52]

(59)

with

Figure 9 Configuration of hollow circular plate under internal pressure

In the practical computation Poissonrsquos ratio and elastic modulus at the internal surface

as well as internal pressure respectively are assumed to be 03 ( ) 200 GPaE a

50 MPaap Figure 10 and Figure 11 display the convergent performance of the proposed

meshless method when the PS basis function 3r is used It is found from Figure 10 and Figure

11 that the accuracy increases with an increase in M or N

In order to investigate the variation of radial and hoop stresses along the radial direction

for various graded parameters 32 boundary nodes and 140 interior interpolation points are

used Comparisons between analytical solutions and numerical results are shown in Figure

12

0( ) ( )E r E r a

r

12 2 2 1

2

12 2 2 1

22 2

2 2

kk

k k

r ak k

kk k k

ak k

a ra b r p

b a

k r k ba ra p

b a k k

2 4 4k


Figure 10 Convergent performance vs M with 2 and N=32

Figure 11 Convergent performance vs N with 2 and M=140

It is found that regardless of the value of radial stress increases monotonously from

the inner to the outer surface whereas hoop stress does not As increases the value of

radial stress decreases at any point in the cylinder except for the points on the boundary

whereas the maximum hoop stress occurs on the inner surface when 0 and on the outer

surface when 3


Figure 12 Distribution of radial and hoop stresses with various graded parameter when 32 boundary

nodes and 140 interior interpolation points are used

The variation in the hoop stress looks like rotation around a center when increases It

is also found that the variation in hoop stress in FGMs becomes worse when increases

Therefore to avoid material instability the graded parameter should be smaller than some

critical values

In this example effects of the types and orders of RBF are also tested for the case of the

high graded parameter 4


Figure 13 Effect of orders of radial basis functions when 32 boundary nodes and 220 interior

interpolation points are used

Figure 13 shows the average relative error distributions It is evident that a higher order

of RBF does not always result in better accuracy The calculation indicates that 3r and

5r in

PS and 2 lnr r and

4 lnr r in TPS seem to be able to produce relatively high accuracy in this

example

Moreover TPS has better accuracy than that of PS Therefore 4 lnr r is used in the

remaining computation

Example 542 Functionally graded elastic beam under sinusoidal transverse load

An elastic beam shown in Figure 14 is considered in this example which is made of two-

phase AlSiC composite The elastic modulus varying exponentially in the z direction is given

by 0( ) zE z E e The left and right end faces of the FG beam are assumed to be simply

supported such that


(60)

The top surface of the beam is assumed to be free of mechanical force and the bottom

surface is subjected to a distributed load p as shown in Figure 14

(61)

The problem is solved under a plane strain assumption with the length 100 mmL and

thickness 40 mmh The material properties of aluminum and SiC are respectively

70 GPaAlE and 427 GPaSiCE ( [ln( )]SiC AlE E h ) The maximum transverse load

0p is

equal to 10MPa Total 34 boundary nodes and 169 interior interpolation points are selected in

the analysis

Figure 14 Functionally graded beam subjected to symmetric sinusoidal transverse loading

Figure 15 (continued)

(0 ) ( ) 0

(0 ) ( ) 0x x

w z w L z

t z t L z

( 0) ( ) 0

( 0) ( ) 0

x x

z z

t x t x h

t x p t x h


Figure 15 Transverse displacement and stress components along the line 2z h

Figure 16 Variation of stress components along the cross section 5x L

Figure 15 and Figure 16 respectively show the variation of transverse displacement and

stress components along the line 2z h and 5x L Good agreement can be observed

between the numerical results and analytical solutions [53] Furthermore the shapes of cross-

sections after deformation are provided in Figure 17 from which it can be seen that for

smaller ratios of thickness and length for example 110h L the cross section

approximately maintains plane after deformation This phenomenon demonstrates the validity

of the cross-section assumption in classic thin beam bending theory


Figure 17 Shape of transverse cross section after deformation with various ratio of thickness and

length

Example 543 Symmetrical thermoelastic problem in a long cylinder

Consider a thick hollow cylinder with same geometries and mechanical boundary

conditions as in Figure 9 The same power-law assumptions are used to define the elastic

modulus and coefficient of thermal expansion that is 0( ) ( )E r E r a and 0( ) ( )r r a

The temperature change in the entire domain is given in a closed form as

(62)

with ( )aT T a and ( )bT T b

The two-phase aluminumceramic FGM is examined here The metal aluminum

constituent is arranged on the inner surface while the ceramic constituent is on the outer

surface The related material properties are 70 GPaAlE 6 o12 10 CAl

151 GPaceramicE 6 o259 10 Cceramic Poissonrsquos ratio is taken to be 03 The inner

and outer boundary temperature changes respectively are o10 CaT and o0 CbT

for 0

ln ln

for 0

ln

a b

a b

T b r T r a

b a

b rT T Tr a

b

a


Figure 18 Stresses and radial displacement distributions in FGM and homogeneous material with

N=32 M=220

Analytical solutions of displacements and stresses for the case of plane strain state are

provided by Jabbari et al [54] The results in Figure 18 show good agreement between the

analytical solutions and the numerical results in FGM and homogeneous material which

corresponds to 0 Furthermore we again find that after graded treatment the maximum

value of hoop stress decreases from 826MPa to 53MPa Additionally the radial displacement

in FGM also decreases compared to the response in homogeneous media Since the value of

radial displacement is very small radial deformation can be neglected in practical analysis

CONCLUSIONS

The paper presents an efficient meshless method for transient heat transfer and

thermoelastic analysis of FGMs in which the combination of MFS and RBF provides a

powerful numerical procedure Numerical experiments show that a good agreement is

achieved between the results obtained from the proposed meshless method and available


analytical solutions It is clear that the temperature and stress responses in FGMs differ

substantially from those in their homogeneous counterparts The appropriate graded

parameter can lead to different temperature distribution low stress concentration and little

change in the distribution of stress fields in the domain under consideration

Additionally from the solution procedure we can see that the construction of the full

displacement variables is independent of the type of problem of interest This characteristic

means that the proposed method can be easily extended to other spatial variations of the

material parameters of FGM and other engineering problems instead of restrictions on

exponential variation of the material properties with Cartesian coordinates

On the other hand we must note that the proposed method still has some disadvantages

such as the linear system of equations formed at length being dense and possibly being ill-

conditioned for large and complex domains which are also the inherent disadvantages of

conventional MFS approximation

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[5] J Jirousek and QH Qin Application of Hybrid-Trefftz element approach to transient

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[9] QH Qin Nonlinear analysis of Reissner plates on an elastic foundation by the BEM

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[10] C Saizonou R Kouitat-Njiwa and J von Stebut Surface engineering with

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[11] A Sutradhar and GH Paulino The simple boundary element method for transient heat

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[12] BN Rao and S Rahman Mesh-free analysis of cracks in isotropic functionally graded

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[13] KY Dai GR Liu X Han and KM Lim Thermomechanical analysis of functionally

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[14] HK Ching and SC Yen Meshless local Petrov-Galerkin analysis for 2D functionally

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[15] HK Ching and SC Yen Transient thermoelastic deformations of 2-D functionally

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[16] J Sladek V Sladek and Ch Zhang Stress analysis in anisotropic functionally graded

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[17] DF Gilhooley JR Xiao RC Batra MA McCarthy and JW Gillespie Jr Two-

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[18] J Sladek V Sladek and Ch Zhang A meshless local boundary integral equation

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[19] V Sladek J Sladek M Tanaka and Ch Zhang Transient heat conduction in

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Analysis with Boundary Elements 29 (2005) 1047-1065

[20] L Marin Numerical solution of the Cauchy problem for steady-state heat transfer in

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[21] L Marin and D Lesnic The method of fundamental solutions for nonlinear

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6878-6890

[22] JR Berger PA Martin V Mantiˇc and LJ Gray Fundamental solutions for steady-

state heat transfer in an exponentially graded anisotropic material Zeitschrift fuuml r

Angewandte Mathematik und Physik 56 (2005) 293-303

[23] HC Sun LZ Zhang Q Xu and YM Zhang Nonsingularity Boundary Element

Methods Dalian University of Technology Press (in Chinese) Dalian 1999

[24] MA Golberg CS Chen and JA Fromme Discrete projection methods for integral

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[25] K Balakrishnan and PA Ramachandran A particular solution Trefftz method for non-

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150 (1999) 239-267

[26] LC Nitsche and H Brenner Hydrodynamics of particulate motion in sinusoidal pores

via a singularity method AIChE Journal 36 (1990) 1403-1419

[27] YS Chan LJ Gray T Kaplan and GH Paulino Greens function for a two-

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460 (2004) 1689-1706

[28] JT Katsikadelis The analog equation method- A powerful BEM-based solution

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182


[29] D Nardini and CA Brebbia A new approach to free vibration analysis using

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[30] G Bao and L Wang Multiple cracking in functionally graded ceramicmetal coatings


[31] F Delale and F Erdogan The crack problem for a nonhomogeneous plane Journal of

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[32] G Fairweather and A Karageorghis The method of fundamental solutions for elliptic

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[34] WA Yao and H Wang Virtual boundary element integral method for 2-D

piezoelectric media Finite Elements in Analysis and Design 41 (2005) 875-891

[35] H Wang and QH Qin A meshless method for generalized linear or nonlinear

Poisson-type problems Engineering Analysis with Boundary Elements 30 (2006) 515-

521

[36] H Wang and QH Qin Some problems with the method of fundamental solution using

radial basis functions Acta Mechanica Solida Sinica 20 (2007) 21-29

[37] H Wang QH Qin and YL Kang A meshless model for transient heat conduction in

functionally graded materials Computational Mechanics 38 (2006) 51-60

[38] H Wang and QH Qin Meshless approach for thermo-mechanical analysis of

functionally graded materials Engineering Analysis with Boundary Elements 32 (2008)

704-712

[39] MA Golberg CS Chen H Bowman and H Power Some comments on the use of

Radial Basis Functions in the Dual Reciprocity Method Computational Mechanics 21

(1998) 141-148

[40] PW Partridge CA Brebbia and LC Wrobel The dual reciprocity boundary element

method Computational Mechanics Publications Southampton 1992

[41] AHD Cheng Particular solutions of Laplacian Helmholtz-type and polyharmonic

operators involving higher order radial basis functions Engineering Analysis with

Boundary Elements 24 (2000) 531-538

[42] CS Chen and YF Rashed Evaluation of thin plate spline based particular solutions

for Helmholtz-type operators for the DRM Mechanics Research Communications 25

(1998) 195-201

[43] M Golberg Recent developments in the numerical evaluation of particular solutions in

the boundary element method Applied Mathematics and Computation 75 (1996) 91-

101

[44] PA Ramachandran and K Balakrishnan Radial basis functions as approximate

particular solutions review of recent progress Engineering Analysis with Boundary

Elements 24 (2000) 575-582

[45] RA Bialecki P Jurgas and G Kuhn Dual reciprocity BEM without matrix inversion

for transient heat conduction Engineering Analysis with Boundary Elements 26 (2002)

227-236

[46] P Mitic and YF Rashed Convergence and stability of the method of meshless

fundamental solutions using an array of randomly distributed sources Engineering



[47] R Schaback Error estimates and condition numbers for radial basis function

interpolation Advances in Computational Mathematics 3 (1995) 251-264

[48] J Sladek V Sladek and Ch Zhang Transient heat conduction analysis in functionally

graded materials by the meshless local boundary integral equation method

Computational Materials Science 28 (2003) 494-504

[49] CA Brebbia and J Dominguez Boundary elements an introductory course

Computational Mechanics Publications Southampton 1992

[50] APS Selvadurai Partial Differential Equations in Mechanics Springer Berlin 2000

[51] AHD Cheng CS Chen MA Golberg and YF Rashed BEM for theomoelasticity

and elasticity with body force--a revisit Engineering Analysis with Boundary Elements

25 (2001) 377-387

[52] CO Horgan and AM Chan The pressurized hollow cylinder or disk problem for

functionally graded isotropic linearly elastic materials Journal of Elasticity 55 (1999)

43-59

[53] BV Sankar An elasticity solution for functionally graded beams Composites Science

and Technology 61 (2001) 689-696

[54] M Jabbari S Sohrabpour and MR Eslami Mechanical and thermal stresses in a

functionally graded hollow cylinder due to radially symmetric loads International

Journal of Pressure Vessels and Piping 79 (2002) 493-497

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Additional color graphics may be available in the e-book version of this book

Library of Congress Cataloging-in-Publication Data

Functionally graded materials editor Nathan J Reynolds p cm Includes index ISBN 97S-1-6 I 209-616-2 (hardcover) I Functionally gradient materials 1 Reynolds Nathan 1 TA4IS9FS5FS42011 620 J IS--dc23 2011027544

Published by Nova Science Publishers Inc t New York


319

CONTENTS

vii









tm or mical

~sed or - No out of

ial ~ of or

~ Icated

~~d in anlage

- nise

ni 10 the

F THE

Preface

Chapter 1

Chapter 2

Chapter 3

Chaptemiddot 4

Chapter 5

bapter 6

C pteI 7

pt ) 8



Chapter 3



GRADED MATERIALS

Hui Wang12

and Qing-Hua Qin3







ABSTRACT













numerical results





1 INTRODUCTION



































































































(1)

or

(2)









(3)


(4)

0T t

k T t f t c tt

xx x x x x x

2


t

xx x x x x x x

2

11 22

1

2

3

T t T t

Tq t k q t

n

q t h T T

x x x

x x x

x x

00T Tx x



2 respectively h



2 and 3 are


2 3 1 3 and

1 2 3






(5)







(6)

with







contraction

n

0 Ωij j ib x

ij

2ij ij kk ij ijm T

2

1 2

2 1

E

1 2

Em

2

for plane strain


1 1 21

E E

E E

x





(7)



(8)




(9)



t For a well-















respectively

ij

1( )

2ij i j j iu u


i i u

i ij j i t

u u

t n t

x

x






(10)




desirable



(11)





(12)


(13)



( ) V x x L

1 2( ) 1 ( ) ( )P x V x P V x P

( )x

LV x e

1 1( ) ( )x

LP x V x P Pe


(14)








ANALYSIS



2

1

lnP

P




function namely

2 ( ) ( ) t tT b x x x (15)



( )tb x is known




2 ( ) 0t

hT x (16)


(17)


by

(18)

where ( )t

hq x and ( )t


respectively









i As a result

the potential ( )t




( )t

pT x

2 ( ) ( )t t

pT b x x

( ) ( ) ( ) ( ) ( ) ( )t t t t t t


x


1

1

( ) ( )

( ) ( )

Nt t

h i si si

i

Nt t

h i si si

i

T T

q Q

x x x x x

x x x x x

(19)



(20)

with


boundaries







1 1 1 2 2 22

1( ) ln

2

( ) 1( )

2

sj

sisj si si

T r

TQ k k x x n x x n

n r

x x

x xx x

2 2



1

( ) ( ) M

t t

j j j

j

b

x x x x x (21)












(22)



(23)


relationship



RBF PSK

( )

( )j x x

jx

( )j x x

2 2r c

r

2 2

1 1 2 2j jr x x x x

( )t

pT x ( )t

pq x

1

1

Mt t

p j j

j

Mt t

p j j

j

T

q

x x x

x x x

2 11 nr 1n

2 2 1

24 2 1

nr r

n

2 lnr r4 41 1

ln16 32

r r r

2 2r c 3 2 2 2ln 4

3 9

c c c r c


(24)





as

(25)


(26)




(27)


(28)

2 ( )j j x x x x

( )j x x

1 1

1 1

( ) ( ) s

N Mt t t

i si j j

i j

N Mt t t

i si j j

i j

T T

q Q

x x x x x

x x x x x

1 1

t N Mjsit t

i j

i jk k k

T T

x x x

x xx x x

1

1

1

1

1

k k

k k

k k

T t u u

f t f f

T TT

t t

x x x

x x x

x x

1 1

2 1

2

1

1

1 1

k k

k

k k

k

k k

k T c TT

k k t

k T c TT

k k t

f fk

x x x x xx

x x

x x x x xx

x x

x xx











(29)


(30)




(31)

where 1 1N

2 2N 3 3N and



(32)

1 1 1

2 1

k k k k

kk T c T c T f

Tk k t k t k

x x x x x x x x xx

x x x x

1 1

1

1 1

2

1 1

3

on

on

on

k k

k k

k k

T u t

q q t

h T q h T

x x

x x

x x

1

1

1 1

1 1

1

1

1

k kN Mm mk k

i m si j m j

i j m m

Nk

i n si

i

f c TT

k k t

m M

T

x x x xL x x L x x

x x

x x

1 1

2 2 2

3 3 3 3

1 1

1 1

1

1 1 1

2 2

1 1

1 1

1 1

1

1

Mk k

j n j n

j

N Mk k k

i n si j n j n

i j

N Mk k


i j

u n N

Q q n N

h T Q h

x x x

x x x x x

x x x y x x x x

3 3 1

h u

n N

2

k c

k k t

x x xL I

x x












(33)


bx and cx










2

0

4( ) 1 ( 1) cos( )exp( )

(2 1)

i

i i

i

T x t x ti

(34)

with (2 1) 2i i














( )( 1) ( 1)s b b c b c x x x x x x












1 15 2 25 3 35 4 45 5 0

1

2

3

4

5

6

7

Similarity ratio

Re

lative

err

or

in

te

mp

era

ture

t = 05s

t = 10s

0 01 02 03 04 05 06 07 08 09 1 0

1

2

3

4

5

6

7

8

9

x (m)

Re

lative

err

or

in

te

mp

era

ture

t = 05s t = 01s

t = 05s t = 001s

t = 10s t = 01s

t = 10s t = 001s


(35)



05 cm-1






given as

2 2

21

2 cos( ) sin exp

n


L n L L

(36)





0 10 20 30 40 50 60

-01

0

01

02

03

04

05

06

07

08

Time t (second)

Te

mp

era

ture

(

)

Meshless x=001

x=002

x=003

Analytical x=001

x=002

x=003





respectively The





obtained as

0 0( ) ( )x xk x k e a x a e


(37)







right to left












and x = 001 m


LBIEM 01871 03281 03800 03986 04019 04053 04581

MFS 03915 04497 04546 04550 04551 04551 04551

Exact 04551

1( ) ( with 0)

1

x

L

e xT x T

e L











elastic system as

(38)







iu that is

(39)



(40)

0k ki i kk iu u b

( ) ( ) ( )h p

i i iu u u x x x

0h h

k ki i kku u




equation

(41)












1

sN

h

i nl li sn

n

u U

x x x (42)




(43)




(44)



overlap in the MFS

0p p

k ki i kk iu u b

x

x


sn x x

1 1 (3 4 ) ln


v r

x y

snx x





(45)






coefficient set

(46)




relations

(47)

holds true



(48)


(49)


