Engineering Analysis with Boundary Elements 37 (2013) 812–817

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Engineering Analysis with Boundary Elements

0955-79

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n Corr

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

A Kriging interpolation-based boundary face method for 3Dpotential problems

J.H. Lv a, Y. Miao a,n, H.P. Zhu a, Y.P. Li b

a School of Civil Engineering and Mechanics, Huazhong University of Science and Technology, Wuhan 430074, Chinab State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, The Chinese Academy of Sciences,

Wuhan 430071, China

a r t i c l e i n f o

Article history:

Received 3 September 2012

Accepted 14 February 2013Available online 27 March 2013

Keywords:

Meshless method

Boundary face method

Moving least squares approximation

Kriging interpolation method

97/$ - see front matter & 2013 Elsevier Ltd. A

x.doi.org/10.1016/j.enganabound.2013.02.006

esponding author. Tel.: þ86 27 87540172; fa

ail address: my_miaoyu@163.com (Y. Miao).

a b s t r a c t

In this paper, a new implementation of the boundary face method (BFM) is presented and developed for

solving 3D potential problems. The BFM is implemented directly based on the boundary representation

data structure for geometry modeling to eliminate geometry errors. This study combines the BFM with

Kriging interpolation method and the corresponding formulae are derived. The Kriging interpolation is

applied instead of the traditional moving least squares (MLS) approximation to overcome the lack of

Kronecker delta function property, then essential boundary conditions can be imposed directly and

easily. Several numerical examples with different geometry and boundary conditions are presented to

illustrate the performance of the present method. The comparisons of accuracy between MLS

approximation and Kriging interpolation are studied.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

The mesh generation of mesh-based numerical methods, suchas the finite element method (FEM), is generally recognized to bearduous, time-consuming and prone to errors, especially invol-ving moving boundaries, large deformations or crack propagation.Hence, meshless methods have attracted a wide interest due tothe versatility for complex geometry and flexibility for differentengineering problems [1] in the past two decades. Many kinds ofmeshless methods have been proposed so far. The previouslydeveloped meshless methods can be roughly grouped into twocategories: the domain type and the boundary type.

The domain-type meshless methods are represented by theelement-free Galerkin method (EFG) [2] which couples the mov-ing least-squares (MLS) interpolation with the global symmetricweak form. Atluri and his coworkers combined the local weakform over local sub-domains with MLS interpolation and pro-posed two other meshless methods of domain type: the meshlesslocal boundary integral equation (MLBIE) [3] and the meshlesslocal Petrov-Galerkin (MLPG) [4] approach. There are also someother meshless methods developed [5], such as the point inter-polation method (PIM) [6] and the radial point interpolationmethod (RPIM) [7], etc.

The boundary-type meshless methods can be represented bythe boundary node method (BNM) which couples the MLSinterpolation and the boundary integral BIE [8,9]. Zhang proposed

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x: þ86 27 87542231.

the hybrid boundary node method (HBNM) [10,11] using thehybrid displacement variational formulae. This method has beendeveloped by Miao et al. [12,13] and applied to elastodynamicsproblems [14], Helmholtz problems [15], etc. Li developed aGalerkin boundary node method (GBNM) [16] to over the dis-advantage of the MLS shape functions. Recently, an attractivemethod-the boundary face method (BFM) has been proposed byZhang [17,18], which introduces the boundary representationdata structure to eliminate geometry errors. In all the abovemeshless methods, the MLS interpolation is employed to get theshape function. Although the MLS interpolation has been appliedin many meshless methods, some disadvantages still exist such asthe lack of Kronecker delta function property, which leads to thedifficulty of imposing essential boundary conditions accurately.Hence, to find an efficient, stable and preferably conformablemethod to construct the shape function is still one of the keyissues in the development of meshless methods.

In the present work, the Kriging interpolation is adopted toconstruct the shape function in the implementation of the BFM.The Kriging interpolation is a form of generalized linear regres-sion for the formulation of an optimal estimator in a minimummean square error sense, which derived its name from Krige [19].It has many advantages compared with the MLS interpolation.As it is a passing node interpolation, the Kronecker delta functionproperty is satisfied automatically. In addition, since it is based onthe statistician theories for minimum of mean square error,interpolation accuracy can be ensured. Hence, some studies havebeen conducted to explore the possible application in meshlessmethods. The Kriging interpolation has been introduced into theEFG method and applied in structural analysis [20,21], dynamic

J.H. Lv et al. / Engineering Analysis with Boundary Elements 37 (2013) 812–817 813

analysis [22], and large deformation analysis [23]. The compar-ison between the radial point interpolation and the Kriginginterpolation has been carried out by Liu [24]. Most applicationsof the Kriging interpolation presented are restricted to 2D EFGmethod.

