Formulation of the MFS for the two-dimensional Laplace equation
with an added constant and constraint
Jeng-Tzong Chena,b,*, Jheng-Lin Yanga, Ying-Te Leea, Yu-Lung Changa
a
Department of Harbor and River Engineering, National Taiwan
Ocean University, Keelung 20024, Taiwan
b
Department of Mechanical and Mechatronic Engineering,
National Taiwan Ocean University, Keelung 20024, Taiwan
Abstract
Motivated by the incompleteness of single-layer potential approach for the interior
problem with a degenerate-scale domain and the exterior problem with bounded
potential at infinity, we revisit the method of fundamental solutions (MFS). Although
the MFS is an easy method to implement, it is not complete for solving not only the
interior 2D problem in case of a degenerate scale but also the exterior problem with
bounded potential at infinity for any scale. Following Fichera’s idea for the boundary
integral equation, we add a free constant and an extra constraint to the traditional
MFS. The reason why the free constant and extra constraint are both required is
clearly explained by using the degenerate kernel for the closed-form fundamental
solution. Since the range of the single-layer integral operator lacks the constant term
in the case of a degenerate scale for a two dimensional problem, we add a constant to
provide a complete base. Due to the rank deficiency of the influence matrix in the
case of a degenerate scale, we can promote the rank by simultaneously introducing a
constant term and adding an extra constraint to enrich the MFS. For an exterior
problem, the fundamental solution does not contain a constant field in the degenerate
kernel expression. To satisfy the bounded potential at infinity, the sum of all source
strengths must be zero. The formulation of the enriched MFS can solve not only the
degenerate-scale problem for the interior problem but also the exterior problem with
bounded potential at infinity. Finally, three examples, a circular domain, an infinite
domain with two circular holes and an eccentric annulus were demonstrated to see
the validity of the enriched MFS.
Keywords: Method of fundamental solutions, Fichera’s method, degenerate scale,
degenerate kernel
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Corresponding author at: Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung 20224, Taiwan.
Tel.:+886-2-24622192x6177; fax:+886-2-24632375. E-mail address: jtchen@mail.ntou.edu.tw (Jeng-Tzong Chen)
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1. Introduction
The method of fundamental solutions (MFS) is a popular method because it is
meshless and easy to implement. As a result, the MFS has been applied to many areas
such as potential problems [1], acoustics [2], elasticity [3] and biharmonic problems
[4-6]. The basic idea of the MFS is to distribute sources outside the domain of interest
and the field potential is approximated by a linear combination of fundamental
solutions. From another point of view, the MFS can be seen as one kind of
discretization of the indirect boundary element method (BEM) by using the
concentrated source to substitute the boundary density distribution. Therefore, some
rank-deficiency problems such as degenerate scale, spurious eigenvalue and fictitious
frequency appearing in the BEM may be also inherent in the MFS since the
formulation of the MFS as well as the indirect BEM [7] is not necessary and sufficient.
Chen et al. [8] used the MFS to solve the problems of multiply-connected membranes
and they found that the spurious eigenvalues exist. Numerical experiments show that
spurious eigensolution depends on the location of the inner boundary where the
sources are distributed. Later, Chen [9] applied the MFS to solve radiation and
scattering problems of exterior acoustics. Numerical results show that there are
irregular frequencies when the MFS is used. However, these irregular frequencies do
not exist in physics and they are called the fictitious frequencies. Whether a
degenerate scale appears in the MFS or not, there are few papers on this topic.
Following the experience of the BEM, we may wonder whether the degenerate scale
may exist or not when the MFS is employed to solve the 2D Laplace problem. To
enrich a formulation of the MFS to overcome the degenerate scale problem is our
main goal.
It is well known that a Fredholm integral equation of the first kind is often used
to solve potential problems. However, as the scale of the domain size dilates or
shrinks to a specific size, it causes failure in the BEM implementation [10] and may
yield unacceptable results. This specific size is termed “degenerate scale”. For this
degenerate scale, there are various terminologies such as critical value [11, 12],
transfinite diameter [13], Gamma contour [14] and the unit logarithmic capacity [15].
Chen et al. [16] used complex variables to derive the degenerate scale of an elliptical
geometry for the elasticity problem and found two degenerate scales. Later, not only
circular boundary but also elliptical, square and triangular boundaries are considered
for the 2D Laplace problems. They also found the degenerate scales when the BEM is
