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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 Using Nonuniform Nodal Space in Meshless Groundwater Modeling 1

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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

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Types of meshless method Common used meshless methods (1) Domain-type - 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) Node-distribution for RBFCM (Hu et al. 2007) 3

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Radial basis functions (RBF) 4 ? R 11 R 13 R 15 R 12 R 14

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Algorithm of RBFCM 123 x=L x=0 5

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Governing equations 6

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Boundary conditions (1) 2 Dirichlet boundaries (2) 1 Dirichlet + 1 Neumann boundary Unconfined aquifer Aquitard Confined aquifer Aquitard Confined aquifer Unconfined aquifer 7

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Nodal configuration (1) Uniformly-distributed nodes 8

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Nodal configuration (2) Non-uniformly distributed nodes Linear type -node dense at partial boundary 9

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Nodal configuration (3) Non-uniformly distributed nodes Linear type -node dense at all boundary 10

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Steady-state flow in Cartesian coord. with 2 Dirichlet BCs (Case1) Uniformly-distributed nodes (Configuration 1) 11

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Steady-state flow in Cartesian coor. with 1 Dirichlet & 1 Neumann BCs (Case 2) Uniformly-distributed nodes (Configuration 1) 12

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Steady-state radial flow with 2 Dirichlet BCs (Case 3) Uniformly-distributed nodes (Configuration 1) 13

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Steady-state radial flow in with 2 Dirichlet BCs (Case 3) NMAE(m) 4980.12.70.011.39811.96 6140.11.90.011.3955.67 9560.11.10.011.4510.48 11490.10.90.011.3920.28 11890.10.90.0051.260.35 14690.10.90.0011.2610.22 16880.10.70.0011.2410.24 21630.10.50.0011.2160.21 24050.050.450.0010.6630.48 29880.050.350.0010.6960.18 40830.050.250.0010.6820.084 91460.010.110.0010.1410.85 111490.010.090.0010.2020.064 143180.010.070.0010.1560.052 Cases be tested in linear-type non-uniformly-distributed nodes dense at partial boundary (Configuration 2) 14

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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) NMAE(m) 6210.12.90.011.489.8 6510.12.30.011.467.43 8030.11.50.011.412.23 11930.10.90.011.140.65 12570.10.90.0051.140.58 18330.10.90.0011.140.28 19550.10.70.0011.070.24 23250.10.50.0011.010.22 25850.050.450.0010.610.215 31230.050.350.0010.590.058 41650.050.250.0010.560.027 91910.010.110.0010.170.136 111930.010.090.0010.150.067 143390.010.070.0010.160.002 15

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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) NMAE(m) 23250.10.50.0011.239.69 100010.1 01.0810.74 41650.050.250.0010.661.76 200010.05 00.6011.865 143390.010.070.0010.160.027 10001 and 20001 points of nodes are uniformly-distributed for the purpose of comparison with non-uniformly-distributed nodes. 16

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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

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Shape parameter c and the nodal interval Cases for uniformly-distributed nodes (Configuration 1) 18

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Shape parameter c and the nodal interval Cases for linearly increasing nodal space (Configuration 2) 19

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Shape parameter c and the nodal interval Cases for linearly increasing nodal space with symmetric distribution (Configuration 3) 20

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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

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22 Finite element method Meshless method 刪繁就簡三秋樹 領異標新二月花 鄭板橋

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Thank you for your attention ~ 23

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Groundwater flow equation (1) Mass conservation : Taylor series expansion : REV for deriving groundwater flow equations (from Schwartz and Zhang, 2003) 24

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Groundwater flow equation (2) Assume fluid density is constant in spatial Definition of Specific storage : Darcy’s law : (Storage Equation) 25

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Nodal arrangement (4) Non-uniformly distributed nodes Exponential type -node dense at partial boundary 26

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Nodal arrangement (5) Non-uniformly distributed nodes Exponential type -node dense at all boundary 27

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Sensitivity analysis (Transient problem for example) - effect of shape parameter c 28

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Sensitivity analysis (Transient problem for example) - effect of θ and time step (Δt) effect of θ effect of Δt 29

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Sensitivity analysis (Transient problem for example) -effect of domain size (R) R=1 (m) R=10 (m) R=100 (m) 30

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Steady-state flow in cylindrical coor. with 2 Dirichlet BCs (Case 3) (Configuration2.) (Configuration3.) Why not kept using configuration2. ? Reason (1). 31

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Steady-state flow in cylindrical coor. with 2 Dirichlet BCs (Case 3) (Configuration2.) (Configuration3.) Why not kept using configuration2. ? Reason (2). 32

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Steady-state flow in cylindrical coor. with 2 Dirichlet BCs (Case 3) Cases be tested in exponential-type non-uniformly-distributed nodes (nodes dense at partial boundary) (nodes dense at all boundary) 33

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Literature review Hardy (1971) developed the algorithm for scattered data 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 problem- dependent. Zerroukat et al. (1998) applied Kansa’s algorithm for heat transfer problem and found the system became “ill- conditioned” by increasing number of nodes and c. 34

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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 randomly- distributed node) Amaziane et al. (2004) found that Neumann boundary condition seriously affect hydraulic head. 35

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