a new error estimation approach based on fuzzy logic system for h-r adaptive boundary element method

Applied Mathematics and Computation 218 (2011) 2167–2177

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

A new error estimation approach based on fuzzy logic system for H-Radaptive boundary element method

Yiming Chen ⇑, Zhiquan Zhou, Jing Zhang, Yulian Li, Xiaojuan WangCollege of Sciences, Yanshan University, Qinhuangdao, Hebei 066004, China

a r t i c l e i n f o

Keywords:Fuzzy logic systemH-R adaptive boundary elementError estimationAdaptive mesh refinementElasticity problems

0096-3003/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.amc.2011.07.033

⇑ Corresponding author.E-mail address: [email protected] (Y. Chen).

a b s t r a c t

Conventional adaptive boundary element method cannot be universally applied to solvemany more problems than the subject it discussed, and different error estimation formulasneed to be designed for varied problems. This paper put forward a new error analysismethod based on the fuzzy logic system, which is able to make error estimation effectivelyusing human expert experience, and solve the two classical elasticity problems in conjunc-tion with the H-R adaptive boundary element method. Numerical examples have illus-trated the effectiveness, superiority and potential of a fuzzy logic approach in theadaptive boundary element method.

� 2011 Elsevier Inc. All rights reserved.

1. Introduction

Boundary element method is an alternate method for obtaining approximate solutions of partial differential equations inthe computational mechanics and computational mathematics, so it is as powerful a tool as the finite element method. Yetthe boundary element method is easier in that its discrete grid relatively reduces one dimension compared to the finite ele-ment method [1,2]. In order to improve efficiency and prevent subjective errors, the so-called adaptive boundary elementmethod (ABEM) is developed [1]; the basic idea of ABEM is to use computer to automatically judge the accuracy of theboundary element solution and then decide whether to improve it or not. In this process, the error estimation of the solutionis the most important part, which is not only used to determine the accuracy of the solution, but also is used to drive theadaptive subdivision process of the discrete grid [3,4], so it can greatly meet the need of engineering practice. Therefore,the method of error estimation for boundary element solution has been widely studied and become a research focus in thisfield [4–11]. However, the error estimation formulas of previous studies are all designed to target towards certain types ofproblems, and therefore cannot be universally applied and promoted.

This paper presents a new error estimation method which is based on fuzzy logic system (FLS). Fuzzy logic theory is orig-inated in the field of artificial intelligence, which can effectively use expert experience, and has strong universality. At pres-ent, fuzzy logic systems have been successfully applied in control systems [12–14], system identification [15] and powersystems [16]. Fuzzy systems also have been successfully applied in the finite element method to solve nonlinear magneticproblems and have obtained better results than conventional methods [17–19].

Ref. [8] is a representative sample of conventional error estimation methods because its author introduces a new but tra-ditional adaptive error analysis method in its adaptive process. Compared to the approach proposed by Ref. [8] better studyresults are obtained by the error estimation method with the application of fuzzy logic in this paper. In addition, ten fuzzylogic rules are summarized based on the results of the given numerical experiment, and an error estimation formula isgenerated [14]. There are five parameters in the formula, which in actual application, can be used selectively according to

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2168 Y. Chen et al. / Applied Mathematics and Computation 218 (2011) 2167–2177

different problems, and thus generates solutions that meet different accuracy requirements. In order to verify the effective-ness of the proposed error estimation method, we compile a computer program with FORTRAN in which the H-R adaptiveboundary element is used to solve two classical elasticity problems. Examples illustrate that the proposed method is feasible,effective and universal.

