1. Degenerate scale for a torsion bar problem using BEM
J. T. Chen, Ph.D., Prof.
Department of Harbor and River Engineering
National Taiwan Ocean University, Keelung, Taiwan
Sep.10-12, Beijing, 2002

2. Degenerate problems in computational mechanics
Stress analysis of incompressible materials
Shear locking
Hourglass mode
FEM
Degenerate boundary
Degenerate scale
Spurious eigenvalue (Multiply-connected domain)
Spurious eigenvalue (Simply-connected domain)
Fictitious frequency (exterior acoustics)
Flexibility
BEM

3. Mathematical essence—rank deficiency
Motivation
Four pitfalls in BEM
Why numerical instability occurs in BEM ?
(1) degenerate scale
(2) degenerate boundary
(3) fictitious frequency
Why spurious eigenvalues appear ?
(4) true and spurious eigenvalues
Mathematical essence—rank deficiency
(How to deal with ?)

4. The degenerate scale for torsion bar using BEM
Error (%) of
torsional
rigidity
125 5 a
Previous approach : Try and error on a
Present approach : Only one trial

5. Determination of the degenerate scale by trial and error
Direct searching for the degenerate scale
Trial and error---detecting zero singular value by using SVD
[Chen et al. EABE(2001), EABE(2002), IJNME(2002)]

6. Degenerate scale for torsion bar problems with arbitrary cross sections
An analytical way to determine the degenerate scale
The existence of degenerate scale for the two-dimensional Laplace problem xd x sd s Bd B

7. where is boundary density function
Degenerate scale for torsion bar problems with arbitrary cross sections
where is boundary density function
Mapping
properties
Expansion ratio , where

8. Adding a rigid body term c in the fundamental solution
For arbitrary cross section, expansion ratio is

9. Original degenerate scale
Degenerate scale for torsion bar problems with arbitrary cross sections
0.184
0.50
Unregularized
Regularized c=1.0
Shifting
Normal scale
Original degenerate scale
New degenerate scale

10. Original degenerate scale
Degenerate scale for torsion bar problems with arbitrary cross sections
Unregularized
Regularized
Shifting
0.85
0.31
( ): exact solution of the degenerate scale 2a
0.4
0.8
1.2
1.6 2 a
0.1
0.2
0.3
0.5 1 S q u r e c o s t i n
Conventional BEM (UT formulation)
Adding a rigid body term (c=1.0)
Normal scale
Original degenerate scale
New degenerate scale s

11. Determination of the degenerate scale for the two-dimensional Laplace problems
Cross
Section
Normal scale
Torsional rigidity
Reference equation
, where , x on B, .
1.4480
1.4509
(N.A.)
(N.A.)
(6.1538)
Expansion ratio
(0.5)
(0.5)
(N.A.)
(N.A.)
(0.85)
Degenerate scale
R= (1.0)
= (2.0)
a= (N.A.)
h= (N.A.)
a= (0.85) 2a R a b h
Note: Data in parentheses are exact solutions.
Data marked in the shadow area are derived by using the polar coordinate.

12. Three regularization techniques to deal with degenerate scale problems in BEM
Hypersingular formulation (LM equation)
Adding a rigid body term (U*(s,x)=U(s,x)+c)
CHEEF concept

13. Regularization techniques are not necessary.
Numerical results
Normal scale
( =3.0, =1.0)
Degenerate scale
( =1.5, =0.5)
Analytical
solution
8.4823
0.5301
2.249
1.174
U T
Conventional BEM
(3.30%)
(268.10%)
2.266 (0.76%)
(75.21%)
L M
formulation
(9.22%)
(2.31%)
Add a rigid
body term
c=1.0
(2.26%)
1.1721(0.19%)
c=2.0
(2.36%)
(0.17%)
CHEEF concept
(6.53%)
CHEEF POINT (2.0, 2.0)
(0.18%)
CHEEF POINT
(5.0, 5.0)
cross
section 2a
Torsion
rigidity
Square
Ellipse
method
(a=1.0)
(a=0.85)
Regularization techniques are not necessary.
Regularization techniques are not necessary.
Note: data in parentheses denote error.

14. Regularization techniques are not necessary.
Numerical results
Normal scale
h=3.0
Degenerate scale
h=2.07
Analytical
solution
3.1177
0.7067
0.9609
U T
Conventional BEM
(2.09%)
(57.08%)
(0.83%)
(94.73%)
L M
formulation
(3.25%)
(0.82%)
Add a rigid
body term
c=1.0
(0.45%)
(2.78%)
c=2.0
(0.61%)
(2.84%)
CHEEF concept
(5.46%)
CHEEF POINT (15.0, 15.0)
(3.51%)
CHEEF POINT (20.0, 20.0)
cross
section b a h
Torsion
rigidity
Keyway
Triangle
(a=2.0)
(a=1.05)
method
Regularization techniques are not necessary.
Regularization techniques are not necessary.
Note: data in parentheses denote error.

15. Simply-Connected Problem

16. Multiply-Connected Problem
Without CHIEF method
CHIEF method

17. Conclusions
A more efficient technique was proposed to directly determine the degenerate scale since only one normal scale needs to be computed.
Five cross sections of torsion bar have been demonstrated to see the occuring mechanism of degenerate scale.
Three regularization techniques, addition of a rigid body mode, hypersingular formulation and CHEEF technique were successfully applied to deal with the degenerate scale problem.

18. Related References
J. T. Chen, S. R. Kuo and J. H. Lin, 2002, Analytical study and numerical experiments for degenerate scale problems in the boundary element method for two-dimensional elasticity, Int. J. Numer. Meth. Engng., Vol.54, No.12, pp (SCI and EI)
J. T. Chen, C. F. Lee, I. L. Chen and J. H. Lin, 2002 An alternative method technique for degenerate scale problem in boundary element methods for the two-dimensional Laplace equation, Engineering Analysis with Boundary Elements, Vol.26, No.7, pp (SCI and EI)
J. T. Chen, J. H. Lin, S. R. Kuo and Y. P. Chiu, 2001, Analytical study and numerical experiments for degenerate scale problems in boundary element method using degenerate kernels and circulants, Engineering Analysis with Boundary Elements, Vol.25, No.9, pp (SCI and EI)

19. Acknowledgements Financial support to the authors
from NSF (Taiwan) and Wu Ta-You
Memorial Award is highly appreciated