1. Torsional rigidity of a circular bar with multiple circular inclusions using a null-field integral approach ICOME2006 Ying-Te Lee, Jeng-Tzong Chen and An-Chien Wu Date: November 14-16 Place: Hefei, China

2. Second Asia-Pacific International Conference on Computational Methods in Engineering Nov. 14-16, 2006, Hefei, China 2 Outlines Introduction1. 2. 3. Problem statement Method of solution Numerical examples4. 5.Concluding remarks

3. Second Asia-Pacific International Conference on Computational Methods in Engineering Nov. 14-16, 2006, Hefei, China 3 Motivation 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 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks

4. Second Asia-Pacific International Conference on Computational Methods in Engineering Nov. 14-16, 2006, Hefei, China 4 Motivation BEM / BIEM Improper integral Singularity & hypersingularity Regularity Bump contour Limit process Fictitious boundary Collocation point Fictitious BEM Null-field approach CPV and HPV Ill-posed Guiggiani (1995) Gray and Manne (1993) Waterman (1965) Achenbach et al. (1988) 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks

5. Second Asia-Pacific International Conference on Computational Methods in Engineering Nov. 14-16, 2006, Hefei, China 5 Present approach Fundamental solution No principal value Advantages of present approach 1.No principal value 2.Well-posed model 3.Exponential convergence 4.Free of mesh Degenerate kernel CPV and HPV 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks

6. Second Asia-Pacific International Conference on Computational Methods in Engineering Nov. 14-16, 2006, Hefei, China 6 Literature review Key pointMain applicationAuthor Conformal mappingTorsion 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 coordinateElectrostatic potential Elasticity Lebedev et al. (1965) Howland & Knight (1939) Möbius transformationAnti-plane piezoelectricity & elasticity Honein et al. (1992) Complex potential approachAnti-plane piezoelectricityWang & Shen (2001) Those analytical methods are only limited to doubly connected regions. Analytical solutions for problems with circular boundaries 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks

7. Second Asia-Pacific International Conference on Computational Methods in Engineering Nov. 14-16, 2006, Hefei, China 7 Literature review AuthorMain applicationKey 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. Fourier series approximation 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks

8. Second Asia-Pacific International Conference on Computational Methods in Engineering Nov. 14-16, 2006, Hefei, China 8 Problem statement B0B0 B1B1 B2B2 B3B3 BiBi B4B4 a0a0 a1a1 a2a2 a3a3 a4a4 aiai x y 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks A circular bar with circular inclusions

9. Second Asia-Pacific International Conference on Computational Methods in Engineering Nov. 14-16, 2006, Hefei, China 9 Domain superposition 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks A circular bar with circular holesEach circular inclusion problem B0B0 B1B1 B2B2 B3B3 BiBi B4B4 B0B0 B1B1 B2B2 B3B3 BiBi B4B4 Satisfy

10. Second Asia-Pacific International Conference on Computational Methods in Engineering Nov. 14-16, 2006, Hefei, China 10 Interior case Exterior case Degenerate (separate) form Boundary integral equation and null-field integral equation 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks

11. Second Asia-Pacific International Conference on Computational Methods in Engineering Nov. 14-16, 2006, Hefei, China 11 Degenerate kernel and Fourier series 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks s O x kth circular boundary cosnθ, sinnθ boundary distributions x Expand fundamental solution by using degenerate kernel Expand boundary densities by using Fourier series

12. Second Asia-Pacific International Conference on Computational Methods in Engineering Nov. 14-16, 2006, Hefei, China 12 collocation point r 0, f 0 r 1, f 1 rk,fkrk,fk r2,f2r2,f2 Adaptive observer system 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks

13. Second Asia-Pacific International Conference on Computational Methods in Engineering Nov. 14-16, 2006, Hefei, China 13 Comparisons of conventional BEM and present method Boundary density discretization Auxiliary system Formulation Observer system SingularityConvergence 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 DisappearExponential 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks

14. Second Asia-Pacific International Conference on Computational Methods in Engineering Nov. 14-16, 2006, Hefei, China 14 Case 1: A circular bar with an eccentric inclusion 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks R1R1 R0R0 exex Ratio: Torsional rigidity: G T : total torsion rigidity G M : torsion rigidity of matrix G I : torsion rigidity of inclusion

15. Second Asia-Pacific International Conference on Computational Methods in Engineering Nov. 14-16, 2006, Hefei, China 15 Results of case 1 Torsional rigidity versus number of Fourier series terms Torsional rigidity versus shear modulus of inclusion 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks

16. Second Asia-Pacific International Conference on Computational Methods in Engineering Nov. 14-16, 2006, Hefei, China 16 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

17. Second Asia-Pacific International Conference on Computational Methods in Engineering Nov. 14-16, 2006, Hefei, China 17 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 R 1 =0.3 R 0 =1 e x =0.5

18. Second Asia-Pacific International Conference on Computational Methods in Engineering Nov. 14-16, 2006, Hefei, China 18 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 Results of case 2

19. Second Asia-Pacific International Conference on Computational Methods in Engineering Nov. 14-16, 2006, Hefei, China 19 Stress calculation t 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks tmtm External diameter of the tube D:D: tm:tm:The maxium wall thickness (eccentricity)

20. Second Asia-Pacific International Conference on Computational Methods in Engineering Nov. 14-16, 2006, Hefei, China 20 Stress calculation along outer and inner boundary at boundaries for λ=0.3 and p=0.4 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks (0.0%) (0.1%) (0.0%) (0.4%) (0.0%) (0.3%) (0.0%) (1.5%) (0.6%)

21. Second Asia-Pacific International Conference on Computational Methods in Engineering Nov. 14-16, 2006, Hefei, China 21 Stress calculation for point in the center line alnog lines and for λ=0.3 and p=0.4 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks (0.0%) (0.1%) (0.3%) (0.0%) (0.2%) (0.5%) (0.0%) (0.6%)

22. Second Asia-Pacific International Conference on Computational Methods in Engineering Nov. 14-16, 2006, Hefei, China 22 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. 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 %. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Four gains of our approach, (1) free of calculating principal value, (2) exponential convergence, (3) free of mesh and (4) well-posed model 3. A general-purpose program for multiple circular inclusions of various radii, numbers and arbitrary positions was developed. 4.

23. Second Asia-Pacific International Conference on Computational Methods in Engineering Nov. 14-16, 2006, Hefei, China 23 The End Thanks for your kind attention Welcome to visit the web site of MSVLAB http://ind.ntou.edu.tw/~msvlab

24. Second Asia-Pacific International Conference on Computational Methods in Engineering Nov. 14-16, 2006, Hefei, China 24 Torsion problem 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Displacement fields: Strain components: Stress components: Equilibrium equation: : the shear modulus : angle of twist per unit length Following the theory of Saint-Venant torsion, we assume