ArticlePDF Available

Analyzing Three-Dimensional Laplace Equations Using the Dimension Coupling Method

Authors:

Abstract and Figures

Due to the low computational efficiency of the Improved Element-Free Galerkin (IEFG) method, efficiently solving three-dimensional (3D) Laplace problems using meshless methods has been a longstanding research direction. In this study, we propose the Dimension Coupling Method (DCM) as a promising alternative approach to address this challenge. Based on the Dimensional Splitting Method (DSM), the DCM divides the 3D problem domain into a coupling of multiple two-dimensional (2D) problems which are handled via the IEFG method. We use the Finite Element Method (FEM) in the third direction to combine the 2D discretized equations, which has advantages over the Finite Difference Method (FDM) used in traditional methods. Our numerical verification demonstrates the DCM’s convergence and enhancement of computational speed without losing computational accuracy compared to the IEFG method. Therefore, this proposed method significantly reduces computational time and costs when solving 3D Laplace equations with natural or mixed boundary conditions in a dimensional splitting direction, and expands the applicability of the dimension splitting EFG method.
Content may be subject to copyright.
Citation: Liu, F.; Zuo, M.; Cheng, H.;
Ma, J. Analyzing Three-Dimensional
Laplace Equations Using the
Dimension Coupling Method.
Mathematics 2023,11, 3717.
https://doi.org/10.3390/
math11173717
Academic Editor: Rade Vignjevic
Received: 19 July 2023
Revised: 16 August 2023
Accepted: 21 August 2023
Published: 29 August 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
mathematics
Article
Analyzing Three-Dimensional Laplace Equations Using the
Dimension Coupling Method
Fengbin Liu 1, Mingmei Zuo 2, Heng Cheng 2and Ji Ma 1, *
1College of Mechanical and Vehicle Engineering, Taiyuan University of Technology, Taiyuan 030024, China;
liufengbin@tyut.edu.cn
2School of Applied Science, Taiyuan University of Science and Technology, Taiyuan 030024, China;
mingmei9766@163.com (M.Z.); chengheng@shu.edu.cn (H.C.)
*Correspondence: maji@tyut.edu.cn; Tel.: +86-191-0342-1640
Abstract:
Due to the low computational efficiency of the Improved Element-Free Galerkin (IEFG)
method, efficiently solving three-dimensional (3D) Laplace problems using meshless methods has
been a longstanding research direction. In this study, we propose the Dimension Coupling Method
(DCM) as a promising alternative approach to address this challenge. Based on the Dimensional
Splitting Method (DSM), the DCM divides the 3D problem domain into a coupling of multiple
two-dimensional (2D) problems which are handled via the IEFG method. We use the Finite Element
Method (FEM) in the third direction to combine the 2D discretized equations, which has advantages
over the Finite Difference Method (FDM) used in traditional methods. Our numerical verification
demonstrates the DCM’s convergence and enhancement of computational speed without losing
computational accuracy compared to the IEFG method. Therefore, this proposed method significantly
reduces computational time and costs when solving 3D Laplace equations with natural or mixed
boundary conditions in a dimensional splitting direction, and expands the applicability of the
dimension splitting EFG method.
Keywords:
dimension splitting method; improved element-free Galerkin method; dimension cou-
pling method; finite element method; Laplace equation
MSC: 65N22
1. Introduction
In science and engineering fields, the meshless method has become an important
tool in numerical methods for solving partial differential equations. Compared with the
traditional Finite Element Method (FEM), the meshless method [
1
] is only based on node
distribution, eliminating mesh constraints and allowing for higher-accuracy solutions to
large deformation problems by establishing appropriate shape functions without requiring
grid reconstruction.
The Element-Free Galerkin Method (EFGM) [
2
5
] is a noteworthy meshless method
proposed by Belytschko et al. that utilizes the Moving Least-Squares (MLS) approximation
to construct shape functions. The MLS approximate functions have been improved to
accelerate computational speed, including the Improved Moving Least-Squares (IMLS)
approximation [
6
], interpolating MLS approximation [
7
,
8
], and complex variable MLS
approximation [
9
,
10
]. Using these methods to construct shape functions resulted in the
presentation of the IEFG method [
11
14
], interpolating EFG method [
15
20
], and complex
variable EFG method [10,21,22], respectively.
Due to the complexity of establishing trial functions, the meshless method has been
known to be time-consuming when solving complex 3D mechanics problems. Therefore,
finding ways to analyze complex 3D problems more efficiently has become a significant
research direction in the field of numerical methods. Recently, researchers have made strides
Mathematics 2023,11, 3717. https://doi.org/10.3390/math11173717 https://www.mdpi.com/journal/mathematics
Mathematics 2023,11, 3717 2 of 20
in this area, leading to the development of numerous dimensional split meshless methods,
including the dimensional split complex variable EFG method [
23
26
], dimensional split
EFG method [
27
30
], dimensional split reproducing kernel particle method [
31
34
], and
interpolating dimensional split EFG method [
35
,
36
]. These hybrid meshless methods have
demonstrated the capability to efficiently solve a wide range of 3D partial differential
equations with smaller relative errors.
However, when natural or mixed boundary conditions occur in a dimensional splitting
direction, traditional hybrid meshless methods require the 3D problems to be split into
more layers to achieve higher accuracy. In [
28
], when a natural boundary condition existed
in a split direction, Meng et al. had to choose 50 planes to obtain greater computational
accuracy. In [
33
], the wave propagation problem with mixed boundary conditions was
investigated. The splitting direction of the second numerical example had natural boundary
conditions, so Peng and Cheng had to select 100 planes to achieve a smaller relative error
of 0.62% with a computation time of 15.2 s. In contrast, if the splitting direction is selected
as an essential boundary condition, only 10 layers are required, resulting in an error of
0.24% and a computation time of 1.1 s. Hence, it is worth noting that time consumption
remains a concern when using the FDM in the dimensional split direction to address natural
boundary conditions.