used. Besides, they provided several remedy approaches to improve this drawback of
the BEM. Recently, Chen and his coworkers [17, 18] not only used the null-field
boundary integral equation in conjunction with the degenerate kernel but also
employed the unit logarithmic capacity in conjunction with complex variables to
2
analytically study the degenerate scale of the elliptical boundary. It is known that a
degenerate scale is inherent in the BEM for solving the 2D Laplace problem subject to
the Dirichelet boundary condition. In order to overcome this pitfall, Chen et al. [19]
introduced Fichera’s concept [20] to the indirect BEM. It can not only overcome the
ill-posed problem caused from the degenerate scale, but also can solve the exterior
problem with bounded potential at infinity. Following the Fichera’s idea, they added a
constant term and an extra constraint to deal with the rank-deficiency problem. Based
on this successful experience, we introduce this idea to improve the MFS.
Regarding the traditional MFS, it is possible to obtain an inaccurate solution. As
a result, many investigators proposed to enrich their formulation. Alves and Leitao [21]
introduced the enriched MFS to simulate a crack singularity. Saavedra and Power [22,
23] have added a constant term in their formula and stated that “usually it is necessary
to add a constant term in particular in two dimensions, where it is required for
completeness purposes”. Although the inclusion of the constant in the MFS is
sometimes recommended, it is rarely used in practice as the degenerate cases it
"cures" occur rather rarely. The existence of this critical scale still needs further
investigation. In the literature, several researchers have discussed the inclusion of the
constant [17, 19, 24, 25]. In order to avoid the trouble of choosing a fictitious
boundary, Chen et al. [26] developed the singular boundary method (SBM). They
assumed a test example to calculate the source weighting, and then used this source
weighting to determine the value of diagonal term where the source and field points
can coincide. However, for certain case, the SBM yields an inaccurate approximation.
As a result, they provided an improved formulation of adding a constant term and a
constraint [27, 28]. They called this constraint “moment condition”. However, the role
of the constant and the constraint was not discussed in detail in their papers.
Following Fichera’s idea, the enriched formulation is useful to solve not only the
interior problem, but also the exterior case. Although they [26-30] have successfully
solved the problem, they didn’t deeply examine the role of the constant term and the
constraint. Besides, they also did not discover that the problematic case they
encountered is that their sources are distributed on a degenerate scale. In this article,
we would like to provide an enriched MFS and explain why the constant term and
constraint are required to ensure a unique solution for the degenerate scale. Besides,
the role in exterior problems is also examined.
We examine the incompleteness of the MFS for interior problems containing a
degenerate scale and exterior problems with bounded potential at infinity. Following
Fichera’s idea for the boundary integral equation (BIE), we add a free constant and an
extra constraint to enrich the MFS formulation. The enriched MFS can be used not
only for the interior 2D problem in the case of a degenerate scale but also the exterior
3
problem with bounded potential at infinity for any scale. The extra constraint, the sum
of all source strengths to be zero, is used to make the potential at infinity bounded as
well as suppressing the coefficient of indeterminacy to be zero. However, the constant
term is added to compensate for the range deficiency by a constant field in case of a
degenerate scale. Besides, it can also be used to capture a bounded potential at infinity.
Finally, three examples, a circular domain, an infinite domain with two circular holes
and an eccentric annulus were presented to demonstrate the validity of the enriched
MFS. Besides, a constant potential test in the MFS is designed to verify the range
deficiency instead of the patch test of the constant strain in the finite element method.
2. Formulation
In this section, the MFS is used to solve interior and exterior boundary value
problems for the Laplace equation. A domain D   2 is given, whose boundary B
is simple and closed. The governing equation and boundary condition are shown
below:
2u(x)  0, x  D,
(1)
u (x)  f (x), x  B.
(2)
The fundamental solution of the Laplace equation is
U (s, x)  ln x  s .
(3)
As an alternative to the direct BEM, the indirect BEM solution of interior problem is
expressed in terms of single-layer potential with unknown boundary density of  0
as given below:
u(x)   U (s, x)0 (s)dB(s), x  D,
(4)
f (x)   U (s, x)0 (s)dB(s), x  B.
(5)
B'
B'
Fichera [31] proposed the enriched boundary integral formulation as shown below:
u(x)   U (s, x) (s)dB '(s)  c, x  D,
(6)
f (x)   U (s, x) (s)dB '(s)  c, x  B,
(7)
B'
B'
where B' is the fictitious boundary, and added a constraint equation as shown
below:

