Analysis of circular torsion bar with
circular holes using null-field approach
研究生：沈文成
指導教授：陳正宗 教授
時間：15:10 ~ 15:25
地點：Room 47450-3
九十四年電子計算機於土木水利工
程應用研討會
1
Outlines


Motivation and literature review
Mathematical formulation
Expansions of fundamental solution
and boundary density
Adaptive observer system
Linear algebraic equation


Numerical examples
Conclusions
2
Outlines


Motivation and literature review
Mathematical formulation
Expansions of fundamental solution
and boundary density
Adaptive observer system
Linear algebraic equation


Numerical examples
Conclusions
3
Motivation and literature review
BEM/BIEM
Improper integral
Singular and hypersingular
Bump contour
Limit process
Regular
Fictitious BEM
Fictitious
boundary
Null-field approach
CPV and HPV
Collocation
point
Ill-posed
4
Present approach
a(x)b(s)
j (x) =
i
K (s, x)
ò
ò K (s, x)f (s)dB(s)
j (x) =
B
O(
a(x)b(s)f (s)dB(s)
B
K e (s, x)
1
1
), O(
)
2
x- s
x- s
Degenerate kernel
Fundamental solution
No principal value
CPV and HPV
Advantages of degenerate kernel
1. No principal value
2. Well-posed
5
Engineering problem with arbitrary
geometries
Straight boundary
Degenerate boundary
(Chebyshev polynomial)
(Legendre polynomial)
Circular boundary
(Fourier series)
a
(Mathieu function)
Elliptic boundary
6
Motivation and literature review
Analytical methods for solving Laplace
problems with circular holes
Conformal mapping
Chen and Weng, 2001,
“Torsion of a circular
compound bar with
imperfect interface”,
ASME Journal of
Applied Mechanics
Bipolar coordinate
Special solution
Lebedev, Skalskaya and
Uyand, 1979, “Work
problem in applied
mathematics”, Dover
Publications
Honein, Honein and
Hermann, 1992, “On
two circular inclusions
in harmonic problem”,
Quarterly of Applied
Mathematics
Limited to doubly connected domain
7
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
8
Contribution and goal


However, they didn’t 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.
9
Outlines


Motivation and literature review
Mathematical formulation
Expansions of fundamental solution
and boundary density
Adaptive observer system
Linear algebraic equation


Numerical examples
Conclusions
10
Boundary integral equation and null-field
integral equation
Interior case
Exterior case
U (s, x) = ln x- s = ln r
x
x
x
D
D
x
Dc
Dc
2p u (x) =
t (s) =
ò
¶ U (s, x)
¶ ns
¶ u (s)
¶ ns
ò T (s, x)u(s)dB(s) - ò U (s, x)t (s)dB(s),
B
0=
T (s, x) =
xÎ D
B
T (s, x)u (s)dB(s) -
B
ò
U (s, x)t (s)dB (s), x Î D c
B
Null-field integral equation
11
Outlines


Motivation and literature review
Mathematical formulation
Expansions of fundamental solution
and boundary density
Adaptive observer system
Linear algebraic equation


Numerical examples
Conclusions
12
Expansions of fundamental solution and
boundary density

Degenerate kernel - fundamental solution
ìï i
ïï U ( R, q; r , f ) = ln R ïï
U (s, x) = í
ïï e
ïï U ( R, q; r , f ) = ln r ïî

¥
1 r m
åm= 1 m ( R ) cos m(q - f ), R ³ r
¥
1 R m
åm= 1 m ( r ) cos m(q - f ), r > R
Fourier series expansions - boundary density
M
u (s) = a0 +
å
(an cos nq + bn sin nq), s Î B
n= 1
M
t (s) = p0 +
å
( pn cos nq + qn sin nq), s Î B
n= 1
13
Separable form of fundamental solution
(1D)
Separable property
U s,x
10
ìï 2
ïï å ai (x)bi (s), s ³ x
ï i= 1
U (s, x) = ïí 2
ïï
ïï å ai (s)bi (x), x > s
ïî i= 1
T s,x
0.4
8
6
0.2
s
-10
10
20
4
-0.2
2
-0.4
-10
10
20
continuous
ìï 1
ïï (s- x), s ³ x
1
ï2
U (s, x) = r = í
ïï 1
2
ïï (x- s), x > s
ïî 2
discontinuous
ìï 1
ïï , s > x
ï 2
T (s, x) = í
ïï - 1
, x> s
ïï
2
ïî
14
Separable form of fundamental solution
(2D)
ìï i
ïï U ( R, q; r , f ) = ln R ïï
U (s, x) = í
ïï e
ïï U ( R, q; r , f ) = ln r ïî
¥
1 r m
åm= 1 m ( R ) cos m(q - f ), R ³ r
¥
1 R m
åm= 1 m ( r ) cos m(q - f ), r > R
20
15
s = ( R, q )
10
5
R
i
U o
0
r
-5
x = (r , f )
-10
-15
-20
-20
Ue
-15
x = (r , f )
-10
-5
0
5
10
15
20
15
Boundary density discretization
Fourier series
Ex . constant element
Present method
Conventional BEM
16
Outlines


