a novel boundary-integral based finite element method for...

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A Novel Boundary-Integral Based Finite ElementMethod for 2D and 3D Thermo-Elasticity ProblemsChangyong Cao a , Qing-Hua Qin a & Aibing Yu ba Research School of Engineering, Australian National University, Canberra, Australiab School of Materials Science and Engineering, University of New South Wales, Sydney, NSW,AustraliaVersion of record first published: 08 Nov 2012.

To cite this article: Changyong Cao, Qing-Hua Qin & Aibing Yu (2012): A Novel Boundary-Integral Based Finite Element Methodfor 2D and 3D Thermo-Elasticity Problems, Journal of Thermal Stresses, 35:10, 849-876

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Journal of Thermal Stresses, 35: 849–876, 2012Copyright © Taylor & Francis Group, LLCISSN: 0149-5739 print/1521-074X onlineDOI: 10.1080/01495739.2012.720204

A NOVEL BOUNDARY-INTEGRAL BASED FINITEELEMENT METHOD FOR 2D AND 3DTHERMO-ELASTICITY PROBLEMS

Changyong Cao1, Qing-Hua Qin1, and Aibing Yu21Research School of Engineering, Australian National University,Canberra, Australia2School of Materials Science and Engineering, University of New South Wales,Sydney, NSW, Australia

In this article a new hybrid boundary integral-based (HBI) finite element method(FEM) is presented for analyzing two-dimensional (2D) and three-dimensional (3D)thermoelastic problems with arbitrary distribution of body force and temperaturechanges. The method of particular solution is used to decompose the displacement fieldinto homogeneous part and particular part. The homogeneous solution is obtained byusing the HBI-FEM with fundamental solutions, yet the particular solution relatedto the body force and temperature change is approximated by radial basis function(RBF). The detailed formulation for both 2D and 3D HBI-FEM for thermoelasticproblems are given, and two different approaches for treating the inhomogenous termsare presented and compared. Five numerical examples are presented to demonstrate theaccuracy and performance of the proposed method. When compared with the existinganalytical solutions or ABAQUS results, it is found that the proposed method workswell for thermoelastic problems and also when using a very coarse mesh, results withsatisfactory accuracy can be obtained.

Keywords: Body force; Fundamental solution; Hybrid FEM; Particular solution; Radial basisfunction; Temperature; Thermoelasticity.

INTRODUCTION

Problems of thermoelasticity arise in many practical designs and structuressuch as the design of steam and gas turbines, jet engines, rocket motors and nuclearreactors. Thermal stresses induced in these structures are one of the importantconcerns in product design and analysis. A general thermoelastic problem isgoverned by two time-dependent coupled differential equations: the heat conductionequation and the Navier-Cauchy equation with body force induced by temperaturechange [1]. In most engineering applications, the coupling term of the heat equationand the inertia term in the Navier-Cauchy equation are generally negligible [1].

Received 6 October 2011; Accepted 2 November 2011.Changyong Cao would like to express his thanks to Dr Hui Wang at Henan University of

Technology for his helpful discussion.Address correspondence to Qing-Hua Qin, Department of Engineering, Australian National

University, Canberra ACT 0200, Australia; E-mail: [email protected]

849

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850 C. CAO ET AL.

Consequently, most of the analyses employ the uncoupled thermoelasticity theory,which is adopted in this work.

Currently, numerical methods such as the finite element method (FEM) havebeen widely employed to study thermoelasticity problems [2–6]. Despite manyattractive features of FEM, it is still a non-trivial and time consuming taskfor problems with complicated domains, particularly for three-dimensional (3D)problems. One alternative to alleviate this difficulty is to use the boundary elementmethod (BEM), which requires only boundary discretization rather than the domaindiscretization [7–9]. However, the treatment of singular or near-singular boundaryintegrals is usually quite tedious and inefficient, and an extra boundary integralequation is also required to evaluate the interior fields inside the domain. Inaddition, for multi-domain problems, the implementation of the BEM becomesquite complicated and the nonsymmetrical coefficient matrix of the resultingequations may weaken the advantages of BEM [10–13]. Recently, the combinationof fundamental solution (MFS) and dual reciprocity method (DRM) was alsoutilized to solve two-dimensional thermoelasticity with general body forces [14] andthree-dimensional thermoelasticity [15].