2. The boundary element formulations

The boundary integral equation for the two-dimensional linear elastic problem is given in any standard text books[1,10,11]. Neglecting the body force, in an isotropic and homogeneous elastic solid, the relation between the displacementuk and the traction pk can be expressed in terms of the following integral equation:

clk xi� �

uk xi� �þZ

Cp�lk xi; xj� �

uk xj� �

dC ¼Z

Cu�lk xi; xj� �

pk xj� �

dC l; k ¼ 1;2ð Þ; ð1Þ

where X and C denote the domain and the boundary of the solid, respectively. xi and xj represent the field points, u�lk and p�lkare the fundamental solutions of the displacement and the traction, respectively, known as Kelvin’s solution. The constantclk(xi) is defined as follows:

clkðxiÞ ¼dlk xi 2 X

� �;

dlk2 xi 2 C on the smooth boundary

� �;

0 xi R X� �

;

8><>: ð2Þ

and clk(xi) takes different values at the non-smooth boundary. Dividing the domain boundary into many elements enables usto discretize the integral equation using the boundary elements. A linear system of algebraic equations is then constructedby appropriate lower order interpolation functions for the displacements and the tractions (here we employ quadratic inter-polation functions). We express xi(n) using quadratic interpolation shape functions um(n) and the nodal coordinates xi

m:

xiðnÞ ¼X3

m¼1

umðnÞxim ði ¼ 1;2Þ; ð3Þ

where, quadratic interpolation shape functions um(n)(m = 1,2,3) can be defined as:

u1ðnÞ ¼ 12 nðn� 1Þ;

u2ðnÞ ¼ 1� n2;

u3ðnÞ ¼ 12 nðnþ 1Þ;

8><>: ð4Þ

where n denotes the local coordinate along the element.We have finally:

HU ¼ GP; ð5Þ

where H and G are the coefficient matrixes. With appropriate boundary conditions, the linear system of equations is solved,and the displacements and tractions at the collocation points are found. But how to design the effective error estimation for-mula in order to get the most accurate solution is the topic of this paper.

The accuracy of the solution of the boundary element method can be improved by the following methods:

1. Subdividing the large error elements, the total number of elements increases but the order of interpolation functionsremain unchanged, i.e. the H-process;

2. Adding the order of the interpolation functions along the large error elements, while the total number of elements keepsthe same, namely P-process;

3. Re-arranging the collocation points along the large error elements, this is R-process;4. The combination of the above methods, such as the HP-process, HR-process, etc.

Considering the engineering practice, in the non-critical area, the most commonly used method is H-process, in that itsalgorithm is simple and easier facility of program; in the critical areas, such as high stress areas, in order to get more accuratesolution, it is better to use the R-process during the adaptive process, which is more consistent with the needs of engineer-ing. This is the idea of the HR-process.

3. Fuzzy logic system

Fuzzy logic system can provide a systematic and efficient framework to incorporate linguistic fuzzy information fromhuman expert. The basic configuration of a fuzzy logic system consists of a fuzzifier, some fuzzy IF–THEN rules, a fuzzyinference engine and a defuzzifier, as shown in Fig. 1. The fuzzy inference engine uses the fuzzy IF–THEN rules to performa mapping from an input vector xT = [x1,x2, . . . ,xn] 2 Rn to an output f 2 Rn.

x

Fuzzy Rules Base

Fuzzifier Defuzzifier

Fuzzy Inference Engine

f

Fig. 1. The basic configuration of a fuzzy logic system.

xb a

1

µ

xa b

1

µ

x c b a

µ1

(A) (B) (C)

Fig. 2. Membership function: (A) triangular, (B) L-shaped and (C) Gamma.

Y. Chen et al. / Applied Mathematics and Computation 218 (2011) 2167–2177 2169

The ith fuzzy rule is written as:

RðiÞ : if x1 is Ai1 and � � � and xn is Ain then f is Bi; ð8Þ

where Ai1, Ai2, � � �, and Ain are fuzzy sets and Bi is the fuzzy singleton for the output in the ith rule. By using the singleton fuzz-ifier, product inference, and center-average defuzzifier, the output of the fuzzy system can be expressed as follows:

f ðxÞ ¼Pm

i¼1ziQn

j¼1lAijðxjÞ

� �Pm

i¼1

Qnj¼1lAij

ðxjÞ� � ; ð9Þ

where lAijðxjÞ is the degree of membership of xj to Aij, m is the number of fuzzy rules, zi is the output quantity, namely the

center of Bi. It is worth noting that the fuzzy system (9) is the most frequently used in practical applications. Following theuniversal approximation results [14], the fuzzy system (9) is able to approximate any nonlinear smooth function to an arbi-trary degree of accuracy. Of particular importance, it is assumed that the structure of the fuzzy system and the membershipfunction type are properly specified in advance by the designers.