In this paper, the Kriging interpolation is applied to the BFM for3D potential problems as a substitution of the MLS interpolation.The paper is organized as follows. In Section 2, the formulae of thelocal Kriging interpolation on a generic surface in parametric formsare developed. Section 3 briefly gives the well-known BIE for 3Dpotential problems. Numerical examples for 3D potential problemsare demonstrated in Section 4, and the paper ends with conclu-sions and discussions on future work in Section 5.

2. Kriging interpolation on a generic 3D surface

The Kriging interpolation is widely used to estimate field valueat a point of a problem domain, for which the variogram isknown. Several forms of Kriging formulae, such as simple,ordinary, and general Kriging, etc., have been developed for theestimation. In case that the drift of the function is not a constantmean, the universal Kriging concept may be an alternative inwhich a polynomial drift model is commonly applied. The presentwork is based on the universal Kriging.

2.1. Kriging interpolation

The first step for BFM is to choose a proper coordinate system.For most solid models, surfaces can be represented in parametricforms

x1 ¼ x1ðs1,s2Þ, x2 ¼ x2ðs1,s2Þ, x3 ¼ x3ðs1,s2Þ ð1Þ

where the parametric coordinates are defined in the range,s1,s2A ½0,1�, which is consistent with the boundary representationdata structure used in most CAD packages. The potential and itsnormal gradient can also be expressed in parametric forms as

uðx1,x2,x3Þ ¼ uðs1,s2Þ

qðx1,x2,x3Þ ¼ qðs1,s2Þ ð2Þ

It is assumed that the boundary surface is the union ofpiecewise smooth segments called faces. Suppose a set of scat-tered nodes xi ði¼ 1,2, . . . ,nÞ defined by a random function uðxÞare distributed on a face. Only the surrounding nodes on the sameface have effect on evaluation point x0. The estimated value offunction unðx0Þ can be obtained [25]

unðx0Þ ¼Xn

i ¼ 1

liuðxiÞ ð3Þ

where unðx0Þ is the value at node xi, li is the weight of theneighborhood nodes, which is determined by minimizing thesquared variance of the estimation error. Suppose the drift m

is assumed to be a polynomial model, the above equationis subjected to

mðx0Þ ¼Xm

l ¼ 0

mlplðxÞ, p0ðxÞ ¼ 1 ð4Þ

where ml is the coefficient for the polynomial basis plðxÞ.Based on the no-bias property of Kriging interpolation, one can

obtain

E½uðx0Þ� ¼ E½unðx0Þ� ¼Xn

i ¼ 1

liE½uðxiÞ� ð5Þ

The squared variance of the estimation error can be written as

s2E ¼ E½uðx0Þ�unðx0Þ�

2 ¼ E uðx0Þ�Xn

i ¼ 1

liuðxiÞ

" #2

¼ E½uðx0Þ�2�Xn

i ¼ 1

2liE½uðx0Þ�E½uðxiÞ�

þXn

i ¼ 1

Xn

j ¼ 1

liljE½uðxiÞ�E½uðxjÞ� ð6Þ

By minimizing Eq. (6) with respect to the coefficients li

subjected to mþ1 linear constraints, the solution is characterizedby a linear system of nþmþ1 equations with respect to nþmþ1unknowns

Xn

j ¼ 1

E½uðxiÞuðxjÞ�liþXmj ¼ 0

mlplðxiÞ ¼ E½uðx0ÞuðxiÞ�, i¼ 1, . . . ,n ð7Þ

Xn

j ¼ 1

ljplðxjÞ ¼ plðx0Þ, l¼ 0, . . . ,m ð8Þ

The scattered nodes are assumed to be defined by a randomfunction uðxÞ with the semivariogram gðhÞ, which is in the form of

gðx0,xiÞ ¼ gðhÞ ¼ 12E½uðxiÞ�uðx0Þ�

2 ð9Þ

where h is the Euclidean distance between xi and x0. Using theintrinsic hypothesis [26,27], the covariance E½uðx0ÞuðxiÞ� can bereplaced by gðhÞ. Eqs. (7) and (8) can be rewritten in matrix form