On the other hand, hybrid Trefftz FEM (HT-FEM) as an importantalternative to traditional FEM, has become popular over the past three decadesand has been increasingly used to analyze various engineering problems [16–24].In contrast to conventional FEM, HT-FEM is based on a hybrid method, whichincludes the use of an independent auxiliary inter-element frame field defined onelement boundary and an independent internal field chosen as a prior satisfyingthe homogeneous governing differential equations by means of a suitable truncatedT-complete function set of homogeneous solutions. The property of nonsingularelement boundary integral in HT-FEM, enables us to easily construct an arbitrarilyshaped element. However, the terms of truncated T-complete functions should becarefully selected when achieving desired results, and the T-complete functions forsome physical problems are difficult to develop [12, 25]. To remove the drawback ofHT-FEM, a novel hybrid boundary integral-based FEM (HBI-FEM) was developedfor solving two dimensional heat conduction problems in single and multilayer-materials [26, 27] and isotropic and orthotropic elastic problems [28–30]. In thisapproach, the intra-element field is approximated by a linear combination of thefundamental solutions, analytically satisfying the related governing equations. Theindependent frame field defined along the element boundary and the correspondingvariational functional are employed to guarantee the inter-element continuity,generate the stiffness equation and establish the linkage between the boundaryframe field and internal field at the element level. In the HBI-FEM, the domainintegrals in the hybrid functional can be directly converted into boundary integralswithout any appreciable increase in computational effort, and no singular integralsare involved by locating the source point outside the element of interest and do notoverlap with field point during the computation [28]. Moreover, the features of twoindependent interpolation fields and element boundary integral in the HBI-FEMmake the algorithm have potential applications in the aspect of mesh reductionby constructing special-purpose elements such as functionally graded element, holeelement, crack element, and so on [23, 31, 32].

In this article, a new solution procedure based on the HBI-FEM is proposedto solve 2D and 3D thermoelastic problems with arbitrary body forces and

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NOVEL FINITE ELEMENTS METHOD FOR THERMO-ELASTICITY PROBLEMS 851

temperature changes. The method of particular solution is used to decomposethe displacement solution into two parts: homogeneous solution and particularsolution. The homogeneous solution is obtained by using the HBI-FEM withelastic fundamental solutions, yet the particular solution associated with the bodyforce and temperature effects is approximated by using the radial basis functioninterpolation. Five numerical examples are presented to demonstrate the accuracyand versatility of the proposed method. Compared with the existing closed-formsolutions or ABAQUS results, it is shown that even with a very coarse mesh,relatively accurate results can still be obtained by the proposed method.

The article is organized as follows; first, we review the basic equations ofthe thermoelasticity, then describe the methods of particular solution and radialbasis function approximation which are employed to deal with the temperaturechange and the body force. Two different approaches are presented to treat thisproblem, followed by the detailed derivation of the HBI-FEM formulations. Fivenumerical examples are presented to demonstrate the validity and performance ofthe approach, and conclusions are presented.

BASIC EQUATIONS FOR THERMOELASTICITY

Consider a finite isotropic material in domain � (see Figure 1), and let (x1,x2, x3) denote the coordinates in Cartesian coordinate system. The equilibriumgoverning equation of the thermoelasticity with the body force bi is expressed as

�ij�j = −bi (1)

where �ij is the stress tensor, bi the body force vector and i� j = 1� 2� 3. Thegeneralized thermoelastic stress-strain relations and the generalized kinematicalrelation are given as

�ij�j =2G�

1− 2��ije+ 2Geij −m�ijT (2)

eij =12�ui�j + uj�i� (3)

Figure 1 Geometrical definitions and boundary conditions for general 3D problems. (color figureavailable online.)

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852 C. CAO ET AL.

in which eij is the strain tensor, ui the displacement vector, T the temperaturechange, G the shear modulus, � the Poisson’s ratio, �ij the Kronecker delta, and

m = 2G��1+ v�/�1− 2v� (4)

is the thermal constant with � being the coefficient of linear thermal expansion.Substituting Eqs. (2) and (3) into Eq. (1), the equilibrium equations may be rewrittenin terms of displacements as

Gui�jj +G

1− 2�uj�ji = mT�i − bi (5)

For a well-posed boundary value problem, the following boundary conditions,either displacement or traction boundary condition, are given as

ui = ui on u� (6)

ti = ti on t� (7)

where u ∪ t = is the boundary of the solution domain �, ui and ti are theprescribed boundary values, and

ti = �ijnj� (8)

is the boundary traction, in which nj denotes the boundary outward normal.

THE METHOD OF PARTICULAR SOLUTION

It is important to evaluate the inhomogeneous term, mT�i − bi of Eq. (5).A widely used approach is to evaluate it by employing the method of particularsolution [1, 20, 28, 33]. Using superposition principle, we can decompose thedisplacement ui into two major parts, the homogeneous solution uh

i and theparticular solution u

pi as follows

ui = uhi + u

pi (9)

in which the particular solution upi satisfies the governing equation

Gupi�jj +

G

1− 2�upj�ji = mT�i − bi (10)

but does not necessarily satisfy any boundary conditions. It should be pointed outthat its solution is not unique and can be obtained by various numerical techniques.However, the homogeneous solution should satisfy

Guhi�jj +

G

1− 2�uhj�ji = 0 (11)

with modified boundary conditions

uhi = ui − u

pi � on u (12)

thi = ti +mTni − tpi � on t (13)

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It can be seen in Eqs. (12) and (13) that once the particular solution upi is

known, the homogeneous solution uhi defined in Eqs. (11)–(13) can be determined.

In the following two sections, radial basis function approximation is introducedto obtain the particular solution, and the corresponding HBI-FEM is derived forsolving the boundary-value problem defined by Eqs. (11)–(13).