There are many types of membership functions, some of them are smooth functions (namely Gaussian, Sigmoid, etc.membership functions) and others are non-smooth (namely triangular and trapezoidal, etc. membership functions). Thechoice of the type of membership function used in specific problems is not unique. The most commonly used membershipfunction is triangular membership function. A triangular membership function depends on three parameters a, b, c, and canbe described as shown in Eqs. (10)–(12) and Fig. 2:

lAðxÞ ¼

0; x < a;

ðx� aÞ=ðb� aÞ; a 6 x 6 b;

ðc � aÞ=ðc � bÞ; b 6 x 6 c;

0; x > c:

8>>><>>>:

ð10Þ

lAðxÞ ¼1; x < a;

ðx� aÞ=ðb� aÞ; a 6 x 6 b;

0; x > c:

8><>: ð11Þ

lAðxÞ ¼0; x < a;

ða� xÞ=ðb� aÞ; a 6 x 6 b;

1; x > b:

8><>: ð12Þ

4. Adaptive error analysis

Firstly, corner points, inflection points, and nodes that are prone to transform under the force are all referred to as sin-gular points in this article. Elements which contain these points are referred to as singular elements. We use continuous er-ror analysis at the singular element and use iterative error analysis at the non-singular element. According to a large numberof references [1–11], the error of the boundary element is always related to the order of the interpolation function, the length


of the element and the relative change of the numerical results. Therefore based on fuzzy logic theory [14,19] ten rules aresummarized as follows:

Rule 1: if Dsolutioni is Negative Big and hj is Small then DRi is Negative Big.Rule 2: if Dsolutioni is Negative Small and hj is Small then DRi is Negative Small.Rule 3: if Dsolutioni is Very Small and hj is Small then DRi is Very Small.Rule 4: if Dsolutioni is Positive Small and hj is Small then DRi is Positive Small.Rule 5: if Dsolutioni is Positive Big and hj is Small then DRi is Positive Big.Rule 6: if Dsolutioni is Negative Big and hj is Big then DRi is Negative Big.Rule 7: if Dsolutioni is Negative small and hj is Big then DRi is Negative Big.Rule 8: if Dsolutioni is Very Small and hj is Big then DRi is Very Small.Rule 9: if Dsolutioni is Positive Small and hj is Big then DRi is Positive Big.Rule 10: if Dsolutioni is Positive Big and hj is Big then DRi is Positive Big.

We note: DRi = si � si represents the error at the node i (si and si are the exact solution and the solution of the boundaryelement method, respectively), Dsolutioni represents the relative change of the numerical results as follows:

D solutioni ¼sb � sa; at the singular element;sl � sl�1; at the non-singular element;

�ð13Þ

wheresb and sa are the numerical solutions of singular nodal point i as the final node of former element and the initial node oflatter element; sl�1 and sl are numerical solutions of non-singular nodal point i at the (l � 1) th iterations and the lth itera-tions, where sl is replaced by ul when displacement is taken into account and sl is replaced by tl when traction is taken intoaccount; hjrepresents the length of the elementj. To simplify calculation, we employ the triangular membership functions forD solutioni, DRi and hj as shown in Figs. 3–5. Then according to Ref. [14] we obtained the formula as follows:

S4S11

xm51 m3

1m21 m1

1

1

µ

m41

S21 S3

1S5

Fig. 3. Membership functions of Dsolutioni.

