National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering Dual BEM since 1986 with emphasis on vibrations and acoustics J T Chen (陳正宗特聘教授) Department of Harbor and River Engineering National Taiwan Ocean University, Keelung, Taiwan Jan. 08, 15:10-15:40, 2007 台日聲學研討會 (Tai-Jap-aco2007.ppt)

Outlines Overview of BEM and dual BEM Mathematical formulation Hypersingular BIE Nonuniqueness and its treatments Degenerate scale True and spurious eigensolution (interior prob.) Fictitious frequency (exterior acoustics) Conclusions

Top ten countries of BEM and dual BEM USA (3423) , China (1288) , UK, Japan, Germany, France, Taiwan (551), Canada, South Korea, Italy (No.7) Dual BEM (Made in Taiwan) UK (119), USA (90), Taiwan (69), China (46), Germany, France, Japan, Australia, Brazil, Slovenia (No.3) (ISI information Nov.06, 2006) 台灣加油 FEM Taiwan (No.9/1311)

Top ten countries of FEM, FDM and Meshless methods USA, China, Japan, France, Germany, England, South Korea, Canada, Taiwan, Italy Meshless methods USA, China, Singapore, Japan, Spain, Germany, Slovakia, England, France, Taiwan FDM USA, Japan, China, England, France, Germany, Canada, Taiwan, South Korea, Italy (ISI information Nov.06, 2006)

Top three scholars on BEM and dual BEM Aliabadi M H (UK, Queen Mary College) Mukherjee S (USA, Cornell Univ.) Chen J T (Taiwan, Ocean Univ.) 56 篇 Tanaka M (Japan, Shinshu Univ.) Dual BEM (Made in Taiwan) Aliabadi M H (UK, Queen Mary Univ. London) Chen J T (Taiwan, Ocean Univ.) 43 篇 Power H (UK, Univ Nottingham) (ISI information Nov.06, 2006) NTOU/MSV 加油

Top 25 scholars on BEM/BIEM since 2001 NTOU/MSV Taiwan 北京清華姚振漢教授提供(Nov., 2006)

Overview of numerical methods Domain Boundary MFS Trefftz method MLS, EFG DE PDE- variational IE 7

Number of Papers of FEM, BEM and FDM 6 2 1 (Data form Prof. Cheng A. H. D.)

Growth of BEM/BIEM papers (data from Prof. Cheng A.H.D.)

Advantages of BEM Disadvantages of BEM 北京清華 Discretization dimension reduction Infinite domain (half plane) Interaction problem Local concentration Disadvantages of BEM 北京清華 Integral equations with singularity Full matrix (nonsymmetric)

BEM and meshless methods can be seen as a supplement of FEM. BEM and FEM BEM and meshless methods can be seen as a supplement of FEM. (2) BEM utilizes the discretization concept of FEM as well as the limitation. Whether the supplement is needed or not depends on its absolutely superior area than FEM. Ｃrack & large scale problems Ó NTUCE

What Is Boundary Element Method ? Finite element method Boundary element method What Is Boundary Element Method ? 5 4 1 2 6 3 1 2 geometry node the Nth constant or linear element N Ó NTUCE

Artifical boundary introduced ! Dual integral equations needed ! Dual BEM Why hypersingular BIE is required (potential theory) Degenerate boundary 7 4 7 4 6 5 6 5 8 3 8 3 10 9 1 2 1 2 Artifical boundary introduced ! BEM Dual integral equations needed ! Dual BEM Ó NTUCE

Some researchers on Dual BEM Chen(1986) 412 citings in total Hong and Chen (1988） 71 citings ASCE Portela and Aliabadi (1992) 188 citings IJNME Mi and Aliabadi (1994) Wen and Aliabadi (1995) Chen and Chen (1995) 新竹清華 黎在良等---斷裂力學邊界數值方法(1996) 周慎杰(1999) Chen and Hong (1999) 76 citings ASME Niu and Wang (2001) Yu D H, Zhu J L, Chen Y Z, Tan R J … cite Ó NTUCE

1969 2006 1986 Dual Integral Equations by Hong and Chen(1984-1986) Singular integral equation Hypersingular integral equation Cauchy principal value Hadamard principal value Boundary element method Dual boundary element method 1969 2006 1986 normal boundary degenerate boundary Ó NTUCE

Degenerate boundary geometry node the Nth constant or linear element 7 (-1,0.5) (1,0.5) 7 4 the Nth constant or linear element N 6 5 8 3 (0,0) 1 2 (-1,-0.5) (1,-0.5) 5(+) 6(+) 5(+) 6(-) 5(+) 6(+) 5(+) 6(+) 5(+) 6(+) 5(+) 6(-) 5(+) 6(-) 5(+) 6(-)

How to get additional constraints The constraint equation is not enough to determine coefficients of p and q, Another constraint equation is required

Original data from Prof. Liu Y J BEM Cauchy kernel singular DBEM Hadamard kernel hypersingular crack 1888 Integral equation (1984) (2000) FMM Large scale Degenerate kernel Original data from Prof. Liu Y J Desktop computer fauilure

Fundamental solution Field response due to source (space) Green’s function Casual effect (time) K(x,s;t,τ)

Green’s function, influence line and moment diagram Force Force s x s s=1/2 x=1/4 G(x,s) G(x,s) x s Moment diagram s:fixed x:observer Influence line s:moving x:observer(instrument)

Two systems u and U U(x,s) Domain(D) u(x) s source Boundary (B) Infinite domain

Dual integral equations for a domain point (Green’s third identity for two systems, u and U) Primary field Secondary field where U(s,x)=ln(r) is the fundamental solution.

