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BEM introduction. J. T. Chen. Department of Harbor and River Engineering National Taiwan Ocean University, Keelung, Taiwan Sep. 21, 2006 First-class introduction for NTPU/MSV BEM course (lecture1-2006.ppt). Outlines. Overview of BIE and BEM Mathematical tools
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BEM introduction J. T. Chen Department of Harbor and River Engineering National Taiwan Ocean University, Keelung, Taiwan Sep. 21, 2006 First-class introduction for NTPU/MSV BEM course (lecture1-2006.ppt)
Outlines • Overview of BIE and BEM • Mathematical tools Hypersingular BIE Degenerate kernel Circulants SVD updating term SVD updating document Fredholm alternative theorem • Nonuniqueness and its treatments Degenerate scale Degenerate boundary True and spurious eigensolution (interior prob.) Fictitious frequency (exterior acoustics) Corner • Conclusions and further research
Overview of numerical methods Domain Boundary MFS Trefftz method MLS DE PDE- variational IE 3
Numeber 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 • 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 Cauchy kernel singular DBEM Hadamard kernel hypersingular crack (1984) (2000) FMM Large scale Degenerate kernel Original data from Prof. Liu Y J Desktop computer fauilure
Why engineers should learn mathematics ? • Well-posed ? • Existence ? • Unique ? • Mathematics versus Computation • Some examples
Error (%) of torsional rigidity 125 5 0 a Numerical phenomena(Degenerate scale) Commercial ode output ? Previous approach : Try and error on a Present approach : Only one trial
t(a,0) Numerical phenomena(Fictitious frequency) A story of NTU Ph.D. students
Hypersingular integral equation Singular integral equation Cauchy principal value Hadamard principal value Dual boundary element method Boundary element method normal boundary degenerate boundary Numerical phenomena(Degenerate boundary) 1967-1984-2006
Numerical phenomena(Corner) Boundary
Motivation Five pitfalls in BEM • Numerical instability occurs in BEM ? (1) degenerate scale (2) degenerate boundary (3) fictitious frequency (4) corner • Spurious eigenvalues appear ? (5) true and spurious eigenvalues Mathematical essence—rank deficiency ? (How to deal with ?) nonuniqueness ?
Mathematical tools Hypersingular BIE Degenerate kernel Circulants SVD updating term SVD updating document Fredholm alternative theorem
Mathematical tools Hypersingular BIE (potential theory) Degenerate kernel Circulants SVD updating term SVD updating document Fredholm alternative theorem
Two systems u and U U(x,s) u(x) Domain(D) 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 Moment Force U:moment diagram T:moment diagram L:shear diagram M:moment diagram
Order of pseudo-differential operator • 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
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)
NTUCE Ó 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)
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) U T s s 0 x x 0 L s M s x x
Bump contribution (3-D) 0` s s x x 0 s s x x
Hypersingular BIE Degenerate kernel
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)
Fundamental solution • Field response due to source (space) • Green’s function • Casual effect (time) K(x,s;t,τ)
Separable form of fundamental solution (1D) Separable property continuous discontinuous
Degenerate kernel (step1) x Step 1 x: variable s: fixed R S
Degenerate kernel (step2, step3) Step 2 x Step 3 s s RA RB A A B B x
Mathematical tools Hypersingular BIE Degenerate kernel Circulants SVD updating term SVD updating document Fredholm alternative theorem
oblique incident water wave Seepage with sheetpiles Thin-airfoill Aerodynamics Top view y 4 7 Free water surface 6 5 S x 8 3 O Pseudo boundary Pseudo boundary II III d z I 1 2 O S x Engineering problems
Degeneracy of the Degenerate Boundary geometry node (1,0.5) (-1,0.5) 4 7 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(-)
Theory of dual integral equations The constraint equation is not enough to determine the coefficient p and q, Another constraint equation is required