(50)



only that is

1 1

( ) ( ) ( )M M

m m

i m i li m l

m m

b

x x x x x

1

( ) ( )M

p m

i li m l

m

u

x x x


1 1

2li li mm mi mlF F

4

1 = 11 2


4 1

1li liF


(51)





(1) Conical spline

(52)

with


(53)

with




1 1

( ) ( ) ( )N M

n m m

i li n l li l

n m

u U

x x y x (54)


(7) and (6) as

4 2 2 1 d d 1 d d

d d d dr r r r

r r r r r r

mr x x

2 1

1 2 2

1 1

2 1 2 1 2 3

n


1

2

4 5 2 2 3

2 1

A n n

A n

2 2

1 2 3 2

1

32 1 1 2

n

li il i l

rA A r r

n n

22

1

2

8 29 27 8 2 2 1 2 4 7 4 2 ln

2 1 2 3 2 1 2 ln

A n n n n n n n r

A n n n n r


1 1

( ) ( ) ( )N M

n m


n m

S m T

x x y x x (55)

where

(56)




(57)











( )Arerr defined by

(58)





S U U U

H A B

T

1 1 1 1

1 2 1 2 1 2 1 2

N N M M A

2

1

2

1

L

j j

j

L

j

j

Arerr









(59)

with










12

0( ) ( )E r E r a

r

12 2 2 1

2

12 2 2 1

22 2

2 2

kk

k k

r ak k

kk k k

ak k

a ra b r p

b a

k r k ba ra p

b a k k

2 4 4k








surface when 3







critical values








5r in

PS and 2 lnr r and


example







supported such that


(60)



(61)




0p is


the analysis



(0 ) ( ) 0

(0 ) ( ) 0x x

w z w L z

t z t L z

( 0) ( ) 0

( 0) ( ) 0

x x

z z

t x t x h

t x p t x h













length






(62)







for 0

ln ln

for 0

ln

a b

a b

T b r T r a

b a

b rT T Tr a

b

a



N=32 M=220








CONCLUSIONS



















REFERENCES








2000









69 (2002) 502-514








and Engineering 193 (2004) 4511-4539






Structures 83 (2005) 1487-1502



Engineering 36 (2005) 223-240



(2006) 381-393



(2005) 597-609












Structures 42 (2005) 4338-4351



6878-6890










150 (1999) 239-267





460 (2004) 1689-1706




182












Mechanics 74 (2005) 563-579





521







704-712



(1998) 141-148








(1998) 195-201



101



Elements 24 (2000) 575-582



227-236















25 (2001) 377-387



43-59







319

CONTENTS

vii









tm or mical

~sed or - No out of

ial ~ of or

~ Icated

~~d in anlage

- nise

ni 10 the

F THE

Preface

Chapter 1

Chapter 2

Chapter 3

Chaptemiddot 4

Chapter 5

bapter 6

C pteI 7

pt ) 8



Chapter 3



GRADED MATERIALS

Hui Wang12

and Qing-Hua Qin3







ABSTRACT













numerical results





1 INTRODUCTION



































































































(1)

or

(2)









(3)


(4)

0T t

k T t f t c tt

xx x x x x x

2


t

xx x x x x x x

2

11 22

1

2

3

T t T t

Tq t k q t

n

q t h T T

x x x

x x x

x x

00T Tx x



2 respectively h



2 and 3 are


2 3 1 3 and

1 2 3






(5)







(6)

with







contraction

n

0 Ωij j ib x

ij

2ij ij kk ij ijm T

2

1 2

2 1

E

1 2

Em

2

for plane strain


1 1 21

E E

E E

x





(7)



(8)




(9)



t For a well-















respectively

ij

1( )

2ij i j j iu u


i i u

i ij j i t

u u

t n t

x

x






(10)




desirable



(11)





(12)


(13)



( ) V x x L

1 2( ) 1 ( ) ( )P x V x P V x P

( )x

LV x e

1 1( ) ( )x

LP x V x P Pe


(14)








ANALYSIS



2

1

lnP

P




function namely

2 ( ) ( ) t tT b x x x (15)



( )tb x is known




2 ( ) 0t

hT x (16)


(17)


by

(18)

where ( )t

hq x and ( )t


respectively









i As a result

the potential ( )t




( )t

pT x

2 ( ) ( )t t

pT b x x

( ) ( ) ( ) ( ) ( ) ( )t t t t t t


x


1

1

( ) ( )

( ) ( )

Nt t

h i si si

i

Nt t

h i si si

i

T T

q Q

x x x x x

x x x x x

(19)



(20)

with


boundaries







1 1 1 2 2 22

1( ) ln

2

( ) 1( )

2

sj

sisj si si

T r

TQ k k x x n x x n

n r

x x

x xx x

2 2



1

( ) ( ) M

t t

j j j

j

b

x x x x x (21)












(22)



(23)


relationship



RBF PSK

( )

( )j x x

jx

( )j x x

2 2r c

r

2 2

1 1 2 2j jr x x x x

( )t

pT x ( )t

pq x

1

1

Mt t

p j j

j

Mt t

p j j

j

T

q

x x x

x x x

2 11 nr 1n

2 2 1

24 2 1

nr r

n

2 lnr r4 41 1

ln16 32

r r r

2 2r c 3 2 2 2ln 4

3 9

c c c r c


(24)





as

(25)


(26)




(27)


(28)

2 ( )j j x x x x

( )j x x

1 1

1 1

( ) ( ) s

N Mt t t

i si j j

i j

N Mt t t

i si j j

i j

T T

q Q

x x x x x

x x x x x

1 1

t N Mjsit t

i j

i jk k k

T T

x x x

x xx x x

1

1

1

1

1

k k

k k

k k

T t u u

f t f f

T TT

t t

x x x

x x x

x x

1 1

2 1

2

1

1

1 1

k k

k

k k

k

k k

k T c TT

k k t

k T c TT

k k t

f fk

x x x x xx

x x

x x x x xx

x x

x xx











(29)


(30)




(31)

where 1 1N

2 2N 3 3N and



(32)

1 1 1

2 1

k k k k

kk T c T c T f

Tk k t k t k

x x x x x x x x xx

x x x x

1 1

1

1 1

2

1 1

3

on

on

on

k k

k k

k k

T u t

q q t

h T q h T

x x

x x

x x

1

1

1 1

1 1

1

1

1

k kN Mm mk k

i m si j m j

i j m m

Nk

i n si

i

f c TT

k k t

m M

T

x x x xL x x L x x

x x

x x

1 1

2 2 2

3 3 3 3

1 1

1 1

1

1 1 1

2 2

1 1

1 1

1 1

1

1

Mk k

j n j n

j

N Mk k k

i n si j n j n

i j

N Mk k


i j

u n N

Q q n N

h T Q h

x x x

x x x x x

x x x y x x x x

3 3 1

h u

n N

2

k c

k k t

x x xL I

x x












(33)


bx and cx










2

0

4( ) 1 ( 1) cos( )exp( )

(2 1)

i

i i

i

T x t x ti

(34)

with (2 1) 2i i














( )( 1) ( 1)s b b c b c x x x x x x












1 15 2 25 3 35 4 45 5 0

1

2

3

4

5

6

7

Similarity ratio

Re

lative

err

or

in

te

mp

era

ture

t = 05s

t = 10s

0 01 02 03 04 05 06 07 08 09 1 0

1

2

3

4

5

6

7

8

9

x (m)

Re

lative

err

or

in

te

mp

era

ture

t = 05s t = 01s

t = 05s t = 001s

t = 10s t = 01s

t = 10s t = 001s


(35)



05 cm-1






given as

2 2

21

2 cos( ) sin exp

n


L n L L

(36)





0 10 20 30 40 50 60

-01

0

01

02

03

04

05

06

07

08

Time t (second)

Te

mp

era

ture

(

)

Meshless x=001

x=002

x=003

Analytical x=001

x=002

x=003





respectively The





obtained as

0 0( ) ( )x xk x k e a x a e


(37)







right to left












and x = 001 m


LBIEM 01871 03281 03800 03986 04019 04053 04581

MFS 03915 04497 04546 04550 04551 04551 04551

Exact 04551

1( ) ( with 0)

1

x

L

e xT x T

e L











elastic system as

(38)







iu that is

(39)



(40)

0k ki i kk iu u b

( ) ( ) ( )h p

i i iu u u x x x

0h h

k ki i kku u




equation

(41)












1

sN

h

i nl li sn

n

u U

x x x (42)




(43)




(44)



overlap in the MFS

0p p

k ki i kk iu u b

x

x


sn x x

1 1 (3 4 ) ln


v r

x y

snx x





(45)






coefficient set

(46)




relations

(47)

holds true



(48)


(49)


(50)



only that is

1 1

( ) ( ) ( )M M

m m

i m i li m l

m m

b

x x x x x

1

( ) ( )M

p m

i li m l

m

u

x x x


1 1

2li li mm mi mlF F

4

1 = 11 2


4 1

1li liF


(51)





(1) Conical spline

(52)

with


(53)

with




1 1

( ) ( ) ( )N M

n m m

i li n l li l

n m

u U

x x y x (54)


(7) and (6) as

4 2 2 1 d d 1 d d

d d d dr r r r

r r r r r r

mr x x

2 1

1 2 2

1 1

2 1 2 1 2 3

n


1

2

4 5 2 2 3

2 1

A n n

A n

2 2

1 2 3 2

1

32 1 1 2

n

li il i l

rA A r r

n n

22

1

2

8 29 27 8 2 2 1 2 4 7 4 2 ln

2 1 2 3 2 1 2 ln

A n n n n n n n r

A n n n n r


1 1

( ) ( ) ( )N M

n m


n m

S m T

x x y x x (55)

where

(56)




(57)











( )Arerr defined by

(58)





S U U U

H A B

T

1 1 1 1

1 2 1 2 1 2 1 2

N N M M A

2

1

2

1

L

j j

j

L

j

j

Arerr









(59)

with










12

0( ) ( )E r E r a

r

12 2 2 1

2

12 2 2 1

22 2

2 2

kk

k k

r ak k

kk k k

ak k

a ra b r p

b a

k r k ba ra p

b a k k

2 4 4k








surface when 3







critical values








5r in

PS and 2 lnr r and


example







supported such that


(60)



(61)




0p is


the analysis



(0 ) ( ) 0

(0 ) ( ) 0x x

w z w L z

t z t L z

( 0) ( ) 0

( 0) ( ) 0

x x

z z

t x t x h

t x p t x h













length






(62)







for 0

ln ln

for 0

ln

a b

a b

T b r T r a

b a

b rT T Tr a

b

a



N=32 M=220








CONCLUSIONS



















REFERENCES








2000









69 (2002) 502-514








and Engineering 193 (2004) 4511-4539






Structures 83 (2005) 1487-1502



Engineering 36 (2005) 223-240



(2006) 381-393



(2005) 597-609












Structures 42 (2005) 4338-4351



6878-6890










150 (1999) 239-267





460 (2004) 1689-1706




182












Mechanics 74 (2005) 563-579





521







704-712



(1998) 141-148








(1998) 195-201



101



Elements 24 (2000) 575-582



227-236















25 (2001) 377-387



43-59








Chapter 3



GRADED MATERIALS

Hui Wang12

and Qing-Hua Qin3







ABSTRACT













numerical results





1 INTRODUCTION



































































































(1)

or

(2)









(3)


(4)

0T t

k T t f t c tt

xx x x x x x

2


t

xx x x x x x x

2

11 22

1

2

3

T t T t

Tq t k q t

n

q t h T T

x x x

x x x

x x

00T Tx x



2 respectively h



2 and 3 are


2 3 1 3 and

1 2 3






(5)







(6)

with







contraction

n

0 Ωij j ib x

ij

2ij ij kk ij ijm T

2

1 2

2 1

E

1 2

Em

2

for plane strain


1 1 21

E E

E E

x





(7)



(8)




(9)



t For a well-















respectively

ij

1( )

2ij i j j iu u


i i u

i ij j i t

u u

t n t

x

x






(10)




desirable



(11)





(12)


(13)



( ) V x x L

1 2( ) 1 ( ) ( )P x V x P V x P

( )x

LV x e

1 1( ) ( )x

LP x V x P Pe


(14)








ANALYSIS



2

1

lnP

P




function namely

2 ( ) ( ) t tT b x x x (15)



( )tb x is known




2 ( ) 0t

hT x (16)


(17)


by

(18)

where ( )t

hq x and ( )t


respectively









i As a result

the potential ( )t




( )t

pT x

2 ( ) ( )t t

pT b x x

( ) ( ) ( ) ( ) ( ) ( )t t t t t t


x


1

1

( ) ( )

( ) ( )

Nt t

h i si si

i

Nt t

h i si si

i

T T

q Q

x x x x x

x x x x x

(19)



(20)

with


boundaries







1 1 1 2 2 22

1( ) ln

2

( ) 1( )

2

sj

sisj si si

T r

TQ k k x x n x x n

n r

x x

x xx x

2 2



1

( ) ( ) M

t t

j j j

j

b

x x x x x (21)












(22)



(23)


relationship



RBF PSK

( )

( )j x x

jx

( )j x x

2 2r c

r

2 2

1 1 2 2j jr x x x x

( )t

pT x ( )t

pq x

1

1

Mt t

p j j

j

Mt t

p j j

j

T

q

x x x

x x x

2 11 nr 1n

2 2 1

24 2 1

nr r

n

2 lnr r4 41 1

ln16 32

r r r

2 2r c 3 2 2 2ln 4

3 9

c c c r c


(24)





as

(25)


(26)




(27)


(28)

2 ( )j j x x x x

( )j x x

1 1

1 1

( ) ( ) s

N Mt t t

i si j j

i j

N Mt t t

i si j j

i j

T T

q Q

x x x x x

x x x x x

1 1

t N Mjsit t

i j

i jk k k

T T

x x x

x xx x x

1

1

1

1

1

k k

k k

k k

T t u u

f t f f

T TT

t t

x x x

x x x

x x

1 1

2 1

2

1

1

1 1

k k

k

k k

k

k k

k T c TT

k k t

k T c TT

k k t

f fk

x x x x xx

x x

x x x x xx

x x

x xx











(29)


(30)




(31)

where 1 1N

2 2N 3 3N and



(32)

1 1 1

2 1

k k k k

kk T c T c T f

Tk k t k t k

x x x x x x x x xx

x x x x

1 1

1

1 1

2

1 1

3

on

on

on

k k

k k

k k

T u t

q q t

h T q h T

x x

x x

x x

1

1

1 1

1 1

1

1

1

k kN Mm mk k

i m si j m j

i j m m

Nk

i n si

i

f c TT

k k t

m M

T

x x x xL x x L x x

x x

x x

1 1

2 2 2

3 3 3 3

1 1

1 1

1

1 1 1

2 2

1 1

1 1

1 1

1

1

Mk k

j n j n

j

N Mk k k

i n si j n j n

i j

N Mk k


i j

u n N

Q q n N

h T Q h

x x x

x x x x x

x x x y x x x x

3 3 1

h u

n N

2

k c

k k t

x x xL I

x x












(33)


bx and cx










2

0

4( ) 1 ( 1) cos( )exp( )

(2 1)

i

i i

i

T x t x ti

(34)

with (2 1) 2i i














( )( 1) ( 1)s b b c b c x x x x x x












1 15 2 25 3 35 4 45 5 0

1

2

3

4

5

6

7

Similarity ratio

Re

lative

err

or

in

te

mp

era

ture

t = 05s

t = 10s

0 01 02 03 04 05 06 07 08 09 1 0

1

2

3

4

5

6

7

8

9

x (m)

Re

lative

err

or

in

te

mp

era

ture

t = 05s t = 01s

t = 05s t = 001s

t = 10s t = 01s

t = 10s t = 001s


(35)



05 cm-1






given as

2 2

21

2 cos( ) sin exp

n


L n L L

(36)





0 10 20 30 40 50 60

-01

0

01

02

03

04

05

06

07

08

Time t (second)

Te

mp

era

ture

(

)

Meshless x=001

x=002

x=003

Analytical x=001

x=002

x=003





respectively The





obtained as

0 0( ) ( )x xk x k e a x a e


(37)







right to left












and x = 001 m


LBIEM 01871 03281 03800 03986 04019 04053 04581

MFS 03915 04497 04546 04550 04551 04551 04551

Exact 04551

1( ) ( with 0)

1

x

L

e xT x T

e L











elastic system as

(38)







iu that is

(39)



(40)

0k ki i kk iu u b

( ) ( ) ( )h p

i i iu u u x x x

0h h

k ki i kku u




equation

(41)












1

sN

h

i nl li sn

n

u U

x x x (42)




(43)




(44)



overlap in the MFS

0p p

k ki i kk iu u b

x

x


sn x x

1 1 (3 4 ) ln


v r

x y

snx x





(45)






coefficient set

(46)




relations

(47)

holds true



(48)


(49)


(50)



only that is

1 1

( ) ( ) ( )M M

m m

i m i li m l

m m

b

x x x x x

1

( ) ( )M

p m

i li m l

m

u

x x x


1 1

2li li mm mi mlF F

4

1 = 11 2


4 1

1li liF


(51)





(1) Conical spline

(52)

with


(53)

with




1 1

( ) ( ) ( )N M

n m m

i li n l li l

n m

u U

x x y x (54)


(7) and (6) as

4 2 2 1 d d 1 d d

d d d dr r r r

r r r r r r

mr x x

2 1

1 2 2

1 1

2 1 2 1 2 3

n


1

2

4 5 2 2 3

2 1

A n n

A n

2 2

1 2 3 2

1

32 1 1 2

n

li il i l

rA A r r

n n

22

1

2

8 29 27 8 2 2 1 2 4 7 4 2 ln

2 1 2 3 2 1 2 ln

A n n n n n n n r

A n n n n r


1 1

( ) ( ) ( )N M

n m


n m

S m T

x x y x x (55)

where

(56)




(57)











( )Arerr defined by

(58)





S U U U

H A B

T

1 1 1 1

1 2 1 2 1 2 1 2

N N M M A

2

1

2

1

L

j j

j

L

j

j

Arerr









(59)

with










12

0( ) ( )E r E r a

r

12 2 2 1

2

12 2 2 1

22 2

2 2

kk

k k

r ak k

kk k k

ak k

a ra b r p

b a

k r k ba ra p

b a k k

2 4 4k








surface when 3







critical values








5r in

PS and 2 lnr r and


example







supported such that


(60)



(61)




0p is


the analysis



(0 ) ( ) 0

(0 ) ( ) 0x x

w z w L z

t z t L z

( 0) ( ) 0

( 0) ( ) 0

x x

z z

t x t x h

t x p t x h













length






(62)







for 0

ln ln

for 0

ln

a b

a b

T b r T r a

b a

b rT T Tr a

b

a



N=32 M=220








CONCLUSIONS



















REFERENCES








2000









69 (2002) 502-514








and Engineering 193 (2004) 4511-4539






Structures 83 (2005) 1487-1502



Engineering 36 (2005) 223-240



(2006) 381-393



(2005) 597-609












Structures 42 (2005) 4338-4351



6878-6890










150 (1999) 239-267





460 (2004) 1689-1706




182












Mechanics 74 (2005) 563-579





521







704-712



(1998) 141-148








(1998) 195-201



101



Elements 24 (2000) 575-582



227-236















25 (2001) 377-387



43-59









1 INTRODUCTION



































































































(1)

or

(2)









