1 Some problems in BEM applications J. T. Chen 陳正宗 特聘教授 Department of Harbor and River Engineering National Taiwan Ocean University, Keelung, Taiwan March 1, 2005 Presentation for 台科大營建系

2 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 Outlines

3 Related works since 1984 Research topics of NTOU MSV LAB ( )

4 Numerical phenomena (Degenerate scale) Error (%) of torsional rigidity a Previous approach : Try and error on a Present approach : Only one trial

5 Numerical phenomena (Fictitious frequency) t(a,0)

6 Numerical phenomena (Spurious eigensolution)

7 Numerical phenomena (Degenerate boundary) Singular integral equation Hypersingular integral equation Cauchy principal valueHadamard principal value Boundary element method Dual boundary element method degenerate boundary normal boundary

8 Numerical phenomena (Corner) Boundary

9 Motivation Numerical instability occurs in BEM ? (1) degenerate scale (2) degenerate boundary (3) fictitious frequency (4) corner Spurious eigenvalues appear ? (5) true and spurious eigenvalues Five pitfalls in BEM Mathematical essence—rank deficiency ? (How to deal with ?) nonuniqueness ?

10 Mathematical tools Hypersingular BIE Degenerate kernel Circulants SVD updating term SVD updating document Fredholm alternative theorem

11 Mathematical tools Hypersingular BIE Degenerate kernel Circulants SVD updating term SVD updating document Fredholm alternative theorem

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

13 Mathematical tools Hypersingular BIE Degenerate kernel Circulants SVD updating term SVD updating document Fredholm alternative theorem

14 Degenerate kernel (step1) Step 1 S x R x: variable s: fixed

15 Degenerate kernel (step2, step3) x s A B Step 2 RARA Step 3 x B A s RBRB

16 Mathematical tools Hypersingular BIE Degenerate kernel Circulants SVD updating term SVD updating document Fredholm alternative theorem

17 Circulant

18 Mathematical tools Hypersingular BIE Degenerate kernel Circulants SVD updating term SVD updating document Fredholm alternative theorem

19 SVD Diagonal matrix Unitary matrix,

20 Direct method for Dirichlet B. C. : SVD updating terms SVD updating terms Singular equation (UT method) Hypersingular equation (LM method)

21 Mathematical tools Hypersingular BIE Degenerate kernel Circulants SVD updating term SVD updating document Fredholm alternative theorem

22 For double-layer potential approach: SVD updating documents A b x

23 Mathematical tools Hypersingular BIE Degenerate kernel Circulants SVD updating term SVD updating document Fredholm alternative theorem

24 Fredholm ’ s alternative theorem: For solving an algebraic system: Fredholm alternative theorem Alternative theorem Infinite solution No solution : the transpose conjugate matrix of where  satisfies

25 1.Degenerate scale for torsion bar problems 2.Degenerate boundary problems 3.True and spurious eigensolution for interior eigenproblem 4.Fictitious frequency for exterior acoustics 5. Corner Five pitfalls in BEM

26 The degenerate scale for torsion bar using BEM Error (%) of torsional rigidity a Previous approach : Try and error on a Present approach : Only one trial

27 Direct searching for the degenerate scale Trial and error---detecting zero singular value by using SVD [Lin (2000) and Lee (2001)] Determination of the degenerate scale by trial and error

28 Cross Section Normal scale Torsional rigidity Reference equation, where, x on B, (N.A.) (N.A.) (6.1538) Expansion ratio (0.5) (0.5) (N.A.) (N.A.) (0.85) Degenerate scale R= (1.0) = (2.0)a= (N.A.)h= (N.A.)a= (0.85) h R a b 2a2a 2a2a Determination of the degenerate scale for the two-dimensional Laplace problems Note: Data in parentheses are exact solutions. Data marked in the shadow area are derived by using the polar coordinate.

29 Three regularization techniques to deal with degenerate scale problems in BEM Hypersingular formulation (LM equation) Adding a rigid body term (U*(s,x)=U(s,x)+c) CHEEF concept

30 Degenerate scale for torsion bar problems with arbitrary cross sections Unregularized Regularized c=1.0 Shifting Normal scale Original degenerate scale New degenerate scale

31 Degenerate scale for torsion bar problems with arbitrary cross sections 

32 Normal scale ( =3.0, =1.0) Degenerate scale ( =1.5, =0.5) Normal scaleDegenerate scale Analytical solution U T Conventional BEM (3.30%) (268.10%)2.266 (0.76%) (75.21%) L M formulation (9.22%) (2.31%) Add a rigid body term c= (2.26%)1.1721(0.19%) c= (2.36%) (0.17%) CHEEF concept (6.53%) CHEEF POINT (2.0, 2.0) (0.18%) CHEEF POINT (5.0, 5.0) Ellipse 2a2a 2a2a Square cross section Torsion rigidity method Regularization techniques are not necessary. (a=1.0)(a=0.85) Numerical results Note: data in parentheses denote error.

