Analysis of two-spheres radiation problems by using the null-field integral equation approach The 32 nd Conference of Theoretical and Applied Mechanics Ying-Te Lee ( 李應德 ) and Jeng-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 日 地 點 : 國立中正大學

The 32 nd Conference of Theoretical and Applied Mechanics November 28-29, 2008, National Chung Cheng University, Chia-Yi 2 Outline Introduction Problem statement Method of solution Numerical examples4. 5.Concluding remarks

The 32 nd Conference of Theoretical and Applied Mechanics November 28-29, 2008, National Chung Cheng University, Chia-Yi 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

The 32 nd Conference of Theoretical and Applied Mechanics November 28-29, 2008, National Chung Cheng University, Chia-Yi 4 Advantages of BEM 1. Mesh reduction 2. Solve infinite problem 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks FEMBEM Area (2D)Line (1D) 2-D problem: Volume (3D)Surface (2D) 3-D problem: Only boundary discretization is needed and without the DtN map.

The 32 nd Conference of Theoretical and Applied Mechanics November 28-29, 2008, National Chung Cheng University, Chia-Yi 5 BEMFEM 5 DtN interface BEM and FEM 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks

The 32 nd Conference of Theoretical and Applied Mechanics November 28-29, 2008, National Chung Cheng University, Chia-Yi 6 Singular and hypersingular integrals 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks x s x s Conventional approach to calculate singular and hypersingular integral (Bump contour) Present approach x s x s

The 32 nd Conference of Theoretical and Applied Mechanics November 28-29, 2008, National Chung Cheng University, Chia-Yi 7 Fictitious frequency (1) CHIEF method (Schenck, JASA, 1968) (2) Burton and Miller method (Burton and Miller, PRS, 1971) (3) SVD updating term technique (Chen et al., JSV, 2002) 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Additional constraint (CHIEF point) Non-unique solution: t(a,0)

The 32 nd Conference of Theoretical and Applied Mechanics November 28-29, 2008, National Chung Cheng University, Chia-Yi 8 Successful experiences in 2-D problems using the present approach Fundamental solution Advantages of present approach: 1.No principal value 2.Well-posed model 3.Exponential convergence 4.Free of mesh generation Degenerate kernel 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks The proposed approach will be extended to deal with 3-D problem. (Laplace) (Helmholtz)

The 32 nd Conference of Theoretical and Applied Mechanics November 28-29, 2008, National Chung Cheng University, Chia-Yi 9 3-D radiation problem 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks

The 32 nd Conference of Theoretical and Applied Mechanics November 28-29, 2008, National Chung Cheng University, Chia-Yi 10 Interior caseExterior 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

The 32 nd Conference of Theoretical and Applied Mechanics November 28-29, 2008, National Chung Cheng University, Chia-Yi 11 UEUE x UIUI x Expansions 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Expand fundamental solution by the using degenerate kernel Expand boundary densities by using the spherical harmonics M term in the real implementation s

The 32 nd Conference of Theoretical and Applied Mechanics November 28-29, 2008, National Chung Cheng University, Chia-Yi 12 Degenerate kernels 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Addition theorem

The 32 nd Conference of Theoretical and Applied Mechanics November 28-29, 2008, National Chung Cheng University, Chia-Yi 13 U(s,x)U(s,x)T(s,x)T(s,x) L(s,x)L(s,x)M(s,x)M(s,x) Relationship of kernel functions 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks

The 32 nd Conference of Theoretical and Applied Mechanics November 28-29, 2008, National Chung Cheng University, Chia-Yi 14 Degenerate kernels 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks

The 32 nd Conference of Theoretical and Applied Mechanics November 28-29, 2008, National Chung Cheng University, Chia-Yi 15 Adaptive observer system 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks

The 32 nd Conference of Theoretical and Applied Mechanics November 28-29, 2008, National Chung Cheng University, Chia-Yi 16 Problem with spherical boundaries Null-field BIE Expansion Degenerate kernel for the fundamental solution Spherical harmonics for boundary density Collocating the collocation point and matching the boundary conditions Boundary integration in adaptive observer system Linear algebraic system Obtain the unknown spherical harmonics coefficients Velocity potential Flowchart 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Sound pressure

The 32 nd Conference of Theoretical and Applied Mechanics November 28-29, 2008, National Chung Cheng University, Chia-Yi 17 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 Numerical examples 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks

The 32 nd Conference of Theoretical and Applied Mechanics November 28-29, 2008, National Chung Cheng University, Chia-Yi 18 a Case 1: A sphere pulsating with uniform radial velocity 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Uniform radial velocity U 0 (Seybert et al., JASA, 1985) O y z x Analytical solution:

The 32 nd Conference of Theoretical and Applied Mechanics November 28-29, 2008, National Chung Cheng University, Chia-Yi 19 Results 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Uniform radial velocity U 0

The 32 nd Conference of Theoretical and Applied Mechanics November 28-29, 2008, National Chung Cheng University, Chia-Yi 20 Comparison 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Exact solution (Seybert et al.) Spherical Hankel function of series form: Present approach

The 32 nd Conference of Theoretical and Applied Mechanics November 28-29, 2008, National Chung Cheng University, Chia-Yi 21 Distribution of collocation points 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks

The 32 nd Conference of Theoretical and Applied Mechanics November 28-29, 2008, National Chung Cheng University, Chia-Yi 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.