µ1

x 1

h21h1

1

n21 n1

1

Fig. 4. Membership functions of hj.

R41R1

1

xa01 a1

1a-11a-2

1

1

µ

a21

R21 R3

1R5

Fig. 5. Membership function of DRi.

(A) Definition of problem (B) Quarter plate model

P

P A B

C

D

Fig. 6. Hollow cylinder under internal pressure.


DRi ¼a�1ls1lh1 þ a�1ls1lh2 þ a�1ls2lh2 þ a1ls4lh1 þ a1ls3lh2 þ a1ls4lh2 þ a�2ls2lh1 þ a2ls3lh2

ls1lh1 þ ls2lh1 þ ls3lh1 þ ls4lh1 þ ls5lh1 þ ls1lh2 þ ls2lh2 þ ls3lh2 þ ls4lh2 þ ls5lh2; ð14Þ

where ls1, ls2, ls3, 0ls4 and ls5 are membership functions and they represent that D solutioni are Negative Big, NegativeSmall, Positive Small, Positive Big and Very Small, respectively (see s1 � s5 in Fig. 3). lh1 and lh2 are membership functionsand they represent that hj are Small and Big, respectively ( see h1 and h2 in Fig. 4). a�2 and a2represent the center of the mem-bership function of DRi when DRi are Negative Big and Positive Big. a�1 and a1 represent the center of the membership func-tion of DRi when DRi are Negative Small and Positive Small. And the center of the membership function of DRi is zero whenDRi is Very Small (see Fig. 5).

In addition, m1, m2, m3, m4 and m5 (m5 = 0) represent the centers of ls1, ls2, ls3, ls4 and ls5, respectively. n1 (n1 = 0) and n2

represent the centers of lh1 and lh2, respectively. Let a�1 = �a1 and a�2 = �a2. Then there are five parameters to be selecteddepending on the considered problem [14] (i.e., m1, m2, n2, a�2, a�1). Moreover, in order to avoid dimension disaster [14] andto increase calculation efficiency, the number of memberships of the Dsolutioni, hj and DRi (see Figs. 3–5) is appropriateaccording to the common usage [14,19]. This can be verified by examples in the 5th section.

The ten fuzzy logic rules we stated above can be verified by a practical example.This problem represents the case of a hollow cylinder under internal pressure, as shown in Fig. 6(A). The pressure is as-

sumed to be rr = P = 100 MPa, while the internal and external radii are a = 10 mm and b = 25 mm, respectively.E = 200000 MPa,t = 0.25. This is a typical plane-strain problem, its analytical solution expressed in polar coordinates as fol-lows [11]:

rr ¼a2P

b2 � a21� b2

r2

!; rh ¼

a2P

b2 � a21þ b2

r2

!: ð15Þ

According to the shape of the object and symmetry of loading, we take one quarter of the domain (Fig. 6(B)) as a com-putational model. Under the condition of excellent discrete element of boundary, we use ordinary boundary element methodwith the constant element, linear element and quadratic element to calculate this problem. The obtained results are com-pared with the analytical solution, which shows the relationship between the error and the relative change of the numericalresults, as well as the relationship between the error and the element length. The two relationships are shown in Figs. 7 and8. Fig. 7 shows the relationship between the error and the relative change of the displacement in the boundary AB, and wecan similarly obtain the relationship between the error and the relative change of the traction. It shows in Figs. 7 and 8 thatthe relationship between the error of the boundary element and the length of the element, and the relationship between theerror and the relative change of the numerical results agree with the ten fuzzy rules mentioned above.

