• 沒有找到結果。

A square domain (1×1) subjected to the Dirchlet B.C. is considered as a

The problem sketch is shown in Fig. 3-2 which has been solved by Young et al. [14], and an analytical solution is found as follows:

(3-22)

The R.M.S error curves (results) versus the number of elements with different number of terms of Trefftz complete set function comparing with quasi-analytical solution are shown in Fig. 3-3. By observing the error curves, the optimal number of elements is 200 at the corner of the curve. By observing Fig. 3-4, we can obtain the convergence and efficient result by using the BEM with the optimal number of elements.

Case 2: Quarter tube cross-section domain

We consider a rather standard problem, subjected to the mixed-type boundary condition as shown in Fig. 3-5 which has been solved by Chen et al. [5]. The analytical solution of the radial temperature distribution is given by

)

where and denote the inner and outer radii, respectively. The R.M.S error curves (results) are shown in Fig. 3-6. By observing the error curves, the optimal number of elements

R1 R2

is 80. The field solution along the radius, r=2.43, by using the different elements are shown in Fig. 3-7, and the potential at the point u(x=0.248, y=0.391) by using the different elements are

own in Fig. 3-8.

Cas

in Fig. 3-9(a) and (b), respectively. In this case, an analytical solution is found as follows:

(3-26)

Case 3-1: Dirichlet boundary condition

The R.M.S error curves (results) comparing with quasi-analytical solution are shown in Fig. 3-10. By observing the error curves, the optimal number of elements is 100.

Case 3-2: Mixed type boundary condition

The R.M.S error curves (results) comparing with quasi-analytical solution are shown in Fig. 3-11. By observing the error curves, the optimal number of elements is 100.

3.5 Conclusion

In this thesis, a new estimation technique is developed to obtain the optimal number of elements for the BEM, we successfully applied the systematic error estimation scheme to the BEM to solve 2-D potential problems. The numerical examination verifies the validity of the error estimator technique. The technique plays a role in determining the optimal number of elements which can be viewed as an objective way to obtain the relative errors in different number of elements without having analytical solution, and we can obtain the convergent results efficiently.

sh

e 3: arbitrary domain

The case 3-1 and case 3-2 are different B.C.s, as shown

y e

u= 0.5xsin

Chapter4 Conclusions

.1 Conclusions

esults of numerical example in the thesis, some concluding remarks are

and a quasi-analytical solution is

ue. By observing the error carve, we can obtain the optimal

roblem including interior problems,

e parameters on the numerical accuracy in the MFS, and we

on condition and hence the convergent numerical solution 4

In this study, we developed a novel estimation technique to obtain the optimal parameter of the method of fundamental solution MFS and the optimal number of elements in the BEM without having analytical solution. The convergent numerical solutions of the MFS and the BEM can be obtained in unavailable analytic solution condition. Based on the proposed formulation and the r

itemized as follows:

1. We present a novel error estimation technique for detecting the error quantities in the numerical solution without having analytical solution,

simulated to substitute for the real analytical solution.

2. The error curve of MFS versus different parameter can be derived by using the novel error estimation techniq

parameter in the MFS.

3. The estimation scheme that we proposed has been applied successfully to estimate the error magnitude of MFS for solving the 2D Laplace p

exterior problems and multiply connected problems.

4. We studied the effect of shap find the optimal parameters.

5. In the conventional boundary element analysis, but the mesh number depends solely on an analyst’s experience and his/her intuition. By applying the error estimation technique that proposed in this thesis, the optimal number of elements in the BEM can be gained in unavailable analytical soluti

can be efficiently obtained.

.2 Further research

Laplace problem. Then,

problem. We may also

ccessful experiences in 2D problems, it is possible to extend this concept

e multiple inclusions problem or the more real world

d BEM, we may also employ the estimation scheme in the finite element method (FEM).

4

In this thesis, the systematic error estimation scheme that we proposed has been successfully applied to the MFS and BEM, respectively, to solve 2D

there are several interested issues which need further investigations:

1. This article mainly discusses the Laplace problem. According to our successful experiences of novel estimation technique for solving Laplace

employ the estimation scheme to solve the Helmholtz equation.

2. Based on our su to 3-D cases.

3. The possible applications to th problem deserve further study.

4. According to our successful experiences of novel estimation technique in the MFS an

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Estimation the optimal parameter in the MFS

(Chapter 2)

Estimation the optimal number of elements in the the BEM

(Chapter 3)

Interior domain Exterior domain Multiply-connected domain

Conclusions and further research

Application of the novel error estimation technique

Laplace problem Estimation the optimal parameter

in the MFS (Chapter 2)

Estimation the optimal number of elements in the the BEM

(Chapter 3)

Interior domain Exterior domain Multiply-connected domain

Conclusions and further research

Application of the novel error estimation technique

Laplace problem

Fig.1-1.The frame of this thesis

Define G.E., contour and B.C. type of new problem

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