33 Normal scale h=3.0 Degenerate scale h=2.07 Normal scaleDegenerate scale Analytical solution U T Conventional BEM (2.09%) (57.08%) (0.83%) (94.73%) L M formulation (3.25%) (0.82%) Add a rigid body term c= (0.45%) (2.78%) c= (0.61%) (2.84%) CHEEF concept (5.46%) CHEEF POINT (15.0, 15.0) (3.51%) CHEEF POINT (20.0, 20.0) Triangle Keyway cross section Torsion rigidity method Regularization techniques are not necessary. (a=2.0)(a=1.05) Numerical results Note: data in parentheses denote error. h a b

34 Error using three methods

35 Multiply-connected problem

36 1.Degenerate scale for torsion bar problems 2.Degenerate boundary problems 3.True and spurious eigensolution for interior eigenproblem 4.Fictitious frequency for exterior acoustics 5. Corner Five pitfalls in BEM

37 Engineering problems Thin-airfoill Aerodynamics Seepage with sheetpiles oblique incident water wave Free water surface I IIIII Pseudo boundary O S x S y d z  x Top view O

38 Degeneracy of the Degenerate Boundary geometry node the Nth constant or linear element (0,0) (-1,0.5) (-1,-0.5) (1,0.5) (1,-0.5) N 5(+) 6(+) 5(+) 6(-) 5(+) 6(+) 5(+) 6(+) 5(+) 6(+) 5(+) 6(-) 5(+) 6(-) 5(+) 6(-)

39 The constraint equation is not enough to determine the coefficient p and q, Another constraint equation is required Theory of dual integral equations

40 Successful experiences

41 X-ray detection

42 FEM simulation

43 Stress analysis

44 BEM simulation

45 V-band structure (Tien-Gen missile)

46 FEM simulation

47

48 Shong-Fon II missile

49 IDF

50 Flow field

51 Seepage flow

52 Meshes of FEM and BEM

53 Screen in acoustics b a e c t=0

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

55 Reflection and Transmission

56 Cracked torsion bar

57

58

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

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

61 Dual BEM Multi-domain BEM UT BEM + SVD (Present method) versus k Determinant versus k

62 Two sources of rank deficiency (k=3.09) N d= 5 None trivial sol. EigensolutionDegenerate boundary

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

64 1.Degenerate scale for torsion bar problems 2.Degenerate boundary problems 3.True and spurious eigensolution for interior eigenproblem 4.Fictitious frequency for exterior acoustics 5. Corner Five pitfalls in BEM r2r2 r1r1

65 11 k r2r2 r1r1 r 1 =0.5 m r 2 =2.0 m Spurious eigenvalue of membrane

66 Treatments SVD updating term Burton & Miller method CHIEF method

67 11 k SVD updating term for true eigenvalue True eigenvalues s s s

68 Burton & Miller method

69 Two CHIEF points for spurious eigenvalues of multiplicity two 11 k s

70 Spurious eigenvalue of plate

71 Treatments SVD updating term Burton & Miller method CHIEF method

72 SVD updating term

73 The Burton & Miller concept

74 The CHIEF concept

75 1.Degenerate scale for torsion bar problems 2.Degenerate boundary problems 3.True and spurious eigensolution for interior eigenproblem 4.Fictitious frequency for exterior acoustics 5. Corner Five pitfalls in BEM ka t(a,0)

76 On the Mechanism of Fictitious Eigenvalues in Direct and Indirect BEM incident wave radiation Physical resonance Numerical resonance

77 CHIEF and Burton & Miller method ka

78 1.Degenerate scale for torsion bar problems 2.Degenerate boundary problems 3.True and spurious eigensolution for interior eigenproblem 4.Fictitious frequency for exterior acoustics 5. Corner Five pitfalls in BEM

79 Boundary Theory of Dual Integral Equations for a Corner

80 Domain Boundary Domain Boundary Domain Boundary Domain Boundary (1) U(s,x) (4) M(s,x) (2) T(s,x) (3) L(s,x) The related symbols around the corner

81 kernelLaplace equationHelmholtz equation U(s,x)U(s,x) T(s,x) L(s,x) M(s,x) Free terms

82 Conclusions The nonuniqueness in BIEM and BEM were reviewed and its treatment was addressed. The role of hypersingular BIE was examined. The numerical problems in the engineering applications using BEM were demonstrated. Several mathematical tools, SVD, degenerate kernel, …, were employed to deal with the problems.

83 The End Thanks for your kind attention

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