1 The 27 th Conference on Theoretical and Applied Mechanics --- 學生論文競賽 National Cheng-Kung University, Tainan, Taiwan, R. O. C. On the equivalence of the Trefftz method and the method of fundamental solutions for plate problem Ching-Sen Wu ( 吳清森 ) National Taiwan Ocean University Dec

2 The 27 th Conference on Theoretical and Applied Mechanics --- 學生論文競賽 National Cheng-Kung University, Tainan, Taiwan, R. O. C. Outlines 1.Literature review 2.Trefftz method and Method of fundamental solutions (MFS) 3.Description of the biharmonic equation 4.Connection between the Trefftz method and the MFS for plate problem 5.On the complete set of the Trefftz method and the MFS using the degenerate kernel 6.Concluding remarks

3 The 27 th Conference on Theoretical and Applied Mechanics --- 學生論文競賽 National Cheng-Kung University, Tainan, Taiwan, R. O. C. 1.Literature review 2.Trefftz method and Method of fundamental solutions (MFS) 3.Description of the biharmonic equation 4.Connection between the Trefftz method and the MFS for plate problem 5.On the complete set of the Trefftz method and the MFS using the degenerate kernel 6.Concluding remarks

4 The 27 th Conference on Theoretical and Applied Mechanics --- 學生論文競賽 National Cheng-Kung University, Tainan, Taiwan, R. O. C. Literature review Trefftz method Trefftz (1926) Cheung et al IJNME Hemholtz equation Jin et al IJNMEElasticity (Navier equation) Jin et al IJNME Biharmonic equation Kita et al Adv. Engng. Soft. overview Method of fundamental solutions (MFS) Kupradze (1964) Fairweather et al Adv. Comp. Math. Potential problem Karageophis et al IJNME Potential problem Poullikkas et al JASA Acoustic problem Poullikkas et al Comp. Mech. Harmonic and biharmonic BVP Karageophis et al Appl. Math. Letters Helmholtz equation Motivation 1 relation ?

5 The 27 th Conference on Theoretical and Applied Mechanics --- 學生論文競賽 National Cheng-Kung University, Tainan, Taiwan, R. O. C. Literature review EngineersYearProblemJournalTreatment He et al. (China) D potential and plane elasticity problem Computer & Structures CNME Necessary and sufficient boundary integral formulation (NSBIE) Zhou et al. (China) D elasticity problem CNME Boundary contour method (BCM) He, W. J. (China) 2000BIE---thin plate Computer & Structures Equivalent BIE Chen et al. (Taiwan) 2002Plane elasticityIJNMEHypersingular formulation Chen et al. (Taiwan) D Laplace equationEABE Combined Helmholtz exterior integral equation formulation (CHEEF) Chen et al. (Taiwan) D Laplace and Navier problem IJNME (Accepted) Addition of rigid body term Hypersingular formulation CHEEF concept Does the degenerate scale will occur using the Trefftz method and MFS? Motivation 2:

6 The 27 th Conference on Theoretical and Applied Mechanics --- 學生論文競賽 National Cheng-Kung University, Tainan, Taiwan, R. O. C. 1.Literature review 2.Trefftz method and Method of fundamental solutions (MFS) 3.Description of the biharmonic equation 4.Connection between the Trefftz method and the MFS for plate problem 5.On the complete set of the Trefftz method and the MFS using the degenerate kernel 6.Concluding remarks

7 The 27 th Conference on Theoretical and Applied Mechanics --- 學生論文競賽 National Cheng-Kung University, Tainan, Taiwan, R. O. C. Trefftz method and MFS Method Trefftz methodMFS Definition Base u j (x) (T-complete function) U(x,s)=U(r), r=|x-s| G. E. L u(x)=0, L U(x,s)=0, (singularity at s) B. C.match c j match v j is the number of complete functions is the number of source points in the MFS s D u(x) r D

8 The 27 th Conference on Theoretical and Applied Mechanics --- 學生論文競賽 National Cheng-Kung University, Tainan, Taiwan, R. O. C. 1.Literature review 2.Trefftz method and Method of fundamental solutions (MFS) 3.Description of the biharmonic equation 4.Connection between the Trefftz method and the MFS for plate problem 5.On the complete set of the Trefftz method and the MFS using the degenerate kernel 6.Concluding remarks