To address the above-mentioned issue of time consumption and explore a novel ap-
proach to enhance the convenience and efficiency of applying natural or mixed boundary
conditions, we propose a modification that replaces the FDM with the FEM in the dimen-
sional splitting direction. The FEM [
37
,
38
] is widely recognized for its advantages over the
FDM in numerical simulations. Therefore, compared with the Dimensional Splitting EFG
Method presented by Meng et al. [
28
], our proposed method can reduce CPU time when
solving partial differential equations with mixed or natural boundary conditions in the
splitting direction.
Due to the low computational efficiency of the EFG for solving 3D problems, re-
searchers have been exploring alternative efficient meshless methods. The Dimension
Coupling Method (DCM) [
39
] as a potential approach for solving the 3D Laplace equation
is presented in this study. By introducing the Dimensional Split Method (DSM), the problem
domain of the 3D Laplace equation is divided into a coupling of multiple 2D problems, and
the IEFG method is employed to handle 2D forms. Two-dimensional discrete equations are
combined by the FEM in the third direction and the final algebraic equation of the original
3D Laplace’s equation is derived.
The error formula of the DCM is given in numerical examples, and we discuss the
influence of meshes of the FEM in the third direction, weight function, other parameters,
and node distribution in 2D domains on precision. Furthermore, the convergence of the
proposed method in this study is numerically verified. Through three numerical examples,
it is demonstrated that the DCM for 3D Laplace equations can significantly improve
computational efficiency without reducing computational accuracy.
2. IMLS Approximation
For a point
x
, the approximation of the corresponding function
u(x)
can be written as
uh(x) = e
Φu=
n
I=1e
ΦI(x)uI,(x), (1)
uT= (u1,u2,··· ,un), (2)
e
Φis the shape function and its form is
e
Φ= ( e
Φ1(x),e
Φ2(x),··· ,e
Φn(x)) = pT(x)A(x)B(x), (3)
Mathematics 2023,11, 3717 3 of 20
pT(x)is the vector of the basis function,
A(x) =
1
(p1,p1)0··· 0
01
(p2,p2)0 0
.
.
..
.
.....
.
.
0 0 ··· 1
(pn,pn)
, (4)
B(x) = PTW, (5)
P=
p1(x1)p2(x1)··· pm(x1)
p1(x2)p2(x2)··· pm(x2)
.
.
..
.
.....
.
.
p1(xn)p2(xn)··· pm(xn)
, (6)
W=
w(xx1)0··· 0
0w(xx2)··· 0
.
.
..
.
.....
.
.
0 0 ··· w(xxn)
, (7)
and w(xxI)is the weighting function.
Equations (1)–(7) are the IMLS approximation [6].
3. Dimension Coupling Method for 3D Laplace Equations
When solving 3D Laplace equations using the IEFG method, the numerical solution
obtained has low computational speed due to the complexity of shape functions compared
to 2D problems.
In this paper, we propose using the DCM to solve this issue. The 3D problem can be
transformed into multiple 2D problems as shown in Figure 1, which are then discretized
using the IEFG method. In the splitting direction, 2D discretized equations are coupled via
the FEM.
Mathematics 2023, 11, x FOR PEER REVIEW 4 of 22
Figure 1. Layers in the direction 3
x of the domain Ω.
The formula of the DCM for 3D Laplace equations is derived in this section. Consid-
ering the governing equation [23]
0=Δu, Ω),,( 321 xxx ; (8)
with boundary conditions
),,( 321 xxxuu =, u
xxx
Γ
),,( 321 , (9)
),,( 3213
3
2
2
1
1
xxxqn
x
u
n
x
u
n
x
u
q=
+
+
=, q
xxx
Γ
),,( 321 ; (10)
q and u are given values, and i
n is the unit outward normal to the boundary Γ
in direction i
x, qu ΓΓΓ =, =
qu ΓΓ .
The governing equation of 2D form in the k-th layer based on the DCM is
2
3
)(2
2
2
)(2
2
1
)(2
x
u
x
u
x
ukkk
=
+
, )(
21 Ω),( k
xx , )(
33
k
xx =, (11)
where )(
Ωk is the 2D domain of the k-th layer of Ω, and
{}
1)(
1
1)(
3
)(
3
)( Ω],[ΩΩ +
=
+
×= L
L
k
kkk xx
, (12)
),,( )(
321
)( kk xxxuu =, (13)
with boundary conditions
),,(),( )(
32121
)()( kkk xxxuxxuu == , )(
21 Γ),( k
u
xx , (14)
),,(),( )(
32121
)()( kkk xxxqxxqq == , )(
21 Γ),( k
q
xx , (15)
where )(
Γk
u and )(
Γk
q are essential and natural boundaries. )()()( ΓΓΓ k
q
k
u
k= , and
=
)()( ΓΓ k
q
k
u.
Figure 1. Layers in the direction x3of the domain .
The formula of the DCM for 3D Laplace equations is derived in this section. Consider-
ing the governing equation [23]
u=0, (x1,x2,x3); (8)
Mathematics 2023,11, 3717 4 of 20
with boundary conditions
u=u(x1,x2,x3),(x1,x2,x3)Γu, (9)
q=u
x1
n1+u
x2
n2+u
x3
n3=q(x1,x2,x3),(x1,x2,x3)Γq; (10)
q
and
u
are given values, and
ni
is the unit outward normal to the boundary
Γ
in
direction xi,Γ=ΓuΓq,ΓuΓq=.
