Scattering of sound from axisymetric sources by multiple circular cylinders using addition theorem and superposition technique The 32 nd National Conference on Theoretical and Applied Mechanics Authors : Yi-Jhou Lin, Ying-Te Lee, I-Lin Chen and Jeng-Tzong Chen Date: November 28-29, 2008 Place: National Chung Cheng University, Chia-Yi Reporter : Yi-Jhou Lin National Taiwan Ocean University Department of Harbor and River Engineering

The 32 nd National Conference on Theoretical and Applied Mechanics 2 Outlines Introduction Problem statement Method of solution Mathematical Equivalence  Mathematical equivalence between the solution of Green’s third identity and that of superposition technique Numerical examples Concluding remarks Introduction

The 32 nd National Conference on Theoretical and Applied Mechanics 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

The 32 nd National Conference on Theoretical and Applied Mechanics 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)

The 32 nd National Conference on Theoretical and Applied Mechanics 5 Present approach Fundamental solution No principal value Advantages of present approach 1.mesh-free generation 2.well-posed model 3.principal value free 4.elimination of boundary-layer effect 5.exponential convergence Degenerate kernel CPV and HPV

The 32 nd National Conference on Theoretical and Applied Mechanics 6 Green’s third identity BIE for Green’s function

The 32 nd National Conference on Theoretical and Applied Mechanics 7 Outlines Introduction Problem statement Method of solution Mathematical Equivalence  Mathematical equivalence between the solution of Green’s third identity and that of superposition technique Numerical examples Concluding remarks Problem statement

The 32 nd National Conference on Theoretical and Applied Mechanics 8 Problem statement Original Problem Free field Radiation field (typical BVP) (soft)

The 32 nd National Conference on Theoretical and Applied Mechanics 9 Flowchart Original problem Decompose two parts Free fieldRadiation field Expansion Fourier series of boundary densities Degenerate kemel For fundamental solution Collocate of the real boundary Linear algebraic system Calculation of the unknown Fourier BIE for the domain point Superposing the solution of two parts Total field

The 32 nd National Conference on Theoretical and Applied Mechanics 10 Outlines Introduction Problem statement Method of solution Mathematical Equivalence  Mathematical equivalence between the solution of Green’s third identity and that of superposition technique Numerical examples Concluding remarks Method of solution

The 32 nd National Conference on Theoretical and Applied Mechanics 11 Method of solution Boundary integral equation and null-field integral equation Interior case Exterior case Degenerate (separate) form

The 32 nd National Conference on Theoretical and Applied Mechanics 12 Degenerate kernel and Fourier series 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

The 32 nd National Conference on Theoretical and Applied Mechanics 13 Degenerate kernels U(s,x) T(s,x) L(s,x) M(s,x)

The 32 nd National Conference on Theoretical and Applied Mechanics 14 Degenerate kernels

The 32 nd National Conference on Theoretical and Applied Mechanics 15 Adaptive observer system Source point Collocation point

The 32 nd National Conference on Theoretical and Applied Mechanics 16 Linear algebraic system x y

The 32 nd National Conference on Theoretical and Applied Mechanics 17 Outlines Introduction Problem statement Method of solution Mathematical Equivalence  Mathematical equivalence between the solution of Green’s third identity and that of superposition technique Numerical examples Concluding remarks Mathematical Equivalence

The 32 nd National Conference on Theoretical and Applied Mechanics 18 Mathematical equivalence between the solution of Green’s third identity and that of superposition technique += Green’s third identity Superposition technique

The 32 nd National Conference on Theoretical and Applied Mechanics 19 Outlines Introduction Problem statement Method of solution Mathematical Equivalence  Mathematical equivalence between the solution of Green’s third identity and that of superposition technique Numerical examples Concluding remarks Numerical examples

The 32 nd National Conference on Theoretical and Applied Mechanics 20 An infinite plane with two equal circular cylinders subject to a point sound source. Governing equation: Dirichlet Boundary condition: (soft) Fundamental solution:

The 32 nd National Conference on Theoretical and Applied Mechanics 21 Distribution potential on the artificial boundaries in the free field

The 32 nd National Conference on Theoretical and Applied Mechanics 22 Case 1 parameter use size and cylinder B1B1 B2B2 b b y Probe(soft)

The 32 nd National Conference on Theoretical and Applied Mechanics 23 Distribution potential on the artificial boundaries in the free field versus polar angle.

The 32 nd National Conference on Theoretical and Applied Mechanics 24 Relative amplitude of total field versus the probe location y (M=20). Total field Free field B1B1 B2B2 b b Probe (soft) B1B1 B2B2 b b Probe (soft) Versus

The 32 nd National Conference on Theoretical and Applied Mechanics 25 Relative amplitude of total field versus the probe location (M=20). Total field Free field B1B1 B2B2 b b (soft) B1B1 B2B2 b b Probe Versus

The 32 nd National Conference on Theoretical and Applied Mechanics 26 Relative amplitude of total field versus (M=20). B1B1 B2B2 b b Probe (soft) B1B1 B2B2 b b Probe (soft) Versus Total field Free field

The 32 nd National Conference on Theoretical and Applied Mechanics 27 Convergence test of Parseval’s sum for (real part).

The 32 nd National Conference on Theoretical and Applied Mechanics 28 Convergence test of Parseval’s sum for (imaginary part).

The 32 nd National Conference on Theoretical and Applied Mechanics 29 Case 2 parameter use cylinder center- to-center B1B1 B2B2 b b Probe (soft)

The 32 nd National Conference on Theoretical and Applied Mechanics 30 Relative amplitude of total field versus (M=20). B1B1 B2B2 b b Probe (soft)

The 32 nd National Conference on Theoretical and Applied Mechanics 31 Relative amplitude of total field versus (M=20). B1B1 B2B2 b b Probe (soft)

The 32 nd National Conference on Theoretical and Applied Mechanics 32 Relative amplitude of total field versus (M=20). B1B1 B2B2 b b Probe (soft)

The 32 nd National Conference on Theoretical and Applied Mechanics 33 Relative amplitude of total field versus (M=20). b b Probe (soft) B1B1 B2B2

The 32 nd National Conference on Theoretical and Applied Mechanics 34 Outlines Introduction Problem statement Method of solution Mathematical Equivalence  Mathematical equivalence between the solution of Green’s third identity and that of superposition technique Numerical examples Concluding remarks

The 32 nd National Conference on Theoretical and Applied Mechanics 35 Concluding remarks A general-purpose program for solving the problems with arbitrary number, size and various locations of circular cavities was developed. We have proposed a BIEM formulation by using degenerate kernels, null-field integral equation and Fourier series in companion with adaptive observer system.

The 32 nd National Conference on Theoretical and Applied Mechanics 36 The end Welcome to visit the web site of MSVLAB Thanks for your kind attention