4.1. Continuous error analysis

By definition:

D solutionmik ¼ tmi

bk � tmiak ; ðk ¼ x; yÞ; ð16Þ

where tmibk and tmi

ak are the traction of nodal point i as the final node of former element and the initial node of latter element,the subscript k represents components in the x and y directions of the tmi

b and tmia at the mth iteration. DRmi

tkis the residual, so

DRmitk

can be represented as Eq. (17) combined with hj, which conforms to Eq. (14), i.e.,

DRmitk¼ a�1lmi

kt1lmikh1 þ a�1lmi


kt2lmikh2 þ a1lmi

kt4lmikh1 þ a1lmi

kt3lmikh2 þ a1lmi


kt2lmikh1 þ a2lmi

kt3lmikh2

lmikt1lmi

kh1 þ lmikt2lmi

kh1 þ lmikt3lmi

kh1 þ lmikt4lmi

kh1 þ lmikt5lmi

kh1 þ lmikt1lmi

kh2 þ lmikt2lmi

kh2 þ lmikt3lmi

kh2 þ lmikt4lmi

kh2 þ lmikt5lmi

kh2

:

ð17Þ

The residual of the displacement can be defined as:

DRmiuk¼ 0: ð18Þ

0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

The length of element/mm

The

erro

r of o

rdin

ary

boun

dary

ele

men

t met

hod/

mm

Quadratic elementLinear elementConstant element

Fig. 8. The relationship between the error and the element length.

-0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-0.4

-0.2

0

0.2

0.4

0.6

0.8

The relative change of the displacement in the element length boundary AB/mm

The

erro

r of o

rdin

ary

boun

dary

ele

men

t met

hod/

mm

Quadratic elementLinear elementConstant element

Fig. 7. The relationship between the error and the relative change of the displacement in the boundary AB.


The residual of the traction becomes:

DRmtk¼Z

Cu�lk � DRmi

tkdC ¼

XNE

j¼1

ZCj

u�lk � DRmitk

dC ¼ CRmtk

� �ij; ð19Þ


where CRmtk

� �ij

is the kth component of the residual at the mth iteration for the element j when the source point is at xi.The error indicator factor for element j is defined by

Ckj ¼max CRmtx

� �ij

�� þ CRmty

� �ij

��

� ; i ¼ 1;2; . . . ;N

� : ð20Þ

Consecutive errors CRmt can be measured by the norm:

CRmt

�� 1 ¼ CRm

txðx1Þ

�� þ CRmtyðx1Þ

�� þ � � � þ CRmtxðxNÞ

�� þ CRmtyðxNÞ

�� ¼XNE

j¼1

XN

i¼1

CRmtx

� �ij

��þ

XN

i¼1

CRmty

� �ij

��

( )

6

XNE

j¼1

XN

i¼1

CRmtx

� �ij

�� þ CRmty

� �ij

��

� 6

XNE

j¼1

Ckj: ð21Þ

Therefore, error estimates factor Cg at the singular element can be defined as:

Cg ¼XNE

j¼1

Ckj: ð22Þ

4.2. Iterative error analysis

By definition:

Duk ¼ ~ulk � ~ul�1

k ;

Dtk ¼ ~tlk � ~tli�1

k ;

(ð23Þ

Then the residual of the displacement and traction is respectively defined as:

uk �Q

uk � DRuk;

tk �Q

tk � DRtk;

�ð24Þ

where ~ulk; ~u

l�1k are numerical solutions at the lth iterations and the (l � 1) th iterations, respectively, when the source point is

at xi,Q

uk are the interpolation functions. DRukis the residual, so DRuk

can be represented as Eq. (25) combined with hj, whichconforms to Eq. (14), i.e.

DRuk¼ a�1lku1lkh1 þ a�1lku1lkh2 þ a�1lku2lkh2 þ a1lku4lkh1 þ a1lku3lkh2 þ a1lku4lkh2 þ a�2lku2lkh1 þ a2lku3lkh2

lku1lkh1 þ lku2lkh1 þ lku3lkh1 þ lku4lkh1 þ lku5lkh1 þ lku1lkh2 þ lku2lkh2 þ lku3lkh2 þ lku4lkh2 þ lku5lkh2:

ð25Þ

The case of traction is similar.Interpolation residual vector Rite

k ðxiÞ can be expressed as:

