Development of Modiﬁed Extended Boundary-Node Method-New Approach for Determining Data PointsKenta Miyashita1 , Ayumu Saitoh1 , Taku Itoh2 , Atsushi Kamitani3 ,Naotake Kamiura1 , and Nobuyuki Matsui1
1

Graduate School of Enginnering, University of Hyogo, Japan
Faculty of Science and Technology, Seikei University, Japan
3
Graduate School of Science and Enginnering, Yamagata University, Japan
21Introduction

The Boundary-Node Method (BNM) [1] has been so far
proposed for solving the boundary-value problem of a partial diﬀerential equation and has yielded excellent resultsin the ﬁeld of engineering [1, 2]. It is a primary meritof the BNM that elements of a geometrical structure are
no longer necessary. However, the BNM has the following
diﬃculty: a boundary must be divided into a set of integration cells to calculate contour integrals. If the BNM
were extended to the method without integration cells, thedemerit of the BNM can be resolved completely.
For the purpose of resolving the above demerit, Saitoh
et al. proposed the eXtended BNM (X-BNM) and investigated its performance numerically [3]. As a result, theyshowed that the accuracy of the X-BNM is much higher
than that of the standard BNM for the Poisson problem.
However, when the boundary shape is strongly concave,
the accuracy of the X-BNM is drastically degraded.
The purpose of the present study is to modify the XBNM for suppressing the accuracy degradation and tocompare the performance of the modiﬁed X-BNM withthat of the conventional one.

2

Extended Boundary-Node Method

Throughout the present study, we adopt a twodimensional (2-D) Laplace problem:−∆u = 0,in Ω(1)u = ū,on ΓD(2)q ≡ ∂u/∂n = q,¯on ΓN(3)where Ω denotes a domain bounded by a simple closedcurve ∂Ω. Here, ∂Ω consists of two parts, ΓD and ΓN ,which satisfy the following relations: ΓD ∪ ΓN = ∂Ω andΓD ∩ ΓN = φ. In addition, ū and q̄ denote known functions
on ΓD and on ΓN , respectively, and n indicates an outward
unit normal on ∂Ω.
As is well known, (1) can be easily transformed to thefollowing boundary integral equation:I∂w∗ (x(s), y)[u(x(s)) − u(y)] ds∂n∂ΩIw∗ (x(s), y) q(x(s)) ds = 0,−In order to discretize (4) and its associated boundaryconditions, N boundary nodes, z1 , z2 , · · · , zN , are placedon ∂Ω. Furthermore, shape functions, Φ1 , Φ2 , · · · , ΦN , aredeﬁned by means of the Moving Least-Squares (MLS) approximation [3]. In addition, u and q are assumed asu (x(s)) =NX

Φj (s) u∗j ,j=1

q (x(s)) =NX

Φj (s) qj∗ .

(5)

j=1

where u∗j and qj∗ (j = 1, 2, · · · , N ) are all constants.
Under the above assumptions, (4) and its associatedboundary conditions can be discretized to a linear system.
In this way, the 2-D Laplace problem is reduced to theproblem in which the linear system is solved.
In the X-BNM, contour integrals of (4) are directly calculated by use of the vector equation of ∂Ω. To this end,
the vector equation is determined as follows. First, theimplicit-function representation f (x) = 0 is determined forthe curve passing through all boundary nodes. Next, we
numerically solve the following ordinary diﬀerential equation:„«dxπ∇f=R·,(6)
ds2
|∇f |
where R(θ) denotes a tensor representing a rotation
through an angle θ. Finally, the resulting P data points,x(1) , x(2) , · · · , x(P ) , are interpolated with the cubic spline
to get the vector equation x = g(s). In the next section,
we explain the numerical solution of (6) in detail.

