11 advanced fundamental-solution-based computational...

331

Ashutosh Tiwari, N. Arul Murugan, and Rajeev Ahuja (eds.) Advanced Engineering Materials and Modeling, (331–368) © 2016 Scrivener Publishing LLC

11

Advanced Fundamental-solution-based Computational Methods for Thermal Analysis of Heterogeneous Materials

Hui Wang1 and Qing-Hua Qin2*

1Zhengzhou Key Laboratory of Scientific & Engineering Computation, Henan University of Technology, Zhengzhou, China

2Research School of Engineering, Australian National University, Acton, Australia

AbstractHeterogeneous materials such as composites and functionally graded materi-als (FGMs) have been widely used in space engineering, and thus their efficient numerical analysis becomes important and popular in practical design. This chapter presents an overview on the development of fundamental-solution-based numerical methods including the meshless method of fundamental solutions (MFS) and the hybrid finite-element method (FEM) and their applications in thermal analysis of FGMs and fiber-reinforced composites. Basic concepts of the two methods are briefly described, such as fundamental solution, meshless collo-cations, modified variational functional, intraelement fields, frame fields, and spe-cial polygonal elements. Related formulations are derived for thermal analysis of such heterogeneous materials, and some typical numerical solutions are discussed to show the application of the MFS and the fundamental-solution-based hybrid FEM. Finally, a brief summary of these two approaches are presented, and future trends in the analysis of heterogeneous materials with the fundamental-solution-based computational methods are identified.

Keywords: Functionally graded materials, fiber-reinforced composites, fundamental solutions, method of fundamental solutions, hybrid finite-element method, special elements

*Corresponding author: [email protected]

332 Advanced Engineering Materials and Modeling

11.1 Introduction

Heterogeneous materials such as functionally graded materials (FGMs) [1, 2] and fiber-reinforced composites [3–6] have been widely used in the space engineering to achieve some special purposes. For instance, FGMs, which may be characterized by a spatial gradation variation in composi-tion and structure, were first considered in 1984 by material scientists in the Sendai area in Japan during a space plane project for the special pur-pose of preparing materials with high thermal barrier capability [7]. Fiber-reinforced composite materials represent a radical approach to designing new aerospace structural materials with desirable physical and chemical properties including high stiffness and strength and lightweight when compared with traditional materials such as metals and ceramics [5, 8]. It is noted that numerical methods have been extensively employed for solv-ing thermal and mechanical problems in these heterogeneous materials [9–26], because the related analytical solutions are difficult to obtain.

Among available numerical methods, the methods related to fundamen-tal solutions of the problem under consideration have been considerably developed and have shown to be a highly efficient computational tool for the solution of complex boundary value problems, including the method of fundamental solutions (MFS) [27–31], the boundary element method (BEM) [12, 32–34], and the fundamental-solution-based hybrid finite-element method (HFS-FEM) [35–38]. Different from the BEM, both the MFS and the HFS-FEM use the fundamental solutions of the problem of interest as trial functions for approximating the essential field (usually, the displacement or temperature fields), and the fictitious sources are placed outside the elemental domain to avoid the singularity induced when the field point approaches the source point. The MFS can be viewed as a global approximating method, whereas the HFS-FEM is a local approximating method. In this chapter, particular attention is paid to the meshless MFS and the HFS-FEM and an overview of them is made.

In contrast to the conventional finite element method (FEM) [39], the MFS is a truly meshless method, in which the sought fields are globally approximated by the linear combination of fundamental solutions of the problem of interest in the full domain by placing all singularities in the approximation of the sought fields outside the domain so that they satisfy the governing partial differential equations of the problem at any point within the domain. As a result, only boundary collocation discretization is required and the satisfaction of the boundary conditions at boundary col-locations can form the final solving system of equations, whose unknown variables have usually no physical meaning and the system equation has

Advanced Fundamental-solution-based Computational Methods 333

non-symmetric coefficient matrix. The first attempt of the MFS was made by Kupradze for solving some second order elliptical equations in 1964 [40], and later the question of the uniqueness of the solution in Kupradze’s method is addressed by Christiansen [41]. Since then, the concept of the MFS has become increasingly popular, attracting a growing number of researchers into this field [28, 42–57]. More recently, the MFS has been applied to heat conduction and elastic problems of heterogeneous materi-als. For instance, Berger et al. applied the MFS to heat conduction prob-lems and elastic problems in two-layered composite materials using special fundamental solution that satisfies the interface continuity conditions [58, 59]. Dong et al. developed the MFS formulation for analyzing the inverse heat conduction problems in anisotropic medium [60]. Marine and Lesnic established the MFS formulation for nonlinear FGMs [61]. Poullikkas et al. solved the inhomogeneous problem by the MFS [62].

As noted in literature above, the procedure of the MFS is very simple and the accuracy is high, especially for those with domains having simple geometry. However, the MFS has some inherent drawbacks [44, 63]: (1) the coefficient matrix of the resulting linear equation system is usually full and unsymmetric, (2) the coefficient matrix becomes ill-conditioned as the number of singularities increases or when the boundary shape is com-plicated, (3) it is inefficient for multi-domain problems, and (4) the fun-damental solutions of some physical problems may not be available. The first three shortcomings can be overcome by introducing the weak-formed HFS-FEM stated below, whereas the last shortcoming can be partially handled by means of the analog equation method (AEM) [64, 65] and the radial basis functions (RBFs) [66–70].

As one of the hybrid FEMs [71–77], the HFS-FEM was developed to inherit the advantages of the MFS and BEM and overcome some disad-vantages of the MFS and BEM in multi-domain problems. The method includes the use of auxiliary inter-element frame fields to link the inter-nal fields between elements, and the local internal fields are approximated by a linear combination of fundamental solutions of the problem inside the local element domain. In the formulation of HFS-FEM based on local weak form, the inter-element continuity is enforced by using a modified variational principle together with the independent frame fields defined on each element boundary, and the element formulation, during which the internal parameters are eliminated at the element level, leads to the stan-dard force–displacement relationship with a symmetric positive definite stiffness matrix. So far, the HFS-FEM has been well established for heat/bioheat conduction [35, 78–80], discontinuous loads [81], isotropic elas-ticity with holes [82, 83], anisotropic elasticity [36], elastic cracks [84], and


composites [85, 86]. More recently, the special polygonal fiber/matrix ele-ments and interphase/fiber elements have been developed for heat transfer and elasticity in fiber-reinforced composites [14, 15, 87–90]. Besides, spe-cially purposed graded elements (SPGEs) have been formulated for simu-lating thermal and elastic behaviors in FGMs [91–93].

Generally, the approximated fields are chosen so as to a priori satisfy the governing partial differential equations of the problem in both the MFS and the HFS-FEM so that the computation involved is only along the domain boundary in the MFS or along the element boundary in the HFS-FEM. This is one of advantages of the MFS and HFS-FEM over the conventional finite-element (FE) formulation. Besides, different from the conventional FE, the HFS-FEM can generate arbitrary polygonal or even curve-sided elements for computation because the formulation calls for integration along the element boundary only. From this point, it may be considered as a special, symmet-ric, substructure-oriented boundary solution approach and thus possessed the advantages of the BEM. In contrast to the conventional boundary ele-ment formulation, however, the HFS-FEM model avoids the introduction of singular integral equations, because all source points are placed outside the element, as done in the MFS. Moreover, the HFS-FEM model offers the attractive possibility of developing special hole, inclusion, or crack elements to achieve the balance of accuracy and computing efficiency with fewer elements than the conventional FEM, simply by using appropriate known special fundamental solutions as the trial functions of internal fields.

Following this introduction, the present review consists of six sections. Basic formulations of the MFS and the HFS-FEM are described in Sections 2 and 3, respectively. Sections 4 and 5 focus, respectively, on the application of the MFS and the HFS-FEM to FGMs and fiber-reinforced composite mate-rials. Finally, a brief summary of the developments of the MFS and the HFS-FEM is provided, and areas that need further research are identified.

