Using Nonuniform Nodal Space in
Meshless Groundwater Modeling
Kuo-Chin Hsu, Wen-Han Tsai, and Der-Liang Young
Department of Resources Engineering, National Cheng Kung University,
Tainan 70101,Taiwan, R.O.C.
Department of Civil Engineering, National Taiwan University,Taipei,
106,Taiwan
1
Background and Motivation
 Mesh-dependent numerical methods are
commonly used.
 However, meshing and remeshing in traditional
numerical methods is a tedious work.
 Can numerical methods be done without
meshing? Meshless method!
 How does the meshless method apply to
groundwater modeling?
2
Types of meshless method
 Common used meshless methods
Node-distribution for RBFCM
(1) Domain-type
(Hu et al. 2007)
- Global Radial basis function collocation method (RBFCM)
- Localized radial basis function collocation method (LRBFCM)
(2) Boundary-type
-Method of fundamental solution (MFS)
-Method of particular solution (MPS)
-Trefftz method (TM)
Node distribution for MFS
(Young et al. 2006)
3
R11
R12
Radial basis functions (RBF)
 𝜑𝑖 =
𝑁
𝑗=1 𝛼𝑗 𝑅𝑖𝑗 (𝑟𝑖𝑗 )
R13
R14
?
R15
𝜑𝑖 : unknown , 𝛼𝑗 : undetermined coefficient , 𝑅𝑖𝑗 : radial basis function (RBF)
𝑟𝑖𝑗 : distance between ith point of interest and jth node, 𝑁 : number of source nodes
𝑟𝑖𝑗 = 𝑥𝑖 − 𝑥𝑗 (1D, Cartesian)
𝑟𝑖𝑗 = (𝑥𝑖 , 𝑦𝑖 ) − (𝑥𝑗 , 𝑦𝑗 ) (2D, Cartesian)
 Types of common used RBF
-
𝑟𝑖𝑗2
+ 𝒄2
- 𝑟𝑖𝑗2 +
- 𝑒
2
−𝑐𝑟𝑖𝑗
1
2
, Multiquardrics (MQ)
1
−
𝑐2 2
, Inverse Multiquardrics
, Gaussian (EXP)
- 𝑟𝑖𝑗2 log(𝑟𝑖𝑗 ) , Thin plate splines (TPS)
4
Algorithm of RBFCM
𝜕2𝜑

𝜕𝑥 2
Let 𝜑𝑖 =
𝑟13
= 0, 𝜑0 = 𝑏0 , 𝜑𝐿 = 𝑏𝐿
𝑁=3
𝑗=1 𝛼𝑗 𝑅𝑖𝑗 (𝑟𝑖𝑗 ), 𝑖
𝑟12
= 1, 2, 3
For G.E.
 𝑅21 (𝑟21 )′′𝛼1 + 𝑅22 (𝑟22 )′′𝛼2 + 𝑅23 (𝑟23 )′′𝛼3 = 0
For B.C.s
 𝑅11 (𝑟11 )𝛼1 + 𝑅12 (𝑟12 )𝛼2 + 𝑅13 (𝑟13 )𝛼3 = 𝑏0
𝑅31 (𝑟31 )𝛼1 + 𝑅32 (𝑟32 )𝛼2 + 𝑅33 (𝑟33 )𝛼3 = 𝑏𝐿

2
3
x=0
x=L
𝑅11 𝛼1 + 𝑅12 𝛼2 + 𝑅13 𝛼3 = 𝑏0
𝑅21 𝛼1 + 𝑅22 𝛼2 + 𝑅23 𝛼3 = 0
𝑅31 𝛼1 + 𝑅32 𝛼2 + 𝑅33 𝛼3 = 𝑏𝐿
𝑅11
 𝑅21
𝑅31
5
1
𝑅12
𝑅22
𝑅32
𝑅13
𝑅23
𝑅33
𝛼1
𝑏0
𝛼2 = 0
𝛼3
𝑏𝐿
𝛼1
𝜶 = 𝑨−𝟏 𝒃 = 𝛼2
𝛼3
For any 𝝋 ∗= 𝜶R= = 𝛼1 𝛼2 𝛼3
𝑅∗1
𝑅∗2
𝑅∗3
Governing equations
Assume isotropic and homogenous confined aquifer
 1D steady-state flow in Cartesian coordinate :
𝜕2𝜑
=0
2
𝜕𝑥
 Qusi-3D steady-state radial flow in cylindrical coordinate :
𝜕 2 𝜑 1 𝜕𝜑
+
=0
2
𝜕𝑟
𝑟 𝜕𝑟
 Qusi-3D transient radial flow in cylindrical coordinate
𝜕 2 𝜑 1 𝜕𝜑 𝑆𝑠 𝜕𝜑
+
=
2
𝜕𝑟
𝑟 𝜕𝑟
𝐾 𝜕𝑡
6
Boundary conditions
(1) 2 Dirichlet boundaries
𝜑 = 𝜑1
𝜑 = 𝜑2
Unconfined aquifer
Aquitard
Confined aquifer
(2) 1 Dirichlet + 1 Neumann boundary
Unconfined aquifer
Aquitard