RADIAL BASIS FUNCTION APPROXIMATION

It is usually impossible to find an analytical solution for Eq. (10) except forsome special cases. As a consequence, Radial Basis Functions (RBF) [34, 35] will beused in our analysis to approximate the body force bi and the temperature field T inorder to obtain a particular solution. There are two different ways to implement thisapproximation: one is to treat body force bi and the temperature field T separatelyas done by Tsai [15]. The other is to treat mT�i − bi as a whole. It is found thatthe performance of the two approaches is different from one another which will bediscussed in the numerical examples presented later.

Approach 1

The body force bi and temperature T are assumed in the following forms

bi =N∑j=1

�ji

j�i = 1� 2in �2 and i = 1� 2� 3 in �3� (14)

and

T =N∑j=1

�jj (15)

where N is the number of interpolation points, �2 and �3 denote 2D and 3D realspace, respectively, j are the basis functions, and �

ji and �j are the coefficients to be

determined by collocation. Subsequently, the approximate particular solution canbe written as follows

upi =

3∑k=1

N∑j=1

�ji�

jik +

N∑j=1

�j ji (16)

where �jik and

ji is the approximated particular solution kernels. Once the RBF

is selected, the problem of finding a particular solution will be reduced to solve thefollowing equations

G�il�kk +G

1− 2��kl�ki = −�il (17)

G i�kk +G

1− 2� k�ki = m�i (18)

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854 C. CAO ET AL.

To solve Eq. (17), the displacement is expressed in terms of the Galerkin-Papkovich vectors as [34, 36]

�ik =1− �

GFik�mm − 1

2GFmk�mi (19)

Substituting Eq. (19) into Eq. (17), yields following bi-harmonic equation

�4Fil = − 11− �

�il (20)

If the Spline Type RBF = r2n−1 is used, we can obtain following solutions

Fli = − �li1− v

r2n+3

�2n+ 1�2�2n+ 3�2��2� for n = 1� 2� 3 � � � (21)

�li = A0�A1�li + A2r�ir�l� ��2� for n = 1� 2� 3 � � � (22)

A0 = − 12G�1− v�

r2n+1

�2n+ 1�2�2n+ 3�

A1 = 5+ 4n− 2v�2n+ 3� (23)

A2 = −�2n+ 1�

for a two-dimensional problem and

Fli = − �li1− v

r2n+3

�2n+ 1��2n+ 2��2n+ 3��2n+ 4��3� for n = 1� 2� 3 � � � (24)

�li = A0�A1�li + A2r�ir�l� ��3� for n = 1� 2� 3 � � � (25)

A0 = − 18G�1− v�

r2n+1

�n+ 1��n+ 2��2n+ 1�

A1 = 7+ 4n− 4v�n+ 2� (26)

A2 = −�2n+ 1�

for a three-dimensional problem, where rj represents the Euclidean distance betweena field point (x, y� and a given point (xj , yj) in the solution domain. Thecorresponding particular stress solution can be obtained as

Slij = G��li�j +�lj�i�+ ��ij�lk�k (27)

where � = 2v1−2vG. Substituting Eq. (25) into (27), we obtain

Slij = B0

{B1�r�j�li + r�i�jl�+ B2�ijr�l + B3r�ir�jr�l

}��2� for n = 1� 2� 3 � � �

B0 = − 1�1− v�

r2n

�2n+ 1��2n+ 3�

B1 = �2n+ 2�− v�2n+ 3� (28)

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B2 = v�2n+ 3�− 1

B3 = 1− 2n

for a two-dimensional problem, one can obtain

Slij = B0

{B1

(r�j�li + r�i�jl

)+ B2�ijr�l + B3r�ir�jr�l}

��3� for n = 1� 2� 3 � � �

B0 = − 14 �1− v�

r2n

�n+ 1� �n+ 2�

B1 = 3+ 2n− 2v�n+ 2� (29)

B2 = 2v�n+ 2�− 1

B3 = 1− 2n

for a the three-dimensional problem.To solve Eq. (18), a function i is introduced as the gradient of a scalar

function

i = U�i (30)

Substituting Eq. (30) into Eq. (18) yields Poisson’s equation

�2U = m�1− 2��2G�1− ��

(31)

Thus, assuming = r2n−1, its particular solution can be obtained as

U = m�1− 2��2G�1− ��

r2n+1

�2n+ 1�2��2� for n = 1� 2� 3 � � � (32)

U = m�1− 2��2G�1− ��

r2n+1

�2n+ 1��2n+ 2��3� for n = 1� 2� 3 � � � (33)

Then from Eq. (30) i is obtained as [36]

i =m�1− 2��2G�1− ��

r�ir2n

2n+ 1��2� for n = 1� 2� 3 � � � (34)

i =m�1− 2��2G�1− ��

r�ir2n

�2n+ 2��3� for n = 1� 2� 3 � � � (35)

The corresponding particular stress solution can be obtained as

Sij = G� i�j + j�i�+ ��ij k�k (36)

Then we have

Sij =mr2n−1

�1− v� �2n+ 1�

{�1+ 2nv��ij + �1− 2v��2n− 1�r�ir�j

}��2� for n = 1� 2� 3 � � � (37)

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856 C. CAO ET AL.