Dual integral equations for a boundary point (x push to boundary) Singular integral equation Hypersingular integral equation where U(s,x) is the fundamental solution.

Potential theory Single layer potential (U) Double layer potential (T) Normal derivative of single layer potential (L) Normal derivative of double layer potential (M)

Physical examples for potentials Force Moment U:moment diagram T:moment diagram L:shear diagram M:shear diagram

Order of pseudo-differential operators Single layer potential (U) --- (-1) Double layer potential (T) --- (0) Normal derivative of single layer potential (L) --- (0) Normal derivative of double layer potential (M) --- (1) Pseudo differential operator Real differential operator

Calderon projector

How engineers avoid singularity 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)

Definitions of R.P.V., C.P.V. and H.P.V. using bump approach R.P.V. (Riemann principal value) C.P.V.(Cauchy principal value) H.P.V.(Hadamard principal value) Definitions of R.P.V., C.P.V. and H.P.V. using bump approach Ó NTUCE

Principal value in who’s sense Common sense Riemann sense Lebesgue sense Cauchy sense Hadamard sense (elasticity) Mangler sense (aerodynamics) Liggett and Liu’s sense The singularity that occur when the base point and field point coincide are not integrable. (1983)

Two approaches to understand HPV Differential first and then trace operator (Limit and integral operator can not be commuted) Trace first and then differential operator (Leibnitz rule should be considered)

Bump contribution (2-D) s T s x x L s M s x x

Bump contribution (3-D) 0` s s x x s s x x

Successful experiences since 1986

Solid rocket motor (Army工蜂火箭)

X-ray detection (三溫暖測試) Crack initiation crack growth Stress reliever

FEM simulation

Stress analysis

BEM simulation

Shong-Fon II missile (Navy)

IDF (Air Force)

Flow field

V-band structure (Tien-Gen missile)

FEM simulation

Seepage flow

FEM (iteration No.49) BEM(iteration No.13) Free surface seepage flow using hypersingular formulation FEM (iteration No.49) BEM(iteration No.13) Initial guess Initial guess After iteration After iteration Remesh area Remesh line

Meshes of FEM and BEM

Incomplete partition in room acoustics b a e c t=0

Water wave problem with breakwater Free water surface S x Top view O y z breakwater oblique incident water wave

Reflection and Transmission

Cracked torsion bar

IEEE J MEMS

Radiation and scattering problems Nonuniform radiaton scattering 2

Adaptive Mesh BEM FEM DtN interface 5

Strategy of adaptive BEM Mesh       Singular Equation Hypersingular Equation   Error estimator Error estimator Solution u,t u,t 21

Nonuniform radiation： Dirichlet problem Numerical solution: BEM Numerical solution: FEM 64 ELEMENTS 2791 ELEMENTS Nonuniform radiation： Dirichlet problem 9

Is it possible ! No hypersingularity ! No subdomain !

Degenerate boundary problems Multi-domain BEM u=0 r=1 interface Subdomain 1 Subdomain 2 Dual BEM

Conventional BEM in conjunction with SVD Singular Value Decomposition Rank deficiency originates from two sources: (1). Degenerate boundary (2). Nontrivial eigensolution Nd=5 Nd=4 Nd=5

UT BEM + SVD (Present method) Sub domain Dual BEM versus k Determinant versus k Dual BEM Determinant versus k

k=3.14 k=3.82 k=4.48 UT BEM+SVD k=3.09 k=3.84 k=4.50 FEM (ABAQUS)

BEM trap ? Why engineers should learn mathematics ? Well-posed ? Existence ? Unique ? Mathematics versus Computation equivalent ? Some examples

Numerical and physical resonance Numerical resonance radiation incident wave

Numerical phenomena (Fictitious frequency) t(a,0) A story of NTU Ph.D. students

Numerical phenomena (Spurious eigensolution)

Treatments SVD updating term Burton & Miller method CHIEF method Mathematical analysis and numerical study for free vibration of plate using BEM-69

Conclusions Introduction of dual BEM The role of hypersingular BIE was examined Successful experiences in the engineering applications using BEM were demonstrated The traps of BIEM and BEM were shown

Thanks for your kind attention The End Thanks for your kind attention

E-mail: jtchen@mail.ntou.edu.tw 歡迎參觀海洋大學力學聲響振動實驗室 烘焙雞及捎來伊妹兒 http://ind.ntou.edu.tw/~msvlab/ E-mail: jtchen@mail.ntou.edu.tw