𝑞
7
Confined aquifer
𝜑 = 𝜑3
Nodal configuration (1)
 Uniformly-distributed nodes
𝑑𝑐
8
𝑑𝑐(𝑎𝑣𝑔) =
𝐿
𝑁−1
𝑁 ∶ Number of total used nodes
Nodal configuration (2)
 Non-uniformly distributed nodes
Linear type-node dense at partial boundary
𝑑𝑐
𝑚𝑖𝑛 1
𝑑𝑐
𝑚𝑖𝑛 2
𝑑𝑐
𝑚𝑎𝑥 1
𝑑𝑐
𝑚𝑎𝑥 2
𝑅𝑖𝑐𝑟 ∶ distance Increasing rate
9
Nodal configuration (3)
 Non-uniformly distributed nodes
Linear type-node dense at all boundary
10
Steady-state flow in Cartesian coord.
with 2 Dirichlet BCs (Case1)
Uniformly-distributed nodes (Configuration 1)
A n a l y t ic a l s o l u ti o n
M Q _ 1 1 p ts _ d c ( a v g ) = 1 0 0 (m ) _ c = 1 0 0 0
10
2 0 0 -h (m )
8
6
4
2
0
11
0
2 00
4 00
60 0
x (m )
80 0
1 00 0
Steady-state flow in Cartesian coor. with 1
Dirichlet & 1 Neumann BCs (Case 2)
Uniformly-distributed nodes (Configuration 1)
A n a ly tic a l so lu ti o n
M Q _ 1 1 p t s _ d c ( av g ) = 1 0 0 m _ c = 1 0 0 0
M Q _ 2 1 p t s _ d c ( av g ) = 5 0 m _ c = 4 7 0
M Q _ 5 1 p t s _ d c ( av g ) = 2 0 m _ c = 1 9 0
M Q _ 1 0 1 p t s _ d c(a v g)= 1 0 m _ c = 9 5
50
2 0 0 -h ( m )
40
30
20
10
0
12
0
2 00
4 00
60 0
x (m )
80 0
1 00 0
Steady-state radial flow with 2 Dirichlet
BCs (Case 3)
Uniformly-distributed nodes (Configuration 1)
10
A n a l y t ic a l s o l u t i o n
M Q _ 1 1 p t s _ d c ( a v g ) = 1 0 0 m _ c = 1 1 2 5 .8
M Q _ 1 0 1 p t s _ d c ( a v g ) = 1 0 m _ c = 9 5 .6
8
M Q _ 1 0 0 1 p t s _ d c ( a v g ) = 1 m _ c = 1 0 .3
2 0 0 -h (m )
M Q _ 2 0 0 1 p t s _ d c ( a v g ) = 0 .5 m _ c = 5 . 4
M Q _ 5 0 0 1 p t s _ d c ( a v g ) = 0 .2 m _ c = 2 . 1 9
6
M Q _ 1 0 0 0 1 p t s _ d c ( a v g ) = 0 . 1 m _ c = 1 .0 4
M Q _ 2 0 0 0 1 p t s _ d c ( a v g ) = 0 . 0 5 m _ c = 0 .5 2
4
2
0
2 00
4 00
60 0
r (m )
13
80 0
1 00 0
Steady-state radial flow in with 2 Dirichlet
BCs (Case 3)
Cases be tested in linear-type non-uniformly-distributed nodes dense at partial
boundary (Configuration 2)
14
N
𝑑𝑐(min) (m)
𝑑𝑐(max) (m)
𝑅𝑖𝑐𝑟 (m)
𝑐
𝛿(%)
MAE(m)
498
0.1
2.7
0.01
1.398
11.96
1.30 × 10−1
614
0.1
1.9
0.01
1.395
5.67
6.18 × 10−2
956
0.1
1.1
0.01
1.451
0.48
5.26 × 10−3
1149
0.1
0.9
0.01
1.392
0.28
3.08 × 10−3
1189
0.1
0.9
0.005
1.26
0.35
3.82 × 10−3
1469
0.1
0.9
0.001
1.261
0.22
2.45 × 10−3
1688
0.1
0.7
0.001
1.241
0.24
2.59 × 10−3
2163
0.1
0.5
0.001
1.216
0.21
2.26 × 10−3
2405
0.05
0.45
0.001
0.663
0.48
5.18 × 10−3
2988
0.05
0.35
0.001
0.696
0.18
2.01 × 10−3
4083
0.05
0.25
0.001
0.682
0.084
9.16 × 10−4
9146
0.01
0.11
0.001
0.141
0.85
9.27 × 10−3
11149
0.01
0.09