Gc¼ g ð10Þ

where

G¼R P

PT 0

� �

¼

gðx1,x1Þ . . . gðx1,xnÞ 1 p1ðx1Þ . . . pmðx1Þ

^ ^ ^ ^ ^ ^ ^

gðxn,x1Þ . . . gðxn,xnÞ 1 p1ðxnÞ . . . pmðxnÞ

1 . . . 1 0 0 . . . 0

p1ðx1Þ . . . p1ðxnÞ 0 0 . . . 0

^ ^ ^ ^ ^ ^ ^

pmðx1Þ . . . pmðxnÞ 0 0 . . . 0

2666666666664

3777777777775

ð11Þ

c¼ ½l1 . . . ln m0 m1 . . . mm�T ð12Þ

g¼ ½gðx0,x1Þ . . . gðx0,xnÞ 1 p1ðx0Þ . . . pmðx0Þ�T ð13Þ

The weights li can be got by solving Eq. (10) and the estimatedvalue unðx0Þ computed using Eq. (3). For simplicity, the estimatedvalue can be written in the form of [28]

unðx0Þ ¼ pðxÞTAuþgðxÞTBu¼Uu ð14Þ

where

A¼ ðPTR�1PÞ�1PTð15Þ

B¼R�1ðI�PAÞ ð16Þ

u¼ ½uðx1Þ uðx2Þ . . . uðxnÞ�T ð17Þ

2.2. Semivariogram model

In Eq. (10), the values of matrix G and g are taken fromsemivariogram. Semivariogram plays an important role in thecomputation. There are several theoretical models available [25],such as spherical, Gaussian, exponential and power models.The Kriging interpolation method is in fact exactly the same asthe radial point interpolation method (RPIM) if the same basis or

J.H. Lv et al. / Engineering Analysis with Boundary Elements 37 (2013) 812–817814

semivariogram is used [24]. Several basis functions in RPIM havebeen used successfully to structural [29] and free vibrationanalyses [30]. Proper model should be chosen to get a betterresult. Gaussian semivariogram model is widely used in compu-tational mechanics and leads to good results in EFG. In the presentwork, the Gaussian model same as [24] is employed as follows:

gðhÞ ¼ c0 1�exp �h2

a20

! !ð18Þ

where h is the lag, and c0 and a0 are the sill and range,respectively. The range of a0 is taken as

a0 ¼ aðs1þs2 Þ=2 ð19Þ

where s1 and s2 are the average sizes of background cells in s1

and s2 directions, respectively. Suitable values of a and c0 will bediscussed in Section 4.

Fig. 1. Nodes distribution on a cube.

3. Boundary integral equations and discretization

The potential problems in three dimensions governed byLaplace’s equation along with prescribed boundary conditions iswritten as

u,ii ¼ 0 8xAO

u¼ u 8xAGu

u,ini � q¼ q 8xAGq ð20Þ

where the domain O is enclosed by G¼GuþGq, u and q are theprescribed potentials and the normal flux on the essentialboundary Gu and the flux boundary Gq, respectively, andni ði¼ 1,2,3Þ, are the components of the outward normal directionto the boundary G.

The problem can be recast into an integral equation on theboundary. The well-known self-regular BIE for potential problemsin 3D is

0¼

ZGðuðsÞ�uðyÞÞqsðs,yÞ dG�

ZG

qðsÞusðs,yÞ dG ð21Þ

where y is the source point and s the source point on theboundary, q¼ @u=@n. usðs,yÞ and qsðs,yÞ are the fundamentalsolutions. For 3D potential problems,

usðs,yÞ ¼1

4p1

rðs,yÞð22Þ

qsðs,yÞ ¼@usðs,yÞ

@nð23Þ

with rðs,yÞ being the Euclidean distance between the source andthe field points.

The Kriging interpolation derived in Section 2 will be used toapproximate u and q on the boundary. The boundary surface isdiscretized into cells face by face. Substituting Eq. (14) intoEq. (21) and dividing the boundary surface into Nc cells, we have

0¼�XNc

j ¼ 1

ZGj

qsðs,yÞXN

I ¼ 1

ðUIðsÞ�UIðyÞÞuI dG

þXNc

j ¼ 1

ZGj

usðs,yÞXN

I ¼ 1

UIðsÞqI dG ð24Þ

where UIðyÞ and UIðsÞ are the contributions from the Ith node tothe collocation point y and field point s. As s is very close to y,special integration techniques should be employed. Even when sand y belong to different cells, they can still be very close to eachother. Special integration schemes have been developed in detailin Refs. [17,18].

Eq. (24) can be put in a matrix form as

Hu�Gq¼ 0 ð25Þ

where u and q contain the approximations to the nodal values ofu and q on the boundary. A well-posed boundary value problemcan be solved using Eq. (25). Similar to the BFM, the integrationquantities, such as the coordinates of Gauss points, the outwardnormal and Jacobian are calculated directly from the surfacerepresented in parametric form as real geometry, and thus nogeometry errors will be introduced, which will provide betteraccuracy.