Sij =mr2n−1

�1− v� �2n+ 2�

{�1− 2��2n− 1�r�ir�j + �ij�1+ 2nv�

}��3� for n = 1� 2� 3 � � � (38)

Approach 2

Considering the temperature gradient as a part of the body force, mT�i − bican be evaluated by the following equation

mT�i − bi =N∑j=1

�ji

j (39)

Thus, the particular solution can be written as

upi =

3∑k=1

N∑j=1

�ji�

jik (40)

Consequently, the follow-up treatment is the same as that in Approach 1, i.e.,making use of Eq. (17) and Eqs. (19)–(28) to obtain the desired �

jik and Slij , which

are the same as that of body force bi only. Once the particular solutions of Eq. (9)are established, they can be used to obtain the modified boundary conditions inEq. (12) to solve the homogeneous solution of Eq. (11). In the following section,we will employ the proposed HBI-FEM to determine the homogeneous part of thesolution.

FORMULATIONS OF THE HBI-FEM

Assumed Fields for 2D Problems

In the HBI-FEM approach, two different assumed fields are employed:intra-element field and frame field. The intra-element continuity is enforced on anonconforming internal displacement field chosen to be the fundamental solutionof the problem [28]. In this approach, the intra-element displacement field isapproximated in terms of a linear combination of fundamental solutions of theproblem as

u�x� ={u1�x�u2�x�

}=

ns∑j=1

[u∗11�x� ysj� u∗

12�x� ysj�u∗21�x� ysj� u∗

22�x� ysj�

]{c1jc2j

}= Nece �x ∈ �e� ysj � �e�

(41)

where ns is the number of source points outside the element domain, which is equalto the number of nodes of an element in the present work, ce an unknown coefficientvector (not nodal displacements), Ne the fundamental solution matrix, which iswritten as

Ne =[u∗11�x� ys1� u∗

12�x� ys1� · · · u∗11�x� ysns � u∗

12�x� ysns �u∗21�x� ys1� u∗

22�x� ys1� · · · u∗21�x� ysns � u∗

22�x� ysns �

](42)

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ce = � c11 c21 · · · · · · c1n c2n �T (43)

where x and ysj are, respectively, the field point and source point in the localcoordinate system (X1, X2�. The component u∗

ij�x� ysj� is the fundamental solution,i.e., induced displacement component in i-direction at the field point x due to a unitpoint load applied in j-direction at the source point ysj , which are given by [13]

u∗ij�x� ysj� =

−18��1− ��G

{�3− 4��ij ln r − r�ir�j

}(44)

where ri = xi − xis,r =√r21 + r22 .

In our analysis, the number of source points is seen to be the same as thenumber of element nodes, which is free of spurious energy modes and keep thematrix H (see Eq. (73)) in full rank, as indicated in [12]. The source point ysj�j =1� 2� � � � � ns�can be generated by means of the following procedure [28]

ys = x0 + ��x0 − xc� (45)

where � is a dimensionless coefficient, x0 is the related nodal point on theelement boundary and xc the geometrical centroid of the element (see Figure 2).Determination of � was already discussed in [27, 28] and � = 5 is used in thefollowing analysis.

Making use of the assumption of intra-element field (41), the correspondingstress fields can be obtained by the constitutive Eq. (1) as

��x� = [�11 �22 �12

]T = Tece (46)

Figure 2 Intra-element field and frame field of an HBI-FEM element for 2D thermoelastic problems.(color figure available online.)

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858 C. CAO ET AL.

where

Te =�∗

111�x� y1� �∗211�x� y1� · · · · · · �∗

111�x� yns � �∗211�x� yns �

�∗122�x� y1� �∗

222�x� y1� · · · · · · �∗122�x� yns � �∗

222�x� yns ��∗112�x� y1� �∗

212�x� y1� · · · · · · �∗112�x� yns � �∗

212�x� yns �

(47)

As a consequence, the traction is written as{t1t2

}= n� = Qece (48)

in which

Qe = nTe� n =[n1 0 n2

0 n2 n1

](49)

The components �∗ijk�x� y� for plain strain problems is given as

�∗ijk�x� y� =

−14��1− ��r

[�1− 2��r�k�ij + r�j�ki − r�i�jk�+ 2r�ir�jr�k

](50)

The unknown ce in Eq. (41) can be calculated using a hybrid technique [27], inwhich the elements are linked through an auxiliary conforming displacement framewhich has the same form as in conventional FEM (see Figure 3). This means thatin the HBI-FEM, a conforming displacement field should be independently definedon the element boundary to enforce the field continuity between elements and alsoto establish a relationship between the unknown c and the nodal displacement de.Thus, the frame field is defined as

u�x� ={u1

u2

}={N1

N2

}de = Nede� �x ∈ e� (51)

Figure 3 Typical quadratic interpolation for the frame fields. (color figure available online.)

