Null-field integral equation approach for structure problems with circular boundaries

1. 第九屆結構工程研討會 Null-field integral equation approach for structure problems with circular boundaries Jeng-Tzong Chen, Ying-Te Lee, Wei-Ming Lee and I-Lin Chen • 時 間: 2008年08月22~24日 • 地 點: 高雄國賓大飯店

2. Outline 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. 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. 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

9. 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

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

11. 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

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

13. Expansion SVD 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Flowchat Problem with circular boundaries Mode shape Null-field BIE Obtain the unknown Fourier coefficients Frequency parameter Degenerate kernel for Fundamental solution Fourier series for fictitious boundary densities Linear algebraic system Collocating the collocation point and matching the boundary conditions Boundary integration in observer system

14. Numerical examples • Case 1: A circular bar with an eccentric hole • Case 2: A circular plate with two holes • Case 3: Stress concentration factor problem

15. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Case 1: A circular bar with an eccentric hole tm t D: External diameter of the tube tm: The maxium wall thickness (eccentricity)

16. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Stress calculationalong outer and inner boundary at boundaries for λ=0.3 and p=0.4 (0.0%) (0.3%) (0.1%) (0.0%) (0.0%) (0.0%) (0.0%) (1.5%) (0.4%) (0.6%)

17. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Stress calculationfor point in the center line alnog lines and for λ=0.3 and p=0.4 (0.0%) (0.0%) (0.1%) (0.2%) (0.1%) (0.5%) (0.1%) (0.5%) (0.1%) (0.0%) (0.3%) (0.6%)

18. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Case 2: A circular plate with two holes

19. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Results The minimum singular value versus the frequency parameter Natural frequency parameter versus terms of Fourier series

20. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks The first five eigenvalues and eigenmodes

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

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

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

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

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

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

27. 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. Fouradvantages of our approach, (1) free of calculating principal value, (2) exponential convergence, (3) free of meshand (4) well-posed model 4.

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