0.001
0.202
0.064
6.97 × 10−4
14318
0.01
0.07
0.001
0.156
0.052
𝟓. 𝟕𝟐 × 𝟏𝟎−𝟒
Steady-state radial flow with 2 Dirichlet
BCs (Case 3)
Cases be tested in linear-type non-uniformly-distributed nodes dense at all
boundary (Configuration 3)
N
𝑑𝑐(min) (m)
𝑑𝑐(max) (m
𝑅𝑖𝑐𝑟 (m)
𝑐
𝛿(%)
MAE(m)
)
15
621
0.1
2.9
0.01
1.48
9.8
1.07 × 10−1
651
0.1
2.3
0.01
1.46
7.43
8.1 × 10−2
803
0.1
1.5
0.01
1.41
2.23
2.42 × 10−2
1193
0.1
0.9
0.01
1.14
0.65
7.08 × 10−3
1257
0.1
0.9
0.005
1.14
0.58
6.27 × 10−3
1833
0.1
0.9
0.001
1.14
0.28
3.10 × 10−3
1955
0.1
0.7
0.001
1.07
0.24
2.65 × 10−3
2325
0.1
0.5
0.001
1.01
0.22
2.43 × 10−3
2585
0.05
0.45
0.001
0.61
0.215
2.35 × 10−3
3123
0.05
0.35
0.001
0.59
0.058
6.33 × 10−4
4165
0.05
0.25
0.001
0.56
0.027
2.99 × 10−4
9191
0.01
0.11
0.001
0.17
0.136
1.49 × 10−3
11193
0.01
0.09
0.001
0.15
0.067
7.27 × 10−4
14339
0.01
0.07
0.001
0.16
0.002
𝟐. 𝟐𝟑 × 𝟏𝟎−𝟓
Steady-state radial flow with 1 Dirichlet & 1
Neumann BCs (Case 4)
Cases be tested in linear-type non-uniformly-distributed nodes dense at all
boundary (Configuration 3)
16
N
𝑑𝑐(min) (m)
𝑑𝑐(max) (m)
𝑅𝑖𝑐𝑟 (m)
𝑐
𝛿(%)
MAE(m)
2325
0.1
0.5
0.001
1.23
9.69
7.74 × 10−2
10001
0.1
0.1
0
1.08
10.74
8.57 × 10−2
4165
0.05
0.25
0.001
0.66
1.76
1.4 × 10−2
20001
0.05
0.05
0
0.601
1.865
1.49 × 10−2
14339
0.01
0.07
0.001
0.16
0.027
𝟐. 𝟏𝟑 × 𝟏𝟎−𝟒
 10001 and 20001 points of nodes are uniformly-distributed for the purpose of
comparison with non-uniformly-distributed nodes.
Transient radial flow with 1 Dirichlet & 1
Neumann BCs (Case 5)
Cases be tested in linear-type non-uniformly-distributed nodes dense at all boundary
(Configuration3)
17
Shape parameter c and the nodal interval
Cases for uniformly-distributed nodes (Configuration 1)
c  11.273  d c ( avg )  2.8578
1200
y = 11.273x - 2.8578
R² = 0.9998
1000
800
c
600
400
200
0
0
20
40
60
dc_(avg)
18
80
100
120
Shape parameter c and the nodal interval
Cases for linearly increasing nodal space (Configuration 2)
c  1.45  [1  exp(2.3* d c ( avg ) )]
1.6
1.4
1.2
c
1
0.8
c=1.45*(1-exp(2.3*d_(avg)))
0.6
0.4
0.2
0
0
0.5
1
1.5
dc_(avg)
19
2
2.5
Shape parameter c and the nodal interval
Cases for linearly increasing nodal space with symmetric distribution
(Configuration 3)
c  1.5  [1  exp(2* d c ( avg ) )]
1.6
1.4
1.2
c
1
0.8
c=1.5*(1-exp(2*d_(avg)))
0.6
0.4
0.2
0
0
0.5
1
dc_(avg)
20
1.5
2
Conclusions
1. Meshless method can be a useful tool for groundwater
modeling
2. The shape parameter c has a strong relation with nodal
arrangement.
