Null field integral equation approach for engineering problems with circular boundaries

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Null field integral equation approach for
engineering problems with circular boundaries
?????
National Taiwan Ocean University MSVLAB Department
of Harbor and River Engineering

• J. T. Chen Ph.D.
• ??? ??????
• Taiwan Ocean University
• Keelung, Taiwan
• June 22-23, 2007
• ???? ??

cmc2007.ppt
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Research collaborators

• Dr. I. L. Chen Dr. K. H. Chen
• Dr. S. Y. Leu Dr. W. M. Lee
• Mr. Y. T. Lee
• Mr. W. C. Shen Mr. C. T. Chen Mr. G. C. Hsiao
• Mr. A. C. Wu Mr.P. Y. Chen
• Mr. J. N. Ke Mr. H. Z. Liao

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Research topics of NTOU / MSV LAB on null-field
BIEs (2003-2007)
Null-field BIEM
NUMPDE revision
Navier Equation
Laplace Equation
Helmholtz Equation
Biharmonic Equation
BiHelmholtz Equation
ASME JAM 2006
JSV
MRC,CMES
EABE
Elasticity Crack Problem
(Plate with circulr holes)
(Potential flow) (Torsion) (Anti-plane
shear) (Degenerate scale)
(Free vibration of plate) Indirect BIEM
Screw dislocation
(Stokes flow)
CMAME 2007
JCA
JoM
ASME
EABE
(Free vibration of plate)
Direct BIEM
(Inclusion) (Piezoleectricity)
(Beam bending)
Green function for an annular plate
SDEE
ICOME 2006
SH wave Impinging canyons
Degenerate kernel for ellipse
(Flexural wave of plate)
Torsion bar (Inclusion) Imperfect interface
CMC
Image method (Green function)
Added mass
SH wave Impinging hill
Green function ofcircular inclusion (special
casestatic)
Green function of half plane (Hole and
inclusion)
??? Water wave impinging circular cylinders
Effective conductivity
URL http//ind.ntou.edu.tw/msvlab E-mail
jtchen_at_mail.ntou.edu.tw ???????????????????
nullsystem2007.ppt
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Overview of numerical methods
????????????????
Domain
Boundary
IE
MFS,Trefftz method MLS, EFG
PDE- variational
DE
? ?
? ?
??
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Prof. C B Ling (1909-1993)Fellow of Academia
Sinica
C B Ling (mathematician and expert in mechanics)
He devoted himself to solve BVPs with holes.
PS short visit (J T Chen) of Academia Sinica
2006 summer
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Outlines

• Motivation and literature review
• Mathematical formulation
• Expansions of fundamental solution
• and boundary density
• Adaptive observer system
• Vector decomposition technique
• Linear algebraic equation
• Numerical examples
• Study of spurious solution
• SVD technique
• Conclusions

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Motivation
Numerical methods for engineering problems
FDM / FEM / BEM / BIEM / Meshless method
BEM / BIEM (mesh required)
Treatment of singularity and hypersingularity
Boundary-layer effect
Ill-posed model
Convergence rate
Mesh free for circular boundaries ?
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Motivation and literature review
BEM/BIEM
Improper integral
Singular and hypersingular
Regular
Fictitious BEM
Bump contour
Limit process
Fictitious boundary
Null-field approach
CPV and HPV
Collocation point
Ill-posed
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Present approach
Degenerate kernel
Fundamental solution
CPV and HPV
No principal value
Advantages of degenerate kernel

1. No principal value

2. Well-posed
3. No boundary-layer effect
4. Exponetial convergence 5. Meshless
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Engineering problem with arbitrary geometries
Straight boundary
Degenerate boundary
(Chebyshev polynomial)
(Legendre polynomial)
Circular boundary
(Fourier series)
(Mathieu function)
Elliptic boundary
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Motivation and literature review
Analytical methods for solving Laplace problems
with circular holes
Special solution
Conformal mapping
Bipolar coordinate
Chen and Weng, 2001, Torsion of a circular
compound bar with imperfect interface, ASME
Journal of Applied Mechanics
Honein, Honein and Hermann, 1992, On two
circular inclusions in harmonic problem,
Quarterly of Applied Mathematics
Lebedev, Skalskaya and Uyand, 1979, Work problem
in applied mathematics, Dover Publications
Limited to doubly connected domain
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Fourier series approximation

• Ling (1943) - torsion of a circular tube
• Caulk et al. (1983) - steady heat conduction with
  circular holes
• Bird and Steele (1992) - harmonic and biharmonic
  problems with circular holes
• Mogilevskaya et al. (2002) - elasticity problems
  with circular boundaries

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Contribution and goal

• However, they didnt employ the null-field
  integral equation and degenerate kernels to fully
  capture the circular boundary, although they all
  employed Fourier series expansion.
• To develop a systematic approach for solving
  Laplace problems with multiple holes is our goal.