Ritek ðxiÞ ¼

ZC

t�lk uk �Y

uk

� �dC�

ZC

u�lk tk �Y

tk

� �dC ¼

XNE

j¼1

ZCj

t�lk uk �Y

uk

� �dC�

ZCj

u�lk tk �Y

tk

� �dC

( )

�XNE

j¼1

ZCj

t�lk � DRudC�Z

Cj

u�lk � DRtdC

( )¼XNE

j¼1

RKð Þij; ð26Þ

where (Rk)ij is the kth component of the residual for the element j when the source point is at xi.The error indicator factor ki for element j is defined by

ki ¼max ½ ðRxÞij�� þ ðRyÞij

�� ; i ¼ 1;2; . . . ;Nn o

: ð27Þ

Residual vector R can be measured by the following model:

Rk k1 ¼ Ritex ðx1Þ

�� þ Ritey ðx1Þ

�� þ � � � þ Ritex ðxNÞ

�� þ Ritey ðxNÞ

�� ¼XNE

j¼1

XN

i¼1

ðRxÞij

��þ

XN

i¼1

ðRyÞij

��

( )

6

XNE

j¼1

XN

i¼1

ðRxÞij�� þ ðRyÞij

�� n o6

XNE

j¼1

kj: ð28Þ

Therefore, error estimates factor g at the non-singular element can be defined as:

g ¼XNE

j¼1

kj: ð29Þ


5. Numerical examples

In this section, the numerical results of two benchmark problems by the H-R adaptive program are presented to test theeffectiveness of the proposed fuzzy error estimation method. For comparison purpose, the numerical solutions of the pro-posed method are compared with the analytical solutions and the boundary element solutions using the error estimationmethod in Ref. [8]. We know that Ref. [8] is a representative sample of conventional error estimation methods. The twoH-R adaptive programs employ the isoparametric quadratic elements. The following two-dimensional elastostatic problemsare studied:

1. Hollow cylinder under uniform internal pressure.2. Square plate with circular hole under uniform tension.

10 15 20 25-140

-120

-100

-80

-60

-40

-20

Coordinate of the boundary AB /mm

The

norm

al tr

actio

n of

the

bou

ndar

y A

B/M

Pa

The analytical solutionThe numerical result of this paperThe numerical result of the methodproposed by [8]

Fig. 9. The traction normal to the boundary AB of Example 1.

0 2 4 6 8 10 12 14

7.86

7.88

7.9

7.92

7.94

7.96

7.98

8

8.02

8.04

8.06

zsymumber/degree

A-x

1-di

spla

cem

ent/m

m

The analytical solutionThe numerical result of this paperThe Fitting curve

Fig. 10. The displacement to the point A of Example 1.


5.1. Example 1 – hollow cylinder under uniform internal pressure

This example represents the case of a plane-strain hollow cylinder under internal pressure, as shown in Fig. 6 of the 4thsection. The initial mesh is constructed by 10 quadratic elements. Choosing five proper parameters in the fuzzy logic systemjust as m1 = �m4 = �0.21, m2 = �m3 = �0.105, n2 = 0.83, a�2 = �a2 = �0.0048 and a�1 = �a1 = �0.0024, we solve this exampleusing the method in this paper and the method proposed by Ref. [8], respectively, thus obtain numerical results that areshown in Figs. 9 and 10.

As is shown in Fig. 9, the numerical result of this paper approximates the analytical solution, and is significantly betterthan the result using the method proposed by Ref. [8]. As is shown in Fig. 10, the numerical results of this paper remainedstable basically, with well stability and convergence. Finally, Table 1 contains the main indicators of the results, which showsthat the calculation results are more accurate during the adaptive process employing the method proposed by this paperthan the method proposed by [8] under the same initial conditions, therefore the error estimation method proposed isproved to be effective. However, from Table 1 we can see that the proposed method needs a little longer time. Our computerconfiguration is that the chip is Pentium (R) 4, CPU is 1.50 GHz, and the memory is 512 MB.