The governing equation of 2D form in the k-th layer based on the DCM is
2u(k)
x2
1
+2u(k)
x2
2
=2u(k)
x2
3
,(x1,x2)(k),x3=x(k)
3, (11)
where (k)is the 2D domain of the k-th layer of , and
=L
k=1n(k)×[x(k)
3,x(k+1)
3]o(L+1), (12)
u(k)=u(x1,x2,x(k)
3), (13)
with boundary conditions
u(k)=u(k)(x1,x2) = u(x1,x2,x(k)
3),(x1,x2)Γ(k)
u, (14)
q(k)=q(k)(x1,x2=q(x1,x2,x(k)
3),(x1,x2)Γ(k)
q, (15)
where
Γ(k)
u
and
Γ(k)
q
are essential and natural boundaries.
Γ(k)=Γ(k)
uΓ(k)
q
, and
Γ(k)
uΓ(k)
q=.
Equations (11), (14) and (15) are then analyzed using the IEFG method. The discretiza-
tion of the second-order partial derivative in the splitting direction is performed using the
FEM, then we obtain the discretized equations.
Π=Z(k)"u 2u
x2
3!#d(k)Z(k)
1
2"u
x12
+u
x22#d(k)ZΓ(k)
q
uqdΓ(k). (16)
Equation (16) is the equivalent functional, and the penalty method is selected for
exerting boundary conditions, hence we can obtain that the modified functional of each 2D
form is
Π=Π+α
2ZΓ(k)
u
(uu)(uu)dΓ(k). (17)
Let
δΠ=0, (18)
hence we have
Z(k)δu·2u
x2
3
d(k)Z(k)δ(Lu)T·(Lu)d(k)ZΓ(k)
q
δu·qdΓ(k)
+αZΓ(k)
u
δu·udΓ(k)αZΓ(k)
u
δu·udΓ(k)=0, (19)
where
L(·) = "
x1
x2#(·). (20)
Mathematics 2023,11, 3717 5 of 20
In the 2D domain
(k)
, we select
M
nodes
xI
; thus, we can obtain the following form:
u(x(k)
I) = u(x(k)
I,x(k)
3) = uI. (21)
From Section 2, the expression of the approximate function of Equation (21) is
u(x,x(k)
3) = e
Φu=
n
I=1e
ΦI(x)uI, (22)
with
u= (u1,u2,··· ,un)T. (23)
Thus,
2u(x,x(k)
3)
x2
3
=2
x2
3
n
I=1e
ΦIu(xI,x(k)
3) =
n
I=1e
ΦI
2uI
x2
3
=e
Φ(x)u”, (24)
Lu(x,x(k)
3) =
n
I=1"
x1
x2#e
ΦIuI=
n
I=1
BIuI=Bu, (25)
where
u= 2u(x1,x(k)
3)
x2
3
,2u(x2,x(k)
3)
x2
3
,··· ,2u(xn,x(k)
3)
x2
3!T
, (26)
B= (B1,B2,··· ,Bn), (27)
BI=e
ΦI,1(x)
e
ΦI,2(x). (28)
Substituting Equations (22), (24) and (25) into Equation (19),
Z(k)δ[e
Φu]T·[e
Φu]d(k)ZΓ(k)
q
δ[e
Φu]T·qdΓ(k)Z(k)δ[Bu]T[Bu]d(k)
+αZΓ(k)
u
δ[e
Φu]T·[e
Φu]dΓ(k)αZΓ(k)
u
δ[e
Φu]T·udΓ(k)=0. (29)
Next, we write Equation (29) in its matrix form
Z(k)δ[e
Φu]T·[e
Φu00 ]d(k)=δuT·[Z(k)e
ΦTe
Φd(k)]·u00 =δuT·C·u, (30)
Z(k)δ[Bu]T·[Bu]d(k)=δuT·[Z(k)BTBd(k)]·u=δuT·K·u, (31)
ZΓ(k)
q
δ[e
Φu]T·qdΓ(k)=δuT·ZΓ(k)
qe
ΦTqdΓ(k)=δuT·fq, (32)
αZΓ(k)
u
δ[e
Φu]T·[e
Φu]dΓ(k)=δuT·[αZΓ(k)
ue
ΦTe
ΦdΓ(k)]·u=δuT·Kα·u, (33)
αZΓ(k)
u
δ[e
Φu]T·udΓ(k)=δuT·αZΓ(k)
ue
ΦTudΓ(k)=δuT·fα, (34)
where
K=Z(k)BTBd(k), (35)
Mathematics 2023,11, 3717 6 of 20
C=Z(k)e
ΦTe
Φd(k), (36)
Kα=αZΓ(k)
ue
ΦTe
ΦdΓ(k), (37)
fq=ZΓ(k)
qe
ΦTqdΓ(k), (38)
fα=αZΓ(k)
ue
ΦTudΓ(k). (39)
Substituting Equations (30)–(34) into Equation (29),
δuT·(Cu+KαuKu fαfq) = 0. (40)
Let
F=fq+fα, (41)
ˆ
K=KαK. (42)
Hence, Equation (40) can be transformed as
Cu+ˆ
Ku=F. (43)
Let
u(x(2)
3) = u(2), (44)
u(x(3)
3) = u(3), (45)
.
.