4. Illustrative numerical results

Three types of 3D geometrical objects have been taken to verifythe current method: a cube, a torus and a pull ring. To compare thecurrent method with the BNM, the following three analytical fieldstaken from Ref. [9] are used to assess the accuracy of the presentmethod: (i) Linear solution: u¼ xþyþz, (ii) Quadratic solution-1:u¼ xyþyzþzx, (iii) Quadratic solution-2: u¼�2x2þy2þz2, (iv)Cubic solution: u¼ x3þy3þz3�3yx2�3xz2�3zy2.

For the purpose of error estimation and convergence study, aglobal L2 norm error, normalized by 9v9max is defined as follows:

e¼1

9v9max

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

N

XN

i ¼ 1

ðvðeÞi �ðvðnÞi Þ

2

vuut ð26Þ

where 9v9max is the maximum value of u and q over N samplepoints, the superscripts (e) and (n) refer to the exact andnumerical solutions, respectively.

4.1. Dirichlet problems on a cube

The case of the field for a 2�2�2 cubic domain governed byLaplace’s equation as shown in Fig. 1 is presented as the firstexample. The cubic faces are x¼ 71, y¼ 71 and z¼ 71,respectively. The linear, quadratic and cubic Dirichlet boundaryconditions are imposed on the surface of the cube.

The 6�6 nodes on each face as shown in Fig. 1 are used tostudy the influence of parameters in Eq. (18). When a varies from3 to 8 and c0 equals 1, the L2 errors of nodal value q are presentedin Table 1. As a varies from 5 to 8, results with high accuracy canbe obtained. The boldfaced errors are the minimum ones fordifferent displacement fields. The proper values of a depend on

Table 1L2 errors of nodal value q for various values of a (%).

a Linear field Quadratic-1 field Quadratic-2 field Cubic field

3 0.2655 0.1315 0.4741 0.4354

4 0.04893 0.02925 0.1649 0.1624

5 0.03617 0.01365 0.06486 0.06818

6 0.04156 0.01238 0.03327 0.0335

7 0.04384 0.01244 0.02877 0.02309

8 0.0447 0.01259 0.03083 0.02199

100 200 300 400

0.00

0.05

0.10

0.15

0.20

Rel

ativ

e er

rors

at p

oint

s ins

ide

the

dom

ain

(%)

Number of nodes

L2 error of u

L2 error of q

Fig. 2. Relative errors and convergence rate at points inside the domain.

Table 2L2 errors of nodal value q for various locations of collocation point in a cell (%).

e Linear field Quadratic-1 field Quadratic-2 field Cubic field

0.1 0.3197 0.1813 0.8628 0.8841

0.3 0.09783 0.06646 0.3269 0.3224

0.5 0.04441 0.03012 0.1536 0.1509

0.7 0.02972 0.01537 0.0766 0.07646

0.9 0.02716 0.0089 0.03497 0.03775

1.0 0.04156 0.01238 0.03327 0.0335

Table 3L2 errors of nodal value q for various values of a (%).

a Linear field Quadratic-1 field Quadratic-2 field Cubic field

1 1.325 0.9822 0.9303 12 1.224 1.109 1.079 1.242

3 1.212 1.088 1.035 1.212

4 1.185 1.065 0.9909 1.405

5 1.175 1.06 0.9736 2.008

Fig. 3. Nodes distribution on the torus.

J.H. Lv et al. / Engineering Analysis with Boundary Elements 37 (2013) 812–817 815

the node distributions and boundary conditions. When parameterc0 is taken as 1, 10, 50 and 100, there is no difference observed inthe results. Therefore, parameter c0 has no influence on theaccuracy and c0 is always taken to be 1 in the later computation.

In order to investigate the convergence, three sets of nodes:(a) 4�4 nodes, (b) 6�6 nodes, (c) 8�8 nodes under cubic fieldwith a being 6 have been studied. The L2 errors of points insidethe domain for different sets of nodes are plotted in Fig. 2. It canbe seen that our method yields very accurate results and has highconvergence rate.

It has been observed in BNM that the locations of the colloca-tion nodes on each cell have some influence on the accuracy.To compare with the BNM, the influence of the locations of nodeson the accuracy of our method has also been studied. The locationof nodes on a cell is determined by a parameter e

sP ¼ sLþeðsR�sLÞ=2 ð27Þ

where sP is the collocation point, sL and sR are the lower-left andthe upper-right corner points, respectively. Computations havebeen performed for all the analytical field using 6�6 nodes witha being 6. The L2 errors of nodal value q for various values of e arepresented in Table 2. It can be found that the accuracy of ourmethod is much less sensitive to the location of the collocationpoints in a cell than that of the BNM.