3

New Approch for Determing Data Points

By solving (6), we can determine P data points. However, even if the higher-order Runge-Kutta method is applied to (6), the numerical solution does not always satisfy
f (x) = 0. So as to resolve this problem, Saitoh et al.
proposed the algorithm in which x(n+1) is calculated fromx(n) by use of the following three steps (see Fig. 1(a)):(i) An approximate solution of x∗ at the (n + 1)th step
is modiﬁed by« »–„∇fπ·δs(7)x∗ = x(n) + R2|∇f | x (n)Here, δs is a constant.

(4)

∂Ω

where w∗ (x, y) ≡ − log |x − y|/2π. Moreover, s indicatesan arclength along ∂Ω.

(ii) In order to calculate an intersection of the straightline x = x∗ + λ(∇f )x ∗ and the curve f (x) = 0, the` ´`´nonlinear equation G λ ≡ f x∗ + λ(∇f )x ∗ = 0 issolved by using the Newton method.

(a)(b)Fig. 1. Schematic view of the approach for determining
data points for the case where a boundary node (a) is notcontained and (b) is contained in the δs-neighborhood ofx(n) . Here, the symbol, ◦, denotes boundary nodes.

Fig. 2. Dependence of the relative error ε on the numberN of boundary nodes (∆ = 3).

(iii) The numerical solution x(n+1) is determined by` ´x(n+1) = x∗ + λ ∇f x ∗ .
By using the above approach, although the curve x =g(s) passes through all data points, boundary nodes do notalways locate on the curve. This is because any boundary
nodes may not be contained in data points to be interpolated. In order to overcome the above problem, we use
the following approach for determining data points: if aboundary node exists in the δs-neighborhood of x(n) , it isemployed as x(n+1) instead of solving (6) numerically (seeFig. 1(b)). Throughout the present study, the X-BNM
with the above correction is called the modiﬁed X-BNM.

4(a)Numerical Results

In this section, we compare the performance of the modiﬁed X-BNM with the conventional one. As an example
problem,
2-D Laplace oproblem over Ω ≡˛ˆ we adopt then˜2˛(x, y) ˛ x − ∆(y/2)2 + (y/2)2 < 1 with the Dirichletcondition: u = cosh πx sin πy + sinh πx cos πy on ∂Ω.
Let us ﬁrst investigate both the accuracy of the modiﬁed X-BNM and that of the conventional one. The relativeerrors are calculated as a function of N and are plotted inFig. 2. The relative error of the modiﬁed X-BNM is diminished in almost proportion to N −1.35 . On the other hand,
the relative error of the conventional X-BNM decreases
with an increase in N for the case with N . 50, whereas it
drastically increases with N for N & 50. In addition, theaccuracy of the modiﬁed X-BNM is almost equal to thatof the conventional one for the case with N . 50.
Next, we investigate the inﬂuence of the boundary shapeon the accuracy of the numerical solution. To this end, therelative errors are calculated as functions of the triangularity ∆ and are plotted in Fig. 3(a). This ﬁgure indicatesthat the accuracy of the modiﬁed X-BNM is much higher
than that of the conventional one for the case with ∆ & 1.
Finally, we compare the speed of the modiﬁed X-BNM
with that of the conventional one. The ratio τM /τC ofthe CPU time is calculated as a function of ∆ and is depicted in Fig. 3(b). Here, τM and τC are the CPU timerequired for the modiﬁed X-BNM and that for the conventional one, respectively. We see from this ﬁgure that thespeed of the modiﬁed X-BNM is almost equal to that ofthe conventional one.
From these results, we can conclude that the performance of the modiﬁed X-BNM is superior to that of theconventional one.

(b)

Fig. 3. Dependence of (a) the relative error ε and (b) theratio τM /τC of CPU times on the triangularity ∆ (N =256).
5Conclusion

We have modiﬁed the X-BNM so that the accuracy
degradation may not be caused by a boundary shape and,have numerically investigated the performance of the modiﬁed X-BNM by comparing with the conventional X-BNM.
Conclusions obtained in this paper are summarized as follows.
1. the accuracy of the modiﬁed X-BNM is much higher
than that of the conventional one, when the boundary
shape is strongly concave.
2. the speed of the modiﬁed X-BNM is almost equal tothat of the conventional one.
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