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 1 A study of free terms and rigid body modes in the dual BEM 對偶邊界元素法中自由項與剛體運動項之研究 指導教授：陳正宗 研 究 生：陳韋誌 中華民國九十年六月十五日AM10:40~12:00 河工二館307室

The structure of thesis 2 The structure of thesis This thesis Laplace equation、 Navier equation (2-D、3-D) Free term [Chap 2、3] Jump term [Chap 2] Rigid body mode + complementary sol. [Chap 4] Closed-form 2. Series-form 1. Closed-form 1-D rod 1-D beam 2-D circular membrane Bump- contour approach 2. Limiting approach 1. Bump-contour approach

3 PART I : Free terms The dual boundary integral equations for 2-D problem with the interior point x Singular integral equation Free terms Hypersingular integral equation

The methods of deriving the free terms using BEM 4 The methods of deriving the free terms using BEM Method (1) : The limiting process B ' The DBIE for 2-D problem with the domain x B ' The null-field integral equations for 2-D problem

5 The methods of deriving the free terms using BEM Method (2) : The bump-contour approach The DBIE with the domain point x for 2-D problem The null-field integral equations for 2-D problem

6 The methods of deriving the free terms using BEM Method (2) : The bump-contour approach The DBIE with the domain point x for 3-D problem

7 The steps of using bump-contour approach for Laplace problem in the dual BEM Four kernel functions the unknown density functions： Using Taylor’s series expansion to find 1. Deriving the free terms with a smooth boundary 2. Deriving the jump terms for a general boundary

The explicit form of kernels for Laplace problem 8 The explicit form of kernels for Laplace problem The governing equation： The explicit form of kernels： Kernel function 2-D case 3-D case Remark

9 B ' Method (1) Method (2)

10 Free terms of different kinds of potential across smooth boundary for the Laplace problem Kernel function Density function -t u Free term Method (1) (2-D) no jump Method (2) (2-D) Method (1) (3-D) Method (2) (3-D)

11 The steps of using bump-contour approach for elasticity problem in the dual BEM Four kernel functions 1. Taylor’s series expansion to find 2. Using the linear elastic property to find 1. Deriving the free terms with a smooth boundary 2. Deriving the jump terms for a general boundary

The relationship between displacement and traction field 12 The relationship between displacement and traction field G.E. : The strain-displacement relations： The Hook’s law： Cauchy formula： (traction and stress) We can obtain the properties on a smooth boundary : OK OK OK

13 Free terms of different kinds of potential across smooth boundary 2-D elasticity problem Kernel function Density function Free term (2-D) (Navier) i=k=1 no jump i=2, k=1 i=1, k=2 i=k=2 (Laplace)

14 Free terms of different kinds of potential across smooth boundary for 3-D elasticity problem Kernel function   Density function Free term (3-D) (Navier) i=k=1  no jump i=2, k=1  0 i=3, k=1 i=1, k=2 i=k=2

15 Free terms of different kinds of potential across smooth boundary for 3-D elasticity problem Kernel function   Density function Free term (3-D) (Navier) i=3, k=2 no jump i=1, k=3 i=2, k=3 i=k=3 (3-D) (Laplace)

16 PART II : Jump terms The limiting approach in conjunction with the degenerate kernels Domain

17 The limiting approach in conjunction with the degenerate kernels Hypersingular integral equation Jump terms singular integral equation

18 ρ R θ

in the degenerate kernels [1] 19 Symbol and series form in the degenerate kernels [1]

in the degenerate kernels [2] 20 Series form in the degenerate kernels [2]

21 The contour of degenerate kernels for 2-D Laplace equation U kernel T kernel L kernel M kernel Closed-form Series-form

22 converge The free terms and jump terms by using the degenerate kernel in conjunction with the limiting approach[1] converge By using the generalized functions : where are the high order terms with a finite value.

23 diverge The free terms and jump terms by using the degenerate kernel in conjunction with the limiting approach[2] By using the generalized functions : By using the generalized functions : diverge where are the high order terms with a finite value.