(3)


(4)

0T t

k T t f t c tt

xx x x x x x

2


t

xx x x x x x x

2

11 22

1

2

3

T t T t

Tq t k q t

n

q t h T T

x x x

x x x

x x

00T Tx x



2 respectively h



2 and 3 are


2 3 1 3 and

1 2 3






(5)







(6)

with







contraction

n

0 Ωij j ib x

ij

2ij ij kk ij ijm T

2

1 2

2 1

E

1 2

Em

2

for plane strain


1 1 21

E E

E E

x





(7)



(8)




(9)



t For a well-















respectively

ij

1( )

2ij i j j iu u


i i u

i ij j i t

u u

t n t

x

x






(10)




desirable



(11)





(12)


(13)



( ) V x x L

1 2( ) 1 ( ) ( )P x V x P V x P

( )x

LV x e

1 1( ) ( )x

LP x V x P Pe


(14)








ANALYSIS



2

1

lnP

P




function namely

2 ( ) ( ) t tT b x x x (15)



( )tb x is known




2 ( ) 0t

hT x (16)


(17)


by

(18)

where ( )t

hq x and ( )t


respectively









i As a result

the potential ( )t




( )t

pT x

2 ( ) ( )t t

pT b x x

( ) ( ) ( ) ( ) ( ) ( )t t t t t t


x


1

1

( ) ( )

( ) ( )

Nt t

h i si si

i

Nt t

h i si si

i

T T

q Q

x x x x x

x x x x x

(19)



(20)

with


boundaries







1 1 1 2 2 22

1( ) ln

2

( ) 1( )

2

sj

sisj si si

T r

TQ k k x x n x x n

n r

x x

x xx x

2 2



1

( ) ( ) M

t t

j j j

j

b

x x x x x (21)












(22)



(23)


relationship



RBF PSK

( )

( )j x x

jx

( )j x x

2 2r c

r

2 2

1 1 2 2j jr x x x x

( )t

pT x ( )t

pq x

1

1

Mt t

p j j

j

Mt t

p j j

j

T

q

x x x

x x x

2 11 nr 1n

2 2 1

24 2 1

nr r

n

2 lnr r4 41 1

ln16 32

r r r

2 2r c 3 2 2 2ln 4

3 9

c c c r c


(24)





as

(25)


(26)




(27)


(28)

2 ( )j j x x x x

( )j x x

1 1

1 1

( ) ( ) s

N Mt t t

i si j j

i j

N Mt t t

i si j j

i j

T T

q Q

x x x x x

x x x x x

1 1

t N Mjsit t

i j

i jk k k

T T

x x x

x xx x x

1

1

1

1

1

k k

k k

k k

T t u u

f t f f

T TT

t t

x x x

x x x

x x

1 1

2 1

2

1

1

1 1

k k

k

k k

k

k k

k T c TT

k k t

k T c TT

k k t

f fk

x x x x xx

x x

x x x x xx

x x

x xx











(29)


(30)




(31)

where 1 1N

2 2N 3 3N and



(32)

1 1 1

2 1

k k k k

kk T c T c T f

Tk k t k t k

x x x x x x x x xx

x x x x

1 1

1

1 1

2

1 1

3

on

on

on

k k

k k

k k

T u t

q q t

h T q h T

x x

x x

x x

1

1

1 1

1 1

1

1

1

k kN Mm mk k

i m si j m j

i j m m

Nk

i n si

i

f c TT

k k t

m M

T

x x x xL x x L x x

x x

x x

1 1

2 2 2

3 3 3 3

1 1

1 1

1

1 1 1

2 2

1 1

1 1

1 1

1

1

Mk k

j n j n

j

N Mk k k

i n si j n j n

i j

N Mk k


i j

u n N

Q q n N

h T Q h

x x x

x x x x x

x x x y x x x x

3 3 1

h u

n N

2

k c

k k t

x x xL I

x x












(33)


bx and cx










2

0

4( ) 1 ( 1) cos( )exp( )

(2 1)

i

i i

i

T x t x ti

(34)

with (2 1) 2i i














( )( 1) ( 1)s b b c b c x x x x x x












1 15 2 25 3 35 4 45 5 0

1

2

3

4

5

6

7

Similarity ratio

Re

lative

err

or

in

te

mp

era

ture

t = 05s

t = 10s

0 01 02 03 04 05 06 07 08 09 1 0

1

2

3

4

5

6

7

8

9

x (m)

Re

lative

err

or

in

te

mp

era

ture

t = 05s t = 01s

t = 05s t = 001s

t = 10s t = 01s

t = 10s t = 001s


(35)



05 cm-1






given as

2 2

21

2 cos( ) sin exp

n


L n L L

(36)





0 10 20 30 40 50 60

-01

0

01

02

03

04

05

06

07

08

Time t (second)

Te

mp

era

ture

(

)

Meshless x=001

x=002

x=003

Analytical x=001

x=002

x=003





respectively The





obtained as

0 0( ) ( )x xk x k e a x a e


(37)







right to left












and x = 001 m


LBIEM 01871 03281 03800 03986 04019 04053 04581

MFS 03915 04497 04546 04550 04551 04551 04551

Exact 04551

1( ) ( with 0)

1

x

L

e xT x T

e L











elastic system as

(38)







iu that is

(39)



(40)

0k ki i kk iu u b

( ) ( ) ( )h p

i i iu u u x x x

0h h

k ki i kku u




equation

(41)












1

sN

h

i nl li sn

n

u U

x x x (42)




(43)




(44)



overlap in the MFS

0p p

k ki i kk iu u b

x

x


sn x x

1 1 (3 4 ) ln


v r

x y

snx x





(45)






coefficient set

(46)




relations

(47)

holds true



(48)


(49)


(50)



only that is

1 1

( ) ( ) ( )M M

m m

i m i li m l

m m

b

x x x x x

1

( ) ( )M

p m

i li m l

m

u

x x x


1 1

2li li mm mi mlF F

4

1 = 11 2


4 1

1li liF


(51)





(1) Conical spline

(52)

with


(53)

with




1 1

( ) ( ) ( )N M

n m m

i li n l li l

n m

u U

x x y x (54)


(7) and (6) as

4 2 2 1 d d 1 d d

d d d dr r r r

r r r r r r

mr x x

2 1

1 2 2

1 1

2 1 2 1 2 3

n


1

2

4 5 2 2 3

2 1

A n n

A n

2 2

1 2 3 2

1

32 1 1 2

n

li il i l

rA A r r

n n

22

1

2

8 29 27 8 2 2 1 2 4 7 4 2 ln

2 1 2 3 2 1 2 ln

A n n n n n n n r

A n n n n r


1 1

( ) ( ) ( )N M

n m


n m

S m T

x x y x x (55)

where

(56)




(57)











( )Arerr defined by

(58)





S U U U

H A B

T

1 1 1 1

1 2 1 2 1 2 1 2

N N M M A

2

1

2

1

L

j j

j

L

j

j

Arerr









(59)

with










12

0( ) ( )E r E r a

r

12 2 2 1

2

12 2 2 1

22 2

2 2

kk

k k

r ak k

kk k k

ak k

a ra b r p

b a

k r k ba ra p

b a k k

2 4 4k








surface when 3







critical values








5r in

PS and 2 lnr r and


example







supported such that


(60)



(61)




0p is


the analysis



(0 ) ( ) 0

(0 ) ( ) 0x x

w z w L z

t z t L z

( 0) ( ) 0

( 0) ( ) 0

x x

z z

t x t x h

t x p t x h













length






(62)







for 0

ln ln

for 0

ln

a b

a b

T b r T r a

b a

b rT T Tr a

b

a



N=32 M=220








CONCLUSIONS



















REFERENCES








2000









69 (2002) 502-514








and Engineering 193 (2004) 4511-4539






Structures 83 (2005) 1487-1502



Engineering 36 (2005) 223-240



(2006) 381-393



(2005) 597-609












Structures 42 (2005) 4338-4351



6878-6890










150 (1999) 239-267





460 (2004) 1689-1706




182












Mechanics 74 (2005) 563-579





521







704-712



(1998) 141-148








(1998) 195-201



101



Elements 24 (2000) 575-582



227-236















25 (2001) 377-387



43-59

































































(1)

or

(2)









(3)


(4)

0T t

k T t f t c tt

xx x x x x x

2


t

xx x x x x x x

2

11 22

1

2

3

T t T t

Tq t k q t

n

q t h T T

x x x

x x x

x x

00T Tx x



2 respectively h



2 and 3 are


2 3 1 3 and

1 2 3






(5)







(6)

with







contraction

n

0 Ωij j ib x

ij

2ij ij kk ij ijm T

2

1 2

2 1

E

1 2

Em

2

for plane strain


1 1 21

E E

E E

x





(7)



(8)




(9)



t For a well-















respectively

ij

1( )

2ij i j j iu u


i i u

i ij j i t

u u

t n t

x

x






(10)




desirable



(11)





(12)


(13)



( ) V x x L

1 2( ) 1 ( ) ( )P x V x P V x P

( )x

LV x e

1 1( ) ( )x

LP x V x P Pe


(14)








ANALYSIS



2

1

lnP

P




function namely

2 ( ) ( ) t tT b x x x (15)



( )tb x is known




2 ( ) 0t

hT x (16)


(17)


by

(18)

where ( )t

hq x and ( )t


respectively









i As a result

the potential ( )t




( )t

pT x

2 ( ) ( )t t

pT b x x

( ) ( ) ( ) ( ) ( ) ( )t t t t t t


x


1

1

( ) ( )

( ) ( )

Nt t

h i si si

i

Nt t

h i si si

i

T T

q Q

x x x x x

x x x x x

(19)



(20)

with


boundaries







1 1 1 2 2 22

1( ) ln

2

( ) 1( )

2

sj

sisj si si

T r

TQ k k x x n x x n

n r

x x

x xx x

2 2



1

( ) ( ) M

t t

j j j

j

b

x x x x x (21)












(22)



(23)


relationship



RBF PSK

( )

( )j x x

jx

( )j x x

2 2r c

r

2 2

1 1 2 2j jr x x x x

( )t

pT x ( )t

pq x

1

1

Mt t

p j j

j

Mt t

p j j

j

T

q

x x x

x x x

2 11 nr 1n

2 2 1

24 2 1

nr r

n

2 lnr r4 41 1

ln16 32

r r r

2 2r c 3 2 2 2ln 4

3 9

c c c r c


(24)





as

(25)


(26)




(27)


(28)

2 ( )j j x x x x

( )j x x

1 1

1 1

( ) ( ) s

N Mt t t

i si j j

i j

N Mt t t

i si j j

i j

T T

q Q

x x x x x

x x x x x

1 1

t N Mjsit t

i j

i jk k k

T T

x x x

x xx x x

1

1

1

1

1

k k

k k

k k

T t u u

f t f f

T TT

t t

x x x

x x x

x x

1 1

2 1

2

1

1

1 1

k k

k

k k

k

k k

k T c TT

k k t

k T c TT

k k t

f fk

x x x x xx

x x

x x x x xx

x x

x xx











(29)


(30)




(31)

where 1 1N

2 2N 3 3N and



(32)

1 1 1

2 1

k k k k

kk T c T c T f

Tk k t k t k

x x x x x x x x xx

x x x x

1 1

1

1 1

2

1 1

3

on

on

on

k k

k k

k k

T u t

q q t

h T q h T

x x

x x

x x

1

1

1 1

1 1

1

1

1

k kN Mm mk k

i m si j m j

i j m m

Nk

i n si

i

f c TT

k k t

m M

T

x x x xL x x L x x

x x

x x

1 1

2 2 2

3 3 3 3

1 1

1 1

1

1 1 1

2 2

1 1

1 1

1 1

1

1

Mk k

j n j n

j

N Mk k k

i n si j n j n

i j

N Mk k


i j

u n N

Q q n N

h T Q h

x x x

x x x x x

x x x y x x x x

3 3 1

h u

n N

2

k c

k k t

x x xL I

x x












(33)


bx and cx










2

0

4( ) 1 ( 1) cos( )exp( )

(2 1)

i

i i

i

T x t x ti

(34)

with (2 1) 2i i














( )( 1) ( 1)s b b c b c x x x x x x












1 15 2 25 3 35 4 45 5 0

1

2

3

4

5

6

7

Similarity ratio

Re

lative

err

or

in

te

mp

era

ture

t = 05s

t = 10s

0 01 02 03 04 05 06 07 08 09 1 0

1

2

3

4

5

6

7

8

9

x (m)

Re

lative

err

or

in

te

mp

era

ture

t = 05s t = 01s

t = 05s t = 001s

t = 10s t = 01s

t = 10s t = 001s


(35)



05 cm-1






given as

2 2

21

2 cos( ) sin exp

n


L n L L

(36)





0 10 20 30 40 50 60

-01

0

01

02

03

04

05

06

07

08

Time t (second)

Te

mp

era

ture

(

)

Meshless x=001

x=002

x=003

Analytical x=001

x=002

x=003





respectively The





obtained as

0 0( ) ( )x xk x k e a x a e


(37)







right to left












and x = 001 m


LBIEM 01871 03281 03800 03986 04019 04053 04581

MFS 03915 04497 04546 04550 04551 04551 04551

Exact 04551

1( ) ( with 0)

1

x

L

e xT x T

e L











elastic system as

(38)







iu that is

(39)



(40)

0k ki i kk iu u b

( ) ( ) ( )h p

i i iu u u x x x

0h h

k ki i kku u




equation

(41)












1

sN

h

i nl li sn

n

u U

x x x (42)




(43)




(44)



overlap in the MFS

0p p

k ki i kk iu u b

x

x


sn x x

1 1 (3 4 ) ln


v r

x y

snx x





(45)






coefficient set

(46)




relations

(47)

holds true



(48)


(49)


(50)



only that is

1 1

( ) ( ) ( )M M

m m

i m i li m l

m m

b

x x x x x

1

( ) ( )M

p m

i li m l

m

u

x x x


1 1

2li li mm mi mlF F

4

1 = 11 2


4 1

1li liF


(51)





(1) Conical spline

(52)

with


(53)

with




1 1

( ) ( ) ( )N M

n m m

i li n l li l

n m

u U

x x y x (54)


(7) and (6) as

4 2 2 1 d d 1 d d

d d d dr r r r

r r r r r r

mr x x

2 1

1 2 2

1 1

2 1 2 1 2 3

n


1

2

4 5 2 2 3

2 1

A n n

A n

2 2

1 2 3 2

1

32 1 1 2

n

li il i l

rA A r r

n n

22

1

2

8 29 27 8 2 2 1 2 4 7 4 2 ln

2 1 2 3 2 1 2 ln

A n n n n n n n r

A n n n n r


1 1

( ) ( ) ( )N M

n m


n m

S m T

x x y x x (55)

where

(56)




(57)











( )Arerr defined by

(58)





S U U U

H A B

T

1 1 1 1

1 2 1 2 1 2 1 2

N N M M A

2

1

2

1

L

j j

j

L

j

j

Arerr









(59)

with










12

0( ) ( )E r E r a

r

12 2 2 1

2

12 2 2 1

22 2

2 2

kk

k k

r ak k

kk k k

ak k

a ra b r p

b a

k r k ba ra p

b a k k

2 4 4k








surface when 3







critical values








5r in

PS and 2 lnr r and


example







supported such that


(60)



(61)




0p is


the analysis



(0 ) ( ) 0

(0 ) ( ) 0x x

w z w L z

t z t L z

( 0) ( ) 0

( 0) ( ) 0

x x

z z

t x t x h

t x p t x h













length






(62)







for 0

ln ln

for 0

ln

a b

a b

T b r T r a

b a

b rT T Tr a

b

a



N=32 M=220








CONCLUSIONS



















REFERENCES








2000









69 (2002) 502-514








and Engineering 193 (2004) 4511-4539






Structures 83 (2005) 1487-1502



Engineering 36 (2005) 223-240



(2006) 381-393



(2005) 597-609












Structures 42 (2005) 4338-4351



6878-6890










150 (1999) 239-267





460 (2004) 1689-1706




182












Mechanics 74 (2005) 563-579





521







704-712



(1998) 141-148








(1998) 195-201



101



Elements 24 (2000) 575-582



227-236















25 (2001) 377-387



43-59








2 respectively h



2 and 3 are


2 3 1 3 and

1 2 3






(5)







(6)

with







contraction

n

0 Ωij j ib x

ij

2ij ij kk ij ijm T

2

1 2

2 1

E

1 2

Em

2

for plane strain


1 1 21

E E

E E

x





(7)



(8)




(9)



t For a well-















respectively

ij

1( )

2ij i j j iu u


i i u

i ij j i t

u u

t n t

x

x






(10)




desirable



(11)





(12)


(13)



( ) V x x L

1 2( ) 1 ( ) ( )P x V x P V x P

( )x

LV x e

1 1( ) ( )x

LP x V x P Pe


(14)








ANALYSIS



2

1

lnP

P




function namely

2 ( ) ( ) t tT b x x x (15)



( )tb x is known




2 ( ) 0t

hT x (16)


(17)


by

(18)

where ( )t

hq x and ( )t


respectively









i As a result

the potential ( )t




( )t

pT x

2 ( ) ( )t t

pT b x x

( ) ( ) ( ) ( ) ( ) ( )t t t t t t


x


1

1

( ) ( )

( ) ( )

Nt t

h i si si

i

Nt t

h i si si

i

T T

q Q

x x x x x

x x x x x

(19)



(20)

with


boundaries







1 1 1 2 2 22

1( ) ln

2

( ) 1( )

2

sj

sisj si si

T r

TQ k k x x n x x n

n r

x x

x xx x

2 2



1

( ) ( ) M

t t

j j j

j

b

x x x x x (21)












(22)



(23)


relationship



RBF PSK

( )

( )j x x

jx

( )j x x

2 2r c

r

2 2

1 1 2 2j jr x x x x

( )t

pT x ( )t

pq x

1

1

Mt t

p j j

j

Mt t

p j j

j

T

q

x x x

x x x

2 11 nr 1n

2 2 1

24 2 1

nr r

n

2 lnr r4 41 1

ln16 32

r r r

2 2r c 3 2 2 2ln 4

3 9

c c c r c


(24)





as

(25)


(26)




(27)


(28)