3. Numerical errors is higher for meshless modeling with
Neumann boundary condition.
4. Numerical errors is higher for meshless modeling with
cylindrical coordinate.
5. Apply non-uniformly-distributed nodes in MQ-RBFCM,
the total number of node can be dramatically
reduced.
21
Finite element method
Meshless method
刪繁就簡三秋樹
領異標新二月花
鄭板橋
22
Thank you for your attention~
23
Groundwater flow equation (1)
 Mass conservation :
inflow mass flux − outflow mass flux
= change of density with time
𝑀𝑥1 − 𝑀𝑥2 + 𝑀𝑦1 − 𝑀𝑦2 + 𝑀𝑧1 − 𝑀𝑧2
𝜕
= (𝜌𝑓 𝑛∆𝑥∆𝑦∆𝑧)
𝜕𝑡
𝑀𝐷𝑖 = 𝜌𝑓 𝑞𝐷𝑖 𝐴𝐷𝑖
 Taylor series expansion :
𝑀𝐷1 − 𝑀𝐷2 = −
−
24
𝜕(𝜌𝑓 𝑞𝐷 )
∆𝑥∆𝑦∆𝑧
𝜕𝑥
REV for deriving groundwater flow equations
(from Schwartz and Zhang, 2003)
𝜕(𝜌𝑓 𝑞𝑥 ) 𝜕(𝜌𝑓 𝑞𝑦 ) 𝜕(𝜌𝑓 𝑞𝑧 )
𝜕
+
+
∆𝑥∆𝑦∆𝑧 =
𝜌 𝑛∆𝑥∆𝑦∆𝑧
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑡 𝑓
𝐷 ∶ 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑙𝑜𝑤
𝐷𝑖 : unit face in REV
Groundwater flow equation (2)
 Assume fluid density is constant in spatial
𝜕𝑞𝑥 𝜕𝑞𝑦 𝜕𝑞𝑧
1 𝜕
−
+
+
=
𝜌 𝑛
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜌𝑓 𝜕𝑡 𝑓
 Definition of Specific storage :
𝑆𝑠 =
𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟
𝑢𝑛𝑖𝑡 𝑎𝑟𝑒𝑎 𝑢𝑛𝑖𝑡 𝑎𝑞𝑢𝑖𝑓𝑒𝑟 𝑡ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠 𝑢𝑛𝑖𝑡 ℎ𝑒𝑎𝑑 𝑐ℎ𝑎𝑛𝑔𝑒
𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟
=
(𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑅𝐸𝑉)(𝑢𝑛𝑖𝑡 ℎ𝑒𝑎𝑑 𝑐ℎ𝑎𝑛𝑔𝑒)
1 𝜕
𝜕ℎ
𝜌 𝑛 = 𝑆𝑠
𝜌𝑓 𝜕𝑡 𝑓
𝜕𝑡
 Darcy’s law :
𝑞𝐷 = −𝐾𝐷
𝜕ℎ
𝜕𝐷
−
𝜕𝑞𝑥 𝜕𝑞𝑦 𝜕𝑞𝑧
𝜕
𝜕ℎ
𝜕
𝜕ℎ
𝜕
𝜕ℎ
+
+
=
𝐾𝑥
+
𝐾𝑦
+
𝐾𝑧
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑥
𝜕𝑥
𝜕𝑦
𝜕𝑦
𝜕𝑧
𝜕𝑧
𝜕
𝜕ℎ
𝜕
𝜕ℎ
𝜕
𝜕ℎ
𝜕ℎ
𝐾𝑥
+
𝐾𝑦
+
𝐾𝑧
= 𝑆𝑠
𝜕𝑥
𝜕𝑥
𝜕𝑦
𝜕𝑦
𝜕𝑧
𝜕𝑧
𝜕𝑡
25
(Storage
Equation)
Nodal arrangement (4)
 Non-uniformly distributed nodes
Exponential type-node dense at partial boundary
𝑑𝑐
𝑑𝑐
𝑖
𝑖
𝑖=1
=
𝑖
ln 𝐿+1
𝑁−1
e
2≤𝑖≤𝑁−1
=
−1
𝑖
ln 𝐿+1
𝑁−1
e
L
−
𝑖−1
ln 𝐿+1
𝑁−1
e
N
N
0
26
Nodal arrangement (5)
 Non-uniformly distributed nodes
Exponential type-node dense at all boundary
𝑑𝑐
𝑑𝑐
𝑖
𝑖=1
=
𝑖
𝐿
𝑁−1 ln 2+1
e 2
𝑖 2≤𝑖≤𝑁−1
=e
𝑖
𝑁−1
2
𝐿
2
−1
ln +1
−e
𝑖−1
𝑁−1
2
ln
𝐿
+1
2
2
𝑑
27 𝑐
𝑖 𝑁+1
≤𝑖≤𝑁−1
2
=e
𝑁−𝑖+1
𝐿
𝑁−1 ln 2+1
2
−e
𝑁−𝑖
𝐿
𝑁−1 ln 2+1
2
Sensitivity analysis (Transient problem for example)
- effect of shape parameter c
28
Sensitivity analysis (Transient problem for example)
- effect of θ and time step (Δt)
effect of θ
29
effect of Δt
Sensitivity analysis (Transient problem for example)