11.2 Basic Formulation of MFS

11.2.1 Standard MFS

To illustrate procedures of the MFS, the following homogeneous elliptic partial differential equation is considered:

Lu( ) ,x x= ∈0 Ω (11.1)

where L is a linear elliptic partial differential operator, Ω is a bounded domain in two-dimensional space, u is the sought field, and x = (x1, x2) is


an arbitrary point in the domain Ω under consideration, as displayed in Figure 11.1.

The basic ideas for the formulation of the MFS were first proposed by Kupradze [40]. In the MFS, the approximate solution of the sought field u is expressed as a linear combination of fundamental solutions

u c Gnn

N

( ) ( , ), ,x x y x y= ∈ ∉∗=

∑1

n nΩ Ω (11.2)

where the source points { }yn nN

=1 are normally placed outside the domain Ω, as shown in Figure 11.1, to avoid singular interference of fundamental solutions and { }cn n

N=1 are unknown coefficients. G*(x, yn) is the fundamen-

tal solution of the differential equation (11.1) and is defined as the solution of the equation of the form in the infinite space

LG n n∗ + =( , ) ( , )x y x yd 0 (11.3)

where d(x, yn) is the Dirac delta function.It is obvious that the solution G*(x, yn) is defined everywhere except that

x = yn, where it is singular. Thus yn is called the singularity (or source point) of the fundamental solution. The locations of the singularities are either preassigned or are to be determined. If the latter strategy is employed, the resulting problem is nonlinear and can be solved using available software with optimization algorithm.

Moreover, it is found that the approximate solution defined in Eq. (11.2) satisfies exactly the governing equation (11.1) of the problem. Making use of the fundamental solution defined in Eq. (11.3), the unknown coefficients

Source point

Virtual boundary

X

X1

X2Ω

Figure 11.1 Schematic of the standard MFS procedure.


in the approximate solution (11.2) can be determined by satisfying the boundary conditions only. For example, if the boundary condition of the problem has the form

Bu f( ) ( ),x x x= ∈∂Ω (11.4)

the discrete linear system of equations after putting M(M ≥ N) collocations along the boundary can be written as

Bu c BG f m Mm n

n

N

m n m m( ) ( , ) ( ), , ,x x y x x y= = ∈∂ ∉ = →=

∗∑1

1Ω Ωn

(11.5)

or in matrix form

BGBG

BG

BGBG

BGM M

∗

∗

∗

∗

∗

∗

( , )( , )

( , )

( , )( , )

(

x yx y

x y

x yx y

x

1 1

2 1

1

1 2

2 2

� �

,, )

( , )( , )

( , )y

x yx y

x y2

1

2

��

�

BGBG

BG

cNN

M N M N

∗

∗

∗×

11

2

1

1

2

1

c

c

ff

fN N M M

� �

=

× ×

( )( )

( )

xx

x

(11.6)

where B stands for a boundary operator, f(x) is a specified function along the boundary, and M(M ≥ N) is the number of boundary collocations. If M ≠ N, Eq. (11.5) or (11.6) can be solved by least squares fit of the boundary data, else it can be directly solved by the standard Gaussian elimination.

From the procedure of the MFS described above, it is found that the MFS formulation is very simple and has been demonstrated with higher efficiency than most of the numerical methods such as the FEM and the BEM [28]. However, the MFS has some inherent drawbacks as mentioned in the introduction.

11.2.2 Modified MFS

As discussed above, when the fundamental solution of a given problem is unavailable, the AEM and the RBF can be employed to make the MFS workable. In the following, cases that the partial differential equation (11.1) has not explicit expression of fundamental solution are considered.


To solve this problem with MFS, the AEM is firstly introduced to con-vert the original governing partial differential equation (11.1) into the sim-pler Poisson equation whose fundamental solution is available. To this end, applying the Laplace operator to the sought field u(x) yields

∇ = ∈2u b( ) ( ),x x x Ω (11.7)

where b(x) is unknown fictitious source distribution. Eq. (11.7) indicates that the solution of Eq. (11.1) can be established by

solving this linear equation under the same boundary condition (11.4). The solution procedure can be implemented through the RBF interpola-tion, as detailed below.

Since Eq. (11.7) is a linear partial differential equation, its solution can be written as a sum of the homogeneous solution and the particular solu-tion as

u u uh p( ) ( ) ( )x x x= + (11.8)

where uh(x) is the homogeneous solution, which satisfies the following homogeneous partial differential equation

∇ =2 0uh ( )x (11.9)

and up(x) is the particular solution, which satisfies the following inhomo-geneous partial differential equation:

∇ =2u bp( ) ( )x x (11.10)

11.2.2.1 RBF Interpolation for the Particular Solution

Making use of the properties of RBFs, the unknown fictitious source distri-bution b(x) is firstly approximated by

b i ii

I

i( ) ( , ), ,x = ∈=∑a j x x x x

1

Ω (11.11)

where I is the total number of interpolation points inside the domain and on the boundary (see Figure 11.2), ai are the coefficients to be determined, and j(x, xi) = j(r) are a set of approximating RBFs, which are typically in terms of the radial distance r

r i= −| |x x (11.12)


Furthermore, the particular solution can be expressed as

up ii

I

i( ) ( , )x x x==∑a y

1

(11.13)

where y(x, xi) = y(r) is the particular solution kernel (PSK) which has the relation to the RBF j(r)

∇ =2ψ ϕ( ) ( )r r (11.14)

In the RBF interpolation, the effectiveness and accuracy of the inter-polation depend on the type of the RBFs. Besides the ad hoc RBF, which is used almost exclusively and uncritically in the engineering literature, the thin plate splines (TPS) and multiquadrics (MQ) are also commonly used in meshless formulation. Here, the three types of RBFs and the cor-responding PSKs are listed in Table 11.1 for reference.

11.2.2.2 MFS for the Homogeneous Solution

To obtain a weak homogeneous solution of the Laplace equation (11.9), the standard MFS can be employed. Provided that N singularities { }yn n

N=1 are

located outside the domain Ω as illustrated in Figure 11.2, we have

u c Gh n nn

N

n( ) ( , ), ,x x y x y= ∈ ∉∗

=∑

1

Ω Ω (11.15)

where { }cn nN=1 are unknown coefficients and G*(x, yn) is the fundamental

solution of the Laplace partial differential equation, i.e. for the case of two dimension, G*(x, yn) can be written as

Source point

Interpolation point

X

X1

X2Ω

Figure 11.2 Schematic of the modified MFS procedure.


G rn∗ = −( , )x y 1

2pln (11.16)

with

r n= −| |x y (11.17)

11.2.2.3 Complete Solution

Based on the discussion above, the complete solution of Eq. (11.7) can be expressed in terms of the homogeneous solution and the particular solu-tion as

u c G rnn

N

n ii

N

( ) ( , ) ( )x x y= +=

∗

=∑ ∑

1 1

a y (11.18)

To determine all unknowns in Eq. (11.18), M(M ≥ N) collocations are placed along the physical boundary, and Eq. (11.18) is required to satisfy the boundary condition (11.4) at M collocations and satisfy the original govern-ing equation (11.1) at I interpolation points, i.e. for the system consisting of Eqs (11.1) and (11.4), the final discrete linear equations can be written as

c LG L

c BG

n j n i j ii

I

n

N

i j n

n m

∗

==

∗

+ = ∈ ∈ ∉∑∑ (x y x x x x y

x

, ) ( , ) , ,

(

a y 011

Ω Ω Ω

,, ) ( , ) ( ) , ,y x x x x x yn i m i mi

I

n

N

i m nB f+ = ∈ ∈∂ ∉

==

∑∑ a y11

Ω Ω Ω

(11.19)

Table 11.1 Three types of RBFs and the corresponding PSKs.