2 ( )j j x x x x

( )j x x

1 1

1 1

( ) ( ) s

N Mt t t

i si j j

i j

N Mt t t

i si j j

i j

T T

q Q

x x x x x

x x x x x

1 1

t N Mjsit t

i j

i jk k k

T T

x x x

x xx x x

1

1

1

1

1

k k

k k

k k

T t u u

f t f f

T TT

t t

x x x

x x x

x x

1 1

2 1

2

1

1

1 1

k k

k

k k

k

k k

k T c TT

k k t

k T c TT

k k t

f fk

x x x x xx

x x

x x x x xx

x x

x xx











(29)


(30)




(31)

where 1 1N

2 2N 3 3N and



(32)

1 1 1

2 1

k k k k

kk T c T c T f

Tk k t k t k

x x x x x x x x xx

x x x x

1 1

1

1 1

2

1 1

3

on

on

on

k k

k k

k k

T u t

q q t

h T q h T

x x

x x

x x

1

1

1 1

1 1

1

1

1

k kN Mm mk k

i m si j m j

i j m m

Nk

i n si

i

f c TT

k k t

m M

T

x x x xL x x L x x

x x

x x

1 1

2 2 2

3 3 3 3

1 1

1 1

1

1 1 1

2 2

1 1

1 1

1 1

1

1

Mk k

j n j n

j

N Mk k k

i n si j n j n

i j

N Mk k


i j

u n N

Q q n N

h T Q h

x x x

x x x x x

x x x y x x x x

3 3 1

h u

n N

2

k c

k k t

x x xL I

x x












(33)


bx and cx










2

0

4( ) 1 ( 1) cos( )exp( )

(2 1)

i

i i

i

T x t x ti

(34)

with (2 1) 2i i














( )( 1) ( 1)s b b c b c x x x x x x












1 15 2 25 3 35 4 45 5 0

1

2

3

4

5

6

7

Similarity ratio

Re

lative

err

or

in

te

mp

era

ture

t = 05s

t = 10s

0 01 02 03 04 05 06 07 08 09 1 0

1

2

3

4

5

6

7

8

9

x (m)

Re

lative

err

or

in

te

mp

era

ture

t = 05s t = 01s

t = 05s t = 001s

t = 10s t = 01s

t = 10s t = 001s


(35)



05 cm-1






given as

2 2

21

2 cos( ) sin exp

n


L n L L

(36)





0 10 20 30 40 50 60

-01

0

01

02

03

04

05

06

07

08

Time t (second)

Te

mp

era

ture

(

)

Meshless x=001

x=002

x=003

Analytical x=001

x=002

x=003





respectively The





obtained as

0 0( ) ( )x xk x k e a x a e


(37)







right to left












and x = 001 m


LBIEM 01871 03281 03800 03986 04019 04053 04581

MFS 03915 04497 04546 04550 04551 04551 04551

Exact 04551

1( ) ( with 0)

1

x

L

e xT x T

e L











elastic system as

(38)







iu that is

(39)



(40)

0k ki i kk iu u b

( ) ( ) ( )h p

i i iu u u x x x

0h h

k ki i kku u




equation

(41)












1

sN

h

i nl li sn

n

u U

x x x (42)




(43)




(44)



overlap in the MFS

0p p

k ki i kk iu u b

x

x


sn x x

1 1 (3 4 ) ln


v r

x y

snx x





(45)






coefficient set

(46)




relations

(47)

holds true



(48)


(49)


(50)



only that is

1 1

( ) ( ) ( )M M

m m

i m i li m l

m m

b

x x x x x

1

( ) ( )M

p m

i li m l

m

u

x x x


1 1

2li li mm mi mlF F

4

1 = 11 2


4 1

1li liF


(51)





(1) Conical spline

(52)

with


(53)

with




1 1

( ) ( ) ( )N M

n m m

i li n l li l

n m

u U

x x y x (54)


(7) and (6) as

4 2 2 1 d d 1 d d

d d d dr r r r

r r r r r r

mr x x

2 1

1 2 2

1 1

2 1 2 1 2 3

n


1

2

4 5 2 2 3

2 1

A n n

A n

2 2

1 2 3 2

1

32 1 1 2

n

li il i l

rA A r r

n n

22

1

2

8 29 27 8 2 2 1 2 4 7 4 2 ln

2 1 2 3 2 1 2 ln

A n n n n n n n r

A n n n n r


1 1

( ) ( ) ( )N M

n m


n m

S m T

x x y x x (55)

where

(56)




(57)











( )Arerr defined by

(58)





S U U U

H A B

T

1 1 1 1

1 2 1 2 1 2 1 2

N N M M A

2

1

2

1

L

j j

j

L

j

j

Arerr









(59)

with










12

0( ) ( )E r E r a

r

12 2 2 1

2

12 2 2 1

22 2

2 2

kk

k k

r ak k

kk k k

ak k

a ra b r p

b a

k r k ba ra p

b a k k

2 4 4k








surface when 3







critical values








5r in

PS and 2 lnr r and


example







supported such that


(60)



(61)




0p is


the analysis



(0 ) ( ) 0

(0 ) ( ) 0x x

w z w L z

t z t L z

( 0) ( ) 0

( 0) ( ) 0

x x

z z

t x t x h

t x p t x h













length






(62)







for 0

ln ln

for 0

ln

a b

a b

T b r T r a

b a

b rT T Tr a

b

a



N=32 M=220








CONCLUSIONS



















REFERENCES








2000









69 (2002) 502-514








and Engineering 193 (2004) 4511-4539






Structures 83 (2005) 1487-1502



Engineering 36 (2005) 223-240



(2006) 381-393



(2005) 597-609












Structures 42 (2005) 4338-4351



6878-6890










150 (1999) 239-267





460 (2004) 1689-1706




182












Mechanics 74 (2005) 563-579





521







704-712



(1998) 141-148








(1998) 195-201



101



Elements 24 (2000) 575-582



227-236















25 (2001) 377-387



43-59










(7)



(8)




(9)



t For a well-















respectively

ij

1( )

2ij i j j iu u


i i u

i ij j i t

u u

t n t

x

x






(10)




desirable



(11)





(12)


(13)



( ) V x x L

1 2( ) 1 ( ) ( )P x V x P V x P

( )x

LV x e

1 1( ) ( )x

LP x V x P Pe


(14)








ANALYSIS



2

1

lnP

P




function namely

2 ( ) ( ) t tT b x x x (15)



( )tb x is known




2 ( ) 0t

hT x (16)


(17)


by

(18)

where ( )t

hq x and ( )t


respectively









i As a result

the potential ( )t




( )t

pT x

2 ( ) ( )t t

pT b x x

( ) ( ) ( ) ( ) ( ) ( )t t t t t t


x


1

1

( ) ( )

( ) ( )

Nt t

h i si si

i

Nt t

h i si si

i

T T

q Q

x x x x x

x x x x x

(19)



(20)

with


boundaries







1 1 1 2 2 22

1( ) ln

2

( ) 1( )

2

sj

sisj si si

T r

TQ k k x x n x x n

n r

x x

x xx x

2 2



1

( ) ( ) M

t t

j j j

j

b

x x x x x (21)












(22)



(23)


relationship



RBF PSK

( )

( )j x x

jx

( )j x x

2 2r c

r

2 2

1 1 2 2j jr x x x x

( )t

pT x ( )t

pq x

1

1

Mt t

p j j

j

Mt t

p j j

j

T

q

x x x

x x x

2 11 nr 1n

2 2 1

24 2 1

nr r

n

2 lnr r4 41 1

ln16 32

r r r

2 2r c 3 2 2 2ln 4

3 9

c c c r c


(24)





as

(25)


(26)




(27)


(28)

2 ( )j j x x x x

( )j x x

1 1

1 1

( ) ( ) s

N Mt t t

i si j j

i j

N Mt t t

i si j j

i j

T T

q Q

x x x x x

x x x x x

1 1

t N Mjsit t

i j

i jk k k

T T

x x x

x xx x x

1

1

1

1

1

k k

k k

k k

T t u u

f t f f

T TT

t t

x x x

x x x

x x

1 1

2 1

2

1

1

1 1

k k

k

k k

k

k k

k T c TT

k k t

k T c TT

k k t

f fk

x x x x xx

x x

x x x x xx

x x

x xx











(29)


(30)




(31)

where 1 1N

2 2N 3 3N and



(32)

1 1 1

2 1

k k k k

kk T c T c T f

Tk k t k t k

x x x x x x x x xx

x x x x

1 1

1

1 1

2

1 1

3

on

on

on

k k

k k

k k

T u t

q q t

h T q h T

x x

x x

x x

1

1

1 1

1 1

1

1

1

k kN Mm mk k

i m si j m j

i j m m

Nk

i n si

i

f c TT

k k t

m M

T

x x x xL x x L x x

x x

x x

1 1

2 2 2

3 3 3 3

1 1

1 1

1

1 1 1

2 2

1 1

1 1

1 1

1

1

Mk k

j n j n

j

N Mk k k

i n si j n j n

i j

N Mk k


i j

u n N

Q q n N

h T Q h

x x x

x x x x x

x x x y x x x x

3 3 1

h u

n N

2

k c

k k t

x x xL I

x x












(33)


bx and cx










2

0

4( ) 1 ( 1) cos( )exp( )

(2 1)

i

i i

i

T x t x ti

(34)

with (2 1) 2i i














( )( 1) ( 1)s b b c b c x x x x x x












1 15 2 25 3 35 4 45 5 0

1

2

3

4

5

6

7

Similarity ratio

Re

lative

err

or

in

te

mp

era

ture

t = 05s

t = 10s

0 01 02 03 04 05 06 07 08 09 1 0

1

2

3

4

5

6

7

8

9

x (m)

Re

lative

err

or

in

te

mp

era

ture

t = 05s t = 01s

t = 05s t = 001s

t = 10s t = 01s

t = 10s t = 001s


(35)



05 cm-1






given as

2 2

21

2 cos( ) sin exp

n


L n L L

(36)





0 10 20 30 40 50 60

-01

0

01

02

03

04

05

06

07

08

Time t (second)

Te

mp

era

ture

(

)

Meshless x=001

x=002

x=003

Analytical x=001

x=002

x=003





respectively The





obtained as

0 0( ) ( )x xk x k e a x a e


(37)







right to left












and x = 001 m


LBIEM 01871 03281 03800 03986 04019 04053 04581

MFS 03915 04497 04546 04550 04551 04551 04551

Exact 04551

1( ) ( with 0)

1

x

L

e xT x T

e L











elastic system as

(38)







iu that is

(39)



(40)

0k ki i kk iu u b

( ) ( ) ( )h p

i i iu u u x x x

0h h

k ki i kku u




equation

(41)












1

sN

h

i nl li sn

n

u U

x x x (42)




(43)




(44)



overlap in the MFS

0p p

k ki i kk iu u b

x

x


sn x x

1 1 (3 4 ) ln


v r

x y

snx x





(45)






coefficient set

(46)




relations

(47)

holds true



(48)


(49)


(50)



only that is

1 1

( ) ( ) ( )M M

m m

i m i li m l

m m

b

x x x x x

1

( ) ( )M

p m

i li m l

m

u

x x x


1 1

2li li mm mi mlF F

4

1 = 11 2


4 1

1li liF


(51)





(1) Conical spline

(52)

with


(53)

with




1 1

( ) ( ) ( )N M

n m m

i li n l li l

n m

u U

x x y x (54)


(7) and (6) as

4 2 2 1 d d 1 d d

d d d dr r r r

r r r r r r

mr x x

2 1

1 2 2

1 1

2 1 2 1 2 3

n


1

2

4 5 2 2 3

2 1

A n n

A n

2 2

1 2 3 2

1

32 1 1 2

n

li il i l

rA A r r

n n

22

1

2

8 29 27 8 2 2 1 2 4 7 4 2 ln

2 1 2 3 2 1 2 ln

A n n n n n n n r

A n n n n r


1 1

( ) ( ) ( )N M

n m


n m

S m T

x x y x x (55)

where

(56)




(57)











( )Arerr defined by

(58)





S U U U

H A B

T

1 1 1 1

1 2 1 2 1 2 1 2

N N M M A

2

1

2

1

L

j j

j

L

j

j

Arerr









(59)

with










12

0( ) ( )E r E r a

r

12 2 2 1

2

12 2 2 1

22 2

2 2

kk

k k

r ak k

kk k k

ak k

a ra b r p

b a

k r k ba ra p

b a k k

2 4 4k








surface when 3







critical values








5r in

PS and 2 lnr r and


example







supported such that


(60)



(61)




0p is


the analysis



(0 ) ( ) 0

(0 ) ( ) 0x x

w z w L z

t z t L z

( 0) ( ) 0

( 0) ( ) 0

x x

z z

t x t x h

t x p t x h













length






(62)







for 0

ln ln

for 0

ln

a b

a b

T b r T r a

b a

b rT T Tr a

b

a



N=32 M=220








CONCLUSIONS



















REFERENCES








2000









69 (2002) 502-514








and Engineering 193 (2004) 4511-4539






Structures 83 (2005) 1487-1502



Engineering 36 (2005) 223-240



(2006) 381-393



(2005) 597-609












Structures 42 (2005) 4338-4351



6878-6890










150 (1999) 239-267





460 (2004) 1689-1706




182












Mechanics 74 (2005) 563-579





521







704-712



(1998) 141-148








(1998) 195-201



101



Elements 24 (2000) 575-582



227-236















25 (2001) 377-387



43-59











(10)




desirable



(11)





(12)


(13)



( ) V x x L

1 2( ) 1 ( ) ( )P x V x P V x P

( )x

LV x e

1 1( ) ( )x

LP x V x P Pe


(14)








ANALYSIS



2

1

lnP

P




function namely

2 ( ) ( ) t tT b x x x (15)



( )tb x is known




2 ( ) 0t

hT x (16)


(17)


by

(18)

where ( )t

hq x and ( )t


respectively









i As a result

the potential ( )t




( )t

pT x

2 ( ) ( )t t

pT b x x

( ) ( ) ( ) ( ) ( ) ( )t t t t t t


x


1

1

( ) ( )

( ) ( )

Nt t

h i si si

i

Nt t

h i si si

i

T T

q Q

x x x x x

x x x x x

(19)



(20)

with


boundaries







1 1 1 2 2 22

1( ) ln

2

( ) 1( )

2

sj

sisj si si

T r

TQ k k x x n x x n

n r

x x

x xx x

2 2



1

( ) ( ) M

t t

j j j

j

b

x x x x x (21)












(22)



(23)


relationship



RBF PSK

( )

( )j x x

jx

( )j x x

2 2r c

r

2 2

1 1 2 2j jr x x x x

( )t

pT x ( )t

pq x

1

1

Mt t

p j j

j

Mt t

p j j

j

T

q

x x x

x x x

2 11 nr 1n

2 2 1

24 2 1

nr r

n

2 lnr r4 41 1

ln16 32

r r r

2 2r c 3 2 2 2ln 4

3 9

c c c r c


(24)





as

(25)


(26)




(27)


(28)

2 ( )j j x x x x

( )j x x

1 1

1 1

( ) ( ) s

N Mt t t

i si j j

i j

N Mt t t

i si j j

i j

T T

q Q

x x x x x

x x x x x

1 1

t N Mjsit t

i j

i jk k k

T T

x x x

x xx x x

1

1

1

1

1

k k

k k

k k

T t u u

f t f f

T TT

t t

x x x

x x x

x x

1 1

2 1

2

1

1

1 1

k k

k

k k

k

k k

k T c TT

k k t

k T c TT

k k t

f fk

x x x x xx

x x

x x x x xx

x x

x xx











(29)


(30)




(31)

where 1 1N

2 2N 3 3N and



(32)

1 1 1

2 1

k k k k

kk T c T c T f

Tk k t k t k

x x x x x x x x xx

x x x x

1 1

1

1 1

2

1 1

3

on

on

on

k k

k k

k k

T u t

q q t

h T q h T

x x

x x

x x

1

1

1 1

1 1

1

1

1

k kN Mm mk k

i m si j m j

i j m m

Nk

i n si

i

f c TT

k k t

m M

T

x x x xL x x L x x

x x

x x

1 1

2 2 2

3 3 3 3

1 1

1 1

1

1 1 1

2 2

1 1

1 1

1 1

1

1

Mk k

j n j n

j

N Mk k k

i n si j n j n

i j

N Mk k


i j

u n N

Q q n N

h T Q h

x x x

x x x x x

x x x y x x x x

3 3 1

h u

n N

2

k c

k k t

x x xL I

x x












(33)


bx and cx










2

0

4( ) 1 ( 1) cos( )exp( )

(2 1)

i

i i

i

T x t x ti

(34)

with (2 1) 2i i














( )( 1) ( 1)s b b c b c x x x x x x












1 15 2 25 3 35 4 45 5 0

1

2

3

4

5

6

7

Similarity ratio

Re

lative

err

or

in

te

mp

era

ture

t = 05s

t = 10s

0 01 02 03 04 05 06 07 08 09 1 0

1

2

3

4

5

6

7

8

9

x (m)

Re

lative

err

or

in

te

mp

era

ture

t = 05s t = 01s

t = 05s t = 001s

t = 10s t = 01s

t = 10s t = 001s


(35)



05 cm-1






given as

2 2

21

2 cos( ) sin exp

n


L n L L

(36)





0 10 20 30 40 50 60

-01

0

01

02

03

04

05

06

07

08

Time t (second)

Te

mp

era

ture

(

)

Meshless x=001

x=002

x=003

Analytical x=001

x=002

x=003





respectively The





obtained as

0 0( ) ( )x xk x k e a x a e


(37)







right to left












and x = 001 m


LBIEM 01871 03281 03800 03986 04019 04053 04581

MFS 03915 04497 04546 04550 04551 04551 04551

Exact 04551

1( ) ( with 0)

1

x

L

e xT x T

e L











elastic system as

(38)







iu that is

(39)



(40)

0k ki i kk iu u b

( ) ( ) ( )h p

i i iu u u x x x

0h h

k ki i kku u




equation

(41)












1

sN

h

i nl li sn

n

u U

x x x (42)




(43)




(44)



overlap in the MFS

0p p

k ki i kk iu u b

x

x


sn x x

1 1 (3 4 ) ln


v r

x y

snx x





(45)






coefficient set

(46)




relations

(47)

holds true



(48)


(49)


(50)



only that is

1 1

( ) ( ) ( )M M

m m

i m i li m l

m m

b

x x x x x

1

( ) ( )M

p m

i li m l

m

u

x x x


1 1

2li li mm mi mlF F

4

1 = 11 2


4 1

1li liF


(51)





(1) Conical spline

(52)

with


(53)

with




1 1

( ) ( ) ( )N M

n m m

i li n l li l

n m

u U

x x y x (54)


(7) and (6) as

4 2 2 1 d d 1 d d

d d d dr r r r

r r r r r r

mr x x

2 1

1 2 2

1 1

2 1 2 1 2 3

n


1

2

4 5 2 2 3

2 1

A n n

A n

2 2

1 2 3 2

1

32 1 1 2

n

li il i l

rA A r r

n n

22

1

2

8 29 27 8 2 2 1 2 4 7 4 2 ln

2 1 2 3 2 1 2 ln

A n n n n n n n r

A n n n n r


1 1

( ) ( ) ( )N M

n m


n m

S m T

x x y x x (55)

where

(56)