9 The 27 th Conference on Theoretical and Applied Mechanics --- 學生論文競賽 National Cheng-Kung University, Tainan, Taiwan, R. O. C. Biharmonic equation 1. Plane elasticity: 2. Plane of slow viscous flow (Stokes’ Flow): 3. Solid mechanics (Plate problem):

10 The 27 th Conference on Theoretical and Applied Mechanics --- 學生論文競賽 National Cheng-Kung University, Tainan, Taiwan, R. O. C. 1.Literature review 2.Trefftz method and Method of fundamental solutions (MFS) 3.Description of the biharmonic equation 4.Connection between the Trefftz method and the MFS for plate problem 5.On the complete set of the Trefftz method and the MFS using the degenerate kernel 6.Concluding remarks

11 The 27 th Conference on Theoretical and Applied Mechanics --- 學生論文競賽 National Cheng-Kung University, Tainan, Taiwan, R. O. C. Problem statement uniform pressure a B w=constant : flexure rigidity : deflection of the circular plate : uniform distributed load : domain of interest Governing equation: Boundary condition: Splitting method Governing equation: Boundary condition: General form Governing equation: Boundary condition:

12 The 27 th Conference on Theoretical and Applied Mechanics --- 學生論文競賽 National Cheng-Kung University, Tainan, Taiwan, R. O. C. Analytic solution for biharmonic equation For the general form of specified boundary conditions: Analytic solution for biharmonic equation: where

13 The 27 th Conference on Theoretical and Applied Mechanics --- 學生論文競賽 National Cheng-Kung University, Tainan, Taiwan, R. O. C. Trefftz method Field solution : T-complete functions: Analytical solution:

14 The 27 th Conference on Theoretical and Applied Mechanics --- 學生論文競賽 National Cheng-Kung University, Tainan, Taiwan, R. O. C. Match the boundary conditions (Trefftz method) By matching the boundary conditions at Undetermined Trefftz coefficients Specified coefficients of B.C.

15 The 27 th Conference on Theoretical and Applied Mechanics --- 學生論文競賽 National Cheng-Kung University, Tainan, Taiwan, R. O. C. Method of fundamental solutions (MFS) Field solution : By using the degenerate kernels, we have

16 The 27 th Conference on Theoretical and Applied Mechanics --- 學生論文競賽 National Cheng-Kung University, Tainan, Taiwan, R. O. C. Derivation of the analytic solution using the MFS Analytical solution:

17 The 27 th Conference on Theoretical and Applied Mechanics --- 學生論文競賽 National Cheng-Kung University, Tainan, Taiwan, R. O. C. Match the boundary conditions (MFS) (1): (2): (3): (4): (5): (6): (7): (8): (9): (10): Relation of the coefficients using the MFS versus the specified coefficients of boundary condition :

18 The 27 th Conference on Theoretical and Applied Mechanics --- 學生論文競賽 National Cheng-Kung University, Tainan, Taiwan, R. O. C. Relation of coefficient using the MFS Undetermined coefficients for MFS Specified coefficients of B.C.

19 The 27 th Conference on Theoretical and Applied Mechanics --- 學生論文競賽 National Cheng-Kung University, Tainan, Taiwan, R. O. C. K matrix

20 The 27 th Conference on Theoretical and Applied Mechanics --- 學生論文競賽 National Cheng-Kung University, Tainan, Taiwan, R. O. C. Relationship between the Trefftz method and MFS Coefficients of the Trefftz method Coefficients of the MFS Mapping matrix [K]

21 The 27 th Conference on Theoretical and Applied Mechanics --- 學生論文競賽 National Cheng-Kung University, Tainan, Taiwan, R. O. C. Decomposition of the K matrix

22 The 27 th Conference on Theoretical and Applied Mechanics --- 學生論文競賽 National Cheng-Kung University, Tainan, Taiwan, R. O. C. By using the orthogonal property, we have Orthogonal property

23 The 27 th Conference on Theoretical and Applied Mechanics --- 學生論文競賽 National Cheng-Kung University, Tainan, Taiwan, R. O. C. Diagonal matrix T R Existence of the degenerate scales Nonuniqueness (in numerical aspect) Degenerate scale problem O. K.!

