Null-field integral equation approach using degenerate kernels and its engineering applications

1. The 31st Conference Conference of Theoretical and Applied Mechanics Null-field integral equation approach using degenerate kernels and its engineering applications Ying-Te Lee andJeng-Tzong Chen Date: December 21-22 Place: I-Shou Univerity, Kaohsiung

2. Outlines 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks

3. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Motivation Numerical methods for engineering problems FDM / FEM / BEM / BIEM / Meshless method BEM / BIEM Treatment of singularity and hypersingularity Boundary-layer effect Convergence rate Ill-posed model

4. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Motivation 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. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Present approach Degenerate kernel Fundamental solution No principal value CPV and HPV • Advantages of present approach • No principal value • Well-posed model • Exponential convergence • Free of mesh

6. y B0 B2 B1 a2 a1 a0 x B3 Bi ai a3 a4 B4 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Problem statement Circular cavities and/or inclusions bounded in the domain

7. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Problem statement Governing Equation Fundamental solution

8. B0 B0 Satisfy B2 B2 B1 B1 B3 B3 Bi Bi B4 B4 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Domain superposition A circular bar with circular holes Each circular inclusion problem

9. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Boundary integral equation and null-field integral equation Interior case Exterior case Degenerate (separate) form

10. cosnθ, sinnθ boundary distributions kth circular boundary 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Degenerate kernel and Fourier series x Expand fundamental solution by using degenerate kernel s O x Expand boundary densities by using Fourier series

11. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Degenerate kernels Laplace problem Helmholtz problem Elasticity problem

12. collocation point 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Adaptive observer system r2,f2 r0 ,f0 r1 ,f1 rk,fk

13. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Comparisons of conventional BEM and present method

14. ex R0 R1 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Case 1: A circular bar with an eccentric inclusion Ratio: Torsional rigidity: GT : total torsion rigidity GM : torsion rigidity of matrix GI : torsion rigidity of inclusion

15. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Results of case 1 Torsional rigidity versus number of Fourier series terms Torsional rigidity versus shear modulus of inclusion

16. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Results of case 1 Torsional rigidity of a circular bar with an eccentric inclusion

17. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Case 2: Water wave impinging four cylinders

18. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Results of case 2 Force for four cylinders

19. Potential at four north poles Potential (ψ) at the north pole of each cylinder (ka = 1.7)

20. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Case 3: Stress concentration factor problem Boundary conditions: and

21. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Domain superposition = +

22. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Null-field BIE

23. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Results of case 3 Similarly,

24. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Results of case 3 = + and

25. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Results of case 3

26. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Concluding remarks 1. A systematic approach was proposed for engineering problems with circular boundariesby usingnull-field integral equation in conjunction with degenerate kernel and Fourier series. 2. A general-purpose program for multiple circular boundaries of various radii, numbers and arbitrary positions was developed. 3. Onlya few number of Fouries series terms for our exampleswere needed on each boundary. 4. Fourgains of our approach, (1) free of calculating principal value, (2) exponential convergence, (3) free of meshand (4) well-posed model

27. The End Thanks for your kind attention Welcome to visit the web site of MSVLAB http://ind.ntou.edu.tw/~msvlab