(57)











( )Arerr defined by

(58)





S U U U

H A B

T

1 1 1 1

1 2 1 2 1 2 1 2

N N M M A

2

1

2

1

L

j j

j

L

j

j

Arerr









(59)

with










12

0( ) ( )E r E r a

r

12 2 2 1

2

12 2 2 1

22 2

2 2

kk

k k

r ak k

kk k k

ak k

a ra b r p

b a

k r k ba ra p

b a k k

2 4 4k








surface when 3







critical values








5r in

PS and 2 lnr r and


example







supported such that


(60)



(61)




0p is


the analysis



(0 ) ( ) 0

(0 ) ( ) 0x x

w z w L z

t z t L z

( 0) ( ) 0

( 0) ( ) 0

x x

z z

t x t x h

t x p t x h













length






(62)







for 0

ln ln

for 0

ln

a b

a b

T b r T r a

b a

b rT T Tr a

b

a



N=32 M=220








CONCLUSIONS



















REFERENCES








2000









69 (2002) 502-514








and Engineering 193 (2004) 4511-4539






Structures 83 (2005) 1487-1502



Engineering 36 (2005) 223-240



(2006) 381-393



(2005) 597-609












Structures 42 (2005) 4338-4351



6878-6890










150 (1999) 239-267





460 (2004) 1689-1706




182












Mechanics 74 (2005) 563-579





521







704-712



(1998) 141-148








(1998) 195-201



101



Elements 24 (2000) 575-582



227-236















25 (2001) 377-387



43-59







(14)








ANALYSIS



2

1

lnP

P




function namely

2 ( ) ( ) t tT b x x x (15)



( )tb x is known




2 ( ) 0t

hT x (16)


(17)


by

(18)

where ( )t

hq x and ( )t


respectively









i As a result

the potential ( )t




( )t

pT x

2 ( ) ( )t t

pT b x x

( ) ( ) ( ) ( ) ( ) ( )t t t t t t


x


1

1

( ) ( )

( ) ( )

Nt t

h i si si

i

Nt t

h i si si

i

T T

q Q

x x x x x

x x x x x

(19)



(20)

with


boundaries







1 1 1 2 2 22

1( ) ln

2

( ) 1( )

2

sj

sisj si si

T r

TQ k k x x n x x n

n r

x x

x xx x

2 2



1

( ) ( ) M

t t

j j j

j

b

x x x x x (21)












(22)



(23)


relationship



RBF PSK

( )

( )j x x

jx

( )j x x

2 2r c

r

2 2

1 1 2 2j jr x x x x

( )t

pT x ( )t

pq x

1

1

Mt t

p j j

j

Mt t

p j j

j

T

q

x x x

x x x

2 11 nr 1n

2 2 1

24 2 1

nr r

n

2 lnr r4 41 1

ln16 32

r r r

2 2r c 3 2 2 2ln 4

3 9

c c c r c


(24)





as

(25)


(26)




(27)


(28)

2 ( )j j x x x x

( )j x x

1 1

1 1

( ) ( ) s

N Mt t t

i si j j

i j

N Mt t t

i si j j

i j

T T

q Q

x x x x x

x x x x x

1 1

t N Mjsit t

i j

i jk k k

T T

x x x

x xx x x

1

1

1

1

1

k k

k k

k k

T t u u

f t f f

T TT

t t

x x x

x x x

x x

1 1

2 1

2

1

1

1 1

k k

k

k k

k

k k

k T c TT

k k t

k T c TT

k k t

f fk

x x x x xx

x x

x x x x xx

x x

x xx











(29)


(30)




(31)

where 1 1N

2 2N 3 3N and



(32)

1 1 1

2 1

k k k k

kk T c T c T f

Tk k t k t k

x x x x x x x x xx

x x x x

1 1

1

1 1

2

1 1

3

on

on

on

k k

k k

k k

T u t

q q t

h T q h T

x x

x x

x x

1

1

1 1

1 1

1

1

1

k kN Mm mk k

i m si j m j

i j m m

Nk

i n si

i

f c TT

k k t

m M

T

x x x xL x x L x x

x x

x x

1 1

2 2 2

3 3 3 3

1 1

1 1

1

1 1 1

2 2

1 1

1 1

1 1

1

1

Mk k

j n j n

j

N Mk k k

i n si j n j n

i j

N Mk k


i j

u n N

Q q n N

h T Q h

x x x

x x x x x

x x x y x x x x

3 3 1

h u

n N

2

k c

k k t

x x xL I

x x












(33)


bx and cx










2

0

4( ) 1 ( 1) cos( )exp( )

(2 1)

i

i i

i

T x t x ti

(34)

with (2 1) 2i i














( )( 1) ( 1)s b b c b c x x x x x x












1 15 2 25 3 35 4 45 5 0

1

2

3

4

5

6

7

Similarity ratio

Re

lative

err

or

in

te

mp

era

ture

t = 05s

t = 10s

0 01 02 03 04 05 06 07 08 09 1 0

1

2

3

4

5

6

7

8

9

x (m)

Re

lative

err

or

in

te

mp

era

ture

t = 05s t = 01s

t = 05s t = 001s

t = 10s t = 01s

t = 10s t = 001s


(35)



05 cm-1






given as

2 2

21

2 cos( ) sin exp

n


L n L L

(36)





0 10 20 30 40 50 60

-01

0

01

02

03

04

05

06

07

08

Time t (second)

Te

mp

era

ture

(

)

Meshless x=001

x=002

x=003

Analytical x=001

x=002

x=003





respectively The





obtained as

0 0( ) ( )x xk x k e a x a e


(37)







right to left












and x = 001 m


LBIEM 01871 03281 03800 03986 04019 04053 04581

MFS 03915 04497 04546 04550 04551 04551 04551

Exact 04551

1( ) ( with 0)

1

x

L

e xT x T

e L











elastic system as

(38)







iu that is

(39)



(40)

0k ki i kk iu u b

( ) ( ) ( )h p

i i iu u u x x x

0h h

k ki i kku u




equation

(41)












1

sN

h

i nl li sn

n

u U

x x x (42)




(43)




(44)



overlap in the MFS

0p p

k ki i kk iu u b

x

x


sn x x

1 1 (3 4 ) ln


v r

x y

snx x





(45)






coefficient set

(46)




relations

(47)

holds true



(48)


(49)


(50)



only that is

1 1

( ) ( ) ( )M M

m m

i m i li m l

m m

b

x x x x x

1

( ) ( )M

p m

i li m l

m

u

x x x


1 1

2li li mm mi mlF F

4

1 = 11 2


4 1

1li liF


(51)





(1) Conical spline

(52)

with


(53)

with




1 1

( ) ( ) ( )N M

n m m

i li n l li l

n m

u U

x x y x (54)


(7) and (6) as

4 2 2 1 d d 1 d d

d d d dr r r r

r r r r r r

mr x x

2 1

1 2 2

1 1

2 1 2 1 2 3

n


1

2

4 5 2 2 3

2 1

A n n

A n

2 2

1 2 3 2

1

32 1 1 2

n

li il i l

rA A r r

n n

22

1

2

8 29 27 8 2 2 1 2 4 7 4 2 ln

2 1 2 3 2 1 2 ln

A n n n n n n n r

A n n n n r


1 1

( ) ( ) ( )N M

n m


n m

S m T

x x y x x (55)

where

(56)




(57)











( )Arerr defined by

(58)





S U U U

H A B

T

1 1 1 1

1 2 1 2 1 2 1 2

N N M M A

2

1

2

1

L

j j

j

L

j

j

Arerr









(59)

with










12

0( ) ( )E r E r a

r

12 2 2 1

2

12 2 2 1

22 2

2 2

kk

k k

r ak k

kk k k

ak k

a ra b r p

b a

k r k ba ra p

b a k k

2 4 4k








surface when 3







critical values








5r in

PS and 2 lnr r and


example







supported such that


(60)



(61)




0p is


the analysis



(0 ) ( ) 0

(0 ) ( ) 0x x

w z w L z

t z t L z

( 0) ( ) 0

( 0) ( ) 0

x x

z z

t x t x h

t x p t x h













length






(62)







for 0

ln ln

for 0

ln

a b

a b

T b r T r a

b a

b rT T Tr a

b

a



N=32 M=220








CONCLUSIONS



















REFERENCES








2000









69 (2002) 502-514








and Engineering 193 (2004) 4511-4539






Structures 83 (2005) 1487-1502



Engineering 36 (2005) 223-240



(2006) 381-393



(2005) 597-609












Structures 42 (2005) 4338-4351



6878-6890










150 (1999) 239-267





460 (2004) 1689-1706




182












Mechanics 74 (2005) 563-579





521







704-712



(1998) 141-148








(1998) 195-201



101



Elements 24 (2000) 575-582



227-236















25 (2001) 377-387



43-59









function namely

2 ( ) ( ) t tT b x x x (15)



( )tb x is known




2 ( ) 0t

hT x (16)


(17)


by

(18)

where ( )t

hq x and ( )t


respectively









i As a result

the potential ( )t




( )t

pT x

2 ( ) ( )t t

pT b x x

( ) ( ) ( ) ( ) ( ) ( )t t t t t t


x


1

1

( ) ( )

( ) ( )

Nt t

h i si si

i

Nt t

h i si si

i

T T

q Q

x x x x x

x x x x x

(19)



(20)

with


boundaries







1 1 1 2 2 22

1( ) ln

2

( ) 1( )

2

sj

sisj si si

T r

TQ k k x x n x x n

n r

x x

x xx x

2 2



1

( ) ( ) M

t t

j j j

j

b

x x x x x (21)












(22)



(23)


relationship



RBF PSK

( )

( )j x x

jx

( )j x x

2 2r c

r

2 2

1 1 2 2j jr x x x x

( )t

pT x ( )t

pq x

1

1

Mt t

p j j

j

Mt t

p j j

j

T

q

x x x

x x x

2 11 nr 1n

2 2 1

24 2 1

nr r

n

2 lnr r4 41 1

ln16 32

r r r

2 2r c 3 2 2 2ln 4

3 9

c c c r c


(24)





as

(25)


(26)




(27)


(28)

2 ( )j j x x x x

( )j x x

1 1

1 1

( ) ( ) s

N Mt t t

i si j j

i j

N Mt t t

i si j j

i j

T T

q Q

x x x x x

x x x x x

1 1

t N Mjsit t

i j

i jk k k

T T

x x x

x xx x x

1

1

1

1

1

k k

k k

k k

T t u u

f t f f

T TT

t t

x x x

x x x

x x

1 1

2 1

2

1

1

1 1

k k

k

k k

k

k k

k T c TT

k k t

k T c TT

k k t

f fk

x x x x xx

x x

x x x x xx

x x

x xx











(29)


(30)




(31)

where 1 1N

2 2N 3 3N and



(32)

1 1 1

2 1

k k k k

kk T c T c T f

Tk k t k t k

x x x x x x x x xx

x x x x

1 1

1

1 1

2

1 1

3

on

on

on

k k

k k

k k

T u t

q q t

h T q h T

x x

x x

x x

1

1

1 1

1 1

1

1

1

k kN Mm mk k

i m si j m j

i j m m

Nk

i n si

i

f c TT

k k t

m M

T

x x x xL x x L x x

x x

x x

1 1

2 2 2

3 3 3 3

1 1

1 1

1

1 1 1

2 2

1 1

1 1

1 1

1

1

Mk k

j n j n

j

N Mk k k

i n si j n j n

i j

N Mk k


i j

u n N

Q q n N

h T Q h

x x x

x x x x x

x x x y x x x x

3 3 1

h u

n N

2

k c

k k t

x x xL I

x x












(33)


bx and cx










2

0

4( ) 1 ( 1) cos( )exp( )

(2 1)

i

i i

i

T x t x ti

(34)

with (2 1) 2i i














( )( 1) ( 1)s b b c b c x x x x x x












1 15 2 25 3 35 4 45 5 0

1

2

3

4

5

6

7

Similarity ratio

Re

lative

err

or

in

te

mp

era

ture

t = 05s

t = 10s

0 01 02 03 04 05 06 07 08 09 1 0

1

2

3

4

5

6

7

8

9

x (m)

Re

lative

err

or

in

te

mp

era

ture

t = 05s t = 01s

t = 05s t = 001s

t = 10s t = 01s

t = 10s t = 001s


(35)



05 cm-1






given as

2 2

21

2 cos( ) sin exp

n


L n L L

(36)





0 10 20 30 40 50 60

-01

0

01

02

03

04

05

06

07

08

Time t (second)

Te

mp

era

ture

(

)

Meshless x=001

x=002

x=003

Analytical x=001

x=002

x=003





respectively The





obtained as

0 0( ) ( )x xk x k e a x a e


(37)







right to left












and x = 001 m


LBIEM 01871 03281 03800 03986 04019 04053 04581

MFS 03915 04497 04546 04550 04551 04551 04551

Exact 04551

1( ) ( with 0)

1

x

L

e xT x T

e L











elastic system as

(38)







iu that is

(39)



(40)

0k ki i kk iu u b

( ) ( ) ( )h p

i i iu u u x x x

0h h

k ki i kku u




equation

(41)












1

sN

h

i nl li sn

n

u U

x x x (42)




(43)




(44)



overlap in the MFS

0p p

k ki i kk iu u b

x

x


sn x x

1 1 (3 4 ) ln


v r

x y

snx x





(45)






coefficient set

(46)




relations

(47)

holds true



(48)


(49)


(50)



only that is

1 1

( ) ( ) ( )M M

m m

i m i li m l

m m

b

x x x x x

1

( ) ( )M

p m

i li m l

m

u

x x x


1 1

2li li mm mi mlF F

4

1 = 11 2


4 1

1li liF


(51)





(1) Conical spline

(52)

with


(53)

with




1 1

( ) ( ) ( )N M

n m m

i li n l li l

n m

u U

x x y x (54)


(7) and (6) as

4 2 2 1 d d 1 d d

d d d dr r r r

r r r r r r

mr x x

2 1

1 2 2

1 1

2 1 2 1 2 3

n


1

2

4 5 2 2 3

2 1

A n n

A n

2 2

1 2 3 2

1

32 1 1 2

n

li il i l

rA A r r

n n

22

1

2

8 29 27 8 2 2 1 2 4 7 4 2 ln

2 1 2 3 2 1 2 ln

A n n n n n n n r

A n n n n r


1 1

( ) ( ) ( )N M

n m


n m

S m T

x x y x x (55)

where

(56)




(57)











( )Arerr defined by

(58)





S U U U

H A B

T

1 1 1 1

1 2 1 2 1 2 1 2

N N M M A

2

1

2

1

L

j j

j

L

j

j

Arerr









(59)

with










12

0( ) ( )E r E r a

r

12 2 2 1

2

12 2 2 1

22 2

2 2

kk

k k

r ak k

kk k k

ak k

a ra b r p

b a

k r k ba ra p

b a k k

2 4 4k








surface when 3







critical values








5r in

PS and 2 lnr r and


example







supported such that


(60)



(61)




0p is


the analysis



(0 ) ( ) 0

(0 ) ( ) 0x x

w z w L z

t z t L z

( 0) ( ) 0

( 0) ( ) 0

x x

z z

t x t x h

t x p t x h













length






(62)







for 0

ln ln

for 0

ln

a b

a b

T b r T r a

b a

b rT T Tr a

b

a



N=32 M=220








CONCLUSIONS



















REFERENCES








2000









69 (2002) 502-514








and Engineering 193 (2004) 4511-4539






Structures 83 (2005) 1487-1502



Engineering 36 (2005) 223-240



(2006) 381-393



(2005) 597-609












Structures 42 (2005) 4338-4351



6878-6890










150 (1999) 239-267





460 (2004) 1689-1706




182












Mechanics 74 (2005) 563-579





521







704-712



(1998) 141-148








(1998) 195-201



101



Elements 24 (2000) 575-582



227-236















25 (2001) 377-387



43-59







1

1

( ) ( )

( ) ( )

Nt t

h i si si

i

Nt t

h i si si

i

T T

q Q

x x x x x

x x x x x

(19)



(20)

with


boundaries







1 1 1 2 2 22

1( ) ln

2

( ) 1( )

2

sj

sisj si si

T r

TQ k k x x n x x n

n r

x x

x xx x

2 2



1

( ) ( ) M

t t

j j j

j

b

x x x x x (21)












(22)



(23)


relationship



RBF PSK

( )

( )j x x

jx

( )j x x

2 2r c

r

2 2

1 1 2 2j jr x x x x

( )t

pT x ( )t

pq x

1

1

Mt t

p j j

j

Mt t

p j j

j

T

q

x x x

x x x

2 11 nr 1n

2 2 1

24 2 1

nr r

n

2 lnr r4 41 1

ln16 32

r r r

2 2r c 3 2 2 2ln 4

3 9

c c c r c


(24)





as

(25)


(26)




(27)


(28)

2 ( )j j x x x x

( )j x x

1 1

1 1

( ) ( ) s

N Mt t t

i si j j

i j

N Mt t t

i si j j

i j

T T

q Q

x x x x x

x x x x x

1 1

t N Mjsit t

i j

i jk k k

T T

x x x

x xx x x

1

1

1

1

1

k k

k k

k k

T t u u

f t f f

T TT

t t

x x x

x x x

x x

1 1

2 1

2

1

1

1 1

k k

k

k k

k

k k

k T c TT

k k t

k T c TT

k k t

f fk

x x x x xx

x x

x x x x xx

x x

x xx











(29)


(30)




(31)

where 1 1N

2 2N 3 3N and



(32)

1 1 1

2 1

k k k k

kk T c T c T f

Tk k t k t k

x x x x x x x x xx

x x x x

1 1

1

1 1

2

1 1

3

on

on

on

k k

k k

k k

T u t

q q t

h T q h T

x x

x x

x x

1

1

1 1

1 1

1

1

1

k kN Mm mk k

i m si j m j

i j m m

Nk

i n si

i

f c TT

k k t

m M

T

x x x xL x x L x x

x x

x x

1 1

2 2 2

3 3 3 3

1 1

1 1

1

1 1 1

2 2

1 1

1 1

1 1

1

1

Mk k

j n j n

j

N Mk k k

i n si j n j n

i j

N Mk k


i j

u n N

Q q n N

h T Q h

x x x

x x x x x

x x x y x x x x

3 3 1

h u

n N

2

k c

k k t

x x xL I

x x












(33)


bx and cx










2

0

4( ) 1 ( 1) cos( )exp( )

(2 1)

i

i i

i

T x t x ti

(34)

with (2 1) 2i i














( )( 1) ( 1)s b b c b c x x x x x x












1 15 2 25 3 35 4 45 5 0

1

2

3

4

5

6

7

Similarity ratio

Re

lative

err

or

in

te

mp

era

ture

t = 05s

t = 10s

0 01 02 03 04 05 06 07 08 09 1 0

1

2

3

4

5

6

7

8

9

x (m)