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where the symbol “∼” is used to specify that the field is defined on the elementboundary only, Ne is the matrix of shape functions, de is the nodal displacementsof the element under consideration. Taking the side 3-4-5 of a particular 8-nodequadrilateral element (see Figure 2) as an example, Ne and de can be expressed as

Ne =0 · · · 0 N1 0 N2 0 N3 0 0 · · · 00 · · · 0︸︷︷︸

4

0 N1 0 N2 0 N3 0 · · · 0︸︷︷︸6

(52)

de =[u11 u21 u12 u22 · · · u18 u28

]T(53)

where N1, N2 and N3 are expressed by natural coordinate as

N1 = −� �1− ��

2� N2 = 1− �2� N3 =

� �1+ ��

2�� ∈ �−1� 1�� (54)

Assumed Fields for 3D Problems

In the case of a three-dimensional problem, the intra-element displacementfields are approximated by

u�x� =u1�x�u2�x�u3�x�

= Nece �x ∈ �e� ysj � �e� (55)

where the matrix Ne and vector ce for 3D elements can be written as

Ne =u∗

11�x� ys1� u∗12�x� ys1� u∗

13�x� ys1� · · ·u∗12�x� ys1� u∗

22�x� ys1� u∗23�x� ys1� · · ·

u∗13�x� ys1� u∗

32�x� ys1� u∗33�x� ys1� · · ·

· · · u∗11�x� ysns � u∗

12�x� ysns � u∗13�x� ysns �· · · u∗

12�x� ysns � u∗22�x� ysns � u∗

23�x� ysns �· · · u∗13�x� ysns � u∗

32�x� ysns � u∗33�x� ysns �

(56)

ce =[c11 c21 c31 · · · · · · c1n c2n c3n

]T(57)

in which the fundamental solution components u∗ij�x� ysj� are given by

u∗ij�x� ysj� =

116��1− ��Gr

{�3− 4��ij + r�ir�j

}(58)

where ri = xi − xis, r =√r21 + r22 + r23 . In 3D analysis, the number of source points

is also assumed to be the same as the number of element nodes as was done in the2D case.

The corresponding stress fields in 3D coordinate system can be expressed as

��x� = [�11 �22 �33 �23 �31 �12

]T = Tece (59)

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860 C. CAO ET AL.

Figure 4 Intra-element field and frame field of a hexahedron HBI-FEM element for 3D thermoelasticproblem (The source points and centroid of 20-node element are omitted in the figure for clarity andclear view, which is similar to that of the 8-node element). (color figure available online.)

where

Te =

�∗111�x� y1� �∗

211�x� y1� �∗311�x� y1� · · ·

�∗122�x� y1� �∗

222�x� y1� �∗322�x� y1� · · ·

�∗133�x� y1� �∗

233�x� y1� �∗333�x� y1� · · ·

�∗123�x� y1� �∗

223�x� y1� �∗323�x� y1� · · ·

�∗131�x� y1� �∗

231�x� y1� �∗331�x� y1� · · ·

�∗112�x� y1� �∗

212�x� y1� �∗312�x� y1� · · ·

�∗111�x� yns � �∗

211�x� yns � �∗311�x� yns �

�∗122�x� yns � �∗

222�x� yns � �∗322�x� yns �

�∗133�x� yns � �∗

233�x� yns � �∗333�x� yns �

�∗123�x� yns � �∗

223�x� yns � �∗323�x� yns �

�∗131�x� yns � �∗

231�x� yns � �∗331�x� yns �

�∗112�x� yns � �∗

212�x� yns � �∗312�x� yns �

(60)

in which the components �∗ijk�x� y� is given as

�∗ijk�x� y� =

−18��1− ��r2

{�1− 2��r�k�ij + r�j�ki − r�i�jk�+ 3r�ir�jr�k

}(61)

The corresponding traction is expressed ast1t2t3

= n� = Qece (62)

where

Qe = nTe� n =n1 0 0 0 n3 n2

0 n2 0 n3 0 n1

0 0 n3 n2 n1 0

(63)

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The frame field of the element is defined as

u�x� =u1

u2

u3

=

N1

N2

N3

de = Nede� �x ∈ e� (64)

For the face 1-2-3-10-15-14-13-9 of a particular 20-node 3D brick element(Figure 3), Ne and de can be written as

Ne =[�N1

�N2�N3 0 · · · 0︸︷︷︸

5

�N8�N4 0 · · · 0︸︷︷︸

2

�N7�N6

�N5 0 · · · 0︸︷︷︸5

](65)

de =[u11 u21 u31 u12 u22 u32 · · · u120 u220 u320

]T(66)

where the shape functions are defined as

�Ni =Ni 0 00 Ni 00 0 Ni

� 0 =0 0 00 0 00 0 0

(67)

and

Ni =14�1+ �i�� 1+ �i�� i�+ �i�− 1� �i = 1� 3� 5� 7�

Ni =12

(1− �2

)�1+ �i�� i = 2� 6� (68)

Ni =12

(1− �2

)�1+ �i�� i = 4� 8�� ∈ �−1� 1�

where ��i� �i� is the natural coordinate of the i-node of the element (Fig. 5).

