Degenerate scale for a torsion bar problem using

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 1

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

Motivation Four pitfalls in BEM n n 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 ? ) 3

The degenerate scale for torsion bar using BEM Error (%) of 125 torsional rigidity 5 0 a Previous approach : Try and error on a Present approach : Only one trial 4

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)] 5

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

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

n Adding a rigid body term c in the fundamental solution 0 For arbitrary cross section, expansion ratio is 8

Degenerate scale for torsion bar problems with arbitrary cross sections Normal scale Regularized c=1. 0 Original degenerate scale Unregularized New degenerate scale Shifting 0. 184 0. 50 9

0. 5 Degenerate scale for torsion bar problems with arbitrary cross sections Normal scale Square cross section Conventional BEM (UT formulation) Adding a rigid body term (c=1. 0) 0. 4 ( ): exact solution of the degenerate scale 0. 3 Regularized s 1 Original degenerate scale 2 a 0. 2 2 a Unregularized 0. 1 New degenerate scale Shifting 0. 31 0. 85 0 0 0. 4 0. 8 1. 2 a 1. 6 2 10

Determination of the degenerate scale for the two-dimensional Laplace problems Cross Section b R a 2 a h 2 a Normal scale Torsional rigidity Reference equation , where , x on B, . 1. 4480 Expansion ratio Degenerate scale 0. 5020 (0. 5) R=1. 0040 (1. 0) 1. 4509 1. 5539 (N. A. ) 2. 6972 (N. A. ) 6. 1530 (6. 1538) 0. 5019 (0. 5) 0. 5254 (N. A. ) 0. 6902 (N. A. ) 0. 8499 (0. 85) a=1. 0508 (N. A. ) h=2. 0700 (N. A. ) a=0. 8499 (0. 85) = 2. 0058 (2. 0) Note: Data in parentheses are exact solutions. Data marked in the shadow area are derived by using the polar coordinate. 11

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

Torsion rigidity cross section Numerical results 2 a Ellipse Normal scale Degenerate scale ( =3. 0, =1. 0) ( =1. 5, =0. 5) method Analytical solution U T Conventional BEM c=1. 0 c=2. 0 Square Normal scale Degenerate scale (a=1. 0) (a=0. 85) 8. 4823 0. 5301 2. 249 1. 174 8. 7623 (3. 30%) -0. 8911 (268. 10%) 2. 266 (0. 76%) 2. 0570 (75. 21%) LM formulation Add a rigid body term 2 a 0. 4812 (9. 22%) Regularization techniques are not necessary. CHEEF concept Note: data in parentheses denote error. 0. 5181 (2. 26%) 0. 5176 (2. 36%) 0. 5647 (6. 53%) CHEEF POINT (2. 0, 2. 0) 1. 1472 (2. 31%) Regularization techniques are not necessary. 1. 1721(0. 19%) 1. 1723 (0. 17%) 1. 1722 (0. 18%) CHEEF POINT (5. 0, 5. 0) 13

Torsion rigidity cross section Numerical results b h Keyway Triangle method Analytical solution U T Conventional BEM Normal scale h=3. 0 Degenerate scale h=2. 07 Normal scale (a=2. 0) Degenerate scale (a=1. 05) 3. 1177 0. 7067 12. 6488 0. 9609 3. 1829 (2. 09%) 1. 1101 (57. 08%) 12. 5440 (0. 83%) LM formulation Add a rigid body term a c=1. 0 c=2. 0 0. 6837 (3. 25%) Regularization techniques are not necessary. CHEEF concept Note: data in parentheses denote error. 0. 7035 (0. 45%) 0. 7024 (0. 61%) 0. 7453 (5. 46%) CHEEF POINT (15. 0, 15. 0) 1. 8712 (94. 73%) 0. 9530 (0. 82%) Regularization techniques are not necessary. 0. 9876 (2. 78%) 0. 9879 (2. 84%) 0. 9272 (3. 51%) CHEEF POINT (20. 0, 20. 0) 14

Simply-Connected Problem r=2. 000 r=1. 012 15

Multiply-Connected Problem Without CHIEF method 16

Conclusions n n n 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. 17

Related References u u u 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. 1669 -1681. (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. 559 -569. (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. 819 -828. (SCI and EI) 18

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

http: //ind. ntou. edu. tw/~msvlab/ 20