.
u(x(L)
3) = u(L), (46)
suppose x3[a,c], thus we have
Cu0(c)v(x(L+1)
3)Cu0(a)v(x(1)
3)CZc
au0v0dx3+ˆ
KZc
auvdx3=Zc
aFvdx3, (47)
the test function
v=ϕi
(
i=
1, 2,
···
,
N+
1) is selected with the shape function based
on piecewise linear interpolation functions, thus
v(x(1)
3) =
0 and
v(x(L+1)
3) =
0 when
x3=x(2)
3,x(3)
3,··· ,x(L1)
3,x(L)
3. Hence, Equation (47) is changed to the following form
ˆ
K
L+1
k=1
u(k)Zc
aϕkϕidx3C
L+1
k=1
u(k)Zc
aϕ0kϕ0idx3=Zc
aF(k)ϕidx3,i=2, 3, ··· ,N. (48)
Let
Hik = L+1
k=1Zc
aϕkϕidx3!ˆ
K L+1
k=1Zc
aϕ0kϕ0idx3!C,i=2, 3, ··· ,N, (49)
Wi=Zc
aF(k)ϕidx3,i=1, 2, ··· ,N+1. (50)
Therefore, Equation (48) is transformed as
Hiku(k)=Wi,i=2, 3, ··· ,N. (51)
Mathematics 2023,11, 3717 7 of 20
If the Dirichlet boundary conditions are known,
u(L+1)=u(c), (52)
u(1)=u(a). (53)
Hence, Equation (47) is written in the following form:
ˆ
Eu(1)=u(a),x3=x(1)
3; (54)
H21u(1)+H22 u(2)+H22u(3)=W2,x3=x(2)
3; (55)
H32u(2)+H33 u(3)+H34u(4)=W3,x3=x(3)
3; (56)
.
.
.
HL,L1u(L1)+HL,Lu(L)+HL,L+1u(L+1)=WL,x3=x(L)
3; (57)
ˆ
Eu(L+1)=u(c),x3=x(L+1)
3; (58)
Let
H=
ˆ
E
H21 H22 H23
H32 H33 H34
...
HL,L1HL,LHL,L+1
ˆ
E
, (59)
W=(u(a))T,W2T,··· ,(u(c))TT, (60)
U=u(1)T,u(2)T,··· ,u(L+1)TT, (61)
thus the final linear equation of the 3D Laplace equation is
HU=W. (62)
If mixed boundary conditions are known,
u0(x(L+1)
3) = u0(c), (63)
u(x(1)
3) = u(a), (64)
Equations (58), (59), and (60) can be changed as follows:
HL+1,Lu(L)+HL+1,L+1u(L+1)=WL+1Cu0(c),x3=x(L+1)
3, (65)
Mathematics 2023,11, 3717 8 of 20
H=
ˆ
E
H21 H22 H23
H32 H33 H34
...
HL,L1HL,LHL,L+1
HL+1,LHL+1,L+1
, (66)
and
W=(u(a))T,W2T,W3T,··· ,WLT,WL+1Cu0(c)TT, (67)
respectively.
Equations (8)–(67) are the DCM for 3D Laplace equations.
4. Numerical Examples
In this section, four numerical examples are calculated using the DCM.
The formula of error is
uuh
rel
L2()=R(uuh)2d1/2
kukL2()
. (68)
We employ the IEFG method based on some distributed nodes and use linear basis
functions to construct trial functions in the 2D computational domain. Additionally, 4
×
4
Gaussian points are used in each cell with two Gaussian points used in the dimensional
split direction in each mesh.
The first example [40,41] is
2u=0, (69)
and the problem domain is = [0, 1]3,
u=sin(πx2)sin(πx3),(x1=0), (70)
u=2 sin(πx2)sin(πx3),(x1=1), (71)
u=0, (x2=x3=0, x2=x3=1), (72)
Equations (70)–(72) are boundary conditions.
u= [2 sin h(π2x1) + sin h(π2(1x1)] sin(πx2)sin(πx3)
sin h(π2). (73)
Equation (73) is the analytical solution.
In this example, we investigate the convergence of the DCM before obtaining the
greater computational accuracy of the numerical solution.
(1) Weight function.
The impact of weight functions on the relative error of the numerical solutions is
discussed. In the case of using the cubic spline function, we choose 19
×
19 regular nodes
and 18
×
18 integral cells. In the x
3
split direction, we employ the FEM with the mesh
number of 10, with d
max
= 1.48 and
α
= 2.1
×
10
6
. The resulting relative error of our
proposed DCM is 0.1435%. For the case of using the quartic spline function, we maintain
the same background integral grid and node distributions. In the x
3
split direction, the
mesh number of the FEM is set to 18, with d
max
= 1.35 and
α
= 3.9
×
10
6
. As a result, the
relative error is 0.1516%. These results indicate that the relative error of the quartic spline
function is slightly larger than that of the cubic spline function. Consequently, the cubic
spline function is used in the following analysis.
Mathematics 2023,11, 3717 9 of 20
(2) Scale parameter.
We choose 19
×
19 regular nodes and 18
×
18 integral cells. The mesh number of the
FEM in the x
3
split direction is 18 and we set
α
= 2.1
×
10
6
. Figure 2shows the relationship
between the scale parameter d
max
and the error. From Figure 2, we can conclude that when
dmax = 1.4~1.5, the computational accuracy of the numerical solution is higher.
Mathematics 2023, 11, x FOR PEER REVIEW 10 of 22
1.01.11.21.31.41.51.6
0.0010
0.0015
0.0020
0.0025
0.0030
0.0035
0.0040
0.0045
Erro
r
dmax
DCM
Figure 2. The correlation between dmax and the errors.
(3) Penalty factor.
We select 19 × 19 regular nodes and 18 × 18 integral cells. The mesh number of the
FEM in the x3 split direction is 18 with dmax = 1.48. The computational accuracy of the nu-
merical solution is higher when α = 2.1 × 106 in Figure 3.
103104105106
0.001
0.002
0.003
0.004
0.005
0.006
0.007
Error
α
DCM
Figure 3. The correlation between α and the errors.
Dierent node distributions in the k-th plane and meshes in the x3 direction are ana-
lyzed in the following, respectively.
(4) Node distribution.