Figure 5 Typical quadratic interpolation for the frame fields. (color figure available online.)

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862 C. CAO ET AL.

Modified Functional for Hybrid Finite Element Method

The HBI-FEM formulations for 3D thermoelastic problem can be establishedby the variational approach [28]. In the absence of body forces, the variationalfunctional �meused for deriving the present HBI-FEM can be constructed as [25]

�me =12

∫∫�e

�ij�ijd�−∫t

tiuid +∫e

ti�ui − ui�d (69)

where ui and ui are the intra-element displacement field defined within the elementand the frame displacement field defined on the element boundary, respectively.�e and e are the element domain and element boundary, respectively. t, u andI stands respectively for the specified traction boundary, specified displacementboundary and inter-element boundary (e = t + u + I�.

Compared to the functional employed in the conventional FEM, the presenthybrid variational functional is constructed by adding a hybrid integral term relatedto the intra-element and frame displacement fields to guarantee the satisfaction ofdisplacement and traction continuity conditions on the common boundary of twoadjacent elements. By applying the Gaussian theorem, Eq. (69) can be simplified as

�me =12

∫∫�e

�ij�ijd�−∫t

tiuid +∫e

ti�ui − ui�d

= 12�∫e

tiuid −∫∫

�e

�ij�juid��−∫t

tiuid +∫e

ti�ui − ui�d (70)

Considering the equilibrium equation with the constructed intra-element fields,Eq (70) can be simplified as

�me =12

∫e

tiuid −∫t

tiuid +∫e

ti�ui − ui�d

= −12

∫e

tiuid +∫e

tiuid −∫t

tiuid (71)

The functional (71) contains boundary integrals only and will be used to derive HBI-FE formulations for 2D and 3D thermoelastic problems.

Element Stiffness Matrix

The element stiffness equation can be generated by setting ��me = 0.Substitute Eqs. (55), (62) and (64) into the functional (71), we have

�me = −12cTeHece + cTeGede − dTe ge (72)

where

He =∫e

QTeNed� Ge =

∫e

QTe Ned� ge =

∫t

NTe td (73)

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To enforce inter-element continuity on the common element boundary, theunknown vectorce should be expressed in terms of nodal DOF de. The stationarycondition of the functional �me with respect to ce and de yields, respectively,

��me

�cTe= −Hece +Gede = 0 (74)

��me

�dTe= GT

e ce − ge = 0 (75)

Therefore, the relationship between ce and de, and the stiffness equation can beobtained as follows

ce = H−1e Gede (76)

Kede = ge (77)

where Ke = GTeH

−1e Ge is the element stiffness matrix. Consequently, the

homogeneous solution can be obtained by Eq. (41) for 2D problems and Eq. (55)for 3D problems.

Recovery of Rigid-body Motion Terms

It is noted that when the procedure described above is used to obtain ue, therigid body motion modes are not included so as to prevent the singularity of theinversion of matrix He. Therefore, it is necessary to reintroduce the discarded rigid-body motion terms after we have obtained the internal field of an element. The leastsquares method can be employed for this purpose and the missing terms can beeasily recovered by setting for the augmented internal field [30]

ue = Nece +[1 0 x20 1 −x1

]c0 �for �2� (78)

ue = Nece +1 0 0 0 x3 −x20 1 0 −x3 0 x10 0 1 x2 −x1 0

c0 �for �3� (79)

The detailed formulations can be found in some previous papers [28, 30].

NUMERICAL RESULTS AND DISCUSSION

In this section, we presented five different numerical examples to assessthe performance of the proposed methodology, in which 2D and 3D elastic andthermoelastic problems with body force and temperature change are included. Thefirst three examples are given to investigate the capability of the method to treatthe 2D problems with the temperature change, body forces, or the combinationof both. The latter two examples are given to show its ability to deal with 3Dthermoelasticity with arbitrary body force and temperature. The analytical resultsor numerical results from ABAQUS with fine meshes are presented to assess theaccuracy of the method.

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864 C. CAO ET AL.

Example 1. Circular cylinder with axisymmetric temperature change: In thisexample, a long circular cylinder with axisymmetric temperature change in thedomain is considered to show the performance of the proposed method. Bothinside and outside surfaces of the cylinder are assumed to be free from traction.The temperature T changes logarithmically along the radial direction. With thesymmetry condition of the problem, only one quarter of the cylinder is modeled. Thegeometrical configurations and the boundary conditions of the cylinder are shownin Figure 6. Under the assumption of plane strain, the analytical solutions for stresscomponents without body forces are given as [33]

�r = − E�T0

2�1− v�

[ln�b/r�ln �b/a�

− �b/r�2 − 1

�b/a�2 − 1

]

�� = − E�T0

2�1− v�

[ln�b/r�− 1ln �b/a�

+ �b/r�2 + 1

�b/a�2 − 1

]

T = ln�r/b�ln �a/b�

T0

which is taken as a reference. In our computation, the parameters a = 5, b = 20,E = 1000, � = 0�3, � = 0�001, and T0 = 10 are employed. As shown in Figure 7, twodifferent meshes with 168-node elements and 128-node elements are, respectively,employed to show the influence of the mesh density. The two approaches listedpreviously for approximating the body force and temperature are discussed andanalyzed in this example.