Motivation and literature review
Mathematical formulation
Expansions of fundamental solution
and boundary density
Adaptive observer system
Linear algebraic equation


Numerical examples
Conclusions
17
Adaptive observer system
(r , f )
collocation point
18
Outlines


Motivation and literature review
Mathematical formulation
Expansions of fundamental solution
and boundary density
Adaptive observer system
Linear algebraic equation


Numerical examples
Conclusions
19
Outlines


Motivation and literature review
Mathematical formulation
Expansions of fundamental solution
and boundary density
Adaptive observer system
Linear algebraic equation


Numerical examples
Conclusions
20
Linear algebraic equation
[U]{t}= [T]{u}
B1
where
B0
Index of collocation circle
éU 00
ê
êU10
[U ]= êê
ê M
êU
êë N 0
U 01
L
U11
L
M O
U N1 L
U0 N ù
ú
U1N ú
ú
Mú
ú
U NN ú
ú
û
Index of routing circle
ìï t 0 üï
ïï ïï
ïï t1 ïï
ï ï
{t}= ïí t 2 ïý
ïï ïï
ïï Mïï
ïï ïï
ïî t N ïþ
Column vector of Fourier coefficients
(Nth routing circle)
21
Explicit form of each submatrix [Upk] and
vector {tk}
Truncated terms of
Fourier series
éU pk ù=
êë ú
û
0c
1c
1s
é U pk
(
f
)
U
(
f
)
U
1
pk
1
pk (f 1 )
ê
1c
1s
ê U 0 c (f )
U
(
f
)
U
pk
2
pk (f 2 )
ê pk 2
1c
1s
ê U 0 c (f )
U
(
f
)
U
pk
3
pk
3
pk (f 3 )
ê
ê
M
M
M
ê
ê 0c
U 1pkc (f 2 M )
U 1pks (f 2 M )
êU pk (f 2 M )
ê 0c
1c
1s
êëêU pk (f 2 M + 1 ) U pk (f 2 M + 1 ) U pk (f 2 M + 1 )
L
L
L
O
L
L
Ms
U pk
(f 1 ) ù
ú
Mc
Ms
U pk (f 2 )
U pk (f 2 ) ú
ú
Mc
Ms
U pk
(f 3 )
U pk
(f 3 ) ú
ú
ú
M
M
ú
ú
Mc
Ms
U pk
(f 2 M )
U pk
(f 2 M ) ú
ú
Mc
Ms
U pk
(f 2 M + 1 ) U pk
(f 2 M + 1 )ú
ú
û
Mc
U pk
(f 1 )
f
Number of collocation points
{t k }= { p0k
p1k
q1k
L
pMk
T
2
f3
qMk }
f1
Fourier coefficients
f 2M
f 2M + 1
22
Flowchart of present method
0=
ò [T (s, x)u(s) -
U (s, x)t (s)]dB(s)
B
Degenerate kernel
Fourier series
Adaptive observer
system
Collocation point and matching B.C.
Analytical
Linear algebraic equation
Numerical
Potential gradient
Vector
decomposition
Potential of
domain point
Fourier coefficients
23
Comparisons of conventional BEM and
present method
Boundary
density
discretization
Auxiliary
system
Formulation Observer Singularity
system
Constant,
Conventional
Linear,
Fundamental
BEM
Qurdrature…
solution
Boundary
integral
equation
Fixed
observer CPV, RPV
system and HPV
Fourier
series
expansion
Null-field
integral
equation
Adaptive
observer
system
Present
method
Degenerate
kernel
No
principal
value
24
Outlines