Re

lative

err

or

in

te

mp

era

ture

t = 05s t = 01s

t = 05s t = 001s

t = 10s t = 01s

t = 10s t = 001s


(35)



05 cm-1






given as

2 2

21

2 cos( ) sin exp

n


L n L L

(36)





0 10 20 30 40 50 60

-01

0

01

02

03

04

05

06

07

08

Time t (second)

Te

mp

era

ture

(

)

Meshless x=001

x=002

x=003

Analytical x=001

x=002

x=003





respectively The





obtained as

0 0( ) ( )x xk x k e a x a e


(37)







right to left












and x = 001 m


LBIEM 01871 03281 03800 03986 04019 04053 04581

MFS 03915 04497 04546 04550 04551 04551 04551

Exact 04551

1( ) ( with 0)

1

x

L

e xT x T

e L











elastic system as

(38)







iu that is

(39)



(40)

0k ki i kk iu u b

( ) ( ) ( )h p

i i iu u u x x x

0h h

k ki i kku u




equation

(41)












1

sN

h

i nl li sn

n

u U

x x x (42)




(43)




(44)



overlap in the MFS

0p p

k ki i kk iu u b

x

x


sn x x

1 1 (3 4 ) ln


v r

x y

snx x





(45)






coefficient set

(46)




relations

(47)

holds true



(48)


(49)


(50)



only that is

1 1

( ) ( ) ( )M M

m m

i m i li m l

m m

b

x x x x x

1

( ) ( )M

p m

i li m l

m

u

x x x


1 1

2li li mm mi mlF F

4

1 = 11 2


4 1

1li liF


(51)





(1) Conical spline

(52)

with


(53)

with




1 1

( ) ( ) ( )N M

n m m

i li n l li l

n m

u U

x x y x (54)


(7) and (6) as

4 2 2 1 d d 1 d d

d d d dr r r r

r r r r r r

mr x x

2 1

1 2 2

1 1

2 1 2 1 2 3

n


1

2

4 5 2 2 3

2 1

A n n

A n

2 2

1 2 3 2

1

32 1 1 2

n

li il i l

rA A r r

n n

22

1

2

8 29 27 8 2 2 1 2 4 7 4 2 ln

2 1 2 3 2 1 2 ln

A n n n n n n n r

A n n n n r


1 1

( ) ( ) ( )N M

n m


n m

S m T

x x y x x (55)

where

(56)




(57)











( )Arerr defined by

(58)





S U U U

H A B

T

1 1 1 1

1 2 1 2 1 2 1 2

N N M M A

2

1

2

1

L

j j

j

L

j

j

Arerr









(59)

with










12

0( ) ( )E r E r a

r

12 2 2 1

2

12 2 2 1

22 2

2 2

kk

k k

r ak k

kk k k

ak k

a ra b r p

b a

k r k ba ra p

b a k k

2 4 4k








surface when 3







critical values








5r in

PS and 2 lnr r and


example







supported such that


(60)



(61)




0p is


the analysis



(0 ) ( ) 0

(0 ) ( ) 0x x

w z w L z

t z t L z

( 0) ( ) 0

( 0) ( ) 0

x x

z z

t x t x h

t x p t x h













length






(62)







for 0

ln ln

for 0

ln

a b

a b

T b r T r a

b a

b rT T Tr a

b

a



N=32 M=220








CONCLUSIONS



















REFERENCES








2000









69 (2002) 502-514








and Engineering 193 (2004) 4511-4539






Structures 83 (2005) 1487-1502



Engineering 36 (2005) 223-240



(2006) 381-393



(2005) 597-609












Structures 42 (2005) 4338-4351



6878-6890










150 (1999) 239-267





460 (2004) 1689-1706




182












Mechanics 74 (2005) 563-579





521







704-712



(1998) 141-148








(1998) 195-201



101



Elements 24 (2000) 575-582



227-236















25 (2001) 377-387



43-59







1

( ) ( ) M

t t

j j j

j

b

x x x x x (21)












(22)



(23)


relationship



RBF PSK

( )

( )j x x

jx

( )j x x

2 2r c

r

2 2

1 1 2 2j jr x x x x

( )t

pT x ( )t

pq x

1

1

Mt t

p j j

j

Mt t

p j j

j

T

q

x x x

x x x

2 11 nr 1n

2 2 1

24 2 1

nr r

n

2 lnr r4 41 1

ln16 32

r r r

2 2r c 3 2 2 2ln 4

3 9

c c c r c


(24)





as

(25)


(26)




(27)


(28)

2 ( )j j x x x x

( )j x x

1 1

1 1

( ) ( ) s

N Mt t t

i si j j

i j

N Mt t t

i si j j

i j

T T

q Q

x x x x x

x x x x x

1 1

t N Mjsit t

i j

i jk k k

T T

x x x

x xx x x

1

1

1

1

1

k k

k k

k k

T t u u

f t f f

T TT

t t

x x x

x x x

x x

1 1

2 1

2

1

1

1 1

k k

k

k k

k

k k

k T c TT

k k t

k T c TT

k k t

f fk

x x x x xx

x x

x x x x xx

x x

x xx











(29)


(30)




(31)

where 1 1N

2 2N 3 3N and



(32)

1 1 1

2 1

k k k k

kk T c T c T f

Tk k t k t k

x x x x x x x x xx

x x x x

1 1

1

1 1

2

1 1

3

on

on

on

k k

k k

k k

T u t

q q t

h T q h T

x x

x x

x x

1

1

1 1

1 1

1

1

1

k kN Mm mk k

i m si j m j

i j m m

Nk

i n si

i

f c TT

k k t

m M

T

x x x xL x x L x x

x x

x x

1 1

2 2 2

3 3 3 3

1 1

1 1

1

1 1 1

2 2

1 1

1 1

1 1

1

1

Mk k

j n j n

j

N Mk k k

i n si j n j n

i j

N Mk k


i j

u n N

Q q n N

h T Q h

x x x

x x x x x

x x x y x x x x

3 3 1

h u

n N

2

k c

k k t

x x xL I

x x












(33)


bx and cx










2

0

4( ) 1 ( 1) cos( )exp( )

(2 1)

i

i i

i

T x t x ti

(34)

with (2 1) 2i i














( )( 1) ( 1)s b b c b c x x x x x x












1 15 2 25 3 35 4 45 5 0

1

2

3

4

5

6

7

Similarity ratio

Re

lative

err

or

in

te

mp

era

ture

t = 05s

t = 10s

0 01 02 03 04 05 06 07 08 09 1 0

1

2

3

4

5

6

7

8

9

x (m)

Re

lative

err

or

in

te

mp

era

ture

t = 05s t = 01s

t = 05s t = 001s

t = 10s t = 01s

t = 10s t = 001s


(35)



05 cm-1






given as

2 2

21

2 cos( ) sin exp

n


L n L L

(36)





0 10 20 30 40 50 60

-01

0

01

02

03

04

05

06

07

08

Time t (second)

Te

mp

era

ture

(

)

Meshless x=001

x=002

x=003

Analytical x=001

x=002

x=003





respectively The





obtained as

0 0( ) ( )x xk x k e a x a e


(37)







right to left












and x = 001 m


LBIEM 01871 03281 03800 03986 04019 04053 04581

MFS 03915 04497 04546 04550 04551 04551 04551

Exact 04551

1( ) ( with 0)

1

x

L

e xT x T

e L











elastic system as

(38)







iu that is

(39)



(40)

0k ki i kk iu u b

( ) ( ) ( )h p

i i iu u u x x x

0h h

k ki i kku u




equation

(41)












1

sN

h

i nl li sn

n

u U

x x x (42)




(43)




(44)



overlap in the MFS

0p p

k ki i kk iu u b

x

x


sn x x

1 1 (3 4 ) ln


v r

x y

snx x





(45)






coefficient set

(46)




relations

(47)

holds true



(48)


(49)


(50)



only that is

1 1

( ) ( ) ( )M M

m m

i m i li m l

m m

b

x x x x x

1

( ) ( )M

p m

i li m l

m

u

x x x


1 1

2li li mm mi mlF F

4

1 = 11 2


4 1

1li liF


(51)





(1) Conical spline

(52)

with


(53)

with




1 1

( ) ( ) ( )N M

n m m

i li n l li l

n m

u U

x x y x (54)


(7) and (6) as

4 2 2 1 d d 1 d d

d d d dr r r r

r r r r r r

mr x x

2 1

1 2 2

1 1

2 1 2 1 2 3

n


1

2

4 5 2 2 3

2 1

A n n

A n

2 2

1 2 3 2

1

32 1 1 2

n

li il i l

rA A r r

n n

22

1

2

8 29 27 8 2 2 1 2 4 7 4 2 ln

2 1 2 3 2 1 2 ln

A n n n n n n n r

A n n n n r


1 1

( ) ( ) ( )N M

n m


n m

S m T

x x y x x (55)

where

(56)




(57)











( )Arerr defined by

(58)





S U U U

H A B

T

1 1 1 1

1 2 1 2 1 2 1 2

N N M M A

2

1

2

1

L

j j

j

L

j

j

Arerr









(59)

with










12

0( ) ( )E r E r a

r

12 2 2 1

2

12 2 2 1

22 2

2 2

kk

k k

r ak k

kk k k

ak k

a ra b r p

b a

k r k ba ra p

b a k k

2 4 4k








surface when 3







critical values








5r in

PS and 2 lnr r and


example







supported such that


(60)



(61)




0p is


the analysis



(0 ) ( ) 0

(0 ) ( ) 0x x

w z w L z

t z t L z

( 0) ( ) 0

( 0) ( ) 0

x x

z z

t x t x h

t x p t x h













length






(62)







for 0

ln ln

for 0

ln

a b

a b

T b r T r a

b a

b rT T Tr a

b

a



N=32 M=220








CONCLUSIONS



















REFERENCES








2000









69 (2002) 502-514








and Engineering 193 (2004) 4511-4539






Structures 83 (2005) 1487-1502



Engineering 36 (2005) 223-240



(2006) 381-393



(2005) 597-609












Structures 42 (2005) 4338-4351



6878-6890










150 (1999) 239-267





460 (2004) 1689-1706




182












Mechanics 74 (2005) 563-579





521







704-712



(1998) 141-148








(1998) 195-201



101



Elements 24 (2000) 575-582



227-236















25 (2001) 377-387



43-59







(24)





as

(25)


(26)




(27)


(28)

2 ( )j j x x x x

( )j x x

1 1

1 1

( ) ( ) s

N Mt t t

i si j j

i j

N Mt t t

i si j j

i j

T T

q Q

x x x x x

x x x x x

1 1

t N Mjsit t

i j

i jk k k

T T

x x x

x xx x x

1

1

1

1

1

k k

k k

k k

T t u u

f t f f

T TT

t t

x x x

x x x

x x

1 1

2 1

2

1

1

1 1

k k

k

k k

k

k k

k T c TT

k k t

k T c TT

k k t

f fk

x x x x xx

x x

x x x x xx

x x

x xx











(29)


(30)




(31)

where 1 1N

2 2N 3 3N and



(32)

1 1 1

2 1

k k k k

kk T c T c T f

Tk k t k t k

x x x x x x x x xx

x x x x

1 1

1

1 1

2

1 1

3

on

on

on

k k

k k

k k

T u t

q q t

h T q h T

x x

x x

x x

1

1

1 1

1 1

1

1

1

k kN Mm mk k

i m si j m j

i j m m

Nk

i n si

i

f c TT

k k t

m M

T

x x x xL x x L x x

x x

x x

1 1

2 2 2

3 3 3 3

1 1

1 1

1

1 1 1

2 2

1 1

1 1

1 1

1

1

Mk k

j n j n

j

N Mk k k

i n si j n j n

i j

N Mk k


i j

u n N

Q q n N

h T Q h

x x x

x x x x x

x x x y x x x x

3 3 1

h u

n N

2

k c

k k t

x x xL I

x x












(33)


bx and cx










2

0

4( ) 1 ( 1) cos( )exp( )

(2 1)

i

i i

i

T x t x ti

(34)

with (2 1) 2i i














( )( 1) ( 1)s b b c b c x x x x x x












1 15 2 25 3 35 4 45 5 0

1

2

3

4

5

6

7

Similarity ratio

Re

lative

err

or

in

te

mp

era

ture

t = 05s

t = 10s

0 01 02 03 04 05 06 07 08 09 1 0

1

2

3

4

5

6

7

8

9

x (m)

Re

lative

err

or

in

te

mp

era

ture

t = 05s t = 01s

t = 05s t = 001s

t = 10s t = 01s

t = 10s t = 001s


(35)



05 cm-1






given as

2 2

21

2 cos( ) sin exp

n


L n L L

(36)





0 10 20 30 40 50 60

-01

0

01

02

03

04

05

06

07

08

Time t (second)

Te

mp

era

ture

(

)

Meshless x=001

x=002

x=003

Analytical x=001

x=002

x=003





respectively The





obtained as

0 0( ) ( )x xk x k e a x a e


(37)







right to left












and x = 001 m


LBIEM 01871 03281 03800 03986 04019 04053 04581

MFS 03915 04497 04546 04550 04551 04551 04551

Exact 04551

1( ) ( with 0)

1

x

L

e xT x T

e L











elastic system as

(38)







iu that is

(39)



(40)

0k ki i kk iu u b

( ) ( ) ( )h p

i i iu u u x x x

0h h

k ki i kku u




equation

(41)












1

sN

h

i nl li sn

n

u U

x x x (42)




(43)




(44)



overlap in the MFS

0p p

k ki i kk iu u b

x

x


sn x x

1 1 (3 4 ) ln


v r

x y

snx x





(45)






coefficient set

(46)




relations

(47)

holds true



(48)


(49)


(50)



only that is

1 1

( ) ( ) ( )M M

m m

i m i li m l

m m

b

x x x x x

1

( ) ( )M

p m

i li m l

m

u

x x x


1 1

2li li mm mi mlF F

4

1 = 11 2


4 1

1li liF


(51)





(1) Conical spline

(52)

with


(53)

with




1 1

( ) ( ) ( )N M

n m m

i li n l li l

n m

u U

x x y x (54)


(7) and (6) as

4 2 2 1 d d 1 d d

d d d dr r r r

r r r r r r

mr x x

2 1

1 2 2

1 1

2 1 2 1 2 3

n


1

2

4 5 2 2 3

2 1

A n n

A n

2 2

1 2 3 2

1

32 1 1 2

n

li il i l

rA A r r

n n

22

1

2

8 29 27 8 2 2 1 2 4 7 4 2 ln

2 1 2 3 2 1 2 ln

A n n n n n n n r

A n n n n r


1 1

( ) ( ) ( )N M

n m


n m

S m T

x x y x x (55)

where

(56)




(57)











( )Arerr defined by

(58)





S U U U

H A B

T

1 1 1 1

1 2 1 2 1 2 1 2

N N M M A

2

1

2

1

L

j j

j

L

j

j

Arerr









(59)

with










12

0( ) ( )E r E r a

r

12 2 2 1

2

12 2 2 1

22 2

2 2

kk

k k

r ak k

kk k k

ak k

a ra b r p

b a

k r k ba ra p

b a k k

2 4 4k








surface when 3







critical values








5r in

PS and 2 lnr r and


example







supported such that


(60)



(61)




0p is


the analysis



(0 ) ( ) 0

(0 ) ( ) 0x x

w z w L z

t z t L z

( 0) ( ) 0

( 0) ( ) 0

x x

z z

t x t x h

t x p t x h













length






(62)







for 0

ln ln

for 0

ln

a b

a b

T b r T r a

b a

b rT T Tr a

b

a



N=32 M=220








CONCLUSIONS



















REFERENCES








2000









69 (2002) 502-514








and Engineering 193 (2004) 4511-4539






Structures 83 (2005) 1487-1502



Engineering 36 (2005) 223-240



(2006) 381-393



(2005) 597-609












Structures 42 (2005) 4338-4351



6878-6890










150 (1999) 239-267





460 (2004) 1689-1706




182












Mechanics 74 (2005) 563-579





521







704-712



(1998) 141-148








(1998) 195-201



101



Elements 24 (2000) 575-582



227-236















25 (2001) 377-387



43-59
















(29)


(30)




(31)

where 1 1N

2 2N 3 3N and



(32)

1 1 1

2 1

k k k k

kk T c T c T f

Tk k t k t k

x x x x x x x x xx

x x x x

1 1

1

1 1

2

1 1

3

on

on

on

k k

k k

k k

T u t

q q t

h T q h T

x x

x x

x x

1

1

1 1

1 1

1

1

1

k kN Mm mk k

i m si j m j

i j m m

Nk

i n si

i

f c TT

k k t

m M

T

x x x xL x x L x x

x x

x x

1 1

2 2 2

3 3 3 3

1 1

1 1

1

1 1 1

2 2

1 1

1 1

1 1

1

1

Mk k

j n j n

j

N Mk k k

i n si j n j n

i j

N Mk k


i j

u n N

Q q n N

h T Q h

x x x

x x x x x

x x x y x x x x

3 3 1

h u

n N

2

k c

k k t

x x xL I

x x












(33)


bx and cx










2

0

4( ) 1 ( 1) cos( )exp( )

(2 1)

i

i i

i

T x t x ti

(34)

with (2 1) 2i i














( )( 1) ( 1)s b b c b c x x x x x x












1 15 2 25 3 35 4 45 5 0

1

2

3

4

5

6

7

Similarity ratio

Re

lative

err

or

in

te

mp

era

ture

t = 05s

t = 10s

0 01 02 03 04 05 06 07 08 09 1 0

1

2

3

4

5

6

7

8

9

x (m)

Re

lative

err

or

in

te

mp

era

ture

t = 05s t = 01s

t = 05s t = 001s

t = 10s t = 01s

t = 10s t = 001s


(35)



05 cm-1






given as

2 2

21

2 cos( ) sin exp

n


L n L L

(36)





0 10 20 30 40 50 60

-01

0

01

02

03

04

05

06

07

08

Time t (second)

Te

mp

era

ture

(

)

Meshless x=001

x=002

x=003

Analytical x=001

x=002

x=003





respectively The





obtained as

0 0( ) ( )x xk x k e a x a e


(37)







right to left












and x = 001 m


LBIEM 01871 03281 03800 03986 04019 04053 04581

MFS 03915 04497 04546 04550 04551 04551 04551

Exact 04551

1( ) ( with 0)

1

x

L

e xT x T

e L











elastic system as

(38)







iu that is

(39)



(40)

0k ki i kk iu u b

( ) ( ) ( )h p

i i iu u u x x x

0h h

k ki i kku u




equation

(41)












1

sN

h

i nl li sn

n

u U

x x x (42)




(43)