B'
 (s)dB '(s)  0,
4
(8)
where  (s) and c are the unknown boundary density and an unknown constant,
respectively. For the interior BVP, it is reduced to solve the simultaneous equations (7)
and (8). For simplicity, to understand the degenerate scale, a circular domain with a
radius a is given for demonstration as shown in Fig. 1. The radius of the fictitious
boundary is a ' , where a '  a for the interior problem, a '  a for the exterior
problem. By setting the field point x  (  ,  ) and the source point s  ( R,  ) , the
fundamental solution using polar coordinates can be expressed by using the following
degenerate kernel,
m

 I
1
U  ln R     cos(m(   )), R   ,
m 1 m  R 

U (s, x)  ln x  s  
m

1 R
 E
U  ln    m    cos(m(   )), R   .
m 1
 

(9)
The unknown boundary density  (s) and the given boundary condition f (x)
can be expanded by using Fourier series along a circular boundary giving


n 1
n 1


n 1
n 1
 (s)  a0   an cos(n )   bn sin(n ), 0    2 ,
(10)
and
f (x)  p0   pn cos(n )   qn sin(n ), 0    2 ,
(11)
where a n and bn are unknown coefficients, and pn and q n are determined from
the Dirichlet boundary condition. Equation (8) yields
2





n 1
n 1

B'  (s)dB '(s)  0  a0   an cos(n )   bn sin(n ) Rd  0 .
(12)
From Eq. (12), we can determine the constant term a0  0 . By substituting the
degenerate kernel and Eq. (11) into Eq. (7), we have
n
n

 

R  
R  
   an
cos(n )   bn
sin(n )   c




 n 1

n R
n R
n 1




n 1
n 1
(13)
 p0   pn cos(n )   qn sin( n ), 0    2 ,
where R  a ' and   a . Therefore, all the unknown coefficients are uniquely
determined as follows:
5
c  p0 ,

 a0  0,

na
 an  
p,
a' n


na
q.
bn  
a' n

(14)
It is found that c is the constant term of the Dirichlet boundary condition. If there is
no constraint and constant c in Eq. (6), Eq. (5) alone yields a non-unique solution
a0 from
2 a0 ln a '  p0 due to ln a '  0 if a '  1 .
Since the MFS can be seen as a discrete version of the indirect BEM, we can
revisit the MFS by using the degenerate kernel. In the enriched MFS formulation, we
have
N
u (x)   U (x, si )  i ,
(15)
i 1
where N is the number of source points and  i is the strength of the ith source,
si  ( Ri ,i ) . By expanding the closed-form fundamental solution, U I (x, s) , into the
degenerate kernel in Eq. (15) for u (x) , we have
m



1
u (x)  U (x, si ) i   ln R     cos m(i   )  i .
i 1
i 1 
m 1 m  R 


N
N
I
(16)
After rearranging the term, we have
1
u(x)  ln R   i     cos m(i   )  i , x  D.
i 1
i 1 m 1 m  R 
N
N
m

(17)
By comparing Eq. (11) with Eq. (17), it is found that
p0  ln R
N
 ,
i
i 1
(18)
when the source distribution is on R  1 ( a  1 ). Equation (18) becomes
p0  0
N
 .
i 1
i
(19)
According to the Fredholm alternative theorem, p0  0 is the case of infinite
solutions, while p0  0 is the case of no solution. Whatever p0 is, it indicates that
6
Eq. (15) is neither sufficient nor necessary to ensure a unique solution. Equation (19)
N

shows that we cannot determine the term
i 1
i
. Following the mapping relationship
between the unknown coefficient of the Trefftz method and the MFS, we similarly
have
N
2 a ' a0    i .
(20)
i 1
The constraint of the indirect BEM, a0  0 , could be transformed to the discrete form
N
in the MFS for any scale as
  i  0. By forcing
i 1
N

i 1
i
to be zero in a similar way
of the indirect BIE of Eq. (18), the constant field is lost in Eq. (17). The outcome of
not including the constant term leads to no solution if p0  0 in the Dirichlet
N
boundary condition. The constraint

i 1
i
 0 and the addition of a constant field, c,
are both required to ensure a unique solution.
Similarly, the solution of the indirect BEM for the exterior problem is presented in Eq.
(4) with the degenerate kernel U E (x, s) as shown below:
u (x)   U E (x, s) 0 (s)dB '(s)
B'