Radial basis function φ(r) Particular solution kernel ψ(r)

1 + r2n−1 (n ≥ 1) r rn

n2 2 1

24 2 1+

+

+

( )

r r21n 116

1 132

4 4r r rn −

r c2 2+−

++

+c c c r c3 2 2 213

49

n( )φ φ( )


11.3 Basic Formulation of HFS-FEM

To solve heat transfer problem of heterogeneous materials, for example, the fiber-reinforced composites, the MFS usually needs to establish separate equation for each material subdomain and then generate the final linear equation system by coupling them with additional equations satisfying the interfacial conditions of adjacent material subdomains. The complica-tion of the MFS increases as multiple material subdomains are involved. To bypass this problem, the fundamental-solution-based hybrid FEM has been developed for thermal analysis in heterogeneous materials [14, 15, 78, 92, 93]. More interestingly, various special n-sided polygonal elements can be developed to simplify the analysis and reduce the quantity of elements.

11.3.1 Problem Statement

For a 2D thermal conduction problem of heterogeneous materials without internal heat sources, as shown in Figure 11.3, the related governing equa-tions include the following:

1. Energy conservation (heat balance) equation

∂∂

+∂

∂

=qx

qx

1

1

2

2

0 (11.20)

2. Heat transfer constitutive law (Fourier law)

q k ux

i ji ijj

= −∂

∂

=( , , ) 1 2 (11.21)

where qi(i = 1, 2) are the heat flux components along i-direction, u is the temperature field, and kij represents anisotropic thermal conductivity coef-ficient of material constituents, which should obey Onsagar’s reciprocity relation kij = kji and positive determinant

Λ = − >k k k11 22 122 0 (11.22)

Generally, for heterogeneous or inhomogeneous materials, the thermal conductivity may be expressed as function in terms of spatial variable x = (x1, x2), while for homogeneous materials, the thermal conductivity is a constant.


Substituting Eq. (11.21) into Eq. (11.20) yields

∂

∂∂∂

+ ∂∂

+ ∂∂

∂∂

+ ∂∂

=x

k ux

k ux x

k ux

k ux1

111

122 2

211

222

0 (11.23)

Specially, when

k k k k11 22 12 210 0= ≠ = =, (11.24)

the problem becomes isotropic and Eq. (11.23) reduces to

∂

∂

∂

∂

+∂

∂

∂

∂

=xk u

x xk u

x111

1 211

2

0 (11.25)

Besides, the boundary conditions for heat transfer usually include

Dirichlet condition onNeumanncondition : on

: u uq q

u

q

=

=

Γ

Γ (11.26)

where the normal heat flux q is given by

q q n k ux

ni i ijj

i= = −∂

∂

(11.27)

kij = kij (x)

Γq

Γu

X

n

X1

X2

Ω

Figure 11.3 Geometrical definition for the two-dimensional bounded domain.


and ni is the direction cosine of the unit outward normal vector n to the boundary Г.

11.3.2 Implementation of the HFS-FEM

To study the thermal behavior of heterogeneous materials mentioned above, it is necessary to solve the corresponding boundary value problem described by the heat transfer in the bounded domain with specific boundary condi-tions, as illustrated in Figure 11.4a, in which the computing domain Ω is discretized with finite hybrid FEs. Then, the hybrid FE formulation in weak form can be established in a particular n-sided element domain Ωe with the boundary Γe shown in Figure 11.4b, for heat transfer analysis by means of a modified variational functional [15, 35, 80, 94, 95]

∏ = − − −∫ ∫me ij j ik u u d q u u de e

12 Ω Γ

Ω Γ( )� (11.28)

where u is the intraelement temperature field satisfying, a priori, the gov-erning equation of heat transfer and �u is the element frame field which is introduced to enforce conformity on the field variable between adja-cent elements. q and q are the normal heat flux and specified value on the boundary Γ Γ Γqe q e= ∩ , respectively. ni(i = 1, 2) are components of outward unit normal to the boundary Γe.

Applying the divergence theorem

h d hn di iΩ Γ

Ω Γ∫ ∫= (11.29)

to the functional (11.28) yields

Π Γ Γ ΓΓ Γ Γ

e n nq ud qud q ude qe e

= − − +∫ ∫ ∫12

� � (11.30)

In Eq. (11.30), the intraelement temperature u satisfying automatically the governing equation of heat transfer is approximated by the linear com-bination of the fundamental solutions of the problem, as done in the stan-dard MFS,

u G cj

n

sj j

s

= =∗

=

∑1

( , )x x Nc (11.31)


where G*(x, xsj) is the related fundamental solution, x(x1, x2) and xsj js

jsx x( , )1 2

are the field point and source point, respectively. It has to be noted that the so-called source points are placed outside the element domain to avoid computing singular integrals. Such a technique has proved to be efficient [35–38, 78–83].

Furthermore, differentiating the temperate field (11.31) to the coordi-nate variable and then substituting it into Eq. (11.27) yield the following normal derivative of temperature

q k Tn

= −∂

∂

=Qc (11.32)

kij = kij (x)

Γq

Γu

n

X1

X2

Schematic of hybrid nite element discretization of plane elastic domain(a)

Element frame eldIntraelement eld

Node

X1

X2 Ω

Schematic of particular n-sided hybrid nite element

u~

(b)

u = Nc

= Nd~

Figure 11.4 Schematics of hybrid FE discretization of plane body and a particular n-sided hybrid FE.


with

Q

N

N= −

∂

∂

∂

∂

{ }n nk kk k

x

x

1 211 12

12 22

1

2

(11.33)

Meanwhile, the independent element frame field �u is defined over the ele-ment boundary and can be expressed in terms of the nodal temperature vector d

� �u = Nd (11.34)

where �N represents the conventional interpolating shape functions for line element in the FEM or BEM.

Then, substituting Eqs (11.31) and (11.34) into the functional (11.30) produces

ΠeT T T= − − +1

2c Hc d g c Gd (11.35)

in which

H Q N G Q N g N= = =∫ ∫ ∫T Td d qd

e e qe

Γ Γ ΓΓ Γ Γ

T � �

Minimization of the functional Πe with respect to c and d, respectively, yields

∂

∂

= − + =

∂

∂

= − =

Π

Π

eT

eT

T

cHc Gd

dG c g

0

0 (11.36)

from which the optional relationship between c and d

c H Gd= −1 (11.37)

and the element stiffness equation

Kd g= (11.38)


are established. In Eq. (11.38), the symmetric stiffness matrix can be expressed by

K G H G= −T 1 (11.39)

Assembling the element stiffness equation (11.38) element by element yields the final global stiffness equation, which can be used to determine the nodal temperature vector d, and furthermore, the unknown interpola-tion coefficients c can be determined using Eq. (11.37). However, we notice that the rigid-body motion mode should be recovered in the assumed internal temperature field (11.31), because the fundamental solution used does not include any rigid-body motion [73, 96, 97].

11.3.4 Recovery of Rigid-body Motion

For the scale temperature field under consideration, the missing rigid-body mode can be recovered by adding a constant c0 to the assumed inter-nal field (11.31) in each element, that is,

u c= +Nc 0 (11.40)

Then, the least square matching of u and �u at element nodes

( ) minNc + − ==∑ c u ii

n

02

1

�node

(11.41)

can be used to determine the constant c0

cn

u ii

n

01

1= −

=

∑( )� Nc node (11.42)

where n is the total number of element nodes.

11.4 Applications in Functionally Graded Materials

11.4.1 Basic Equations in Functionally Graded Materials

FGMs are a class of relatively new and promising heterogeneous materials that have optimized material properties by combining different material components following a predetermined law. The gradual change in terms


of spatial variables gives place to a gradient of properties and performances [2]. Generally, the thermal conductivities of the FGMs can be written as

k kij ij= ( )x (11.43)

Usually, the following three variations are assumed in the practical engineering:

• Exponential form

k k eij ijx x( ) ( )x = +0 2 1 1 2 2β β (11.44)

• Quadratic form

k k a a x a xij ij( ) ( )x = + +0

0 1 1 2 22 (11.45)

• Trigonometric form

k k a x x a x xij ij( ) [ cos( ) sin( )]x = + + +0

1 1 1 2 2 2 1 1 2 22

β β β β (11.46)

where k a a aij0

0 1 2 1 2, , , , ,β β are material constants.