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Outlines (Direct problem)

• Motivation and literature review
• Mathematical formulation
• Expansions of fundamental solution
• and boundary density
• Adaptive observer system
• Vector decomposition technique
• Linear algebraic equation
• Numerical examples
• Conclusions

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Boundary integral equation and null-field
integral equation
Interior case
Exterior case
Degenerate (separate) form
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Outlines (Direct problem)

• Motivation and literature review
• Mathematical formulation
• Expansions of fundamental solution
• and boundary density
• Adaptive observer system
• Vector decomposition technique
• Linear algebraic equation
• Numerical examples
• Degenerate scale
• Conclusions

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Gain of introducing the degenerate kernel
Degenerate kernel
Fundamental solution
CPV and HPV
interior
exterior
No principal value?
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How to separate the region


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Expansions of fundamental solution and boundary
density

• Degenerate kernel - fundamental solution
• Fourier series expansions - boundary density

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Separable form of fundamental solution (1D)
Separable property
continuous
discontinuous
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Separable form of fundamental solution (2D)
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Boundary density discretization
Fourier series
Ex . constant element
Present method
Conventional BEM
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Outlines

• Motivation and literature review
• Mathematical formulation
• Expansions of fundamental solution
• and boundary density
• Adaptive observer system
• Vector decomposition technique
• Linear algebraic equation
• Numerical examples
• Conclusions

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Adaptive observer system
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Outlines

• Motivation and literature review
• Mathematical formulation
• Expansions of fundamental solution
• and boundary density
• Adaptive observer system
• Vector decomposition technique
• Linear algebraic equation
• Numerical examples
• Conclusions

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Vector decomposition technique for potential
gradient
True normal direction
Non-concentric case
Special case (concentric case)
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Outlines

• Motivation and literature review
• Mathematical formulation
• Expansions of fundamental solution
• and boundary density
• Adaptive observer system
• Vector decomposition technique
• Linear algebraic equation
• Numerical examples
• Conclusions

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Linear algebraic equation
where
Index of collocation circle
Index of routing circle
Column vector of Fourier coefficients (Nth
routing circle)
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Physical meaning of influence coefficient
mth collocation point on the jth circular boundary
jth circular boundary
cosn?, sinn? boundary distributions
Physical meaning of the influence coefficient
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Flowchart of present method
Potential gradient
Vector decomposition
Degenerate kernel
Fourier series
Adaptive observer system
Potential of domain point
Collocation point and matching B.C.
Analytical
Fourier coefficients
Linear algebraic equation
Numerical
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Comparisons of conventional BEM and present
method
Boundary density discretization Auxiliary system Formulation Observer system Singularity Convergence Boundary layer effect
Conventional BEM Constant, linear, quadratic elements Fundamental solution Boundary integral equation Fixed observer system CPV, RPV and HPV Linear Appear
Present method Fourier series expansion Degenerate kernel Null-field integral equation Adaptive observer system Disappear Exponential Eliminate
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Outlines

• Motivation and literature review
• Mathematical formulation
• Expansions of fundamental solution
• and boundary density
• Adaptive observer system
• Vector decomposition technique
• Linear algebraic equation
• Numerical examples
• Conclusions

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Numerical examples

• Laplace equation (EABE 2005, EABE 2007)
• (CMES 2006,
  JAM-ASME 2007, JoM2007)
• (CMA2007,MRC
  2007, NUMPDE revision)
• Membrane eigenproblem (JCA 2007)
• Exterior acoustics (CMAME 2007, SDEE 2007)
• Biharmonic equation (JAM-ASME 2006)
• Plate eigenproblem (JSV 2007)

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Laplace equation

• A circular bar under torque
• (free of mesh generation)

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Torsion bar with circular holes removed

• The warping function
• Boundary condition
• where

Torque
on
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Axial displacement with two circular holes
Dashed line exact solution Solid line
first-order solution
Caulks data (1983) ASME Journal of Applied
Mechanics
Present method (M10)
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Torsional rigidity
?
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Extension to inclusion