 

R R m
R R m
 2 R ln  a0    am
( ) cos(m )   bm
( ) sin(m )  ,
m 
m 
m 1
 m1

(21)
where R  a ' . The term 2 R ln  a0 is not a constant but is a function of ln 
since x  (  ,  ) . In other words, the representation is range deficient for a constant
term to the exterior problem of any scale. It indicates that adding a free constant is
necessary. We rewrite Eq. (21) by adding a constant c,
u(x)   U E (x, s)0 (s)dB '(s)  c
B'
(22)
When x   , the representation of the solution reveals an unbounded potential due
to the fact that ln    as    . Since the potential at infinity is bounded, the
constraint of Eq. (12) is needed. In case of   1, we cannot determine a0 for
matching the boundary condition if   1 is the location to match the boundary
condition. However, the bounded potential at infinity enforces a0 to be zero and we
7
need to add a free constant in the solution representation. The constraint a0  0 is
equivalent to
a0    (s)dB '(s)  0.
(23)
B'
To extend to a general problem instead of the circular case, Han and Wu [32] also
proved that a bounded potential at infinity yields the following constraint:
lim u (x)  lim  ln x  s  0 (s)dB '(s)  lim ln x
x 
0,


,


x  B '


B'
B'
x 

B'
 0 (s)dB '(s)
 (s) dB '(s)  0.
(24)
 (s) dB '(s)  0.
Physically speaking, the distance between the source on the complementary domain
and the observer at infinity are almost the same as given in Eq. (24). If the resultant
source is not zero, this yields an unbounded potential at infinity. Therefore, the
enforced zero resultant results in bounded potential at infinity.
The MFS approximation for the solution can be represented as
N
u (x)  U E (x, si ) i .
(25)
i 1
By expanding the closed-form fundamental solution into the degenerate kernel,
U E (x, s) , we have
m
1 R
u (x)  ln  i     cos m(i   )  i , R   .
i 1
i 1 m 1 m   
N

N
(26)
After comparing with Eq. (9) and Eq. (25), we have
p0 
N
If

i 1
i
N
 ln   i  ln 
i 1
N
 .
i 1
i
(27)
is not zero, the potential at infinity behaves like ln  as shown in Eq. (27).
The discrete type of the corresponding constraint of Eq. (8) is
N

i 1
i
 0.
(28)
After combining Eq. (27) with Eq. (28), the lacking of a constant field in the right
hand side of Eq. (26) appears for the exterior problem. Therefore, adding a free
constant in Eq. (26) is required. Following the work of Chen et al. [33], the mapping
relationship of unknown coefficients between the indirect BEM and the MFS for the
8
exterior problem is:
N

i 1
i
 2 a ' a0 .
N
It is found that the constraint

i 1
i
(29)
 0 for the MFS is equivalent to a0  0 for the
indirect BEM.
The enriched MFS expression in conjunction with a free constant and the constraint is
summarized below:
N

Solution
representation:
u
(
x
)

U (x, s i ) i  c



i 1
(30)

