Analysis of two-spheres radiation problems by using the null-field integral equation approach

1. The 32nd Conference of Theoretical and Applied Mechanics Analysis of two-spheres radiation problems by using the null-field integral equation approach Ying-Te Lee (李應德) andJeng-Tzong Chen (陳正宗) • 學 校: 國立臺灣海洋大學 • 科 系: 河海工程學系 • 時 間: 2008年11月28-29日 • 地 點: 國立中正大學

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 Advantages of BEM 1. Mesh reduction FEM BEM 2-D problem: Area (2D) Line (1D) 3-D problem: Volume (3D) Surface (2D) 2. Solve infinite problem Only boundary discretization is needed and without the DtN map.

5. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks BEM and FEM BEM FEM DtN interface 5

6. s x x s x s s x 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Singular and hypersingular integrals Conventional approach to calculate singular and hypersingular integral (Bump contour) Present approach

7. t(a,0) 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Fictitious frequency Non-unique solution: (1) CHIEF method (Schenck, JASA , 1968) Additional constraint (CHIEF point) (2) Burton and Miller method (Burton and Miller, PRS , 1971) (3) SVD updating term technique (Chen et al., JSV, 2002)

8. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Successful experiences in 2-D problems using the present approach Degenerate kernel Fundamental solution (Laplace) (Helmholtz) • Advantages of present approach: • No principal value • Well-posed model • Exponential convergence • Free of mesh generation The proposed approach will be extended to deal with 3-D problem.

9. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks 3-D radiation problem

10. 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

11. s x UI x UE M term in the real implementation 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Expansions Expand fundamental solution by the using degenerate kernel Expand boundary densities by using the spherical harmonics

12. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Degenerate kernels Addition theorem

13. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Relationship of kernel functions U(s,x) T(s,x) L(s,x) M(s,x)

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

15. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Adaptive observer system

16. Expansion 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Flowchart Problem with spherical boundaries Null-field BIE Sound pressure Velocity potential Obtain the unknown spherical harmonics coefficients Degenerate kernel for the fundamental solution Spherical harmonics for boundary density Linear algebraic system Collocating the collocation point and matching the boundary conditions Boundary integration in adaptive observer system

17. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Numerical examples • Case 1: A sphere pulsating with uniform radial velocity • Case 2: A sphere oscillating with non-uniform radial velocity • Case 3: Two spheres vibrating from uniform radial velocity

18. a 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Case 1: A sphere pulsating with uniform radial velocity z y O x Uniform radial velocity U0 (Seybert et al., JASA, 1985) Analytical solution:

19. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Results Uniform radial velocity U0

20. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Comparison Present approach Exact solution (Seybert et al.) Spherical Hankel function of series form:

21. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Distribution of collocation points

22. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Sound pressure Real part of non-dimensional pressure on the surface Imaginary part of non-dimensional pressure on the surface ka=π exist a fictitious frequency in the result of Seybert et al.

23. a 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Case 2: A sphere oscillating with non-uniform radial velocity z y O x Non-uniform radial velocity U0cosθ (Seybert et al., JASA, 1985) Analytical solution:

24. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Result Uniform radial velocity U0

25. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Exact solution Present approach Exact solution (Seybert et al.) Spherical Hankel function of series form:

26. z a a y O 2a 2a x 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Case 3: Two spheres vibrating from uniform radial velocity Uniform radial velocity U0 (Dokumaci, JSV, 1995)

27. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Distribution of collocation points

28. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Contour of sound pressure z=0 and ka=1 Surface Helmholtz Integral Equation (SHIE) (Dokumaci and Sarigül, JSV, 1995) Present approach

29. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Contour of sound pressure z=0 and ka=2 SHIE (Dokumaci and Sarigül) Present approach

30. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Contour of sound pressure z=0 and ka=0.1 SHIE (Dokumaci and Sarigül) Present approach

31. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Potential of the nearest point SHIE+CHIEF SHIE

32. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Potential of the nearest point Present approach Burton & Miller approach

33. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Potential of the furthest point SHIE+CHIEF SHIE

34. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Potential of the furthest point Present approach Burton & Miller approach

35. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Concluding remarks 1. A systematic approach, null-field integral equation in conjunction with degenerate kernel and spherical harmonics, was successfullyproposed to deal with the three-dimensional radiation problem. 2. Only boundary nodes were needed in the present approach. 3. A general-purpose program for multiple radiators of various number,radii, and arbitrary positions was developed. 4. The Burton and Miller approach was successfully used to remedy the fictitious frequency. The present approach can be seen as one kind of semi-analytical approach, since error comes from the number of truncated term of spherical harmonics. 5.

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

37. Potential of the nearest point

38. Elliptic coordinates Degenerate kernel in the 2-D Laplace problems Degenerate kernel in the 2-D Helmholtz problems P. M. Morse and H. Feshbach, Methods of theoretical physics, 1953.

39. 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)

40. Successful experience Non-uniform radiation problem (2D)– Dirichlet BC BEM (64 elements) FEM (2791 elements)

41. Successful experience Non-uniform radiation problem (2D)– Neumann BC FEM (7816 elements) BEM (63 elements)

42. z a a y O 2a 2a x 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks 3-D radiation problem