Torsional rigidity of a circular bar with multiple circular inclusions using a null-field integral approach

1
ICOME2006
Torsional rigidity of a circular bar with
multiple circular inclusions using a null-field
integral approach
Ying-Te Lee, Jeng-Tzong Chen and An-Chien Wu
Date November 14-16
Place Hefei, China
2
Outlines
Introduction
1.
2.
Problem statement
3.
Method of solution
Numerical examples
4.
5.
Concluding remarks
3
Motivation
1. Introduction 2. Problem statement 3. Method of
solution 4. Numerical examples 5. Concluding
remarks
Numerical methods for engineering problems
FDM / FEM / BEM / BIEM / Meshless method
BEM / BIEM
Treatment of singularity and hypersingularity
Boundary-layer effect
Ill-posed model
Convergence rate
4
Motivation
1. Introduction 2. Problem statement 3. Method of
solution 4. Numerical examples 5. Concluding
remarks
BEM / BIEM
Improper integral
Singularity hypersingularity
Regularity
Fictitious BEM
Bump contour
Limit process
Fictitious boundary
Achenbach et al. (1988)
Null-field approach
Guiggiani (1995)
Gray and Manne (1993)
Collocation point
CPV and HPV
Ill-posed
Waterman (1965)
5
Present approach
1. Introduction 2. Problem statement 3. Method of
solution 4. Numerical examples 5. Concluding
remarks
Fundamental solution
Degenerate kernel
No principal value
CPV and HPV

• Advantages of present approach
• No principal value
• Well-posed model
• Exponential convergence
• Free of mesh

6
Literature review
1. Introduction 2. Problem statement 3. Method of
solution 4. Numerical examples 5. Concluding
remarks
Analytical solutions for problems with circular
boundaries
Key point Main application Author
Conformal mapping Torsion problem In-plane electrostatics Anti-plane elasticity Chen Weng (2001) Emets Onofrichuk (1996) Budiansky Carrier (1984) Steif (1989) Wu Funami (2002) Wang Zhong (2003)
Bi-polar coordinate Electrostatic potential Elasticity Lebedev et al. (1965) Howland Knight (1939)
Möbius transformation Anti-plane piezoelectricity elasticity Honein et al. (1992)
Complex potential approach Anti-plane piezoelectricity Wang Shen (2001)
Those analytical methods are only limited to
doubly connected regions.
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Literature review
1. Introduction 2. Problem statement 3. Method of
solution 4. Numerical examples 5. Concluding
remarks
Fourier series approximation
Author Main application Key point
Ling (1943) Torsion of a circular tube
Caulk et al. (1983) Steady heat conduction with circular holes Special BIEM
Bird and Steele (1992) Harmonic and biharmonic problems with circular holes Trefftz method
Mogilevskaya et al. (2002) Elasticity problems with circular holes or inclusions Galerkin method
However, no one employed the null-field approach
and degenerate kernel to fully capture the
circular boundary.
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Problem statement
1. Introduction 2. Problem statement 3. Method of
solution 4. Numerical examples 5. Concluding
remarks
A circular bar with circular inclusions
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Domain superposition
1. Introduction 2. Problem statement 3. Method of
solution 4. Numerical examples 5. Concluding
remarks
A circular bar with circular holes
Each circular inclusion problem
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Boundary integral equation and null-field
integral equation
1. Introduction 2. Problem statement 3. Method of
solution 4. Numerical examples 5. Concluding
remarks
Interior case
Exterior case
Degenerate (separate) form
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Degenerate kernel and Fourier series
1. Introduction 2. Problem statement 3. Method of
solution 4. Numerical examples 5. Concluding
remarks
x
Expand fundamental solution by using degenerate
kernel
s
O
x
Expand boundary densities by using Fourier series
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Adaptive observer system
1. Introduction 2. Problem statement 3. Method of
solution 4. Numerical examples 5. Concluding
remarks
r0 , f0
r1 , f1
r2 , f2
rk , fk
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Comparisons of conventional BEM and present
method
1. Introduction 2. Problem statement 3. Method of
solution 4. Numerical examples 5. Concluding
remarks
Boundary density discretization Auxiliary system Formulation Observer system Singularity Convergence
Conventional BEM Constant, linear, quadratic elements Fundamental solution Boundary integral equation Fixed observer system CPV, RPV and HPV Linear
Present method Fourier series expansion Degenerate kernel Null-field integral equation Adaptive observer system Disappear Exponential
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Case 1 A circular bar with an eccentric
inclusion
1. Introduction 2. Problem statement 3. Method of
solution 4. Numerical examples 5. Concluding
remarks
Ratio
Torsional rigidity
GT total torsion rigidity
GM torsion rigidity of matrix
GI torsion rigidity of inclusion
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Results of case 1
1. Introduction 2. Problem statement 3. Method of
solution 4. Numerical examples 5. Concluding
remarks
Torsional rigidity versus number of Fourier
series terms
Torsional rigidity versus shear modulus of
inclusion
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Results of case 1
1. Introduction 2. Problem statement 3. Method of
solution 4. Numerical examples 5. Concluding
remarks
Torsional rigidity of a circular bar with an
eccentric inclusion
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Case 2 limiting case A circular bar with one
circular hole
1. Introduction 2. Problem statement 3. Method of
solution 4. Numerical examples 5. Concluding
remarks
18
Results of case 2
1. Introduction 2. Problem statement 3. Method of
solution 4. Numerical examples 5. Concluding
remarks
Torsional rigidity of a circular bar with an
eccentric hole
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Stress calculation
1. Introduction 2. Problem statement 3. Method of
solution 4. Numerical examples 5. Concluding
remarks
tm
t
External diameter of the tube
D
tm
The maxium wall thickness
(eccentricity)
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Stress calculationalong outer and inner boundary
1. Introduction 2. Problem statement 3. Method of
solution 4. Numerical examples 5. Concluding
remarks
at boundaries for ?0.3 and p0.4
(0.0)
(0.3)
(0.1)
(0.0)
(0.0)
(0.0)
(0.0)
(1.5)
(0.4)
(0.6)
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Stress calculationfor point in the center line
1. Introduction 2. Problem statement 3. Method of
solution 4. Numerical examples 5. Concluding
remarks
alnog lines and for ?0.3 and
p0.4
(0.0)
(0.0)
(0.1)
(0.2)
(0.1)
(0.5)
(0.1)
(0.5)
(0.1)
(0.0)
(0.3)
(0.6)
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Concluding remarks
1. Introduction 2. Problem statement 3. Method of
solution 4. Numerical examples 5. Concluding
remarks
A systematic approach was proposed for torsion
problems with circular inclusions by using
null-field integral equation in conjunction with
degenerate kernel and Fourier series.
1.
2.
A general-purpose program for multiple circular
inclusions of various radii, numbers and
arbitrary positions was developed.
Only a few number of Fouries series terms for our
examples were needed on each boundary, and for
more accurate results of torsional rigidity with
error less than 2 .
3.
Four gains of our approach, (1) free of
calculating principal value, (2) exponential
convergence, (3) free of mesh and (4) well-posed
model
4.
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The End
Thanks for your kind attention
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Torsion problem
1. Introduction 2. Problem statement 3. Method of
solution 4. Numerical examples 5. Concluding
remarks
Following the theory of Saint-Venant torsion, we
assume
Displacement fields
Strain components
Stress components
Equilibrium equation