N
subject to
i  0


i 1
To sum up, the enriched MFS can handle both the interior and exterior problems at the
same time. It also provides the sufficient and necessary equations for the unique
solution for the interior and exterior problems.
3. Results and discussions
In this section, three examples, a circular case, an infinite plane problem
containing two circular holes and an eccentric annulus were designed to verify the
validity of the enriched MFS. The roles of the free constant and the extra constraint in
the enriched MFS are also examined.
Case 1: Circular case - test of constant potential
The benchmark example for the enriched MFS is the Laplace problem for a
circular domain subject to constant Dirichlet data as shown in Fig. 1. The analytical
solution of the boundary value problem is
u (x)  1, x  D.
(31)
By distributing the sources along R  a '  1 (the radius of the circle is 0.5),
Figure 2(a) shows the results of the traditional MFS. After comparing with the
analytical solution, the traditional MFS yields an inaccurate solution due to the
degenerate scale for ln1  0 in the fictitious boundary of the source distribution. The
traditional MFS fails for the specific source distribution on the degenerate scale. The
proof is given in the previous section. The reason why it fails is that the solution
representation by using the MFS with a special source distribution (degenerate scale)
is range deficient by a constant field. We can add a free constant in the MFS
formulation. Since one more unknown constant is introduced which makes the
number of unknowns larger than the number of equations by one, we can numerically
9
overcome this problem by adding one more collocation point to match the boundary
condition instead of an extra constraint. However, offset of the angle for the 51
collocation points and 50 source points appears as shown in Fig. 2(b). The result of
the traditional MFS containing a constant and offset collocation is illustrated in Fig.
2(c). To avoid this inconsistency, a more logical and natural approach with a
theoretical base is to use one constraint of the zero sum of all source strengths. By
adding a constant and an effective constraint, the enriched MFS recovers the accuracy
of the MFS. Figure 2(d) shows acceptable results in comparison with the exact
solution.
Case 2: An infinite problem containing two circular cavities
The sketch of problem and boundary condition are shown in Fig. 3. The
analytical solution is obtained by way of bipolar coordinates as follows:
u (x)  2 U (x, s1 )  2 U (x, s 2 )  1, x  D.
(32)
Nine source distributions are illustrated in Fig. 5(a). Three columns denote
various outer distributions, locating sources including, passing and excluding the left
focus. Three rows show various inner distributions, locating sources including,
passing and excluding the right focus. To sum up, if both a free constant and the
constraint are not considered, it yields inaccurate results as shown in Fig. 4(a). By
only adding a constant without using the constraint, it yields inaccurate results as
shown in Fig. 4(b). The enriched MFS shows the acceptable result in agreement of the
analytical solution as shown in Fig. 4(c). It is found that all types of source
distribution (Types A, B, C,…., I) yield a constant term, c , near the exact value 1 in
Fig. 5(b). It indicates that the content of constant one in the exact solution is fully
represented by the base of a free constant c only. Figure 6 shows that all the
singularity strengths are found to be zero except two nonzero strengths of amplitude 2
and -2 at the left and right foci, respectively. This matches the exact solution very well
from the view point of bipolar coordinates. The reason is that the fundamental
solution ( ln x  s ) contains no constant part in the exterior expression of the
degenerate kernel as shown in Eq. (9). By adding the constraint, the solution of the
MFS yields a bounded potential at infinity.
Case 3: An eccentric annulus
An eccentric annulus of the Laplace problem was solved by using the MFS [33,
34]. However, they did not employ the enriched idea. Although they solve the
problem, their case does not contain the degenerate scale, a '  1 . Here we design a
10
degenerate scale case of source distribution to show the power of an addition constant
and an extra constraint.
For a doubly-connected domain, an eccentric annulus is considered as shown in
Fig. 7 [33, 34]. In this case, the analytical solution is shown below:
u (x)  2.88539 U (x, s1 )  2.88539 U (x, s 2 )  1, x  D.
(33)
Nine kinds of source distributions are shown in Fig. 8. Three columns denote various