Figures 8 and 9 present the variation of the radial and the circumferentialthermal stresses with the cylinder radius for both coarse mesh and fine mesh,

Figure 6 Geometry and boundary conditions of a quarter of long cylinder with axisymmetrictemperature change. (color figure available online.)

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Figure 7 Mesh configurations of a quarter of the circular cylinder: Coarse Mesh (Left: 16 elements)and Fine Mesh (Right: 128 elements). (color figure available online.)

respectively, in which the theoretical values are given for comparison. It is seenfrom these two figures that the results from Method 2 are much better than thoseobtained from Method 1 for both coarse mesh and fine mesh. When using coarsemesh with Approach 2, the radial stress close to the outer surface of the cylindershows very large errors, yet when refining the mesh the error decreases dramaticallyand the results agree well with the analytical findings. It can be inferred that theerror may be to a large extent due to the RBF interpolation, for which the numberof interpolation points has a significant influence on its accuracy. However, it is alsofound that there is no significant improvement observed for Method 1 when refiningthe mesh to 128 individual 8-node elements.

The circumferential thermal stresses obtained from coarse mesh (left) and finemesh (right) in Figure 7 using Approach 1 show a small error compared to the radialstresses. However, these errors are still larger than those in Approach 2, which canbe seen clearly from Figure 9. Figure 10 displays the contour plots of (a) radial and(b) circumferential thermal stresses (The meshes used for contour plot is differentfrom that employed above due to the use of quadratic elements). It demonstratesthat treating temperature gradient and body force together is better than dealingwith them separately. Consequently, in the latter examples Approach 2 will be

Figure 8 Variation of the radial thermal stresses with the cylinder radius using coarse mesh (a) andfine mesh (b). (color figure available online.)

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866 C. CAO ET AL.

Figure 9 Variation of the circumferential thermal stresses with the cylinder radius using (a) coarsemesh and (b) fine mesh. (color figure available online.)

employed in our analysis. It should be pointed out that Approach 1 applies whenthe temperature change is in discrete distribution or the gradient of the temperaturefield is not available.

Example 2. Long beam under gravity: In the second example, we consider along beam with rectangular cross-section subjected to gravity. The geometry of therectangular cross-section and the corresponding boundary conditions are shown inFigure 11, where g denotes the gravity accelerator, and � is the density of material.The problem can be viewed as a plane strain problem, and the analytical solutionsof displacements and stresses are given by

u1 = 0� u2 = − �1+ v��1− 2v��g2E�1− v�

[L2 − �x2 − L�2

]�11 =

�gv

1− v�x2 − L�� 22 = �g�x2 − L�� 12 = 0

Figure 10 Contour plots of (a) radial and (b) circumferential thermal stresses. (color figure availableonline.)

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Figure 11 Long square cross-section beam under gravity. (color figure available online.)

Let L = 20, E = 1000, � = 0�25, � = 0�001 and �g = 9�8, and four 8-node quadraticelements are used to model the entire square cross-section domain (see Figure 12).

The numerical results for the distribution of nodal displacement componentu2 along x1 = 10 are shown in Figure 13, which shows that the numerical resultsagree with the analytical solutions. Additionally, the stress results at element nodesalong x1 = 10 are also compared with the exact solutions in Figure 13, where it isobvious that the numerical results obtained from HBI-FEM with RBF interpolationhave good accuracy compared with the analytical results. Figure 14 gives the stresscontour plots of the cross-section beam under gravity. It can be seen that the stresseschange linearly with x2 coordinates and independent from the x1 coordinate, whichis consistent with the analytical solution.

Example 3. T-shape domain with body force and temperature change: In thisexample, the proposed numerical method is used to model a T-shape domainwith both the temperature change and body force. The boundary condition and

Figure 12 Mesh configuration of square cross-section beam.

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868 C. CAO ET AL.

Figure 13 Displacement and stresses along x1 = 10 of square cross-section of the beam.

temperature distribution are shown in Figure 15. The material properties areE = 30000, � = 0�15, � = 0�001, � = 2�3 and g = 9�8, respectively.

In this example, 22 separate 8-node quadrilateral elements are utilized tomodel the T-shape domain by the HBI-FEM. Because there is no analytical

Figure 14 Contour plots of the stresses of square cross-section beam. (color figure available online.)

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Figure 15 Dimension and boundary condition of the T-Shape domain.

solution for comparison, the results from ABAQUS using 352 CPE8R elementsare employed as reference. Figure 17 shows the thermal stresses �11 along the linex2 = 14 and �22 along line x1 = 0. It is obvious that the results from the HBI-FEMby coarse mesh shown in Figure 16 agree with the results from ABAQUS with finemesh (see Fig. 16). It should be mentioned that in order to obtain the meaningfulresults at the two sharp corners of the T-shape domain, a refined mesh is necessary,due to the stress concentration at the points near the corner, which is also appliedto the traditional FEM (ABAQUS). The contour plots of both the displacementcomponents and stresses components are presented in Figure 18.