24 The 27 th Conference on Theoretical and Applied Mechanics --- 學生論文競賽 National Cheng-Kung University, Tainan, Taiwan, R. O. C. Special size: : position of the source points The occurrence of the degenerate scales using the MFS numerical fails a

25 The 27 th Conference on Theoretical and Applied Mechanics --- 學生論文競賽 National Cheng-Kung University, Tainan, Taiwan, R. O. C. Review the treatments of the degenerate scale (Mathematicians) MathematicianYearProblemJournal Treatments and logarithm capacity Soren Christiansen (Denmark) 1982 Detect non-unique solution Applicable Analysis Scaling method Restriction method Constanda (U. K.) 1995 Non-unique solution in plane elasticity problem Quart. Appl. Math. First kind integral equations Martin et al. (France) 1996 Invertibility of single layer potential operator Integr. Equat. Oper. Th. Logarithm capacity Soren Christiansen (Denmark) 1998 Investigation of direct BIE for biharmonic problem JCAMLogarithm capacity Soren Christiansen (Denmark) 2001 Detecting non- uniqueness of solution through SVD JCAMLogarithm capacity C. S. Wu (Taiwan) Degenerate scales 2003MFS

26 The 27 th Conference on Theoretical and Applied Mechanics --- 學生論文競賽 National Cheng-Kung University, Tainan, Taiwan, R. O. C. 1.Literature review 2.Trefftz method and Method of fundamental solutions (MFS) 3.Description of the biharmonic equation 4.Connection between the Trefftz method and the MFS for plate problem 5.On the complete set of the Trefftz method and the MFS using the degenerate kernel 6.Concluding remarks

27 The 27 th Conference on Theoretical and Applied Mechanics --- 學生論文競賽 National Cheng-Kung University, Tainan, Taiwan, R. O. C. On the complete set of the Trefftz method and the MFS using the degenerate kernel T-complete functions of the Trefftz method: Degenerate kernel of the MFS: m=0 m=1 m=2, 3…..

28 The 27 th Conference on Theoretical and Applied Mechanics --- 學生論文競賽 National Cheng-Kung University, Tainan, Taiwan, R. O. C. 1.Literature review 2.Trefftz method and Method of fundamental solutions (MFS) 3.Description of the biharmonic equation 4.Connection between the Trefftz method and the MFS for plate problem 5.On the complete set of the Trefftz method and the MFS using the degenerate kernel 6.Concluding remarks

29 The 27 th Conference on Theoretical and Applied Mechanics --- 學生論文競賽 National Cheng-Kung University, Tainan, Taiwan, R. O. C. Concluding remarks 1.The mathematical equivalence for biharmonic equations between the Trefftz method and MFS was proved successfully. 2.It is interesting to find that the T-complete set in the Trefftz method are imbedded in the degenerate kernels of MFS for solving the biharmonic equation. 3.The degenerate scales occur when the fictitious sources of MFS are located at e 0, e -0.5, e -1 for circular case. 4.By means of the derivation of degenerate scales in this paper, the Software designer will be easier to handle the program.

30 The 27 th Conference on Theoretical and Applied Mechanics --- 學生論文競賽 National Cheng-Kung University, Tainan, Taiwan, R. O. C. The End Thanks for your attention

31 The 27 th Conference on Theoretical and Applied Mechanics --- 學生論文競賽 National Cheng-Kung University, Tainan, Taiwan, R. O. C. Degenerate kernel General form of the fundamental solution in the MFS: The superscripts “I” and “E” denote the interior and exterior domain, respectively.

32 The 27 th Conference on Theoretical and Applied Mechanics --- 學生論文競賽 National Cheng-Kung University, Tainan, Taiwan, R. O. C. Degenerate kernel x: field point (variable) s: source point (fixed) S x r x s R U E : R U I : R Closed-form: Series-form:

33 The 27 th Conference on Theoretical and Applied Mechanics --- 學生論文競賽 National Cheng-Kung University, Tainan, Taiwan, R. O. C.

34 The 27 th Conference on Theoretical and Applied Mechanics --- 學生論文競賽 National Cheng-Kung University, Tainan, Taiwan, R. O. C. Derivation of degenerate kernel Use the Complex Variable method to derive the degenerate kernel: Motivation:

35 The 27 th Conference on Theoretical and Applied Mechanics --- 學生論文競賽 National Cheng-Kung University, Tainan, Taiwan, R. O. C. Derivation of degenerate kernel s x Due to: Not important