The Method of Fundamental Solutions and Domain Decomposition Method for Degenerate Seepage Flownet P

1
The Method of Fundamental Solutions and Domain
Decomposition Method for Degenerate Seepage
Flownet Problems(???????????????????)
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96?4?14? 1555-1610

• Der-Liang Young, Chia-Ming Fan, Chia-Cheng
  Tasi, Chi-Wei Chen
• (???,???,???,???)
• Department of Civil Engineering Hydrotech
  Research Institute
• National Taiwan University

2
Outline

• Introduction
• Governing Equations
• Numerical Method (MFS and DDM)
• Results and Discussions
• Conclusions
• Future Works

3
Introduction (Flownet)

• Flownet is one of the simplest tools for
  analyzing groundwater flows and determining flow
  directions.
• Flownet is governed by the solutions of the
  Laplace equations for velocity potential and
  stream function .
• Finite Difference Method (Bramlett and Borden,
  1990)
• Finite Element Method (Aalto, 1984 Fan and
  Tompkins, 1992)
• Boundary Element Method (Rasmussen and Yu, 2003)

4
Introduction (Degenerate problem)

• Numerical treatment of the flownet problems with
  sheet piles normally faces difficulty, since the
  sheet pile is a very small dimension compared to
  the flow domain.
• These problems are generally called degenerate or
  singular boundary problems, meaning that the
  physical presence of such a small structure makes
  the solution approach a numerically complex
  situation.
• Dual real-valued and complex variable BEM (Chen
  et al. 1994 2000)

5
Introduction (Degenerate problem)
6
Introduction (MFS)

• Method of fundamental solutions (MFS), free from
  numerical integration and mesh generation, is a
  boundary-type meshless numerical method.

FEM or FVM mesh
MFS nodes
7
Introduction (MFS)

• The MFS is first proposed by Kupradze and
  Aleksidze in 1964.
• Mathon and Johnston (1977) are among the first to
  provide mathematical fundamentals for the MFS. In
  their work, the sources of MFS are considered as
  unknown variables and solved by nonlinear
  optimization.
• Later, the theoretical developments of MFS are
  advanced that the positions of sources are
  considered as a priori known. (1985)

8
Introduction (MFS)

• The MFS is used to analyze many physical
  problems.
• Seepage problems (Young et al., 2006)
• Eigen-frequencies of membrane (Chen et al.,
  2005)
• Electromagnetic-wave scattering problems
  (Young and Ruan, 2005)
• Waveguide problems (Young et al., 2005)
• Diffusion equation (Young et al., 2004a
  2004b 2006)
• Advection-diffusion equations (Young et al.,
  2006)
• Burgers equations (Young et al., 2006)
• Steady and unsteady Stokes problems (Young et
  al., 2005 2006 Tsai et al., 2006)

9
Introduction (DDM)

• We decompose the computational domain into
  sub-domains to circumvent the singularity near
  the degenerate boundary.
• The DDM not only overcomes the singularity
  problem, but also simultaneously guarantees
  smooth solutions for the entire domain.
• Eigenvalue analysis of membrane with stringers
  (Chen et al., 2005).

10
Governing Equations

• The governing equations of the velocity
  potential and stream function are the
  Laplace equations with suitable boundary
  conditions.
• Velocity Vector

11
Numerical Method
Governing Equation
Boundary Condition
Boundary node Source node
12
Numerical Method
13
Numerical Method
Sheet pile wall
14
Numerical Method
Boundary point
Source point
15
Numerical Method
Domain 1
Fundamental solution
In domain 2, a similar coefficient matrix can be
formed by collocating the known boundary
conditions.
16
Numerical Method

• Then, the two equations for domains 1 and 2 can
  be related by the continuity conditions of the
  common boundaries

Similar procedures can be used for solutions of
stream function.
17
Results and Discussions (Problem 1)

• Validation problem (I) --- flow with a sheet pile

18
Results and Discussions (Problem 1)
MFS-DDM result
BEM, (Chen and Chen, 2000)
19
Results and Discussions (Problem 1)
Potential gradient in the x direction (a) various
numbers of points (b) various source locations
beneath the sheet pile with dual BEM and
analytical solutions
(a)
(b)
20
Results and Discussions (Problem 1)
The comparisons of potential values with exact
solutions (Liggett and Liu, 1983) and BEM results
(Chen and Chen, 2000)
21
Results and Discussions (Problem 2)

• Validation problem (II) --- flow under a dam with
  a sheet pile

22
Results and Discussions (Problem 2)
Boundary point
Distribution of MFS nodes
Source point
23
Results and Discussions (Problem 2)
MFS-DDM result
BEM, (Rasmussen and Yu, 2003)
24
Results and Discussions (Problem 3)

• Seepage flow through a flat soil stratum with a
  dam and a sheet pile

25
Results and Discussions (Problem 3)
MFS-DDM result
Jones et al., 2001
26
Results and Discussions (Problem 4)

• Seepage flow through a semi-circular soil stratum
  with a sheet pile

27
Results and Discussions (Problem 4)
28
Results and Discussions (Problem 4)
29
Results and Discussions (Problem 5)

• Seepage flow through a sloping soil stratum with
  a sheet pile

30
Results and Discussions (Problem 5)
31
Conclusions

• Using the MFS and DDM, the flownets for
  groundwater flow problems with sheet piles, with
  degenerate boundary governed by the Laplace
  equations are simulated.
• Since the MFS is a mesh-free and boundary-type
  method, it will be very powerful for groundwater
  flow problem.

32
Conclusions

• The flownets with a sheet pile show good
  qualitative agreement with the analytical and BEM
  solutions.
• It is expected that the present method (MFS-DDM)
  is also capable of solving other degenerate
  boundary problems such as crack and acoustical
  problems.

33
Future Works
34
Future Works

• To take care of the singularity of pile, we
  consider the solution may be written as the sum
  of a regular and a singular solutions as follow

35

• Thank you
• Professor D.L. Young
• dlyoung_at_ntu.edu.tw
• Dr. C.M. Fan
• d91521006_at_ntu.edu.tw