Study on the degenerate problems in BEM 邊界元素法中退化問題之探討 指導教授：陳正宗 教授 研究生：林書睿 國立臺灣海洋大學河海工程研究所結構組 碩士班畢業論文口試 時間: 5/21, 2002 地點: 河工二館307室

Mathematical essence—rank deficiency Motivation Four pitfalls in BEM Why numerical instability occurs in BEM ? (1) degenerate scale (2) degenerate boundary (3) fictitious frequency Why spurious eigenvalues appear ? (4) true and spurious eigenvalues Mathematical essence—rank deficiency (How to deal with ?)

Outlines Degenerate scale for torsion bar problems Degenerate boundary problems True and spurious eigensolution for interior eigenproblem Fictitious frequency for exterior acoustics Conclusions and further research

Four pitfalls in BEM 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

The degenerate scale for torsion bar using BEM Error (%) of torsional rigidity 125 5 a Previous approach : Try and error on a Present approach : Only one trial

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

Degenerate scale for torsion bar problems with arbitrary cross sections An analytical way to determine the degenerate scale The existence of degenerate scale for the two-dimensional Laplace problem xd x sd s Bd B

where is boundary density function Degenerate scale for torsion bar problems with arbitrary cross sections where is boundary density function Mapping properties Expansion ratio , where

Adding a rigid body term c in the fundamental solution For arbitrary cross section, expansion ratio is

Original degenerate scale Degenerate scale for torsion bar problems with arbitrary cross sections 0.184 0.50 Unregularized Regularized c=1.0 Shifting Normal scale Original degenerate scale New degenerate scale

Original degenerate scale Degenerate scale for torsion bar problems with arbitrary cross sections Unregularized Regularized Shifting 0.85 0.31 ( ): exact solution of the degenerate scale 2a 0.4 0.8 1.2 1.6 2 a 0.1 0.2 0.3 0.5 1 S q u r e c o s t i n Conventional BEM (UT formulation) Adding a rigid body term (c=1.0) Normal scale Original degenerate scale New degenerate scale s

Determination of the degenerate scale for the two-dimensional Laplace problems   Cross Section Normal scale Torsional rigidity Reference equation , where , x on B, . 1.4480 1.4509 1.5539 (N.A.) 2.6972 (N.A.) 6.1530 (6.1538) Expansion ratio 0.5020 (0.5) 0.5019 (0.5) 0.5254 (N.A.) 0.6902 (N.A.) 0.8499 (0.85) Degenerate scale R=1.0040 (1.0) = 2.0058 (2.0) a=1.0508 (N.A.) h=2.0700 (N.A.) a=0.8499 (0.85) 2a R a b h Note: Data in parentheses are exact solutions. Data marked in the shadow area are derived by using the polar coordinate.

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

Regularization techniques are not necessary. Numerical results Normal scale ( =3.0, =1.0) Degenerate scale ( =1.5, =0.5) Analytical solution 8.4823 0.5301 2.249 1.174 U T Conventional BEM 8.7623 (3.30%) -0.8911 (268.10%) 2.266 (0.76%) 2.0570 (75.21%) L M formulation 0.4812 (9.22%) 1.1472 (2.31%) Add a rigid body term c=1.0 0.5181 (2.26%) 1.1721(0.19%) c=2.0 0.5176 (2.36%) 1.1723 (0.17%) CHEEF concept 0.5647 (6.53%) CHEEF POINT (2.0, 2.0) 1.1722 (0.18%) CHEEF POINT (5.0, 5.0) cross section 2a Torsion rigidity Square Ellipse method (a=1.0) (a=0.85) Regularization techniques are not necessary. Regularization techniques are not necessary. Note: data in parentheses denote error.