We s el ect dmax = 1.48, α = 2.1 × 106, and 18 meshes in the x3 split direction. The relation-
ship between the relative error and the number of nodes is shown in Figure 4. We can see
that as the number of nodes increases, the relative error tends to decrease.
Figure 2. The correlation between dmax and the errors.
(3) Penalty factor.
We select 19
×
19 regular nodes and 18
×
18 integral cells. The mesh number of the
FEM in the x
3
split direction is 18 with d
max
= 1.48. The computational accuracy of the
numerical solution is higher when α= 2.1 ×106in Figure 3.
Mathematics 2023, 11, x FOR PEER REVIEW 10 of 22
1.01.11.21.31.41.51.6
0.0010
0.0015
0.0020
0.0025
0.0030
0.0035
0.0040
0.0045
Erro
r
dmax
DCM
Figure 2. The correlation between dmax and the errors.
(3) Penalty factor.
We select 19 × 19 regular nodes and 18 × 18 integral cells. The mesh number of the
FEM in the x3 split direction is 18 with dmax = 1.48. The computational accuracy of the nu-
merical solution is higher when α = 2.1 × 106 in Figure 3.
103104105106
0.001
0.002
0.003
0.004
0.005
0.006
0.007
Error
α
DCM
Figure 3. The correlation between α and the errors.
Dierent node distributions in the k-th plane and meshes in the x3 direction are ana-
lyzed in the following, respectively.
(4) Node distribution.
We s el ect dmax = 1.48, α = 2.1 × 106, and 18 meshes in the x3 split direction. The relation-
ship between the relative error and the number of nodes is shown in Figure 4. We can see
that as the number of nodes increases, the relative error tends to decrease.
Figure 3. The correlation between αand the errors.
Different node distributions in the k-th plane and meshes in the x
3
direction are
analyzed in the following, respectively.
(4) Node distribution.
We select d
max
= 1.48,
α
= 2.1
×
10
6
, and 18 meshes in the x
3
split direction. The
relationship between the relative error and the number of nodes is shown in Figure 4. We
can see that as the number of nodes increases, the relative error tends to decrease.
Mathematics 2023,11, 3717 10 of 20
Mathematics 2023, 11, x FOR PEER REVIEW 11 of 22
Figure 4. The correlation between nodes and the errors.
(5) Mesh number.
We use 19 × 19 regular nodes and 18 × 18 integral cells in each 2D domain, seing dmax
= 1.48 and α = 2.1 × 106. The relationship between the relative error and the mesh number
is shown in Figure 5. We can see that as the number of meshes increases in the x3 spliing
direction, the computational accuracy steadily improves.
8 10121416182022242628
0.000
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
Error
Meshes
DCM
Figure 5. The correlation between meshes and the errors.
As discussed above, the DCM for the 3D Laplace equation is convergent.
Using the proposed DCM in this paper to solve Example 1, we select 19 × 19 regular
nodes and 18 × 18 integral cells. In the x3 spliing direction, we employ the FEM with a
mesh number of 18, dmax = 1.48, and α = 2.1 × 106. As a result, the relative error is 0.1435%
and the computational time is 1.5 s.
Figure 4. The correlation between nodes and the errors.
(5) Mesh number.
We use 19
×
19 regular nodes and 18
×
18 integral cells in each 2D domain, setting
d
max
= 1.48 and
α
= 2.1
×
10
6
. The relationship between the relative error and the mesh
number is shown in Figure 5. We can see that as the number of meshes increases in the x
3
splitting direction, the computational accuracy steadily improves.
Mathematics 2023, 11, x FOR PEER REVIEW 11 of 22
Figure 4. The correlation between nodes and the errors.
(5) Mesh number.
We use 19 × 19 regular nodes and 18 × 18 integral cells in each 2D domain, seing dmax
= 1.48 and α = 2.1 × 106. The relationship between the relative error and the mesh number
is shown in Figure 5. We can see that as the number of meshes increases in the x3 spliing
direction, the computational accuracy steadily improves.
8 10121416182022242628
0.000
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
Error
Meshes
DCM
Figure 5. The correlation between meshes and the errors.
As discussed above, the DCM for the 3D Laplace equation is convergent.
Using the proposed DCM in this paper to solve Example 1, we select 19 × 19 regular
nodes and 18 × 18 integral cells. In the x3 spliing direction, we employ the FEM with a
mesh number of 18, dmax = 1.48, and α = 2.1 × 106. As a result, the relative error is 0.1435%
and the computational time is 1.5 s.
Figure 5. The correlation between meshes and the errors.
As discussed above, the DCM for the 3D Laplace equation is convergent.
Using the proposed DCM in this paper to solve Example 1, we select 19
×
19 regular
nodes and 18
×
18 integral cells. In the x
3
splitting direction, we employ the FEM with a
mesh number of 18, d
max
= 1.48, and
α
= 2.1
×
10
6
. As a result, the relative error is 0.1435%
and the computational time is 1.5 s.
In contrast, when applying the IEFG method to solve Example 1, we choose
19 ×19 ×19
regular nodes, 18
×
18
×
18 integral cells, and the cubic spline weight function. By selecting
d
max
= 1.32 and
α
= 1.5
×
10
3
, the resulting relative error is 0.2134% with a computational
time of 63.9 s.
The comparison of the computational accuracy and time of the DCM and the IEFG
method is shown in Table 1.
Mathematics 2023,11, 3717 11 of 20
Table 1. The comparison of computational accuracy and time of the DCM and the IEFG.
Method Regular Nodes Relative Error CPU Time (s)
DCM 19 ×19 ×19 0.1435% 1.5
IEFG 19 ×19 ×19 0.2134% 63.9
A comparison is performed between the numerical solutions obtained using the DCM
and the IEFG method, along with the exact solutions depicted in Figures 68. It can be
observed that the results of these two methods accord well with the analytical solutions.