outer distributions, locating sources including, passing and excluding the left focus.
Three rows show various inner distributions, locating sources including, passing and
excluding the right focus.
Figure 8(a) shows unacceptable results since the source distribution is on the
degenerate scale. On the other hand, it shows that the result of the enriched MFS
agrees well with the analytical solution in Fig. 8(b). Figure 9 illustrates that three
groups of the value c are found for nine cases of source distribution, c ≒1.50 for
Type A, D, G, c ≒1.00 for Types B, E, H, c ≒0.40 for Types C, F, I. If the outer
source distribution passes the left focus, the value c is found to approach 1.
However, the value c is found to be smaller and larger than one for including and
excluding the left focus, respectively. It is found that the value c is not changed in
the same column. This can be explained that singularities inside the circle could not
contribute the constant field according to the exterior form of the degenerate kernel in
Eq. (21). Therefore, different row by changing the source distribution inside the circle
could not contribute to the constant. Three different columns of outer sources
contribute the constant field by -0.5, 0 and 0.6, for the cases of including, passing and
excluding the left focus, respectively. For the case of passing focus, the constant field
is fully contributed by the free constant c. Figure 10 shows that all the singularity
strengths are found to be zero except two nonzero strengths of amplitude -2.88539
and 2.88539 at the left and right foci, respectively. It also indicates that the value of
the constant term is fully contributed by adding a constant, c , while the source
distribution is along two foci. Figure 11 shows that nine source distributions (Types
A-I) can yield acceptable results.
Although we consider the MFS by adding a free constant and a constraint, this
idea in Fichera’s approach of the indirect BEM can be found [19]. Table 1 shows the
role of the constant and the constraint for interior and exterior problems. The accuracy
of the solution of the single-layer representation is improved in the three cases by
adding a constant and an extra constraint. The enriched MFS is not only free of the
degenerate scale of source distribution for the interior problem but also free of the
unbounded potential at infinity for the exterior problem.
11
4. Conclusions
In this paper, we have demonstrated that the traditional MFS may fail for the
degenerate scale of the source distribution for the interior problem and the exterior
problem with bounded potential at infinity. To enrich the MFS, a free constant and a
corresponding constraint were required. Roles of free constant and the constraint were
also examined. For the exterior problem, the fundamental solution ( ln x  s ) does not
contain the constant field in the exterior expression of degenerate kernel. Therefore, a
free constant should be added. To have a bounded potential at infinity, the constraint
is required. In addition, a bounded potential at infinity yields the constraint that the
sum of boundary densities or source strengths should be zero as mentioned in Han’s
book [32]. Regarding the interior problem, the extra constraint of the sum of source
strengths to be zero is required since it can not be determined due to multiplication to
zero of ln 1  0 in case of the source distribution on the degenerate scale.
Suppression of the coefficient of indeterminacy to be zero yields the rank deficiency
by a constant field. Then, we need to add a free constant. Three examples, a circular
case, an infinite domain containing two circular holes and an eccentric annulus were
illustrated to see the validity of the enriched MFS.
Acknowledgements
The authors wish to thank the support from the National Science Council of
Taiwan under contract NSC 102-2221-E-019-034 for this paper and we also
appreciate Prof. Houde Han for providing us useful information from his book.
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14
Table 1
Table Captions
Why a constant and a constraint are both required in the enriched MFS ?
Figure captions
Fig. 1
Problem sketch of a circular domain. ( a  0.5, a '  1.0 )
N
Fig. 2(a) Unacceptable contour plot of the traditional MFS ( u (x)   U (x, si )  i )
i 1
without the enriched formulation.
Fig. 2(b) Offset of the angle for the collocation point and the source point in case 1.
Fig. 2(c) Contour plot of the traditional MFS containing a constant by using sources
N
with a offset ( u (x)  U (x, si )  i  c ).
i 1
N
Fig. 2(d) Contour plot of the enriched MFS ( u (x)  U (x, si )  i  c and
i 1
N