(44)



overlap in the MFS

0p p

k ki i kk iu u b

x

x


sn x x

1 1 (3 4 ) ln


v r

x y

snx x





(45)






coefficient set

(46)




relations

(47)

holds true



(48)


(49)


(50)



only that is

1 1

( ) ( ) ( )M M

m m

i m i li m l

m m

b

x x x x x

1

( ) ( )M

p m

i li m l

m

u

x x x


1 1

2li li mm mi mlF F

4

1 = 11 2


4 1

1li liF


(51)





(1) Conical spline

(52)

with


(53)

with




1 1

( ) ( ) ( )N M

n m m

i li n l li l

n m

u U

x x y x (54)


(7) and (6) as

4 2 2 1 d d 1 d d

d d d dr r r r

r r r r r r

mr x x

2 1

1 2 2

1 1

2 1 2 1 2 3

n


1

2

4 5 2 2 3

2 1

A n n

A n

2 2

1 2 3 2

1

32 1 1 2

n

li il i l

rA A r r

n n

22

1

2

8 29 27 8 2 2 1 2 4 7 4 2 ln

2 1 2 3 2 1 2 ln

A n n n n n n n r

A n n n n r


1 1

( ) ( ) ( )N M

n m


n m

S m T

x x y x x (55)

where

(56)




(57)











( )Arerr defined by

(58)





S U U U

H A B

T

1 1 1 1

1 2 1 2 1 2 1 2

N N M M A

2

1

2

1

L

j j

j

L

j

j

Arerr









(59)

with










12

0( ) ( )E r E r a

r

12 2 2 1

2

12 2 2 1

22 2

2 2

kk

k k

r ak k

kk k k

ak k

a ra b r p

b a

k r k ba ra p

b a k k

2 4 4k








surface when 3







critical values








5r in

PS and 2 lnr r and


example







supported such that


(60)



(61)




0p is


the analysis



(0 ) ( ) 0

(0 ) ( ) 0x x

w z w L z

t z t L z

( 0) ( ) 0

( 0) ( ) 0

x x

z z

t x t x h

t x p t x h













length






(62)







for 0

ln ln

for 0

ln

a b

a b

T b r T r a

b a

b rT T Tr a

b

a



N=32 M=220








CONCLUSIONS



















REFERENCES








2000









69 (2002) 502-514








and Engineering 193 (2004) 4511-4539






Structures 83 (2005) 1487-1502



Engineering 36 (2005) 223-240



(2006) 381-393



(2005) 597-609












Structures 42 (2005) 4338-4351



6878-6890










150 (1999) 239-267





460 (2004) 1689-1706




182












Mechanics 74 (2005) 563-579





521







704-712



(1998) 141-148








(1998) 195-201



101



Elements 24 (2000) 575-582



227-236















25 (2001) 377-387



43-59

















(33)


bx and cx










2

0

4( ) 1 ( 1) cos( )exp( )

(2 1)

i

i i

i

T x t x ti

(34)

with (2 1) 2i i














( )( 1) ( 1)s b b c b c x x x x x x












1 15 2 25 3 35 4 45 5 0

1

2

3

4

5

6

7

Similarity ratio

Re

lative

err

or

in

te

mp

era

ture

t = 05s

t = 10s

0 01 02 03 04 05 06 07 08 09 1 0

1

2

3

4

5

6

7

8

9

x (m)

Re

lative

err

or

in

te

mp

era

ture

t = 05s t = 01s

t = 05s t = 001s

t = 10s t = 01s

t = 10s t = 001s


(35)



05 cm-1






given as

2 2

21

2 cos( ) sin exp

n


L n L L

(36)





0 10 20 30 40 50 60

-01

0

01

02

03

04

05

06

07

08

Time t (second)

Te

mp

era

ture

(

)

Meshless x=001

x=002

x=003

Analytical x=001

x=002

x=003





respectively The





obtained as

0 0( ) ( )x xk x k e a x a e


(37)







right to left












and x = 001 m


LBIEM 01871 03281 03800 03986 04019 04053 04581

MFS 03915 04497 04546 04550 04551 04551 04551

Exact 04551

1( ) ( with 0)

1

x

L

e xT x T

e L











elastic system as

(38)







iu that is

(39)



(40)

0k ki i kk iu u b

( ) ( ) ( )h p

i i iu u u x x x

0h h

k ki i kku u




equation

(41)












1

sN

h

i nl li sn

n

u U

x x x (42)




(43)




(44)



overlap in the MFS

0p p

k ki i kk iu u b

x

x


sn x x

1 1 (3 4 ) ln


v r

x y

snx x





(45)






coefficient set

(46)




relations

(47)

holds true



(48)


(49)


(50)



only that is

1 1

( ) ( ) ( )M M

m m

i m i li m l

m m

b

x x x x x

1

( ) ( )M

p m

i li m l

m

u

x x x


1 1

2li li mm mi mlF F

4

1 = 11 2


4 1

1li liF


(51)





(1) Conical spline

(52)

with


(53)

with




1 1

( ) ( ) ( )N M

n m m

i li n l li l

n m

u U

x x y x (54)


(7) and (6) as

4 2 2 1 d d 1 d d

d d d dr r r r

r r r r r r

mr x x

2 1

1 2 2

1 1

2 1 2 1 2 3

n


1

2

4 5 2 2 3

2 1

A n n

A n

2 2

1 2 3 2

1

32 1 1 2

n

li il i l

rA A r r

n n

22

1

2

8 29 27 8 2 2 1 2 4 7 4 2 ln

2 1 2 3 2 1 2 ln

A n n n n n n n r

A n n n n r


1 1

( ) ( ) ( )N M

n m


n m

S m T

x x y x x (55)

where

(56)




(57)











( )Arerr defined by

(58)





S U U U

H A B

T

1 1 1 1

1 2 1 2 1 2 1 2

N N M M A

2

1

2

1

L

j j

j

L

j

j

Arerr









(59)

with










12

0( ) ( )E r E r a

r

12 2 2 1

2

12 2 2 1

22 2

2 2

kk

k k

r ak k

kk k k

ak k

a ra b r p

b a

k r k ba ra p

b a k k

2 4 4k








surface when 3







critical values








5r in

PS and 2 lnr r and


example







supported such that


(60)



(61)




0p is


the analysis



(0 ) ( ) 0

(0 ) ( ) 0x x

w z w L z

t z t L z

( 0) ( ) 0

( 0) ( ) 0

x x

z z

t x t x h

t x p t x h













length






(62)







for 0

ln ln

for 0

ln

a b

a b

T b r T r a

b a

b rT T Tr a

b

a



N=32 M=220








CONCLUSIONS



















REFERENCES








2000









69 (2002) 502-514








and Engineering 193 (2004) 4511-4539






Structures 83 (2005) 1487-1502



Engineering 36 (2005) 223-240



(2006) 381-393



(2005) 597-609












Structures 42 (2005) 4338-4351



6878-6890










150 (1999) 239-267





460 (2004) 1689-1706




182












Mechanics 74 (2005) 563-579





521







704-712



(1998) 141-148








(1998) 195-201



101



Elements 24 (2000) 575-582



227-236















25 (2001) 377-387



43-59

















1 15 2 25 3 35 4 45 5 0

1

2

3

4

5

6

7

Similarity ratio

Re

lative

err

or

in

te

mp

era

ture

t = 05s

t = 10s

0 01 02 03 04 05 06 07 08 09 1 0

1

2

3

4

5

6

7

8

9

x (m)

Re

lative

err

or

in

te

mp

era

ture

t = 05s t = 01s

t = 05s t = 001s

t = 10s t = 01s

t = 10s t = 001s


(35)



05 cm-1






given as

2 2

21

2 cos( ) sin exp

n


L n L L

(36)





0 10 20 30 40 50 60

-01

0

01

02

03

04

05

06

07

08

Time t (second)

Te

mp

era

ture

(

)

Meshless x=001

x=002

x=003

Analytical x=001

x=002

x=003





respectively The





obtained as

0 0( ) ( )x xk x k e a x a e


(37)







right to left












and x = 001 m


LBIEM 01871 03281 03800 03986 04019 04053 04581

MFS 03915 04497 04546 04550 04551 04551 04551

Exact 04551

1( ) ( with 0)

1

x

L

e xT x T

e L











elastic system as

(38)







iu that is

(39)



(40)

0k ki i kk iu u b

( ) ( ) ( )h p

i i iu u u x x x

0h h

k ki i kku u




equation

(41)












1

sN

h

i nl li sn

n

u U

x x x (42)




(43)




(44)



overlap in the MFS

0p p

k ki i kk iu u b

x

x


sn x x

1 1 (3 4 ) ln


v r

x y

snx x





(45)






coefficient set

(46)




relations

(47)

holds true



(48)


(49)


(50)



only that is

1 1

( ) ( ) ( )M M

m m

i m i li m l

m m

b

x x x x x

1

( ) ( )M

p m

i li m l

m

u

x x x


1 1

2li li mm mi mlF F

4

1 = 11 2


4 1

1li liF


(51)





(1) Conical spline

(52)

with


(53)

with




1 1

( ) ( ) ( )N M

n m m

i li n l li l

n m

u U

x x y x (54)


(7) and (6) as

4 2 2 1 d d 1 d d

d d d dr r r r

r r r r r r

mr x x

2 1

1 2 2

1 1

2 1 2 1 2 3

n


1

2

4 5 2 2 3

2 1

A n n

A n

2 2

1 2 3 2

1

32 1 1 2

n

li il i l

rA A r r

n n

22

1

2

8 29 27 8 2 2 1 2 4 7 4 2 ln

2 1 2 3 2 1 2 ln

A n n n n n n n r

A n n n n r


1 1

( ) ( ) ( )N M

n m


n m

S m T

x x y x x (55)

where

(56)




(57)











( )Arerr defined by

(58)





S U U U

H A B

T

1 1 1 1

1 2 1 2 1 2 1 2

N N M M A

2

1

2

1

L

j j

j

L

j

j

Arerr









(59)

with










12

0( ) ( )E r E r a

r

12 2 2 1

2

12 2 2 1

22 2

2 2

kk

k k

r ak k

kk k k

ak k

a ra b r p

b a

k r k ba ra p

b a k k

2 4 4k








surface when 3







critical values








5r in

PS and 2 lnr r and


example







supported such that


(60)



(61)




0p is


the analysis



(0 ) ( ) 0

(0 ) ( ) 0x x

w z w L z

t z t L z

( 0) ( ) 0

( 0) ( ) 0

x x

z z

t x t x h

t x p t x h













length






(62)







for 0

ln ln

for 0

ln

a b

a b

T b r T r a

b a

b rT T Tr a

b

a



N=32 M=220








CONCLUSIONS



















REFERENCES








2000









69 (2002) 502-514








and Engineering 193 (2004) 4511-4539






Structures 83 (2005) 1487-1502



Engineering 36 (2005) 223-240



(2006) 381-393



(2005) 597-609












Structures 42 (2005) 4338-4351



6878-6890










150 (1999) 239-267





460 (2004) 1689-1706




182












Mechanics 74 (2005) 563-579





521







704-712



(1998) 141-148








(1998) 195-201



101



Elements 24 (2000) 575-582



227-236















25 (2001) 377-387



43-59







(35)



05 cm-1






given as

2 2

21

2 cos( ) sin exp

n


L n L L

(36)





0 10 20 30 40 50 60

-01

0

01

02

03

04

05

06

07

08

Time t (second)

Te

mp

era

ture

(

)

Meshless x=001

x=002

x=003

Analytical x=001

x=002

x=003





respectively The





obtained as

0 0( ) ( )x xk x k e a x a e


(37)







right to left












and x = 001 m


LBIEM 01871 03281 03800 03986 04019 04053 04581

MFS 03915 04497 04546 04550 04551 04551 04551

Exact 04551

1( ) ( with 0)

1

x

L

e xT x T

e L











elastic system as

(38)







iu that is

(39)



(40)

0k ki i kk iu u b

( ) ( ) ( )h p

i i iu u u x x x

0h h

k ki i kku u




equation

(41)












1

sN

h

i nl li sn

n

u U

x x x (42)




(43)




(44)



overlap in the MFS

0p p

k ki i kk iu u b

x

x


sn x x

1 1 (3 4 ) ln


v r

x y

snx x





(45)






coefficient set

(46)




relations

(47)

holds true



(48)


(49)


(50)



only that is

1 1

( ) ( ) ( )M M

m m

i m i li m l

m m

b

x x x x x

1

( ) ( )M

p m

i li m l

m

u

x x x


1 1

2li li mm mi mlF F

4

1 = 11 2


4 1

1li liF


(51)





(1) Conical spline

(52)

with


(53)

with




1 1

( ) ( ) ( )N M

n m m

i li n l li l

n m

u U

x x y x (54)


(7) and (6) as

4 2 2 1 d d 1 d d

d d d dr r r r

r r r r r r

mr x x

2 1

1 2 2

1 1

2 1 2 1 2 3

n


1

2

4 5 2 2 3

2 1

A n n

A n

2 2

1 2 3 2

1

32 1 1 2

n

li il i l

rA A r r

n n

22

1

2

8 29 27 8 2 2 1 2 4 7 4 2 ln

2 1 2 3 2 1 2 ln

A n n n n n n n r

A n n n n r


1 1

( ) ( ) ( )N M

n m


n m

S m T

x x y x x (55)

where

(56)




(57)











( )Arerr defined by

(58)





S U U U

H A B

T

1 1 1 1

1 2 1 2 1 2 1 2

N N M M A

2

1

2

1

L

j j

j

L

j

j

Arerr









(59)

with










12

0( ) ( )E r E r a

r

12 2 2 1

2

12 2 2 1

22 2

2 2

kk

k k

r ak k

kk k k

ak k

a ra b r p

b a

k r k ba ra p

b a k k

2 4 4k








surface when 3







critical values








5r in

PS and 2 lnr r and


example







supported such that


(60)



(61)




0p is


the analysis



(0 ) ( ) 0

(0 ) ( ) 0x x

w z w L z

t z t L z

( 0) ( ) 0

( 0) ( ) 0

x x

z z

t x t x h

t x p t x h













length






(62)







for 0

ln ln

for 0

ln

a b

a b

T b r T r a

b a

b rT T Tr a

b

a



N=32 M=220








CONCLUSIONS



















REFERENCES








2000









69 (2002) 502-514








and Engineering 193 (2004) 4511-4539






Structures 83 (2005) 1487-1502



Engineering 36 (2005) 223-240



(2006) 381-393



(2005) 597-609












Structures 42 (2005) 4338-4351



6878-6890










150 (1999) 239-267





460 (2004) 1689-1706




182












Mechanics 74 (2005) 563-579





521







704-712



(1998) 141-148








(1998) 195-201



101



Elements 24 (2000) 575-582



227-236















25 (2001) 377-387



43-59







(37)







right to left












and x = 001 m


LBIEM 01871 03281 03800 03986 04019 04053 04581

MFS 03915 04497 04546 04550 04551 04551 04551

Exact 04551

1( ) ( with 0)

1

x

L

e xT x T

e L











elastic system as

(38)







iu that is

(39)



(40)

0k ki i kk iu u b

( ) ( ) ( )h p

i i iu u u x x x

0h h

k ki i kku u




equation

(41)












1

sN

h

i nl li sn

n

u U

x x x (42)




(43)




(44)



overlap in the MFS

0p p

k ki i kk iu u b

x

x


sn x x

1 1 (3 4 ) ln


v r

x y

snx x





(45)






coefficient set

(46)




relations

(47)

holds true



(48)


(49)


(50)



only that is

1 1

( ) ( ) ( )M M

m m

i m i li m l

m m

b

x x x x x

1

( ) ( )M

p m

i li m l

m

u

x x x


1 1

2li li mm mi mlF F

4

1 = 11 2


4 1

1li liF


(51)





(1) Conical spline

(52)

with


(53)

with




1 1

( ) ( ) ( )N M

n m m

i li n l li l

n m

u U

x x y x (54)


(7) and (6) as

4 2 2 1 d d 1 d d

d d d dr r r r

r r r r r r

mr x x

2 1

1 2 2

1 1

2 1 2 1 2 3

n


1

2

4 5 2 2 3

2 1

A n n

A n

2 2

1 2 3 2

1

32 1 1 2

n

li il i l

rA A r r

n n

22

1

2

8 29 27 8 2 2 1 2 4 7 4 2 ln

2 1 2 3 2 1 2 ln

A n n n n n n n r

A n n n n r


1 1

( ) ( ) ( )N M

n m


n m

S m T

x x y x x (55)

where

(56)




(57)











( )Arerr defined by

(58)





S U U U

H A B

T

1 1 1 1

1 2 1 2 1 2 1 2

N N M M A

2

1

2

1

L

j j

j

L

j

j

Arerr









(59)

with










12

0( ) ( )E r E r a

r

12 2 2 1

2

12 2 2 1

22 2

2 2

kk

k k

r ak k

kk k k

ak k

a ra b r p

b a

k r k ba ra p

b a k k

2 4 4k








surface when 3







critical values








5r in

PS and 2 lnr r and


example







supported such that


(60)



(61)




0p is


the analysis



(0 ) ( ) 0

(0 ) ( ) 0x x

w z w L z

t z t L z

( 0) ( ) 0

( 0) ( ) 0

x x

z z

t x t x h

t x p t x h













length






(62)







for 0

ln ln

for 0

ln

a b

a b

T b r T r a

b a

b rT T Tr a

b

a



N=32 M=220








CONCLUSIONS



















REFERENCES








2000









69 (2002) 502-514








and Engineering 193 (2004) 4511-4539






Structures 83 (2005) 1487-1502



Engineering 36 (2005) 223-240



(2006) 381-393



(2005) 597-609












Structures 42 (2005) 4338-4351



6878-6890










150 (1999) 239-267





460 (2004) 1689-1706




182












Mechanics 74 (2005) 563-579





521







704-712



(1998) 141-148








(1998) 195-201



101



Elements 24 (2000) 575-582



227-236















25 (2001) 377-387



43-59
















elastic system as

(38)







iu that is

(39)



(40)

0k ki i kk iu u b

( ) ( ) ( )h p

i i iu u u x x x

0h h

k ki i kku u




equation

(41)












1

sN

h

i nl li sn

n

u U

x x x (42)




(43)




(44)



overlap in the MFS

0p p

k ki i kk iu u b

x

x


sn x x

1 1 (3 4 ) ln


v r

x y

snx x





(45)






coefficient set

(46)




relations

(47)

holds true



(48)


(49)


(50)



only that is

1 1

( ) ( ) ( )M M

m m

i m i li m l

m m

b

x x x x x

1

( ) ( )M

p m

i li m l

m

u

x x x


1 1

2li li mm mi mlF F

4

1 = 11 2


4 1

1li liF


(51)





(1) Conical spline

(52)

with


(53)

with




1 1

( ) ( ) ( )N M

n m m

i li n l li l

n m

u U

x x y x (54)