Figure 6. The numerical results of the DCM, IEFG, and analytical solutions along x1.
Mathematics 2023, 11, x FOR PEER REVIEW 12 of 22
In contrast, when applying the IEFG method to solve Example 1, we choose 19 × 19 ×
19 regular nodes, 18 × 18 × 18 integral cells, and the cubic spline weight function. By se-
lecting dmax = 1.32 and α = 1.5 × 103, the resulting relative error is 0.2134% with a computa-
tional time of 63.9 s.
The comparison of the computational accuracy and time of the DCM and the IEFG
method is shown in Table 1.
Table 1. The comparison of computational accuracy and time of the DCM and the IEFG.
Method Regular Nodes Relative Error CPU Time (s)
DCM 19 × 19 × 19 0.1435% 1.5
IEFG 19 × 19 × 19 0.2134% 63.9
A comparison is performed between the numerical solutions obtained using the
DCM and the IEFG method, along with the exact solutions depicted in Figures 6–8. It can
be observed that the results of these two methods accord well with the analytical solutions.
1/18 3/18 5/18 7 /18 9/18 11/18 13/18 15/18 17/18
0.0
0.1
0.2
0.3
0.4
0.5
u(x1,3/18,15/18)
x1
Analytical
DCM
IEFG
Figure 6. The numerical results of the DCM, IEFG, and analytical solutions along x1.
1/18 3/18 5/18 7/18 9/18 11/ 18 13/18 15/18 17/18
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
u(2/18,x2,10/18)
x2
Analytical
DCM
IEFG
Figure 7. The numerical results of the DCM, IEFG, and analytical solutions along x2.
Mathematics 2023,11, 3717 12 of 20
Mathematics 2023, 11, x FOR PEER REVIEW 13 of 22
Figure 7. The numerical results of the DCM, IEFG, and analytical solutions along x2.
1/18 3/18 5/18 7/18 9/18 11/18 13/18 15/18 17/18
0.0
0.2
0.4
0.6
0.8
1.0
1.2
u(16/18,7/18,x3)
x3
Analytical
DCM
IEFG
Figure 8. The numerical results of the DCM, IEFG, and analytical solutions along x3.
We can see that the DCM not only exhibits superior calculation accuracy but also
signicantly enhances the computational eciency of the IEFG method in the numerical
results of Figures 6–8.
The second example [1] is
0
2= u, (74)
this problem domain is Ω = [0, 1]3, and
0|||| 1,20,21,10,1 2211 ==== ==== xxxx uuuu , (75)
π)2πtanh(2
)πcos()πcos(
|21
0
3
xx
ux=
=, (76)
0| 1,3 2=
=x
u. (77)
Equations (75)–(77) are boundary conditions, and
= π)2πtanh(2
)π2cosh(
π2
)π2sinh(
)πcos()πcos( 33
21
xx
xxu (78)
is the analytical solution.
In Example 2, 15 × 15 regular nodes and 14 × 14 integral cells are selected in each 2D
domain. We employ the FEM in the x3 split direction with a mesh number of 14 and dmax =
1.34. The resulting relative error is 0.3796% and the computational time is 0.6 s.
In contrast, when analyzing Example 2 using the IEFG method, we choose the cubic
spline weight function along with 15 × 15 × 15 regular nodes and 14 × 14 × 14 integral cells.
Moreover, we set dmax = 1.0 and α = 5.8 × 102. The corresponding relative error is 0.3745%
and the computational time is 12.9 s.
The comparison of the computational accuracy and time of the DCM and the IEFG
method is shown in Table 2.
Figure 8. The numerical results of the DCM, IEFG, and analytical solutions along x3.
We can see that the DCM not only exhibits superior calculation accuracy but also
significantly enhances the computational efficiency of the IEFG method in the numerical
results of Figures 68.
The second example [1] is
2u=0, (74)
this problem domain is = [0, 1]3, and
u,1x1=0=u,1 x1=1=u,2x2=0=u,2x2=1=0, (75)
u|x3=0=cos(πx1)cos(πx2)
2πtanh(2π), (76)
u,3x2=1=0. (77)
Equations (75)–(77) are boundary conditions, and
u=cos(πx1)cos(πx2)"sinh(2πx3)
2πcosh(2πx3)
2πtanh(2π)#(78)
is the analytical solution.
In Example 2, 15
×
15 regular nodes and 14
×
14 integral cells are selected in each
2D domain. We employ the FEM in the x
3
split direction with a mesh number of 14 and
dmax = 1.34. The resulting relative error is 0.3796% and the computational time is 0.6 s.
In contrast, when analyzing Example 2 using the IEFG method, we choose the cubic
spline weight function along with 15
×
15
×
15 regular nodes and 14
×
14
×
14 integral
cells. Moreover, we set d
max
= 1.0 and
α
= 5.8
×
10
2
. The corresponding relative error is
0.3745% and the computational time is 12.9 s.
The comparison of the computational accuracy and time of the DCM and the IEFG
method is shown in Table 2.
Table 2. The comparison of computational accuracy and time of the DCM and the IEFG.
Method Regular Nodes Relative Error CPU Time (s)
DCM 15 ×15 ×15 0.3796% 0.6
IEFG 15 ×15 ×15 0.3745% 12.9
Mathematics 2023,11, 3717 13 of 20
The numerical solution of the DCM is compared with that of the IEFG method and the
exact solutions in Figures 911. It can be observed that the results of these two numerical
methods accord well with the exact ones.
Mathematics 2023, 11, x FOR PEER REVIEW 14 of 22
Table 2. The comparison of computational accuracy and time of the DCM and the IEFG.