11.4.2 MFS for Functionally Graded Materials

Theoretically, the thermal conductivities can change in arbitrary form in terms of the spatial variable x, so it is difficult to derive a general form of fundamental solution for such material. To address this issue, we can apply the modified MFS to solve this problem. Here, the transient heat conduc-tion in isotropic FGMs is studied [54]. For this case, the governing equation can be written as

∇⋅ ∇ = ∇ +∇ ⋅∇ =

∂

∂

[ ( ) ( , )] ( ) ( , ) ( ) ( , ) ( , )k u t k u t k u t c u tt

x x x x x x x 2 ρ

(11.47)

with the following boundary condition

Bu t f t( , ) ( , ),x x x = ∈∂Ω (11.48)


and the initial condition

u t g( , ) ( )x x = =0 (11.49)

In order to obtain the temperature field and its flux at any time, the time domain is divided into elements and a simple backward time stepping scheme is used, i.e.

∂

∂

=−

+

+ut

u utt t

t t t

∆

∆

∆

(11.50)

where Δt is the time step, and ut+Δt = u(x, t + Δt), ut = u(x, t).Substituting Eq. (11.50) into Eqs (11.47) and (11.48), we have

k u k u c

tu c

tut t t t t t t( ) ( ) ( ) ( ) ( ) ( )x x x x x x∇ +∇ ⋅∇ − = −+ + +2 ∆ ∆ ∆

∆ ∆

ρ ρ

(11.51)

and

Bu ft t t t+ += ∈∂∆ ∆ Ω( ) ( ),x x x (11.52)

Thus, the operator L in Eq. (11.1) can be written as

Lu k u k u c

tut t t t t t t t+ + + += ∇ +∇ ⋅∇ −∆ ∆ ∆ ∆

∆

( ) ( ) ( ) ( ) ( ) ( )x x x x x x2 ρ

(11.53)

Applying the modified MFS to the system consisting of Eqs (11.51) and (11.52), we finally have

c LG L ct

un j n i j ii

It

j i j nn

∗

==

+ = − ∈ ∈ ∉∑( , ) ( , ) ( ), , ,x y x x x x x ya y r1 ∆

Ω Ω Ω11

11

N

n m n i m i mi

I

n

N

i j nc BG B f

∑

∑∑ ∗==

+ = ∈ ∈∂( , ) ( , ) ( ), , ,x y x x x x x ya y Ω Ω ∉∉

Ω

(11.54)

which can be solved for all unknowns at each time step.


Next, let us consider a square functionally graded finite strip, as dis-played in Figure 11.5, in which the thermal conductivity changes in a uni-directional exponential form

k k e x( )x = 02 1 1b (11.55)

with k0 = 17W/(m ∙ K) and rc = 10−6J/(m3 ∙ K). H(t) is the Heaviside

step function. Usually, the constant coefficient b1 is called as the graded parameter.

In the numerical analysis, the side length of the square is assumed to be 0.4 m, and zero initial temperature is considered. The graded param-eter is assumed to have three different values: b1 = 0.1, 0.1, and 0.25 cm

–1 for comparison. Numerical results are obtained using 36 fictitious source points, 169 interpolation points, and the time step Δt = 1s. Figure 11.6 dis-plays the time variation of temperature in the FGM square strip at different positions for the case of homogeneous case, that is, b1 = 0.0. Figure 11.7 shows the time variation of temperature at position x1 = 0.02 m of the FGM square strip for various graded parameters. Finally, the steady-state tem-perature distributions are illustrated in Figure 11.8 for comparison when the graded parameter has different values.

X1

X2

k (x) = k0e2β1x1

q = 0

q = 0

u = 0 u = H(t)

Figure 11.5 Geometry of a functionally graded finite square strip and boundary conditions.


0

–0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

10 20 30 40

Meshless: x1 = 0.01

Analytical: x1 = 0.01

x1 = 0.02x1 = 0.03

x1 = 0.02x1 = 0.03

Time t50 60

Tem

pera

ture

Figure 11.6 Time variation of temperature in the FGM square strip at different positions with b1 = 0.

10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0

b1 = 0.0b1 = 0.1b1 = 0.25

Time t

Tem

pera

ture

Figure 11.7 Time variation of temperature at position x1 = 0.02 m of the FGM square strip.

11.4.3 HFS-FEM for Functionally Graded Materials

For simplicity, we assume the thermal conductivity varies exponentially with position vector, as defined in Eq. (11.44). Substituting Eq. (11.44) into the governing equation (11.23) yields

k u k uij ij i ij j0 02 0( ) ( )x x+ =β (11.56)


whose fundamental solution satisfies following equation

k u k uij ij i ij j s0 02 0( ) ( ) ( , )x x x x+ + =β δ (11.57)

where x and xs denote arbitrary field point and source point in the infinite domain, respectively.

The closed-form solution to Eq. (11.57) in two dimensions can be expressed as [98]

GK R

s s∗

= − − ⋅ +( , )( )

exp{ ( )}x x x x 002

κ

π

β

Λ

(11.58)

where

Λ0 110

220

120 2 0= − >k k k( ) (11.59)

κ β κ β= ⋅ 0 (11.60)

R is the geodesic distance defined as

R R ks= = ⋅−( , ) ( )x x r r0 1 (11.61)

with r = r(x, xs) = x − xs. K0 is the modified Bessel function of the second kind of zero order.

Figure 11.8 Steady-state distribution of temperature along x1-axis for the FGM square strip.

–0.20

0

0.2

0.4

0.6

0.8

1

1.2

0.005 0.01 0.02 0.025 0.03 0.040.0350.015

x1 (m)

Tem

pera

ture

Meshless: b1 = 0.0

Analytical: b1 = 0.0

b1 = 0.1

b1 = 0.1b1 = 0.25

b1 = 0.25


By means of the fundamental solutions corresponding to the grada-tion variation of the thermal conductivities, we can construct the so-called specially purposed graded element (SPGE), in which the thermal conduc-tivities allow to change as defined in Eq. (11.44). As shown in Figure 11.9, the constructed SPGE is evidently different from the constant FE in which constant material definition is given, and the existing approximately iso-parametric-graded element in which the shape function interpolation is employed to approximate the practical variation of the thermal conductivi-ties [99].

Once the fundamental solution of the problem is known, following the procedure described in Section 3, one can obtain the temperature distri-bution in the solution domain. Next, an example is considered to demon-strate the applicability and efficiency of the proposed SPGE. Figure 11.10 shows the square FG plate with specified boundary conditions. We assume that the plate is isotropic and the thermal conductivity graded along the x2-direction. The side length of the plate to be a = 0.04 m. The analytical solution of the temperature distribution is given by

u ee

x

=−

−

−

−

2

2

2 2

2

11

β

β α (11.62)

In the analysis, the graded parameter b2 = 10, 25, 50 and material con-stant k0 = 17 are used. One 16-node polygonal SPGE is used to model the solution domain, as shown in Figure 11.11a. For comparison, the solu-tion domain is also modelled by the 16 standard quadratic isoparametric FEs with stepwise constant material definition, as shown in Figure 11.11b. Figure 11.12 shows the temperature distribution along direction x2, which is the graded direction for various graded parameter b2. It is found that a higher level of temperature is obtained at the same position for a bigger graded parameter b2. We also observe that an excellent agreement between

(a) Conventional �nite element

(b) Specially purposed graded element

Figure 11.9 Schematic of the conventional FE and SPGE.


X1

X2

k (x) = k0e2β1x2

u = 0

q = 0q = 0

u =1

Figure 11.10 Square FG plate under specified boundary conditions.