Regularization techniques are not necessary. Numerical results Normal scale h=3.0 Degenerate scale h=2.07 Analytical solution 3.1177 0.7067 12.6488 0.9609 U T Conventional BEM 3.1829 (2.09%) 1.1101 (57.08%) 12.5440 (0.83%) 1.8712 (94.73%) L M formulation 0.6837 (3.25%) 0.9530 (0.82%) Add a rigid body term c=1.0 0.7035 (0.45%) 0.9876 (2.78%) c=2.0 0.7024 (0.61%) 0.9879 (2.84%) CHEEF concept 0.7453 (5.46%) CHEEF POINT (15.0, 15.0) 0.9272 (3.51%) CHEEF POINT (20.0, 20.0) cross section b a h Torsion rigidity Keyway Triangle (a=2.0) (a=1.05) method Regularization techniques are not necessary. Regularization techniques are not necessary. Note: data in parentheses denote error.

Four pitfalls in BEM 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

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

Singular Value Decomposition 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) Multi-domain BEM Dual BEM versus k Determinant versus k Dual BEM Determinant versus k

Two sources of rank deficiency (k=3.09) Nd=5 None trivial sol. Degenerate boundary Eigensolution

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

Four pitfalls in BEM 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

True and spurious eigensolution for Interior eigenproblem Complex-valued BEM --- true eigenvalues Spurious eigenvalues? MRM Real-part BEM Imaginary-part BEM

SVD structure for the influence matrices (true) k=kt Dirichlet problem Neumann problem

SVD structure for the influence matrices (spurious) k=ks Singular formulation Hypersingular formulation

True eigensolution for interior eigenproblems Real-part BEM SVD updating technique Dirichlet problem Neumann problem 20 constant elements

True eigensolution for interior eigenproblems Imaginary-part BEM SVD updating technique Dirichlet problem Neumann problem 8 constant elements

Spurious eigensolution for interior eigenproblems Real-part BEM The Fredholm alternative theorem and SVD updating technique Singular formulation Hypersingular formulation 20 constant elements

Spurious eigensolution for interior eigenproblem Imaginary-part BEM The Fredholm alternative theorem and SVD updating technique Singular formulation Hypersingular formulation 8 constant elements

Four pitfalls in BEM 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 t(a,0) ka

Fictitious frequency for exterior acoustics Mathematical structure for updating matrix Source of numerical instability---zero division by zero A criterion to check the validity of CHIEF points Numerical example

Mathematical structure for updating matrix Proof The subscript i denotes the use of interior degenerate kernel for exterior problem

Mathematical structure for updating matrix For the Dirichlet eigenproblem or symmetry or transpose symmetry

Mathematical structure for updating matrix Multiplicity: True boundary mode for the Dirichlet eigenproblem Fictitious boundary mode

Source of numerical instability---zero division by zero For the Dirichlet problem By pre-multiplying the regular modes Solvable

Source of numerical instability---zero division by zero By pre-multiplying the fictitious modes Unsolvable

A criterion to check the validity of CHIEF points Adding CHIEF points New constraints

A criterion to check the validity of CHIEF points The selected P CHIEF points are valid (No change) For the Neumann problem

Numerical example : valid Real-part Imaginary-part : invalid

Conclusions and further research A more efficient technique was proposed to directly determine the degenerate scale since only one normal scale needs to be computed. The conventional BEM in conjunction with SVD was applied to deal with rank-deficiency (degenerate boundary and nontrivial eigensolution) for the degenerate boundary eigenproblem. By using the Fredholm alternative theorem and SVD techniques in conjunction with the dual formulations, the true and spurious eigenvalues in the complex-valued formulation, the real-part, the imaginary-part BEMs and MRM were sorted out successfully. In order to overcome the rank-deficiency problem due to fictitious frequency, the CHIEF method was reformulated in a unified manner by using the Fredholm alternative theorem and SVD technique.

Further research Although the degenerate scale occurs in the Dirichlet problem of simply two-dimensional Laplace problems by using the BEM, there is no proof of the occurrence of degenerate scale for the problem with the mixed-type boundary condition. The main drawback of the imaginary-part BEM seems to produce ill-conditioned matrices. While this is sometimes the case, it is hoped that further research can alleviate the drawback. Whether the spurious (fictitious) modes in the UT and LM formulations are the same or not needs further investigation.