Method Regular Nodes Relative Error CPU Time (s)
DCM 15 × 15 × 15 0.3796% 0.6
IEFG 15 × 15 × 15 0.3745% 12.9
The numerical solution of the DCM is compared with that of the IEFG method and
the exact solutions in Figures 9–11. It can be observed that the results of these two numer-
ical methods accord well with the exact ones.
1/14 3/14 5/14 7 /14 9/14 11/14 13/14
-0.06
-0.04
-0.02
0.00
0.02
0.04
0.06
0.08
u(x1,5/14,2/14)
x1
Analytical
DCM
IEFG
Figure 9. The numerical results of the DCM, IEFG, and analytical solutions along x1.
1/14 3/14 5/14 7/14 9/1 4 11/14 1 3/14
-0.08
-0.06
-0.04
-0.02
0.00
0.02
0.04
0.06
0.08
u(10/14,x2,2/14)
x2
Analytical
DCM
IEFG
Figure 10. The numerical results of the DCM, IEFG, and analytical solutions along x2.
Figure 9. The numerical results of the DCM, IEFG, and analytical solutions along x1.
Mathematics 2023, 11, x FOR PEER REVIEW 14 of 22
Table 2. The comparison of computational accuracy and time of the DCM and the IEFG.
Method Regular Nodes Relative Error CPU Time (s)
DCM 15 × 15 × 15 0.3796% 0.6
IEFG 15 × 15 × 15 0.3745% 12.9
The numerical solution of the DCM is compared with that of the IEFG method and
the exact solutions in Figures 9–11. It can be observed that the results of these two numer-
ical methods accord well with the exact ones.
1/14 3/14 5/14 7 /14 9/14 11/14 13/14
-0.06
-0.04
-0.02
0.00
0.02
0.04
0.06
0.08
u(x1,5/14,2/14)
x1
Analytical
DCM
IEFG
Figure 9. The numerical results of the DCM, IEFG, and analytical solutions along x1.
1/14 3/14 5/14 7/14 9/1 4 11/14 1 3/14
-0.08
-0.06
-0.04
-0.02
0.00
0.02
0.04
0.06
0.08
u(10/14,x2,2/14)
x2
Analytical
DCM
IEFG
Figure 10. The numerical results of the DCM, IEFG, and analytical solutions along x2.
Figure 10. The numerical results of the DCM, IEFG, and analytical solutions along x2.
Mathematics 2023, 11, x FOR PEER REVIEW 15 of 22
1/14 3/14 5/14 7/14 9/14 11/14 13/14
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040
u(6/14,11/14,x3)
x3
Analytical
DCM
IEFG
Figure 11. The numerical results of the DCM, IEFG, and analytical solutions along x3.
The results demonstrate that the DCM exhibits higher eciency compared to the
IEFG method, despite both methods yielding similar errors. Furthermore, the DCM has
the advantage of eciently handling mixed boundary conditions without requiring ex-
cessive layers in the dimensional spliing direction.
The third example is
0
2= u, (79)
this problem domain is Ω = [0, 1]3, and
2
3
2
2xxu += , ( 0
1=x), (80)
2
3
2
2
2xxu ++= , ( 1
1=x), (81)
2
3
2
1
2xxu += , ( 0
2=x), (82)
2
3
2
112 xxu ++= , ( 1
2=x), (83)
2
2
2
1
2xxu += , ( 0
3=x), (84)
12 2
2
2
1++= xxu , ( 1
3=x). (85)
Equations (80)–(85) are boundary conditions.
2
3
2
2
2
1
2xxxu ++= (86)
is the analytical solution.
Then, we apply the DCM to solve Example 3 and analyze three situations in which
the FEM is applied in dierent directions.
(1) The FEM applied in the x1 direction.
The mesh number is 18 with dmax = 1.51 and α = 2.5 × 104. We select 11 × 11 regular
nodes and 10 × 10 integral cells in each 2D domain. The resulting relative error is 0.1025%
and the computational time is 0.3 s.
Figure 11. The numerical results of the DCM, IEFG, and analytical solutions along x3.
Mathematics 2023,11, 3717 14 of 20
The results demonstrate that the DCM exhibits higher efficiency compared to the IEFG
method, despite both methods yielding similar errors. Furthermore, the DCM has the
advantage of efficiently handling mixed boundary conditions without requiring excessive
layers in the dimensional splitting direction.
The third example is
2u=0, (79)
this problem domain is = [0, 1]3, and
u=x2
2+x2
3,(x1=0), (80)
u=2+x2
2+x2
3,(x1=1)(81)
u=2x2
1+x2
3,(x2=0), (82)
u=2x2
1+1+x2
3,(x2=1), (83)
u=2x2
1+x2
2,(x3=0), (84)
u=2x2
1+x2
2+1, (x3=1). (85)
Equations (80)–(85) are boundary conditions.
u=2x2
1+x2
2+x2
3(86)
is the analytical solution.
Then, we apply the DCM to solve Example 3 and analyze three situations in which the
FEM is applied in different directions.
(1) The FEM applied in the x1direction.
The mesh number is 18 with d
max
= 1.51 and
α
= 2.5
×
10
4
. We select 11
×
11 regular
nodes and 10
×
10 integral cells in each 2D domain. The resulting relative error is 0.1025%
and the computational time is 0.3 s.
(2) The FEM applied in the x2or x3direction.
The mesh number is 10 with d
max
= 1.32 and
α
= 4.2
×
10
4
. We select 11
×
11 regular
nodes and 10
×
10 integral cells in each 2D domain. The resulting relative error is 0.1934%
and the computational time is 0.2 s.
Furthermore, we employ the IEFG method for analysis. In this case, we choose
11 ×11 ×11
regular nodes, 10
×
10
×
10 integral cells, and the cubic spline weight function.