(a)0

0

0.01

0.02

0.03

0.04

0.01 0.02 0.03 0.04(b)

00

0.01

0.02

0.03

0.04

0.01 0.02 0.03 0.04

Figure 11.11 The polygonal hybrid graded FE mesh in the proposed HFS-FEM (a) and the conventional FE mesh in the FEM (b).

numerical and analytical results is achieved for different graded param-eters. Besides, Table 11.2 listed the numerical results from the present HFS-FEM using SPGE at some specific locations, in which the analytical solutions and the numerical results from the conventional FEM using the mesh shown in Figure 11.11b are also provided for comparison. It can be seen from Table 11.2 that the proposed SPGE is more efficient than the conventional FE, because it can achieve more accurate results with much less elements.


00

0.005

0.01

0.02

0.03

0.04

0.025

0.035

0.015

Numerical b2 = 10

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Tempeature

X2

b2 = 25b2 = 50

Analytical b2 = 10b2 = 25b2 = 50

Figure 11.12 Temperature distribution along the graded direction with various graded parameters.

11.5 Applications in Composite Materials

11.5.1 Basic Equations of Composite Materials

The heat conduction in composites that usually include two or more mate-rial constituents is more complicated than that in homogeneous materials. Here, for the sake of convenience, the heat conduction in fiber-reinforced composites is reviewed, as shown in Figure 11.13. Assume that x = (x1, x2) denotes the spatial coordinates, Ωm and Ωf are the matrix and fiber domains, respectively, and um and uf are the temperature fields in the corresponding domains. If both the matrix and the fiber are isotropic, then the heat trans-fer equations in fiber-reinforced composites can be written as follows.

1. The governing equations for the fiber and matrix material phases:

∂

∂

+∂

∂

= ∈

∂

∂

+

∂

∂

= ∈

2

12

2

22

2

12

2

22

0

0

ux

ux

ux

ux

m mm

f ff

,

,

x

x

Ω

Ω (11.63)

354 Advanced Engineering Materials and ModelingTa

ble

11.2

Com

paris

on o

f tem

pera

ture

s alo

ng x

2 for

diff

eren

t met

hods

.

x 20.

000

0.00

50.

010

0.01

50.

020

0.02

50.

030.

035

0.04

0

Exac

t0.

0000

0.25

580.

4551

0.61

020.

7311

0.82

520.

8985

0.95

551.

0000

HFS

-FEM

0.00

000.

2556

0.45

530.

6102

0.72

950.

8252

0.89

820.

9556

1.00

00

FEM

0.00

000.

2275

0.45

510.

5931

0.73

110.

8148

0.89

840.

9492

31.

0000


2. The continuity conditions at the interface Γmf between the fiber and the matrix

u u kun

kunm f m

mf

f=

∂

∂

=

∂

∂

, (11.64)

where n is the unit direction normal to the fiber/matrix interface, and km and kf are the thermal conductivities of the matrix and fiber materials, respectively.

Besides, according to Fourier’s law of heat transfer in isotropic media, one has the following relationship between the temperature variable u and the heat flux component qi

q kux

i

q kux

i

mi mm

i

fi ff

i

= −∂

∂

=

= −

∂

∂

=

( , )

( , )

1 2

1 2

(11.65)

Finally, the boundary conditions of the problem defined along the outer boundary of the matrix domain are given by

Bu fm m( ) ( ),x x x= ∈∂Ω 1 (11.66)

Matrix

X2

X1

Fiber

Figure 11.13 Schematic of fiber-reinforced composites.


where B stands for a boundary operator and f(x) is a specified function along the boundary. ∂ = ∂Ω Ω Γm m mf1 / is the outer boundary of the matrix domain.

11.5.2 MFS for Composite Materials

For the heat conduction problem in the composite shown in Figure 11.13, the procedure of MFS is relatively complicated [100, 101]. The MFS needs to establish discrete equation for each material constituent and then link them by the interfacial conditions (11.64). For simplification, let us consider the simplest case that only one fiber is embedded in the matrix.

11.5.2.1 MFS for the Matrix Domain

Following the procedure of the MFS described above, the approximation of temperature field in the matrix domain shown in Figure 11.14 has the form

u c Gm nm

m nm

n

N

m nm

m

m

( ) ( , ), ,x x y x y= ∈ ∉∗=

∑ 1

Ω Ω (11.67)

where Nm is the number of unknown singularities (sources) for the matrix domain, cn

m are unknown real coefficients, and Gm n∗ ( , )x y is the funda-

mental solution related to the matrix material

Gk

rm nm

m

∗

= −( , )x y 12

1π

n (11.68)

with

r nm

= −| |x y (11.69)

11.5.2.2 MFS for the Fiber Domain

Similarly, the MFS approximation of temperature field in the fiber domain shown in Figure 11.15 has the form

u c Gf nf

n

N

f nf

f nf

f

f

( ) ( , ), ,x x y x y= ∈ ∉=

∗∑1

Ω Ω (11.70)

where Nf is the number of unknown singularities (sources) for the fiber domain, cn

f are unknown real coefficients, and G f nf∗ ( , )x y is the funda-

mental solution related to the matrix material


Gk

rf nf

f

∗

= −( , )x y n12

1π

(11.71)

with

r nf

= −| |x y (11.72)

11.5.2.3 Complete Linear Equation System

In Eq. (11.67), there are Nm unknowns, while in Eq. (11.70) there are Nf unknowns. These unknown coefficients can be determined by collocat-ing the boundary conditions and the interfacial conditions at M1 distinct points on the outer boundary ∂ = ∂Ω Ω Γm m mf1 \ and M2 distinct points on the interface Γmf.

Figure 11.14 Collocation discretization of the matrix domain.

Figure 11.15 Collocation discretization of the fiber domain.


Substituting Eq. (11.67) into the boundary condition (11.66), we have

Bu c BG f i Mm i nm

n

N

m i nm

i i m

m

( ) ( , ) ( ), ,x x y x x= = ∈∂ = →=

∗

∑1

1 11Ω (11.73)

Similarly, substituting Eqs (11.67) and (11.70) into the interfacial condi-tion (11.64), we have

c G c G j Mnm

m nm

n

N

nf

n

N

f j nf

j mf

m f∗

= =

∗

∑ ∑= ∈ = →( , ) ( , ), ,x y x y x 1 1

21Γ (11.74)

k c

Gn

k cG

nm nm m j n

m

n

N

f nf

n

Nf j n

f

j

m f∂

∂

=

∂

∂

∈

∗

= =

∗

∑ ∑( , ) ( , )

,x y x y

x

1 1

Γmmf j M, = →1 2

(11.75)

Thus, we have a total number of Nm + Nf unknowns and a total of M1 + 2M2 equations to be solved. The minimization strategy can be carried to solve these equations and then obtain all knowns.

From the procedure above, we find that the MFS is not very suitable for such problem. In particular, when the number of fibers in the matrix becomes large, the computation time of the MFS increases dramatically. In contrast, the HFS-FEM discussed in Section 11.5.3 can handle this prob-lem efficiently.

11.5.3 HFS-FEM for Composite Materials

According to the feature that arbitrarily shaped polygonal hybrid FE can be constructed in the HFS-FEM, the special n-sided fiber/matrix elements can be developed for such analysis, as described below.

11.5.3.1 Special Fundamental Solutions

Fundamental solutions play an important role in the derivation of the special n-sided fiber/matrix elements. With the proposed HFS-FEM, for two-dimensional heat conduction problems in fiber-reinforced compos-ites, the temperature response in an infinite matrix region containing a centered circular fiber is needed under the unit heat source in the matrix. In Figure 11.16, a unit heat source is applied at the source point z0 in the infinite matrix Ωm, then the temperature responses Gm and Gf at any field point z in matrix and fiber regions are, respectively, obtained as [102]


Gk

z zk kk k

Rz

zmm

m f

m f

= − − +

−

+

−

12

1 102

0π

Re[ ] Ren( ) n

∈

= −

+

− ∈

z

Gk k

z z z

m

fm f

f

Ω

Ω1 1 0( )

Re[ ]π

n( )

(11.76)

where R is the radius of the fiber, z = x1 + x2i is a complex number defined in a local coordinate system x = (x1, x2) with its origin coincident with the fiber center, and i = −1 denotes the unit imaginary number.