-effect of domain size (R)
R=100 (m)
R=10 (m)
30
R=1 (m)
Steady-state flow in cylindrical coor. with 2
Dirichlet BCs (Case 3)
Why not kept using configuration2. ? Reason (1).
MQ_1149pts_dc(max)=0.9(m);dc(min)=0.1(m);Ricr=0.01(m)_c=1.392
MQ_1189pts_dc(max)=0.9(m);dc(min)=0.1(m);Ricr=0.005(m)_c=1.26
MQ_1469pts_dc(max)=0.9(m);dc(min)=0.1(m);Ricr=0.001(m)_c=1.261
10
Relative error (%)
1
0.1
0.01
0.001
0.0001
31
200
400
600
r (m)
800
(Configuration2.)
1000
(Configuration3.)
Steady-state flow in cylindrical coor. with 2
Dirichlet BCs (Case 3)
Why not kept using configuration2. ? Reason (2).
MQ_1469pts_dc(max)=0.9(m);dc(min)=0.1(m);Ricr=0.001(m)_c=1.261
MQ_2405pts_dc(max)=0.45(m);dc(min)=0.05(m);Ricr=0.001(m)_c=0.663
MQ_11149pts_dc(max)=0.09(m);dc(min)=0.01(m);Ricr=0.001(m)_c=0.202
1000
100
Relative error (%)
10
1
0.1
0.01
0.001
0.0001
200
32
400
600
800
r (m)
(Configuration2.)
1000
(Configuration3.)
Steady-state flow in cylindrical coor. with 2
Dirichlet BCs (Case 3)
Cases be tested in exponential-type non-uniformly-distributed nodes
500
Average relative error (%)
Average relative error (%)
500
400
300
200
400
300
200
100
100
0
10
33
100
1000
Number of total used nodes (N)
(nodes dense at partial boundary)
10000
10
100
1000
Number of total used nodes (N)
(nodes dense at all boundary)
10000
Literature review
 Hardy (1971) developed the algorithm for scattered data

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interpolation by MQ-RBF.
Tarwater (1985) found that by increasing c in MQ-RBF,
the error dropped to a minimum then increase sharply.
Kansa (1990) first applied MQ-RBF in solving partial
differential equations.
Carlson and Foley (1991) concluded that c is problemdependent.
Zerroukat et al. (1998) applied Kansa’s algorithm for heat
transfer problem and found the system became “illconditioned” by increasing number of nodes and c.
Literature review
 Mai-Duy and Tran-Cong (2001) applied different density of
uniform-distributed nodes in solving differential equations
by MQ-RBFCM.
 Wang and Liu (2002) studied the optimal chose for shape
parameter in different RBFs and in different distribution of
nodes. (uniformly-distributed nodes and randomlydistributed node)
 Amaziane et al. (2004) found that Neumann boundary
condition seriously affect hydraulic head .
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