• Anti-plane elasticity problems
• (free of boundary layer effect)

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Two circular inclusions with centers on the y axis
Equilibrium of traction
Honein et al.sdata (1992)
Present method (L20)
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Convergence test and boundary-layer effect
analysis
boundary-layer effect
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Numerical examples

• Biharmonic equation
• (exponential convergence)

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Plate problems
Geometric data
Essential boundary conditions
on
and
on
and
on
and
on
and
(Bird Steele, 1991)
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Contour plot of displacement
Present method (N101)
Bird and Steele (1991)
(No. of nodes3,462, No. of elements6,606)
FEM mesh
FEM (ABAQUS)
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Stokes flow problem
Governing equation
Angular velocity
Boundary conditions
on
and
(Stationary)
on
and
Eccentricity
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Comparison for
(160)
(28)
Algebraic convergence
u1
(320)
(640)
(36)
Exponential convergence
(8)
(44)
DOF of BIE (Kelmanson)
DOF of present method
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Contour plot of Streamline for
0
-Q/90
Q/20
Q/5
-Q/30
Q/2
Q
Present method (N81)
0
-Q/90
Q/20
Q/5
-Q/30
Q/2
Kelmanson (Q0.0740, n160)
Q
e
Kamal (Q0.0738)
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Outlines

• Motivation and literature review
• Mathematical formulation
• Expansions of fundamental solution
• and boundary density
• Adaptive observer system
• Vector decomposition technique
• Linear algebraic equation
• Numerical examples
• Discussions of spurious eigenvalues
• SVD
• Conclusions

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Disclaimer (commercial code)

• The concepts, methods, and examples using our
  software are for illustrative and educational
  purposes only.
• Our cooperation assumes no liability or
  responsibility to any person or company for
  direct or indirect damages resulting from the use
  of any information contained here.
• inherent weakness ?
• misinterpretation ? User ???

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Eccentric membrane (true and spurious
eignevalues)
Chen et al., 2001, Proc. Royal Soc. London Ser. A
U T formulation Singular integral equations
spurious
spurious
L M formulation Hypersingular formulation
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SVD Technique (Google searching)
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Physical meaning of SVD
Chen et al., 2002, Int. J. Comp. Numer. Anal.
Appl.
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SVD updating terms SVD updating document
(SVD updating terms) Find true eigenvalue
(SVD updating document) Find spurious eigenvalue
Chen et al., 2003, Proc. Royal Soc. London Ser. A
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Eccentric membrane (SVD updating for true
eigenvalues)
Dirichelet case
U L
Neumann case T M
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Eccentric membrane (SVD updating for spurious
eigenvalues)
U T L M
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Eccentric plate
Case 1 Geometric data R11m R20.4m e0.00.5m thickness0.002m Boundary condition Inner circle clamped Outer circle clamped
Figure 1. A clamped-clamped annular-like plate with one circular hole of radius 0.4 m Figure 1. A clamped-clamped annular-like plate with one circular hole of radius 0.4 m
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Eigenvalue versus eccentricity

Figure 2. Effect of the eccentricity e on the natural frequency parameter for the clamped- clamped annular-like plate (R11.0, R2 0.4)
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True boundary eigenmode

Figure 5. Real and imagine part of Fourier
coefficients for first true boundary mode
(
6.1716, e 0.2, R2 0.4m)
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Spurious boundary eigenmode


Figure 6. Real and imagine part of Fourier
coefficients for first spurious
boundary mode ( 7.9906, e 0.2,
R2 0.4m)
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Outlines

• Motivation and literature review
• Mathematical formulation
• Expansions of fundamental solution
• and boundary density
• Adaptive observer system
• Vector decomposition technique
• Linear algebraic equation
• Numerical examples
• Conclusions

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Conclusions

• A systematic approach using degenerate kernels,
  Fourier series and null-field integral equation
  has been successfully proposed to solve boundary
  value problems with circular boundaries.
• Numerical results agree well with available exact
  solutions and FEM (ABAQUS) for only few terms of
  Fourier series.
• Spurious eigenvalues are examined.

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Conclusions

• Free of boundary-layer effect
• Free of singular integrals
• Well posed
• Exponetial convergence
• Mesh-free approach

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The End

• Thanks for your kind attentions.
• Your comments will be highly appreciated.
• URL http//msvlab.hre.ntou.edu.tw/

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