(7) and (6) as

4 2 2 1 d d 1 d d

d d d dr r r r

r r r r r r

mr x x

2 1

1 2 2

1 1

2 1 2 1 2 3

n


1

2

4 5 2 2 3

2 1

A n n

A n

2 2

1 2 3 2

1

32 1 1 2

n

li il i l

rA A r r

n n

22

1

2

8 29 27 8 2 2 1 2 4 7 4 2 ln

2 1 2 3 2 1 2 ln

A n n n n n n n r

A n n n n r


1 1

( ) ( ) ( )N M

n m


n m

S m T

x x y x x (55)

where

(56)




(57)











( )Arerr defined by

(58)





S U U U

H A B

T

1 1 1 1

1 2 1 2 1 2 1 2

N N M M A

2

1

2

1

L

j j

j

L

j

j

Arerr









(59)

with










12

0( ) ( )E r E r a

r

12 2 2 1

2

12 2 2 1

22 2

2 2

kk

k k

r ak k

kk k k

ak k

a ra b r p

b a

k r k ba ra p

b a k k

2 4 4k








surface when 3







critical values








5r in

PS and 2 lnr r and


example







supported such that


(60)



(61)




0p is


the analysis



(0 ) ( ) 0

(0 ) ( ) 0x x

w z w L z

t z t L z

( 0) ( ) 0

( 0) ( ) 0

x x

z z

t x t x h

t x p t x h













length






(62)







for 0

ln ln

for 0

ln

a b

a b

T b r T r a

b a

b rT T Tr a

b

a



N=32 M=220








CONCLUSIONS



















REFERENCES








2000









69 (2002) 502-514








and Engineering 193 (2004) 4511-4539






Structures 83 (2005) 1487-1502



Engineering 36 (2005) 223-240



(2006) 381-393



(2005) 597-609












Structures 42 (2005) 4338-4351



6878-6890










150 (1999) 239-267





460 (2004) 1689-1706




182












Mechanics 74 (2005) 563-579





521







704-712



(1998) 141-148








(1998) 195-201



101



Elements 24 (2000) 575-582



227-236















25 (2001) 377-387



43-59









equation

(41)












1

sN

h

i nl li sn

n

u U

x x x (42)




(43)




(44)



overlap in the MFS

0p p

k ki i kk iu u b

x

x


sn x x

1 1 (3 4 ) ln


v r

x y

snx x





(45)






coefficient set

(46)




relations

(47)

holds true



(48)


(49)


(50)



only that is

1 1

( ) ( ) ( )M M

m m

i m i li m l

m m

b

x x x x x

1

( ) ( )M

p m

i li m l

m

u

x x x


1 1

2li li mm mi mlF F

4

1 = 11 2


4 1

1li liF


(51)





(1) Conical spline

(52)

with


(53)

with




1 1

( ) ( ) ( )N M

n m m

i li n l li l

n m

u U

x x y x (54)


(7) and (6) as

4 2 2 1 d d 1 d d

d d d dr r r r

r r r r r r

mr x x

2 1

1 2 2

1 1

2 1 2 1 2 3

n


1

2

4 5 2 2 3

2 1

A n n

A n

2 2

1 2 3 2

1

32 1 1 2

n

li il i l

rA A r r

n n

22

1

2

8 29 27 8 2 2 1 2 4 7 4 2 ln

2 1 2 3 2 1 2 ln

A n n n n n n n r

A n n n n r


1 1

( ) ( ) ( )N M

n m


n m

S m T

x x y x x (55)

where

(56)




(57)











( )Arerr defined by

(58)





S U U U

H A B

T

1 1 1 1

1 2 1 2 1 2 1 2

N N M M A

2

1

2

1

L

j j

j

L

j

j

Arerr









(59)

with










12

0( ) ( )E r E r a

r

12 2 2 1

2

12 2 2 1

22 2

2 2

kk

k k

r ak k

kk k k

ak k

a ra b r p

b a

k r k ba ra p

b a k k

2 4 4k








surface when 3







critical values








5r in

PS and 2 lnr r and


example







supported such that


(60)



(61)




0p is


the analysis



(0 ) ( ) 0

(0 ) ( ) 0x x

w z w L z

t z t L z

( 0) ( ) 0

( 0) ( ) 0

x x

z z

t x t x h

t x p t x h













length






(62)







for 0

ln ln

for 0

ln

a b

a b

T b r T r a

b a

b rT T Tr a

b

a



N=32 M=220








CONCLUSIONS



















REFERENCES








2000









69 (2002) 502-514








and Engineering 193 (2004) 4511-4539






Structures 83 (2005) 1487-1502



Engineering 36 (2005) 223-240



(2006) 381-393



(2005) 597-609












Structures 42 (2005) 4338-4351



6878-6890










150 (1999) 239-267





460 (2004) 1689-1706




182












Mechanics 74 (2005) 563-579





521







704-712



(1998) 141-148








(1998) 195-201



101



Elements 24 (2000) 575-582



227-236















25 (2001) 377-387



43-59










(45)






coefficient set

(46)




relations

(47)

holds true



(48)


(49)


(50)



only that is

1 1

( ) ( ) ( )M M

m m

i m i li m l

m m

b

x x x x x

1

( ) ( )M

p m

i li m l

m

u

x x x


1 1

2li li mm mi mlF F

4

1 = 11 2


4 1

1li liF


(51)





(1) Conical spline

(52)

with


(53)

with




1 1

( ) ( ) ( )N M

n m m

i li n l li l

n m

u U

x x y x (54)


(7) and (6) as

4 2 2 1 d d 1 d d

d d d dr r r r

r r r r r r

mr x x

2 1

1 2 2

1 1

2 1 2 1 2 3

n


1

2

4 5 2 2 3

2 1

A n n

A n

2 2

1 2 3 2

1

32 1 1 2

n

li il i l

rA A r r

n n

22

1

2

8 29 27 8 2 2 1 2 4 7 4 2 ln

2 1 2 3 2 1 2 ln

A n n n n n n n r

A n n n n r


1 1

( ) ( ) ( )N M

n m


n m

S m T

x x y x x (55)

where

(56)




(57)











( )Arerr defined by

(58)





S U U U

H A B

T

1 1 1 1

1 2 1 2 1 2 1 2

N N M M A

2

1

2

1

L

j j

j

L

j

j

Arerr









(59)

with










12

0( ) ( )E r E r a

r

12 2 2 1

2

12 2 2 1

22 2

2 2

kk

k k

r ak k

kk k k

ak k

a ra b r p

b a

k r k ba ra p

b a k k

2 4 4k








surface when 3







critical values








5r in

PS and 2 lnr r and


example







supported such that


(60)



(61)




0p is


the analysis



(0 ) ( ) 0

(0 ) ( ) 0x x

w z w L z

t z t L z

( 0) ( ) 0

( 0) ( ) 0

x x

z z

t x t x h

t x p t x h













length






(62)







for 0

ln ln

for 0

ln

a b

a b

T b r T r a

b a

b rT T Tr a

b

a



N=32 M=220








CONCLUSIONS



















REFERENCES








2000









69 (2002) 502-514








and Engineering 193 (2004) 4511-4539






Structures 83 (2005) 1487-1502



Engineering 36 (2005) 223-240



(2006) 381-393



(2005) 597-609












Structures 42 (2005) 4338-4351



6878-6890










150 (1999) 239-267





460 (2004) 1689-1706




182












Mechanics 74 (2005) 563-579





521







704-712



(1998) 141-148








(1998) 195-201



101



Elements 24 (2000) 575-582



227-236















25 (2001) 377-387



43-59







(51)





(1) Conical spline

(52)

with


(53)

with




1 1

( ) ( ) ( )N M

n m m

i li n l li l

n m

u U

x x y x (54)


(7) and (6) as

4 2 2 1 d d 1 d d

d d d dr r r r

r r r r r r

mr x x

2 1

1 2 2

1 1

2 1 2 1 2 3

n


1

2

4 5 2 2 3

2 1

A n n

A n

2 2

1 2 3 2

1

32 1 1 2

n

li il i l

rA A r r

n n

22

1

2

8 29 27 8 2 2 1 2 4 7 4 2 ln

2 1 2 3 2 1 2 ln

A n n n n n n n r

A n n n n r


1 1

( ) ( ) ( )N M

n m


n m

S m T

x x y x x (55)

where

(56)




(57)











( )Arerr defined by

(58)





S U U U

H A B

T

1 1 1 1

1 2 1 2 1 2 1 2

N N M M A

2

1

2

1

L

j j

j

L

j

j

Arerr









(59)

with










12

0( ) ( )E r E r a

r

12 2 2 1

2

12 2 2 1

22 2

2 2

kk

k k

r ak k

kk k k

ak k

a ra b r p

b a

k r k ba ra p

b a k k

2 4 4k








surface when 3







critical values








5r in

PS and 2 lnr r and


example







supported such that


(60)



(61)




0p is


the analysis



(0 ) ( ) 0

(0 ) ( ) 0x x

w z w L z

t z t L z

( 0) ( ) 0

( 0) ( ) 0

x x

z z

t x t x h

t x p t x h













length






(62)







for 0

ln ln

for 0

ln

a b

a b

T b r T r a

b a

b rT T Tr a

b

a



N=32 M=220








CONCLUSIONS



















REFERENCES








2000









69 (2002) 502-514








and Engineering 193 (2004) 4511-4539






Structures 83 (2005) 1487-1502



Engineering 36 (2005) 223-240



(2006) 381-393



(2005) 597-609












Structures 42 (2005) 4338-4351



6878-6890










150 (1999) 239-267





460 (2004) 1689-1706




182












Mechanics 74 (2005) 563-579





521







704-712



(1998) 141-148








(1998) 195-201



101



Elements 24 (2000) 575-582



227-236















25 (2001) 377-387



43-59







1 1

( ) ( ) ( )N M

n m


n m

S m T

x x y x x (55)

where

(56)




(57)











( )Arerr defined by

(58)





S U U U

H A B

T

1 1 1 1

1 2 1 2 1 2 1 2

N N M M A

2

1

2

1

L

j j

j

L

j

j

Arerr









(59)

with










12

0( ) ( )E r E r a

r

12 2 2 1

2

12 2 2 1

22 2

2 2

kk

k k

r ak k

kk k k

ak k

a ra b r p

b a

k r k ba ra p

b a k k

2 4 4k








surface when 3







critical values








5r in

PS and 2 lnr r and


example







supported such that


(60)



(61)




0p is


the analysis



(0 ) ( ) 0

(0 ) ( ) 0x x

w z w L z

t z t L z

( 0) ( ) 0

( 0) ( ) 0

x x

z z

t x t x h

t x p t x h













length






(62)







for 0

ln ln

for 0

ln

a b

a b

T b r T r a

b a

b rT T Tr a

b

a



N=32 M=220








CONCLUSIONS



















REFERENCES








2000









69 (2002) 502-514








and Engineering 193 (2004) 4511-4539






Structures 83 (2005) 1487-1502



Engineering 36 (2005) 223-240



(2006) 381-393



(2005) 597-609












Structures 42 (2005) 4338-4351



6878-6890










150 (1999) 239-267





460 (2004) 1689-1706




182












Mechanics 74 (2005) 563-579





521







704-712



(1998) 141-148








(1998) 195-201



101



Elements 24 (2000) 575-582



227-236















25 (2001) 377-387



43-59














(59)

with










12

0( ) ( )E r E r a

r

12 2 2 1

2

12 2 2 1

22 2

2 2

kk

k k

r ak k

kk k k

ak k

a ra b r p

b a

k r k ba ra p

b a k k

2 4 4k








surface when 3







critical values








5r in

PS and 2 lnr r and


example







supported such that


(60)



(61)




0p is


the analysis



(0 ) ( ) 0

(0 ) ( ) 0x x

w z w L z

t z t L z

( 0) ( ) 0

( 0) ( ) 0

x x

z z

t x t x h

t x p t x h













length






(62)







for 0

ln ln

for 0

ln

a b

a b

T b r T r a

b a

b rT T Tr a

b

a



N=32 M=220








CONCLUSIONS



















REFERENCES








2000









69 (2002) 502-514








and Engineering 193 (2004) 4511-4539






Structures 83 (2005) 1487-1502



Engineering 36 (2005) 223-240



(2006) 381-393



(2005) 597-609












Structures 42 (2005) 4338-4351



6878-6890










150 (1999) 239-267





460 (2004) 1689-1706




182












Mechanics 74 (2005) 563-579





521







704-712



(1998) 141-148








(1998) 195-201



101



Elements 24 (2000) 575-582



227-236















25 (2001) 377-387



43-59













surface when 3







critical values








5r in

PS and 2 lnr r and


example







supported such that


(60)



(61)




0p is


the analysis



(0 ) ( ) 0

(0 ) ( ) 0x x

w z w L z

t z t L z

( 0) ( ) 0

( 0) ( ) 0

x x

z z

t x t x h

t x p t x h













length






(62)







for 0

ln ln

for 0

ln

a b

a b

T b r T r a

b a

b rT T Tr a

b

a



N=32 M=220








CONCLUSIONS



















REFERENCES








2000









69 (2002) 502-514








and Engineering 193 (2004) 4511-4539






Structures 83 (2005) 1487-1502



Engineering 36 (2005) 223-240



(2006) 381-393



(2005) 597-609












Structures 42 (2005) 4338-4351



6878-6890










150 (1999) 239-267





460 (2004) 1689-1706




182












Mechanics 74 (2005) 563-579





521







704-712



(1998) 141-148








(1998) 195-201



101



Elements 24 (2000) 575-582



227-236















25 (2001) 377-387



43-59












critical values








5r in

PS and 2 lnr r and


example







supported such that


(60)



(61)




0p is


the analysis



(0 ) ( ) 0

(0 ) ( ) 0x x

w z w L z

t z t L z

( 0) ( ) 0

( 0) ( ) 0

x x

z z

t x t x h

t x p t x h













length






(62)







for 0

ln ln

for 0

ln

a b

a b

T b r T r a

b a

b rT T Tr a

b

a



N=32 M=220








CONCLUSIONS



















REFERENCES








2000









69 (2002) 502-514








and Engineering 193 (2004) 4511-4539






Structures 83 (2005) 1487-1502



Engineering 36 (2005) 223-240



(2006) 381-393



(2005) 597-609












Structures 42 (2005) 4338-4351



6878-6890










150 (1999) 239-267





460 (2004) 1689-1706




182












Mechanics 74 (2005) 563-579





521







704-712



(1998) 141-148








(1998) 195-201



101



Elements 24 (2000) 575-582



227-236















25 (2001) 377-387



43-59











5r in

PS and 2 lnr r and


example







supported such that


(60)



(61)




0p is


the analysis



(0 ) ( ) 0

(0 ) ( ) 0x x

w z w L z

t z t L z

( 0) ( ) 0

( 0) ( ) 0

x x

z z

t x t x h

t x p t x h













length






(62)







for 0

ln ln

for 0

ln

a b

a b

T b r T r a

b a

b rT T Tr a

b

a



N=32 M=220








CONCLUSIONS



















REFERENCES








2000









69 (2002) 502-514








and Engineering 193 (2004) 4511-4539






Structures 83 (2005) 1487-1502



Engineering 36 (2005) 223-240



(2006) 381-393



(2005) 597-609












Structures 42 (2005) 4338-4351



6878-6890










150 (1999) 239-267





460 (2004) 1689-1706




182












Mechanics 74 (2005) 563-579





521







704-712



(1998) 141-148








(1998) 195-201



101



Elements 24 (2000) 575-582



227-236















25 (2001) 377-387



43-59







(60)



(61)




0p is


the analysis



(0 ) ( ) 0

(0 ) ( ) 0x x

w z w L z

t z t L z

( 0) ( ) 0

( 0) ( ) 0

x x

z z

t x t x h

t x p t x h













length






(62)







for 0

ln ln

for 0

ln

a b

a b

T b r T r a

b a

b rT T Tr a

b

a



N=32 M=220








CONCLUSIONS



















REFERENCES








2000









69 (2002) 502-514








and Engineering 193 (2004) 4511-4539






Structures 83 (2005) 1487-1502



Engineering 36 (2005) 223-240



(2006) 381-393



(2005) 597-609












Structures 42 (2005) 4338-4351



6878-6890










150 (1999) 239-267





460 (2004) 1689-1706




182












Mechanics 74 (2005) 563-579





521







704-712



(1998) 141-148








(1998) 195-201



101



Elements 24 (2000) 575-582



227-236















25 (2001) 377-387



43-59


















length






(62)







for 0

ln ln

for 0

ln

a b

a b

T b r T r a

b a

b rT T Tr a

b

a



N=32 M=220








CONCLUSIONS



















REFERENCES








2000









69 (2002) 502-514








and Engineering 193 (2004) 4511-4539






Structures 83 (2005) 1487-1502



Engineering 36 (2005) 223-240



(2006) 381-393



(2005) 597-609












Structures 42 (2005) 4338-4351



6878-6890










150 (1999) 239-267





460 (2004) 1689-1706




182












Mechanics 74 (2005) 563-579





521







704-712



(1998) 141-148








(1998) 195-201



101



Elements 24 (2000) 575-582



227-236















25 (2001) 377-387



43-59








N=32 M=220








CONCLUSIONS



















REFERENCES








2000









69 (2002) 502-514








and Engineering 193 (2004) 4511-4539






Structures 83 (2005) 1487-1502



Engineering 36 (2005) 223-240



(2006) 381-393



(2005) 597-609












Structures 42 (2005) 4338-4351



6878-6890










150 (1999) 239-267





460 (2004) 1689-1706




182












Mechanics 74 (2005) 563-579





521







704-712



(1998) 141-148








(1998) 195-201



101



Elements 24 (2000) 575-582



227-236















25 (2001) 377-387



43-59




















REFERENCES








2000









69 (2002) 502-514








and Engineering 193 (2004) 4511-4539






Structures 83 (2005) 1487-1502



Engineering 36 (2005) 223-240



(2006) 381-393



(2005) 597-609












Structures 42 (2005) 4338-4351



6878-6890










150 (1999) 239-267





460 (2004) 1689-1706




182












Mechanics 74 (2005) 563-579





521







704-712



(1998) 141-148








(1998) 195-201



101



Elements 24 (2000) 575-582



227-236















25 (2001) 377-387



43-59









Structures 83 (2005) 1487-1502



Engineering 36 (2005) 223-240



(2006) 381-393



(2005) 597-609












Structures 42 (2005) 4338-4351



6878-6890










150 (1999) 239-267





460 (2004) 1689-1706




182












Mechanics 74 (2005) 563-579





521







704-712



(1998) 141-148








(1998) 195-201



101



Elements 24 (2000) 575-582



227-236















25 (2001) 377-387



43-59

















Mechanics 74 (2005) 563-579





521







704-712



(1998) 141-148








(1998) 195-201



101



Elements 24 (2000) 575-582



227-236















25 (2001) 377-387



43-59

















25 (2001) 377-387



43-59






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