i 1
Fig. 3
i
 0 ).
Problem sketch of an infinite domain containing two circular holes.
( a1  1.919035, a2  0.469642, c1  1, c2  1 ,
O1 (2.163953, 0), O2 (1.104791, 0) )
N
Fig. 4(a) Unacceptable contour plot of the traditional MFS ( u (x)   U (x, si )  i ) and
i 1
the exact solution. (The black bold circle denotes the real boundary)
Fig. 4(b) Unacceptable contour plot of the traditional MFS containing a constant
N
( u (x)  U (x, si )  i  c ) and the exact solution. (The black bold circle
i 1
denotes the real boundary)
N
Fig. 4(c) Contour plot of the enriched MFS ( u (x)  U (x, si )  i  c and
i 1
N

i 1
i
0)
and the exact solution. (The black bold circle denotes the real boundary)
Fig. 5(a) Sketch of the source distribution Types (A~I) in the infinite domain
containing two circular holes and the value of c . (Real boundaries are
plotted by the solid curve, the dotted curve shows the source distribution
and the red cross denotes the source distribution along the focus)
Fig. 5(b) Contour plot of the solution by using the enriched MFS of the source
distribution Types (A~I) in the domain problem containing two circular
15
holes. (The light blue circle denotes the real boundary)
Fig. 6
Bar chart of the unknown boundary density for the source points in case 2.
(Source distribution is shown at the bottom right corner, and the red point
denotes the focus)
Fig. 7
Problem sketch of an eccentric annulus. ( a1  0.533333, a2  0.2133333,
c1  0.4, c2  0.4 , O1 (0.666667, 0), O2 (0.453333, 0) )
N
Fig. 8(a) Contour plot of the traditional MFS ( u (x)   U (x, si )  i ) and the exact
i 1
solution, where sources are distributed on a degenerate scale, a1  1 . (The
black bold circle denotes the real boundary)
N
Fig. 8(b) Contour plot of the enriched MFS ( u (x)  U (x, si )  i  c and
i 1
N

i 1
i
0)
and the exact solution, where sources are distributed on a degenerate scale,
a1  1 . (The black bold circle denotes the real boundary)
Fig. 9
Sketch of the nine kinds of source distribution in the an eccentric annulus
problem. (Real boundaries are plotted by the solid curve, the dotted curve
shows the source distribution and the red cross denotes the source
distribution along the focus)
Fig. 10 Bar chart of the unknown boundary density for the source points in case 3.
(Source distribution is shown at the bottom right corner, and the red point
denotes the focus)
Fig. 11 Contour plot of the solution by using the enriched MFS of the source
distribution Types (A~I) in the eccentric annulus problem. (The light blue
circle denotes the real boundary)
16
Table 1
Why a constant and a constraint are both required in the enriched MFS ?
Sufficient and necessary
formulation of the MFS
(An enriched MFS)
N
u (x)   U (x, si ) i  c
i 1
N

Source
representation:
u
(
x
)

U (x, s i ) i  c



i 1

N
subject to
i  0


i 1
Interior Domain
Exterior Domain
Range deficiency by a
constant field, c, in case of
degenerate scale
No constant field in the
degenerate (exterior)
kernel for the
fundamental solution for
any scale
N

i 1
i
0
Rank promotion in case of
degenerate scale
Satisfying the bounded
condition at infinity for
any scale
Remark: If the formulation is not appropriate for the solution space, it is impossible to
approach the exact solution even through using many source points. Conversely, if the
formulation is complete for representing the solution space, only a fewer source
points can yield the acceptable solution.
17
Fig. 1 Problem sketch of a circular domain.
( a  0.5, a '  1.0 )
18
N
Fig. 2(a) Unacceptable contour plot of the traditional MFS ( u (x)   U (x, si )  i )
i 1
without the enriched formulation.
19
Source points
Original boundary collocation points
Rearrangement of boundary collocation points
1
0.5
R
’
0
-1
-0.5
0
0.5
1
Original angle
-0.5
Rearranged angle
-1
Fig. 2(b) Offset of the angle for the collocation point and the source point in case 1.
20
0.4
1
1
0.3
0.2
0.1
1
0
-0.1
-0.2
-0.3
1
1
-0.4
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
Fig. 2(c) Contour plot of the traditional MFS containing a constant by using sources
N
with a offset ( u (x)  U (x, si )  i  c ).
i 1
21
0.4
1
1
0.3
0.2
0.1
1
0
-0.1
-0.2
-0.3
1
1
-0.4
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
N
0.4
0.5
Fig. 2(d) Contour plot of the enriched MFS ( u (x)  U (x, si )  i  c and
i 1
N

i 1
i
 0 ).
22
Fig. 3 Problem sketch of an infinite domain containing two circular holes.
( a1  1.919035, a2  0.469642, c1  1, c2  1 , O1 (2.163953, 0), O2 (1.104791, 0) )
23
Exact solution: ———
Numerical solution: – – – – –
N
Fig. 4(a) Unacceptable contour plot of the traditional MFS ( u (x)   U (x, si )  i ) and
i 1
the exact solution. (The black bold circle denotes the real boundary)
24
Exact solution: ———
Numerical solution: – – – – –
Fig. 4(b) Unacceptable contour plot of the traditional MFS containing a constant
N
( u (x)  U (x, si )  i  c ) and the exact solution. (The black bold circle denotes the
i 1
real boundary)
25
Exact solution: ———
Numerical solution: – – – – –
N
Fig. 4(c) Contour plot of the enriched MFS ( u (x)  U (x, si )  i  c and
i 1
N