Figure 16 Mesh configuration of T-shaped domain: (a) HBI-FEM (22 elements); (b) ABAQUS (352elements).

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870 C. CAO ET AL.

Figure 17 The variation of the thermal stresses of T-Shape domain: (a) �11 along line x2 = 14 and (b)�22 along line x1 = 0.

Figure 18 Contour plots of the displacement and stresses of T-shaped domain under temperaturechange and body force. (color figure available online.)

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Figure 19 Schematic of the cube subjected temperature change and body force.

Example 4. A 3D cube under arbitrary temperature distribution and body force:In this example, a three-dimensional cube of 1× 1× 1 with center located at (0.5,0.5, 0.5) is considered, which is shown in Figure 19. The material properties of thecube are: Young’s modulus E = 5000, Poisson’s ratio v = 0�3, and linear thermalexpansion coefficient � = 0�001. The bottom surface is fixed on the ground. Thetemperature distribution and body force are assumed to be

T = 30x3� b1 = 0� b2 = 0� b3 = −2000[�x − 0�5�2 + �y − 0�5�2

]Because there is no analytical solution available, the results from ABAQUS hereinare employed for comparison. The meshes used by the HBI-FEM and ABAQUSare given in Figure 20, in which the coarse mesh consists of 125 individual 20-nodebrick elements and the fine mesh for ABAQUS is 8000 C3D20R elements.

Figure 21 presents the displacement u3 and stress �33 along an edge of the cubewhich coincides with x3 axis. It can be seen that the results from the HBI-FEMagain agree with those by ABAQUS. It is demonstrated that the proposed procedurebased on the HBI-FEM predicts the response of 3D thermoelastic problems underarbitrary temperature and body force. It is also shown that the HBI-FEM with RBFinterpolation gives satisfactory results using very coarse meshes.

Example 5. A heated hollow ball: Finally, we consider a heated hollow ball todemonstrate the capability of the method to solve 3D thermal stress problems withcomplex geometry. As shown in Figure 22, the radius of inner hole is a and theradius of outer surface of the ball is b. The temperature distribution is given by

T = T0�2− 10/r�

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872 C. CAO ET AL.

Figure 20 Mesh configurations of the 3D cube in Example 4: (a) HBI-FEM (125 elements);(b) ABAQUS (8000 element).

Figure 21 Displacement u3 and stress �33 along one cube edge when subjected to specified temperatureand body force.

Figure 22 Schematic and dimension of the heated hollow ball. (color figure available online.)

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Figure 23 Mesh configurations of the heated hollow ball: (a) HBI-FEM (864 elements); (b) ABAQUS(11088 element).

In the calculation, a = 5, b = 10, T0 = 20 are assumed. Considering thesymmetry property of the problem, only one-eighth of the ball is modeled. The meshused for the HBI-FEM is given in Figure 23, in which a total of 864 20-node brickelements are employed. As a reference, the results from ABAQUS are calculatedusing a much finer mesh with 11088 C3D20R elements (see Figure 23).

Figure 24 shows the radial displacement ur and von Mises stress of the hollowball along its radial direction. The radial displacement increases with the radiusfrom 0.071 to 0.143 with a slightly reduction when 5 < r < 6�3. It can be seenfrom Figure 24 that the inner surface of the hollow ball suffers maximum VonMises stress of up to 20 and this value is dramatically reduced to about 0.15 atr = 6�5, then it experiences a moderate increasing to about 8.2 at the outer surface.It is obvious from Figure 24 that the results obtained by the HBI-FEM display agood agreement with the results from ABAQUS. This again demonstrates a goodperformance of the proposed model in predicting the themoelastic response of 3Dproblems with complex geometry.

Figure 24 Radial displacement and Von Mises stress along radius of the heated hollow ball subjectedto temperature change.

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874 C. CAO ET AL.

CONCLUSIONS

In this article a new solution procedure based on the HBI-FEM is proposed tosolve two-dimensional and three-dimensional thermoelastic problems with arbitrarybody forces and temperature changes. The body force and temperature changeare handled by the method of particular solution, in which the homogeneoussolution is obtained by using the HBI-FEM with elastic fundamental solutions,while the particular solution is approximated by the radial basis function. It isfound that treating body force and temperature change as a whole is superior toapproximating them separately. The five numerical examples presented in this articleshow that the proposed method can predict the thermoelastic response of 2D and3D thermoelasticity problems with complex geometry, arbitrary body force andarbitrary temperature changes. It is a promising methodology for mesh reduction,which is capable of obtaining satisfying results with much coarser meshes than thetraditional FEM. It is also possible to improve the results by only increasing theinterpolation points while keeping the meshes at a lower density and the study isunderway.

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a novel boundary-integral based finite element method for...

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