Motivation and literature review
Mathematical formulation
Expansions of fundamental solution
and boundary density
Adaptive observer system
Linear algebraic equation


Numerical examples
Conclusions
25
Torsion bar with circular holes removed
Torque
The warping function j
a
Ñ 2j ( x) = 0, x Î D
b
2p k
N
a
q
Boundary condition
¶j
= xk sin qk - yk cos qk
¶n
a
R
on
Bk
where
xi = b cos
2p i
2p i
, yi = b sin
N
N
26
Torsion problems
Torque
Torque
a
q
b
R
a
b
q
a
R
Eccentric case
Two holes (N=2)
27
Torsion problems
Torque
Torque
a
a
a
a
b
q
a
a
b
q
a
R
R
Three holes (N=3)
Four holes (N=4)
28
Torsional rigidity (Eccentric case)
2G / mp R 4
b
R- a
Exact solution
0.20
0.40
Present method
(M=10)
0.97872
0.95137
0.97872
0.95137
BIE
formulation
0.97872
0.95137
0.60
0.80
0.90
0.90312
0.82473
0.76168
0.90312
0.82473
0.76168
0.90316
0.82497
0.76252
0.98
0.66705
0.66555
0.66732
29
Axial displacement with two circular holes
Dashed line: exact solution
Solid line: first-order solution
2
1.5
1
0.5
0
-0.5
-1
Caulk’s data (1983)
-1.5
ASME Journal of Applied Mechanics
-2
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
Present method (M=10)
30
Axial displacement with three circular
holes
Dashed line: exact solution
Solid line: first-order solution
2
1.5
1
0.5
0
-0.5
-1
Caulk’s data (1983)
-1.5
ASME Journal of Applied Mechanics
-2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Present method (M=10)
31
Axial displacement with four circular holes
Dashed line: exact solution
Solid line: first-order solution
2
1.5
1
0.5
0
-0.5
-1
-1.5
Caulk’s data (1983)
ASME Journal of Applied Mechanics
-2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Present method (M=10)
32
Torsional rigidity (N=2,3,4)
2G / mp R
Number of holes Present method
(M=10)
2
0.8657
4
BIE
formulation
0.8657
First-order
solution
0.8661
3
0.8214
0.8214
0.8224
4
0.7893
0.7893
0.7934
33
Outlines


Motivation and literature review
Mathematical formulation
Expansions of fundamental solution
and boundary density
Adaptive observer system
Linear algebraic equation


Numerical examples
Conclusions
34
Conclusions


A systematic approach using degenerate kernels,
Fourier series and null-field integral equation has
been successfully proposed to solve torsion
problems with circular boundaries.
Numerical results agree well with available exact
solutions and Caulk’s data for only few terms of
Fourier series.
35
Conclusions

Engineering problems with circular boundaries
which satisfy the Laplace equation can be solved
by using the proposed approach in a more efficient
and accurate manner.
36
The end
Thanks for your kind attentions.
Your comments will be highly appreciated.
37
Derivation of degenerate kernel


Graf’s addition theorem
Complex variable
Bessel’s function
s = ( R, q) = zs , x = (r , f ) = zx
ln x- s = ln zx - zs
If zs - zx
¥
1 z
åm= 1 m ( zx )m =
s
1
dx = - ò (1 + x + x 2 + L )dx
x
1
1
= - ( x + x 2 + x3 + L )
2
3
¥
1 m
=- å
x
m= 1 m
ln(1- x) = -
ò 1-
ln R
Real part
z
z
ln( zs - zx ) = ln[( zs )(1- x )] = ln( zs ) + ln(1- x ) = ln( zs ) zs
zs
¥
1 r eif m
åm= 1 m ( R eiq ) =
¥
1 r
åm= 1 m ( R )m[ei (f - q) ]m =
ìï i
ïï U ( R, q; r , f ) = ln R ï
U (s, x) = ïí
ïï e
ïï U ( R, q; r , f ) = ln r ïî
¥
1 zx m
( )
m
zs
m= 1
å
k® 0
¥
1 r m
( ) cos m(q - f )
m= 1 m R
å
Real part
¥
1 r m
( ) cos m(q - f ), R ³ r
m
R
m= 1
å
¥
1 R m
( ) cos m(q - f ), r > R
m
r
m= 1
å
38