11.5.3.2 Special n-Sided Fiber/Matrix Elements

By means of the special fundamental solutions described above, special n-sided fiber/matrix elements can be constructed. Here, one and nine fibers embedded in the unit square matrix region (see Figure 11.17) are consid-ered to demonstrate the performance of the present special n-sided fiber/matrix elements. A uniform heat flux of 100.0 is applied on the left side of the square matrix region, while its opposite side is maintained at a tempera-ture of 0. All other edges of the square are assumed to be insulated.

For simplicity, the thermal conductivities in consistent units are assumed for the matrix and the fiber, that is, kf = 20km = 20. Figures 11.18 and 11.19 show, respectively, the temperature distribution along the middle line x2 = 0.5 for the cases of one and nine fibers. Also, the numerical results from the conventional FEs (ABAQUS) are provided as reference values for com-parison. To obtain steady results, the conventional FE mesh is chosen to be refined enough. It is found that there are good agreements between results

Heat source

Field point zX2

X1R

Interface |z| = R Fiber region |z| < R

In�nite matrix region |z| > R

Z0

Figure 11.16 Special fundamental solutions for heat conduction problems in fiber-reinforced composites.


Single �ber Nine �bers(a) (b)

Figure 11.17 Different fiber numbers in the matrix.

00

10

20

30

40

50

60

70

80

90

100

0.2 0.4

Temperature without �ber

HFS-FEMABAQUS

0.6x1

0.8 1

Tem

pera

tutr

e

Figure 11.18 Temperature variation along the line x2 = 0.5 for one fiber case.

00

10

20

30

40

50

60

0.2 0.4

HFS-FEMABAQUS

0.6x1

0.8 1

Tem

pera

tutr

e

Figure 11.19 Temperature variation along the line x2 = 0.5 for nine-fiber case.


from the present HFS-FEM and ABAQUS for both cases. However, very few elements are employed in the present HFS-FEM, and mesh discretiza-tion inside the fiber domain is avoided.

11.6 Conclusions

On the basis of the preceding discussion, the following conclusions can be made. In contrast to the conventional FEM and the BEM, the fundamen-tal-solution-based MFS and HFS-FEM have some advantages:

1. The formulation of MFS is very simple, and relatively few boundary collocations and singularities are required to produce results with acceptable accuracy. Besides, the MFS has good adaptivity for problems local singulari-ties such as geometric corner and crack. For example, by adjusting the locations of boundary collocations, the cor-ner problem meeting in the BEM is not a specific source of inaccuracy.

2. As for the HFS-FEM, all integrals are performed along the element boundary, and thus we can develop a general for-mulation of n-sided general or special polygonal element to simplify mesh division and reduce numbers of elements, which permits that different elements can have a different number of sides. In addition, it is easy to calculate fields everywhere inside the element.

In the future, there are many possible extensions and areas in need of further development by using the MFS and the HFS-FEM. Among those developments, one could list the following:

• Development of efficient formulations of the MFS and HFS-FEM to thermoelastic analysis of FGM and composites.

• Construction of more special polygonal elements for our use in the HFS-FEM.

• Extensions of the MFS and HFS-FEM to coated composite structures.

• Combination of the MFS and HFS-FEM with topology opti-mization for designing microstructure of materials.

• Development of efficient formulations of the MFS and HFS-FEM for interaction between fluid and structure.


Acknowledgments

The research work was partially supported by the Natural Science Foundation of China (Grant No. 11472099) and the Creative Talents Program of Universities in Henan Province (Grant No. 13HASTIT019).

Conflict of Interest

The authors confirm that there is no conflict of interest related to the con-tent of this article.

References

1. Suresh, S., and Mortensen, A., Fundamentals of functionally graded materials, IOM Communications, London, 1998.

2. Qin, Q.H., and Yang, Q.S., Macro-micro theory on multifield coupling behaivor of hetereogenous materials, Higher Education Press and Springer, Beijing, 2008.

3. Mallick, P.K., Fiber-reinforced composites: materials, manufacturing, and design, CRC Press, Boca Raton, 2007.

4. Gibson, R.F., Principles of composite material mechanics, CRC Press, Boca Raton, 2012.

5. Bunsell, A.R., and Renard, J., Fundamentals of fibre reinforced composite mate-rials, IOP Publishing, London, 2005.

6. Qin, Q.H., Introduction to the composite and its toughening mechanisms, in: Toughening Mechanisms in Composite Materials, Qin, Q.H., and Ye, J.Q., (Ed.), pp. 1–32, Elsevier, Cambridge, 2015.

7. Koizumi, M., FGM activities in Japan. Compos. B. Eng., 28(1), 1–4, 1997. 8. Bakis, C.E., Bank, L.C., Brown, V.L., Cosenza, E.l., Davalos, J.F., Lesko, J.J.,

Machida, A., Rizkalla, S.H., and Triantafillou, T.C., Fiber-reinforced polymer composites for construction-state-of-the-art review. J. Compos. Constr., 6(2), 73–87, 2002.

9. Michel, J.C., Moulinec, H., and Suquet, P., Effective properties of composite materials with periodic microstructure: a computational approach. Comput. Meth. Appl. Mech. Eng., 172(1), 109–143, 1999.

10. Pegoretti, A., Fambri, L., Zappini, G., and Bianchetti, M., Finite element analysis of a glass fibre reinforced composite endodontic post. Biomaterials, 23(13), 2667–2682, 2002.

11. Liu, Y.J., Nishimura, N., Otani, Y., Takahashi, T., Chen, X.L., and Munakata, H., A fast boundary element method for the analysis of fiber-reinforced com-posites based on a rigid-inclusion model. J. Appl. Mech., 72(1), 115–128, 2005.


12. Liu, Y.J., and Chen, X.L., Continuum models of carbon nanotube-based com-posites using the boundary element method. Electron. J. Bound. Elem., 1(2), 2007.

13. Dang, T.D., and Sankar, B.V., Meshless local Petrov-Galerkin micromechani-cal analysis of periodic composites including shear loadings. Comput. Model. Eng. Sci., 26(3), 169, 2008.

14. Wang, H., and Qin, Q.H., Special fiber elements for thermal analysis of fiber-reinforced composites. Eng. Computation, 28(8), 1079–1097, 2011.

15. Wang, H., and Qin, Q.H., A new special coating/fiber element for analyz-ing effect of interface on thermal conductivity of composites. Appl. Math. Comput., 268, 311–321, 2015.

16. Martınez, M.A., Cueto, E., Doblaré, M., and Chinesta, F., Natural element meshless simulation of flows involving short fiber suspensions. J. Non-Newton. Fluid. Mech., 115(1), 51–78, 2003.

17. Yang, Q.S., and Qin, Q.H., Fiber interactions and effective elasto-plastic prop-erties of short-fiber composites. Compos. Struct., 54(4), 523–528, 2001.

18. Yang, Q.S., and Qin, Q.H., Modelling the effective elasto-plastic properties of unidirectional composites reinforced by fibre bundles under transverse ten-sion and shear loading. Mat. Sci. Eng. A-Struct., 344(1), 140–145, 2003.

19. Qin, Q.H., Material properties of piezoelectric composites by BEM and homogenization method. Compos. Struct., 66(1), 295–299, 2004.

20. Ma, H., Xia, L.W., and Qin, Q.H., Computational model for short-fiber com-posites with eigenstrain formulation of boundary integral equations. Appl. Math. Mech., 29, 757–767, 2008.

21. Ma, H., Yan, C., and Qin, Q.H., Eigenstrain formulation of boundary integral equations for modeling particle-reinforced composites. Eng. Anal. Bound. Elem., 33(3), 410–419, 2009.