i 1
and the exact solution. (The black bold circle denotes the real boundary)
26
i
0)
Type A (c=1.000001)
Type B (c=1.000001)
Type C (c=1.000001)
 a'1 =0.663953, a'2 =0.054791
 a'1 =1.163953, a'2 =0.054791
 a'1 =1.663953, a'2 =0.054791
Type D (c=1.000001)
Type E (c=1.000001)
Type F (c=1.000001)
 a'1 =0.663953, a'2 =0.104791
 a'1 =1.163953, a'2 =0.104791
 a'1 =1.663953,a'2 =0.104791
Type G (c=0.999931)
Type H (c=0.999931)
Type I (c=0.999931)
 a'1 =0.663953, a'2 =0.154791
 a'1 =1.163953, a'2 =0.154791
 a'1 =1.663953, a'2 =0.154791
Fig. 5(a) Sketch of the source distribution Types (A~I) in the infinite domain
containing two circular holes and the value of c . (Real boundaries are plotted by the
solid curve, the dotted curve shows the source distribution and the red cross denotes
the source distribution along the focus)
27
Type A (c=1.000001)
Type B (c=1.000001)
Type C (c=1.000001)
Type D (c=1.000001)
Type E (c=1.000001)
Type F (c=1.000001)
Type G (c=0.999931)
Type H (c=0.999931)
Type I (c=0.999931)
Fig. 5(b) Contour plot of the solution by using the enriched MFS of the source
distribution Types (A~I) in the domain problem containing two circular holes. (The
light blue circle denotes the real boundary)
28
Fig. 6 Bar chart of the unknown boundary density for the source points in case 2.
(Source distribution is shown at the bottom right corner, and the red point denotes the
focus)
29
Fig. 7 Problem sketch of an eccentric annulus.
( a1  0.533333, a2  0.2133333, c1  0.4, c2  0.4 ,
O1 (0.666667, 0), O2 (0.453333, 0) )
30
Exact solution: ———
Numerical solution: – – – – –
N
Fig. 8(a) Contour plot of the traditional MFS ( u (x)   U (x, si )  i ) and the exact
i 1
solution, where sources are distributed on a degenerate scale, a1  1 . (The black bold
circle denotes the real boundary)
31
Exact solution: ———
Numerical solution: – – – – –
N
Fig. 8(b) Contour plot of the enriched MFS ( u (x)  U (x, si )  i  c and
i 1
N

i 1
i
0)
and the exact solution, where sources are distributed on a degenerate scale, a1  1 .
(The black bold circle denotes the real boundary)
32
Type A (c=0.40088)
Type B (c=1.00002)
Type C (c=1.49585)
 a'1 =0.86667, a'2 =0.02677
 a'1 =1.06667, a'2 =0.02677
 a'1 =1.26667, a'2 =0.02677
Type D (c=0.40088)
Type E (c=1.00000)
Type F (c=1.49585)
 a'1 =0.86667, a'2 =0.05333
 a'1 =1.06667, a'2 =0.05333
 a'1 =1.26667, a'2 =0.05333
Type G (c=0.40086)
Type H (c=0.99997)
Type I (c=1.49581)
 a'1 =0.86667, a'2 =0.08000
 a'1 =1.06667, a'2 =0.08000
 a'1 =1.26667, a'2 =0.08000
Fig. 9 Sketch of the nine kinds of source distribution in an eccentric annulus problem.
(Real boundaries are plotted by the solid curve, the dotted curve shows the source
distribution and the red cross denotes the source distribution along the focus)
33
Fig. 10 Bar chart of the unknown boundary density for the source points in case 3.
(Source distribution is shown at the bottom right corner, and the red point denotes the
focus)
34
Type A (c=0.40088)
Type B (c=1.00002)
Type C (c=1.49585)
Type D (c=0.40088)
Type E (c=1.00000)
Type F (c=1.49585)
Type G (c=0.40086)
Type H (c=0.99997)
Type I (c=1.49581)
Fig. 11 Contour plot of the solution by using the enriched MFS of the source
distribution Types (A~I) in the eccentric annulus problem. (The light blue circle
denotes the real boundary)
35