4.2. Dirichlet problems on a torus

The second example considers a torus centered at origin,whose exterior and interior radius are 10 and 3 units respectively.The usual ring polar coordinates y and f are used in parametricspace. The torus is distributed with 108 nodes as shown in Fig. 3.The Dirichlet boundary conditions corresponding to differentanalytical solution are imposed on the surface of the torus.

To study the influence of parameter a, various values of a arecomputed and the corresponding L2 errors of nodal value q arepresented in Table 3. It can be seen that results with high

accuracy can be obtained for various values of a. In addition,the results of cubic field are more sensitive to the values of a thanthat of the other fields. The boldfaced errors are the best resultswe have obtained.

The results of the torus under quadratic-1 and quadratic-2 fieldsare presented to show the accuracy at locations inside the domain.Figs. 4 and 5 show the variation of potential and its directionalderivative as a taken as 1. The gradient is dotted with the direction

0 100 200 300 400

-200

-150

-100

-50

0

50

100

Quadratic-1 field Quadratic-2 field Analytical solution

Pote

ntia

l u a

long

the

cent

ral a

xis

φ

Fig. 4. Potential u along the central axis.

0 100 200 300 400

-40

-20

0

20

40

60

Quadratic-1 field Quadratic-2 field Analytical solution

Flux

q a

long

the

cent

ral a

xis

φ

Fig. 5. Flux q along the central axis in direction (1, 0, 0).

Fig. 6. Nodes distribution on the pull ring.

0 50 100 150-4000

-2000

0

2000

4000

Potential inside the domain Flux in direction (1,0,0) Analytical solution

Pote

ntia

l and

Flu

x al

ong

the

cent

ral a

xis

Length along the central axis

Fig. 7. Domain result along the central axis.

liner quadratic-1 quadratic-2 cubic0.05

0.10

0.15

0.20

0.25

Rel

ativ

e er

rors

of n

odal

val

ue q

(%)

Type of analytical field

Kriging interpolation MLS approximation

Fig. 8. Comparison with MLS approximation under different boundary conditions.

J.H. Lv et al. / Engineering Analysis with Boundary Elements 37 (2013) 812–817816

(1, 0, 0) in order to get the directional derivative along this direction.25 sample points are uniformly located along the central axis. It canbe seen that the numerical results are in good agreement withanalytical solutions both on potential and flux. The relative errors ofpotential and flux are 0.4752% and 0.8426%, 0.3357% and 0.9949%,for quadratic-1 and quadratic-2 fields respectively.

4.3. Dirichlet problems on a pull ring

A pull ring model including two half torus surfaces and twocylinder surfaces is taken as the third example. The exterior and

interior radius of the torus are 10 and 3 units and the height of thecylinder is 10 units. The pull ring has been discretized with 316nodes as shown in Fig. 6.

To get the results of locations inside the domain, 29 pointsalong the central axis are chosen as sample points. The results ofpotential and flux under cubic boundary conditions have beenobtained and plotted in Fig. 7 with a being 2. The relative errorsfor domain results of potential are 0.3234% and 0.127% forpotential and flux, respectively. It can be observed that thedomain results for potential and flux have high agreement withexact solution for different boundary conditions.

The comparison with MLS approximation has been performedto demonstrate the accuracy of our method. The results underdifferent boundary conditions are obtained using the MLS approx-imation with same nodes distributions. a is taken as 2 in Kriginginterpolation. The comparison of relative errors of nodal value q isshown in Fig. 8. It can be seen that the Kriging interpolation canget more accurate results than MLS approximation for differentboundary conditions.

5. Conclusions and future work

In present work, the Kriging interpolation has been introducedinto boundary face method as a substitution for the MLS approxima-tion. Several numerical examples with different geometry anddifferent boundary conditions are presented. The results demonstrate

J.H. Lv et al. / Engineering Analysis with Boundary Elements 37 (2013) 812–817 817

the high accuracy and convergence of our method. For eachnumerical example, parameter studies are performed to get a betterresult. The proper value of the parameter depends on the nodesdistributions and boundary conditions. How to get a proper value ofthe parameter is an attractive topic and will be studied in the future.

Acknowledgments

Financial support for the project from the National BasicResearch Program of China (973 Program: 2011CB013800) andthe Natural Science Foundation of Hubei Province of China(No. 2011CDB291).

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