The 32 nd Conference of Theoretical and Applied Mechanics November 28-29, 2008, National Chung Cheng University, Chia-Yi 23 a 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Non-uniform radial velocity U 0 cosθ (Seybert et al., JASA, 1985) O y z x Analytical solution: Case 2: A sphere oscillating with non-uniform radial velocity

The 32 nd Conference of Theoretical and Applied Mechanics November 28-29, 2008, National Chung Cheng University, Chia-Yi 24 Result 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Uniform radial velocity U 0

The 32 nd Conference of Theoretical and Applied Mechanics November 28-29, 2008, National Chung Cheng University, Chia-Yi 25 Exact solution 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Exact solution (Seybert et al.) Spherical Hankel function of series form: Present approach

The 32 nd Conference of Theoretical and Applied Mechanics November 28-29, 2008, National Chung Cheng University, Chia-Yi 26 Case 3: Two spheres vibrating from uniform radial velocity 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Uniform radial velocity U 0 (Dokumaci, JSV, 1995) 2a2a2a2a O y z x aa

The 32 nd Conference of Theoretical and Applied Mechanics November 28-29, 2008, National Chung Cheng University, Chia-Yi 27 Distribution of collocation points 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks

The 32 nd Conference of Theoretical and Applied Mechanics November 28-29, 2008, National Chung Cheng University, Chia-Yi 28 Contour of sound pressure 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks z=0 and ka=1 Present approach (Dokumaci and Sarigül, JSV, 1995) Surface Helmholtz Integral Equation (SHIE)

The 32 nd Conference of Theoretical and Applied Mechanics November 28-29, 2008, National Chung Cheng University, Chia-Yi Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Contour of sound pressure Present approach SHIE (Dokumaci and Sarigül) z=0 and ka=2

The 32 nd Conference of Theoretical and Applied Mechanics November 28-29, 2008, National Chung Cheng University, Chia-Yi 30 Contour of sound pressure 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Present approach SHIE (Dokumaci and Sarigül) z=0 and ka=0.1

The 32 nd Conference of Theoretical and Applied Mechanics November 28-29, 2008, National Chung Cheng University, Chia-Yi 31 Potential of the nearest point 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks SHIE SHIE+CHIEF

The 32 nd Conference of Theoretical and Applied Mechanics November 28-29, 2008, National Chung Cheng University, Chia-Yi 32 Potential of the nearest point 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Present approachBurton & Miller approach

The 32 nd Conference of Theoretical and Applied Mechanics November 28-29, 2008, National Chung Cheng University, Chia-Yi 33 Potential of the furthest point SHIE SHIE+CHIEF 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks

The 32 nd Conference of Theoretical and Applied Mechanics November 28-29, 2008, National Chung Cheng University, Chia-Yi 34 Potential of the furthest point Present approachBurton & Miller approach 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks

The 32 nd Conference of Theoretical and Applied Mechanics November 28-29, 2008, National Chung Cheng University, Chia-Yi 35 Concluding remarks A systematic approach, null-field integral equation in conjunction with degenerate kernel and spherical harmonics, was successfully proposed to deal with the three-dimensional radiation problem 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. 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks 4. A general-purpose program for multiple radiators of various number, radii, and arbitrary positions was developed. 2.Only boundary nodes were needed in the present approach. 5. The Burton and Miller approach was successfully used to remedy the fictitious frequency.

The 32 nd Conference of Theoretical and Applied Mechanics November 28-29, 2008, National Chung Cheng University, Chia-Yi 36 The End Thanks for your kind attention Welcome to visit the web site of MSVLAB/NTOU

The 32 nd Conference of Theoretical and Applied Mechanics November 28-29, 2008, National Chung Cheng University, Chia-Yi 37 Potential of the nearest point

The 32 nd Conference of Theoretical and Applied Mechanics November 28-29, 2008, National Chung Cheng University, Chia-Yi 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.

The 32 nd Conference of Theoretical and Applied Mechanics November 28-29, 2008, National Chung Cheng University, Chia-Yi 39 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 Conference of Theoretical and Applied Mechanics November 28-29, 2008, National Chung Cheng University, Chia-Yi 40 BEM (64 elements)FEM (2791 elements) Non-uniform radiation problem (2D)– Dirichlet BC Successful experience

The 32 nd Conference of Theoretical and Applied Mechanics November 28-29, 2008, National Chung Cheng University, Chia-Yi 41 BEM (63 elements) FEM (7816 elements) Successful experience Non-uniform radiation problem (2D)– Neumann BC

The 32 nd Conference of Theoretical and Applied Mechanics November 28-29, 2008, National Chung Cheng University, Chia-Yi 42 3-D radiation problem 1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks 2a2a2a2a O y z x aa