24 The new definition of Hadamard principal value equivalence The generalized definition of the Hadamard principal value : equivalence The finite part can be exacted.

25 Jump terms of different kinds of potentials across smooth boundary for the Laplace problem Kernel function Density function -t u Jump term (2-D) Method (1) no jump Method (2) (3-D)

PART III : Rigid body mode 26 PART III : Rigid body mode UT formulation : Neumann problem Dirichlet problem Non-unique solution Physically realizable Non-unique solution Mathematically realizable Rigid body mode Spurious mode

The governing equation & fundamental solution 27 The governing equation & fundamental solution G.E. Fundamental Solution 1-D rod 1-D beam 2-D circular membrane

The degenerate scale problem for a rod 28 The degenerate scale problem for a rod UT formulation : Degenerate scale problem A new degenerate scale : Using L’Hopital’s rule

The degenerate scale problem for a beam 29 The degenerate scale problem for a beam UT formulation : A new degenerate scale : Degenerate scale problem

Using L’Hopital’s rule 30 Using L’Hopital’s rule

31 The degenerate scale problem for a circular membrane The original eigenvalues are Degenerate scale problem R=1 Numerical analysis

Overcome the degenerate scale problem for a circular membrane 32 Overcome the degenerate scale problem for a circular membrane The eigenvalues are A new degenerate scale lnR+c=0

The stiffness using SVD technique 33 The stiffness using SVD technique Employing the SVD technique

34 Fredholm’s alternative theorem For solving an algebraic system : If H is a singular matrix, and u has a nontrival solution. alternative theorem Homogeneous sol. where the transpose conjugate matrix of H if H is real

Fredholm’s alternative theorem and SVD updating technique 35 Fredholm’s alternative theorem and SVD updating technique UT formulation : For Dirichlet problem If U is a singular real matrix, Fredholm’s alternative theorem Spurious mode

For Dirichlet problem : 36 A special case for a rod using SVD updating technique and Fredholm’s alternative theorem For Dirichlet problem : The spurious mode

37 A special case for a rod using SVD updating technique and Fredholm’s alternative theorem For Neumann problem The rigid body mode The spurious mode & rigid body mode

38 A special case for a beam using SVD updating technique and Fredholm’s alternative theorem For Dirichlet problem The spurious mode

39 A special case for a beam using SVD updating technique and Fredholm’s alternative theorem For Neumann problem The rigid body mode

SVD updating technique : 40 A circular membrane using SVD updating technique and Fredholm’s alternative theorem SVD updating technique : The spurious mode & rigid body mode

41 Linear algebraic system Using SVD technique, we can find by or UT formulation : Generalized coordinate Based on Fredholm’s alternative theorem

42 Linear algebraic system is singular matrix By adding a matrix to [U] Matrix level We can deal with singular problem by UT formulation. The unique solution is preserved.

43 Conclusions 1. The free terms are contributed one half from L kernel and the other half from M kernel by bump-contour approach for 2-D potential problem. 2. The free terms are contributed one-third from L kernel and two-thirds from M kernel by bump-contour approach for 3-D potential problem. 3. The definition of Hadamard principal value can be generalized,i.e.,

44 Conclusions 4. The free terms for 2-D and 3-D elasticity problems are derived by bump-contour approach . 5. The same stiffness can be derived no matter what the rigid body mode and complementary solution are. 6. Based on the Fredholm’s alternative theorem and SVD updating technique, the rigid body mode and spurious mode can be obtained.

Further research 45 1.How to solve the corner problem and define the normal vector on the boundary deserve research. 2. The jump terms by using the limiting approach in conjunction with the degenerate kernels for 3-D Laplace problem need to study. 3. How to eliminate the rigid body mode for the potential and elasticity problems deserve research. 4. Based on Fredholm’s alternative theorem, the degenerate boundary problem will only be solved by UT formulation.