5.2. Example 2 – square plate with circular hole under uniform tension

This example is a square plate with a circular hole under uniform tension, shown in Fig. 11, which is the famous Kirschproblem (a typical plane-stress problem), its analytical solution expressed in polar coordinates as follows [11]:

rr ¼S2

1� a2

r2

� þ S

21þ 3a4

r4 �4a2

r2

� cos 2h;

rh ¼S2

1þ a2

r2

� � S

21þ 3a4

r4

� cos 2h;

srh ¼ �S2

1� 3a4

r4 þ2a2

r2

� sin 2h:

ð30Þ

Owing to the symmetry, only one quarter of the domain needs to be considered for analysis purpose. Let S = 100 MPa,a = 1 mm, L = 20 mm, E = 80000 MPa, t = 0.25 [10], the initial mesh is constructed by 9 quadratic elements. Choosing fiveproper parameters in fuzzy logic system as m1 = �m4 = �0.23, m2 = �m3 = �0.12, n2 = 1.1, a�2 = �a2 = �0.028 anda�1 = �a1 = �0.014, we solve this example using the method in this paper and the method proposed by Ref. [8]. The obtainednumerical result is shown in Fig. 12. After analyzing Fig. 12, we can see that the numerical result of this paper approximatesthe analytical solution, and is evidently better than the result obtained by the method proposed by Ref. [8].

Table 1Main indicators of the calculation.

Adaptive boundary elementprogram proposed by Ref. [8]

Adaptive boundary elementprogram proposed by this paper

Zsynumber/degree 0 � 15 0 � 15Mostelements/n 82 87Mostnodes/n 164 174Mincoord (y)/mm 1.1984 � 10�4 6.103550 � 10�5

Mindisp (y)/mm �2.379028 � 10�7 �1.811788 � 10�7

Solution time/s 8.516066 9.175164

x1

x2

A B a

C

D

L

L

S

S

Fig. 11. Example 2 – the Kirsch problem.

1 2 3 4 5 6 7 8 9 10-300

-280

-260

-240

-220

-200

-180

-160

-140

-120

-100

Coordinate of the boundary AB/mm

The

norm

al tr

actio

n o

f th

e bo

unda

ry A

B/M

PaThe analytical solutionThe numerical result of this paperThe numerical result of the method proposed by [8]

Fig. 12. The traction normal to the boundary AB of Example 2.

Table 2Main indicators of the calculation.

Adaptive boundary elementprogram proposed by Ref. [8]

Adaptive boundary element programproposed by this paper

Zsynumber/degree 0 � 15 0 � 15Mostelements/n 66 72Mostnodes/n 132 144Mincoord (y)/mm �1.99894 �0.98995Mindisp (y)/mm �0.1801066 � 10�4 �0.103034 � 10�4

Solution time/s 11.527900 12.145533


Finally, Table 2 contains the main indicators of the results. From the table, it can also be found that the calculation resultis more accurate by employing the proposed method than using the method in Ref. [8]. It can also be seen from Table 2 thatthe method of this paper is a litter more time-consuming.

In addition, we have tested the method in other examples such as the cantilever beam and the rectangular plate [10], andthe results are also satisfied.

6. Conclusion

On the basis of available literature, the error of the boundary element is always related to the order of the interpolationfunction, the length of the element and the relative change of the numerical results. Thanks to the fuzzy logic theory, nowwe are able to make full use of expert experience in the field. In this paper, fuzzy logic is successfully applied to improvethe effect of error estimation in adaptive boundary element method, and then the new error estimation method is adoptedinto the HR-adaptive process. The numerical examples illustrate that the proposed error estimation method works well forthe H-R adaptive program, and also demonstrate that the proposed error estimation method is effective and superior.Additionally, the accuracy of the solution is higher, and the adaptive algorithm itself has well stability and convergence.Although the examples are elasticity problems, what is worth noting is that the error analysis method proposed in thispaper can also be extended to the more complex boundary problems such as inflection points and corner points as isshown in Ref. [5].