22. Yang, Q.S., Qin, Q.H., and Zheng, D.H., Analytical and numerical investiga-tion of interfacial stresses of FRP–concrete hybrid structure. Compos. Struct., 57(1), 221–226, 2002.

23. Jiang, W.G., Zhong, R.Z., Qin, Q.H., and Tong, Y.G., Homogenized finite ele-ment analysis on effective elastoplastic mechanical behaviors of composite with imperfect interfaces. Int. J. Mol. Sci., 15(12), 23389–23407, 2014.

24. Kompiš, V., Murčinková, Z., Rjasanow, S., Grzhibovskis, R., and Qin, Q.H., Computational simulation methods for composites reinforced by fibres, in: Proceedings of the International Symposium on Computational Structural Engineering, Yuan, Y., Cui, J.Z., and Mang, Herbert A. (Ed.), pp. 63–69, Springer, 2009

25. Lei, Y.P., Wang, H., and Qin, Q.H., Bio-inspired design of lightweight metal structure based on microstructure of fully ripe loofah. Recent Pat. Mat. Sci., 8(1), 69–73, 2015.

26. Lee, C.Y., Qin, Q.H., and Walpole, G., Numerical modeling on electric response of fibre-orientation of composites with piezoelectricity. Int. J. Res. Rev. Appl. Sci., 16(3), 377–386, 2013.


27. Chen, C.S., Karageorghis, A., and Smyrlis, Y.S., The method of fundamental solutions: a meshless method, Dynamic Publishers, Atlanta, 2008.

28. Fairweather, G., and Karageorghis, A., The method of fundamental solutions for elliptic boundary value problems. Adv. Comput. Math., 9(1–2), 69–95, 1998.

29. Lee, C.Y., Wang, H., and Qin, Q.H., Method of fundamental solutions for 3D elasticity with body forces by coupling compactly supported radial basis func-tions. Eng. Anal. Bound. Elem., 60, 123–136, 2015.

30. Zhang, Z.W., Wang, H., and Qin, Q.H., Meshless method with operator split-ting technique for transient nonlinear bioheat transfer in two-dimensional skin tissues. Int. J. Mol. Sci., 16, 2001–2019, 2015.

31. Zhang, Z.W., Wang, H., and Qin, Q.H., Method of fundamental solutions for nonlinear skin bioheat model. J. Mech. Med. Biol., 14(4), 1450060, 2014.

32. París, F., and Cañas, J., Boundary element method: fundamentals and applica-tions, Oxford Science Publications, New York, 1997.

33. Qin, Q.H., and Huang, Y., BEM of postbuckling analysis of thin plates. Appl. Math. Model., 14(10), 544–548, 1990.

34. Qin, Q.H., and Mai, Y.W., BEM for crack-hole problems in thermopiezoelec-tric materials. Eng. Fract. Mech., 69(5), 577–588, 2002.

35. Wang, H., and Qin, Q.H., Hybrid FEM with fundamental solutions as trial functions for heat conduction simulation. Acta Mech. Solida Sin., 22(5), 487–498, 2009.

36. Wang, H. and Qin, Q.H., Fundamental-solution-based finite element model for plane orthotropic elastic bodies. Eur. J. Mech. Solid, 29(5), 801–809, 2010.

37. Cao, C., Qin, Q.H., and Yu, A., A new hybrid finite element approach for three-dimensional elastic problems. Arch. Mech., 64(3), 261–292, 2012.

38. Wang, H., Gao, Y.T., and Qin, Q.H., Green’s function based finite element for-mulations for isotropic seepage analysis with free surface. Lat. Am. J. Solid. Struct., 12(10), 1991–2005, 2015.

39. Bathe, K.J., Finite element procedures, Prentice Hall, New Jersey, 1996.40. Kupradze, V.D., A method for the approximate solution of limiting problems

in mathematical physics. U.S.S.R. Comput. Maths. Math. Phys., 4(6), 199–205, 1964.

41. Christiansen, S., On Kupradze’s functional equations for plane harmonic problems, in Function theoretic methods in differential equations, Gilbert, R.P., and Weinacht, R.J., (Ed.), pp. 205–243, Pitman Publishing, London, 1976.

42. Bogomolny, A., Fundamental solutions method for elliptic boundary value problems. SIAM J. Numer. Anal., 22(4), 644–669, 1985.

43. Golberg, M.A., and Chen, C.S., (Ed.), The method of fundamental solutions for potential, Helmholtz and diffusion problems, WIT Press, Southampton, 1998.

44. Chen, C.S., Cho, H.A., and Golberg, M.A., Some comments on the ill- conditioning of the method of fundamental solutions. Eng. Anal. Bound. Elem., 30(5), 405–410, 2006.


45. Marin, L., and Lesnic, D., The method of fundamental solutions for the Cauchy problem associated with two-dimensional Helmholtz-type equations. Comput. Struct., 83(4), 267–278, 2005.

46. Poullikkas, A., Karageorghis, A., and Georgiou, G., The method of fundamen-tal solutions for three-dimensional elastostatics problems. Comput. Struct., 80(3), 365–370, 2002.

47. Marin, L., and Lesnic, D., The method of fundamental solutions for the Cauchy problem in two-dimensional linear elasticity. Int. J. Solid. Struct., 41(13), 3425–3438, 2004.

48. Chen, J.T., Wu, C.S., Lee, Y.T., and Chen, K.H., On the equivalence of the Trefftz method and method of fundamental solutions for Laplace and bihar-monic equations. Comput. Math. Appl., 53(6), 851–879, 2007.

49. Barnett, A.H., and Betcke, T., Stability and convergence of the method of fun-damental solutions for Helmholtz problems on analytic domains. J. Comput. Phys., 227(14), 7003–7026, 2008.

50. Alves, C.J.S., On the choice of source points in the method of fundamental solutions. Eng. Anal. Bound. Elem., 33(12), 1348–1361, 2009.

51. Ramachandran, P.A., Method of fundamental solutions: singular value decomposition analysis. Comm. Numer. Meth. Eng., 18(11), 789–801, 2002.

52. Chen, C.S., The method of fundamental solutions for non‐linear thermal explosions. Comm. Numer. Meth. Eng., 11(8), 675–681, 1995.

53. Young, D.L., Jane, S.J., Fan, C.M., Murugesan, K., and Tsai, C.C., The method of fundamental solutions for 2D and 3D Stokes problems. J. Comput. Phys., 211(1), 1–8, 2006.

54. Wang, H., Qin, Q.H., and Kang, Y.L., A meshless model for transient heat con-duction in functionally graded materials. Comput. Mech., 38(1), 51–60, 2006.

55. Wang, H., and Qin, Q.H., A meshless method for generalized linear or non-linear Poisson-type problems. Eng. Anal. Bound. Elem., 30(6), 515–521, 2006.

56. Wang, H., Qin, Q.H., and Kang, Y.L., A new meshless method for steady-state heat conduction problems in anisotropic and inhomogeneous media. Arch. Appl. Mech., 74(8), 563–579, 2005.

57. Wang, H., and Qin, Q.H., Meshless approach for thermo-mechanical analysis of functionally graded materials. Eng. Anal. Bound. Elem., 32(9), 704–712, 2008.

58. Berger, J.R., and Karageorghis, A., The method of fundamental solutions for layered elastic materials. Eng. Anal. Bound. Elem., 25(10), 877–886, 2001.

59. Berger, J.R., and Karageorghis, A., The method of fundamental solutions for heat conduction in layered materials. Int. J. Numer. Meth. Eng., 45(11), 1681–1694, 1999.

60. Dong, C.F., Sun, F.Y., and Meng, B.Q., A method of fundamental solutions for inverse heat conduction problems in an anisotropic medium. Eng. Anal. Bound. Elem., 31(1), 75–82, 2007.

61. Marin, L., and Lesnic, D., The method of fundamental solutions for nonlinear functionally graded materials. Int. J. Solid. Struct., 44(21), 6878–6890, 2007.