The parameters used are d
max
= 1.33 and
α
= 1.4
×
10
3
. The resulting error is 0.1544% with
a calculation time of 7.1 s.
The comparison of the computational accuracy and time of the DCM and the IEFG
method is shown in Table 3.
Table 3. The comparison of computational accuracy and time of the DCM and the IEFG.
Method Regular Nodes Relative Error CPU Time (s)
DCM (split in x1direction) 11 ×11 ×11 0.1025% 0.3
DCM (split in x2or x3direction) 11 ×11 ×11 0.1934% 0.2
IEFG 11 ×11 ×11 0.1544% 7.1
When the FEM is used in the x
1
-axis splitting direction, it can be observed that the
DCM can obtain a smaller error. Comparing the numerical solution of the DCM with those
Mathematics 2023,11, 3717 15 of 20
of the IEFG method and the exact ones in Figures 1214, the results of these two methods
agree well with the exact ones.
Mathematics 2023, 11, x FOR PEER REVIEW 16 of 22
(2) The FEM applied in the x2 or x3 direction.
The mesh number is 10 with dmax = 1.32 and α = 4.2 × 104. We select 11 × 11 regular
nodes and 10 × 10 integral cells in each 2D domain. The resulting relative error is 0.1934%
and the computational time is 0.2 s.
Furthermore, we employ the IEFG method for analysis. In this case, we choose 11 ×
11 × 11 regular nodes, 10 × 10 × 10 integral cells, and the cubic spline weight function. The
parameters used are dmax = 1.33 and α = 1.4 × 103. The resulting error is 0.1544% with a
calculation time of 7.1 s.
The comparison of the computational accuracy and time of the DCM and the IEFG
method is shown in Table 3.
Table 3. The comparison of computational accuracy and time of the DCM and the IEFG.
Method Regular Nodes Relative Error CPU Time (s)
DCM (split in x1 direction) 11 × 11 × 11 0.1025% 0.3
DCM (split in x2 or x3 direction) 11 × 11 × 11 0.1934% 0.2
IEFG 11 × 11 × 11 0.1544% 7.1
When the FEM is used in the x1-axis spliing direction, it can be observed that the
DCM can obtain a smaller error. Comparing the numerical solution of the DCM with those
of the IEFG method and the exact ones in Figures 1214, the results of these two methods
agree well with the exact ones.
0.0 0.2 0.4 0.6 0.8 1.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
u(x1, 0.2, 0.9)
x1
Analytical
DCM
IEFG
Figure 12. The numerical results of the DCM, IEFG, and analytical solutions along x1.
Figure 12. The numerical results of the DCM, IEFG, and analytical solutions along x1.
Mathematics 2023, 11, x FOR PEER REVIEW 17 of 22
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
u(0.5, x2, 0.8)
x2
Analytical
DCM
IEFG
Figure 13. The numerical results of the DCM, IEFG, and analytical solutions along x2.
0.0 0.2 0.4 0.6 0.8 1.0
-0.4
-0.2
0.0
0.2
0.4
0.6
u(0.5, 0.3,
x
3)
x3
Analytical
DCM
IEFG
Figure 14. The numerical results of the DCM, IEFG, and analytical solutions along x3.
The fourth example [27,31] is in an irregular-shaped domain selected as
0
2= u, ( ]1,0[],,0[],2,1[ 3 xr
π
θ
). (87)
The boundary conditions are
33 sin),,1( xxu +=
θ
θ
, (88)
33 ),,2( xxu =
θ
, (89)
33 ),0,( xxru =, (90)
33 ),π,( xxru =, (91)
Figure 13. The numerical results of the DCM, IEFG, and analytical solutions along x2.
Mathematics 2023, 11, x FOR PEER REVIEW 17 of 22
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
u(0.5, x2, 0.8)
x2
Analytical
DCM
IEFG
Figure 13. The numerical results of the DCM, IEFG, and analytical solutions along x2.
0.0 0.2 0.4 0.6 0.8 1.0
-0.4
-0.2
0.0
0.2
0.4
0.6
u(0.5, 0.3,
x
3)
x3
Analytical
DCM
IEFG
Figure 14. The numerical results of the DCM, IEFG, and analytical solutions along x3.
The fourth example [27,31] is in an irregular-shaped domain selected as
0
2= u, ( ]1,0[],,0[],2,1[ 3 xr
π
θ
). (87)
The boundary conditions are
33 sin),,1( xxu +=
θ
θ
, (88)
33 ),,2( xxu =
θ
, (89)
33 ),0,( xxru =, (90)
33 ),π,( xxru =, (91)
Figure 14. The numerical results of the DCM, IEFG, and analytical solutions along x3.
Mathematics 2023,11, 3717 16 of 20
The fourth example [27,31] is in an irregular-shaped domain selected as
2u=0, (r[1, 2],θ[0, π],x3[0, 1]), (87)
The boundary conditions are
u(1, θ,x3) = sin θ+x3, (88)
u(2, θ,x3) = x3, (89)
u(r, 0, x3) = x3, (90)
u(r,π,x3) = x3, (91)
u(r,θ, 0) = 4
31
rr
4sin θ, (92)
u(r,θ, 1) = 4
31
rr
4sin θ+1. (93)
The analytical solution of this problem is
u(r,θ,x3) = 4
31
rr
4sin θ+x3. (94)
For this example, the FEM is applied in the x
3
split direction with a mesh number of 10.
A total of 9
×
31 nodes are distributed in a half-torus domain for the 2D problem. Among
these nodes, 9 nodes are positioned along the radial direction r, and 31 nodes are uniformly
distributed along the angle axis
θ
as shown in Figure 15 given in [
27
]. This means that the
integral node distribution is 9
×
31
×
11 with d
max
= 1.0 and
α
= 0.12. The resulting relative