Acknowledgement

The authors are grateful to the Nature Foundation of Hebei Province, China (E2009000365) for financial support.


References

[1] D.H. Yu, The Mathematical Theory of Adaptive Boundary Element Method, Science Press, Beijing, 1993.[2] E. Kita, N. Kamiya, Error estimation and adaptive mesh refinement in boundary element method, an overview, Engineering Analysis with Boundary

Elements 25 (2001) 479–495.[3] Z.J. Chen, H. Xiao, X. Yang, Error analysis and novel near-field preconditioning techniques for Taylor series multiple-BEM, Engineering Analysis with

Boundary Elements 34 (2010) 173–181.[4] Z.Y. Zhao, X. Wang, Error estimation and h adaptive boundary elements, Engineering Analysis with Boundary Elements 23 (1999) 793–803.[5] A. Charafi, L.C. Wrobel, A new h-adaptive refinement scheme for the boundary element method using local reanalysis, Applied Mathematics and

Computation 82 (1997) 239–271.[6] J.R. Fernández, P. Hild, A posteriori error analysis for the normal compliance problem, Applied Numerical Mathematics 60 (2010) 64–73.[7] C. Erath, S. Ferraz-Leite, S. Funken, Energy norm based a posteriori error estimation for boundary element methods in two dimensions, Applied

Numerical Mathematics 59 (2009) 2713–2734.[8] C.X. Yu, Study on adaptive numerical resolution for elastic problem, Journal of Yanshan University 29 (2005) 17–21.[9] C. Yu, Y. Chen, Y. Mu, Study on adaptive numerical analysis method for elastic problems, International Journal of Innovative Computing, Information

and Control 129 (2005) 227–235.[10] G.X. Shen, H. Xiao, Y.M. Chen, The Boundary Element Method, Mechanical Industry Press, Beijing, 1998.[11] Z.H. Yao, H.T. Wang, The Boundary Element Method, Higher Education Press, Beijing, 2010.[12] C.L. Chen, P.C. Chen, C.K. Chen, Analysis and design of fuzzy control systems, Fuzzy Sets and Systems 57 (1993) 125–140.[13] C.C. Lee, Fuzzy logic in control systems: fuzzy logic controller – part II, IEEE Transactions on Systems, Man Cybernetics 20 (1990) 419–435.[14] L.X. Wang, Adaptive Fuzzy Systems and Control: Design and Analysis, Prentice-Hall, Inc, Englewood Cliffs, NJ, 1994.[15] M. Bouji, A.A. Arkadan, T. Ericsen, Fuzzy inference system for the characterization of SRM under normal and fault conditions, IEEE Transactions on

Magnetics 37 (2001) 3745–3748.[16] T. Hiyama, K. Miyazaki, H. Satoh, A fuzzy logic excitation system for stability enhancement of power systems with multi-mode oscillations, IEEE

Transactions on Energy Conversion 11 (1996) 449–454.[17] K.J. Satsios, D.P. Labridis, P.S. Dokopoulos, An artificial intelligence system for a complex electromagnetic field problem: part I – finite element

calculation and fuzzy logic development, IEEE Transactions on Magnetics 35 (1999) 516–522.[18] K.J. Satsios, D.P. Labridis, P.S. Dokopoulos, An artificial intelligence system for a complex electromagnetic field problem: part II – method

implementation and performance analysis, IEEE Transactions on Magnetics 35 (1999) 523–527.[19] M.A. Arjona, R. Escarela-Perezb, E. Melgoza-Vázquezc, Convergence improvement in two-dimensional finite element nonlinear magnetic problems – a

fuzzy logic approach, Finite Elements in Analysis and Design 41 (2005) 583–598.

a new error estimation approach based on fuzzy logic system for h-r adaptive boundary element method

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