62. Poullikkas, A., Karageorghis, A., and Georgiou, G., The method of funda-mental solutions for inhomogeneous elliptic problems. Comput. Mech., 22(1), 100–107, 1998.

63. Wang, H., and Qin, Q.H., Some problems with the method of fundamental solution using radial basis functions. Acta Mech. Solida Sin., 20(1), 21–29, 2007.

64. Katsikadelis, J.T., The meshless analog equation method: I. Solution of elliptic partial differential equations. Arch. Appl. Mech., 79(6–7), 557–578, 2009.

65. Katsikadelis, J.T., and Nerantzaki, M.S., Non-linear analysis of plates by the analog equation method. Comput. Mech., 14(2), 154–164, 1994.

66. Schaback, R., Error estimates and condition numbers for radial basis function interpolation. Adv. Comput. Math., 3(3), 251–264, 1995.

67. Wu, Z.M., and Schaback, R., Local error estimates for radial basis function interpolation of scattered data. IAM J. Numer. Anal., 13(1), 13–27, 1993.

68. Larsson, E., and Fornberg, B., A numerical study of some radial basis func-tion based solution methods for elliptic PDEs. Comput. Math. Appl., 46(5), 891–902, 2003.

69. Chen, W., Wang, H., and Qin, Q.H., Kernel radial basis functions, in: Proceedings of International Symposium on Computational Mechanics, Yao, Z.H., and Yuan, M.W., (Ed.), pp. 347–347, Tsinghua University Press & Springer, Beijing, 2009

70. Wang, H., Qin, Q.H., and Kang, Y.L., The method of fundamental solutions with radial basis functions approximation for thermoelastic analysis (in Chinese). J. Dalian Univ. Tech., 46(S1), 46–51, 2006.

71. Pian, T.H.H., State-of-the-art development of hybrid/mixed finite element method. Finite Elem. Anal. Des., 21(1), 5–20, 1995.

72. Pian, T.H.H., and Wu, C.C., Hybrid and incompatible finite element methods, Chapman and Hall/CRC, Boca Raton, 2006.

73. Qin, Q.H., The Trefftz finite and boundary element method. WIT Press, Southampton, 2000.

74. Jirousek, J., and Qin, Q.H., Application of hybrid-Trefftz element approach to transient heat conduction analysis. Comput. Struct., 58(1), 195–201, 1996.

75. Qin, Q.H., Hybrid Trefftz finite-element approach for plate bending on an elastic foundation. Appl. Math. Model., 18(6), 334–339, 1994.

76. Jirousek, J., and Zieliński, A.P., Survey of Trefftz-type element formulations. Comput. Struct., 63(2), 225–242, 1997.

77. Wang, H., Qin, Q.H., and Arounsavat, D., Application of hybrid Trefftz finite element method to nonlinear problems of minimal surface. Int. J. Numer. Meth. Eng., 69(6), 1262–1277, 2007.

78. Wang, H., Han, M.Y., Yuan, F., and Xiao, Z.R., Fundamental-solution-based hybrid element model for nonlinear heat conduction problems with temper-ature-dependent material properties. Math. Probl. Eng., 2013, 695458, 2013.

79. Wang, H., and Qin, Q.H., A fundamental solution-based finite element model for analyzing multi-layer skin burn injury. J. Mech. Med. Biol., 12(05), 1250027, 2012.


80. Wang, H., and Qin, Q.H., FE approach with Green’s function as internal trial function for simulating bioheat transfer in the human eye. Arch. Mech., 62(6), 493–510, 2010.

81. Wang, H., and Qin, Q.H., Numerical implementation of local effects due to two-dimensional discontinuous loads using special elements based on bound-ary integrals. Eng. Anal. Bound. Elem., 36(12), 1733–1745, 2012.

82. Wang, H., and Qin, Q.H., A new special element for stress concentration analysis of a plate with elliptical holes. Acta Mech., 223(6), 1323–1340, 2012.

83. Wang, H., and Qin, Q.H., Fundamental-solution-based hybrid FEM for plane elasticity with special elements. Comput. Mech., 48(5), 515–528, 2011.

84. Wang, H., Qin, Q.H., and Yao, W.A., Improving accuracy of opening-mode stress intensity factor in two-dimensional media using fundamental solution based-finite element model. Aust. J. Mech. Eng., 10(1), 41–52, 2012.

85. Cao, C., Yu, A., and Qin, Q.H., Evaluation of effective thermal conductiv-ity of fiber-reinforced composites by boundary integral based finite element method. Int. J. Archit. Eng. Constr., 1(1), 14–29, 2012.

86. Cao, C., Qin, Q.H., and Yu, A., Micromechanical analysis of heterogeneous composites using hybrid Trefftz FEM and hybrid fundamental solution based FEM. J. Mech., 29(4), 661–674, 2013.

87. Qin, Q.H., and Wang, H., Special circular hole elements for thermal analysis in cellular solids with multiple circular holes. Int. J. Comput. Methods, 10(4), 1350008, 2013.

88. Qin, Q.H., and Wang, H., Special elements for composites containing hexago-nal and circular fibers. Int. J. Comput. Methods, 12(3), 1540012, 2015.

89. Wang, H., Zhao, X.J., and Wang, J.S., Interaction analysis of multiple coated fibers in cement composites by special n-sided interphase/fiber elements. Compos. Sci. Technol., 118, 117–126, 2015.

90. Wang, H., Qin, Q.H., and Xiao, Y., Special n-sided Voronoi fiber/matrix ele-ments for clustering thermal effect in natural-hemp-fiber-filled cement com-posites. Int. J. Heat Mass Tran., 92, 228–235, 2016.

91. Wang, H., and Qin, Q.H., Boundary integral based graded element for elastic analysis of 2D functionally graded plates. Eur. J. Mech. Solid, 33, 12–23, 2012.

92. Wang, H., Cao, L.L., and Qin, Q.H., Hybrid graded element model for nonlin-ear functionally graded materials. Mech. Adv. Mater. Struct., 19(8), 590–602, 2012.

93. Cao, L.L., Wang, H., and Qin, Q.H., Fundamental solution based graded ele-ment model for steady-state heat transfer in FGM. Acta Mech. Solida Sin., 25(4), 377–392, 2012.

94. Wang, H., and Zhao, X.J., Anisotropic heat analysis using hybrid finite ele-ment model with element boundary integrals. Adv. Mater. Res., 199, 1613–1616, 2011.

95. Qin, Q.-H., Trefftz finite element method and its applications. Appl. Mech. Rev., 58(5), 316–337, 2005.

96. Qin, Q.H., and Wang, H., Matlab and C programming for Trefftz finite element methods. CRC Press, Boca Raton, 2008.


97. Wang, H., Qin, Q.H., and Liang, X.P., Solving the nonlinear Poisson-type problems with F-Trefftz hybrid finite element model. Eng. Anal. Bound. Elem., 36(1), 39–46, 2012.

98. Berger, J.R., Martin, P.A., Mantič, V., and Gray, L.J., Fundamental solutions for steady-state heat transfer in an exponentially graded anisotropic mate-rial. J. Appl. Math. Phys., 56(2), 293–303, 2005.

99. Kim, J.H., and Paulino, G.H., Isoparametric graded finite elements for non-homogeneous isotropic and orthotropic materials. J. Appl. Mech., 69(4), 502–514, 2002.

100. Karageorghis, A., and Lesnic, D., Steady-state nonlinear heat conduction in composite materials using the method of fundamental solutions. Comput. Meth. Appl. Mech. Eng., 197(33), 3122–3137, 2008.

101. Bin-Mohsin, B., and Lesnic, D., The method of fundamental solutions for Helmholtz-type equations in composite materials. Comput. Math. Appl., 62(12), 4377–4390, 2011.

102. Chao, C.K., and Shen, M.H., On bonded circular inclusions in plane thermo-elasticity. J. Appl. Mech., 64(4), 1000–1004, 1997.

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