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
References
olutions”, otential
dial
Computational
l Chinese Workshop on Boundary Element
breakwater”, Engineering Analysis with
e problems”, Engineering Analysis with Boundary
problem”, Engineering Analysis
using regularization tz problems”
vdlopment”, Engineering Analysis with Boundary
indicators”,
n method”, Engineering Analysis with [1]. C. J. S. Alves, “On the choice of source points in the method of fundamental s
Engineering Analysis with Boundary Elements, Vol. 33, pp. 1348-1361, 2009.
[2]. H. Chen, J. Jin, P. Zhang, P. LÜ, “Multi-Variable Non-Singular BEM for 2-D P Problems”, Tsinghua Science and Technology, Vol. 10, No. 1, pp. 43-50, 2005.
[3]. J. T. Chen, M .H. Chang, K. H. Chen, S. R. Lin, “The boundary collocation method with meshless concept for acoustic eigenanalysis of two-dimensional cavities using ra basis function”, Journal of Sound and Vibration, Vol. 257, No.4, pp. 667-711, 2002.
[4]. J. T. Chen, M. H. Chang, K. H. Chen, “Boundary collocation method for acoustic eigenanalysis of three-dimensional cavities using radial basis function”,
Mechanics, Vol. 29, pp. 392-408, 2002.
[5]. J. T. Chen, I. L. Chen, C. S. Wu, “On the equivalence of MFS and Trefftz method for Laplace problems, Proceedings of the Globa
and Meshless Method”, Hebei, China, 2003.
[6]. K. H. Chen, J. T. Chen, C. R. Chou, C. Y. Yueh, “Dual boundary element analysis of oblique incident wave passing a thin submerged
Boundary Elements, Vol.26, pp.917–928, 2002.
[7]. K. H. Chen, J. H. Kao, J. T. Chen, D.L. Young, M.C. Lu, “Regularized meshless method for multiply-connected-domain Laplac
Elements, Vol. 30, pp. 882-896, 2006.
[8]. K. H. Chen, C. T. Chen, J. F. Lee, “Adaptive error estimation technique of the Trefftz method for solving the over-specified boundary value
with Boundary Elements, Vol. 33, pp. 966-982 , 2009.
[9]. K. H. Chen, J. H. Kao, J. T. Chen, K. L. Wu, “Desingularized meshless method for solving Laplace equation with over-specified boundary conditions
techniques”, Computational Mechanics, Vol. 43, pp. 827-837, 2009.
[10]. W. Chen, “Meshfree boundary particle method applied to Helmhol Engineering Analysis with Boundary Elements, Vol. 26, pp. 577-81, 2002.
[11]. M. A. Golberg, H. Bowman, “Superconvergence and the use of the residual as an error estimator in the BEM. I: theoretical de
Elements, Vol. 25, pp. 511-521, 1999.
[12]. M. A. Golberg, H. Bowman, “Superconvergence and the use of the residual as an error estimator in the BEM. II: Collocation, numerical integration and error
Engineering Analysis with Boundary Elements, Vol. 25, pp. 511-521, 1999.
[13]. C. S. Huang, C. F. Lee, A. H. D. Cheng, “Error estimate, optimal shape factor, and high precision computation of multiquadric collocatio
Boundary Elements, Vol. 31, pp. 614-623, 2007.
[14]. S. C. Huang, R. P. Shaw, “The Trefftz method as an integral equation”, Advances in
problems”, fftz method: an overview”, Advances in Engineering Software,
blems”, Engineering Analysis with Boundary
influence function”, Journal of Sound and Vibration, Vol.
influence function”, Journal of Sound and Vibration, Vol. 234,
ms”, Computational Mathematics ethod”, g, “The Trefftz method for solving
s”, Journal of Computational and Applied Mathematics, Vol. 200, pp. 231-254, uation with degeneracy”, Applied
methods”, Engineering Analysis with
d on BIE”, Engineering Analysis with Boundary Elements,
puter Methods in Applied Mechanics and Engineering, Engineering Software, Vol. 24, pp. 57-63, 1995.
[15]. A. B. Jorge, G. O. Ribeiro, T. S. Fisher, “Application of new error estimators based on gradient recovery and external domain approaches to 2D elastostatics
Engineering Analysis with Boundary Elements, Vol. 29, pp. 963-975 , 2005.
[16]. E. Kita, N. Kamiya, “Tre Vol. 24, pp. 3-12, 1995.
[17]. E. Kita, N. Kamiya, T. Iio, “Application of a direct Trefftz method with domain decomposition to 2D potential pro
Elements, Vol. 23, pp. 539-548, 1999.
[18]. S. W. Kang, J. M. Lee, Y. J. Kang, “Vibration analysis of arbitrary shaped membranes using non-dimensional dynamic
221, No. 1, pp. 117-132, 1999.
[19]. S. W. Kang, J. M. Lee, “Application of free vibration analysis of membranes using the non-dimensional dynamic
No. 3, pp. 455-470, 2000.
[20]. V. D. Kupradze, M. A. Aleksidze, “The method of functional equations for the approximate solution of certain boundary value proble
and Mathematical Physics, Vol. 4, pp. 199-205, 1964.
[21]. M. T. Liang, J. T. Chen, S. S. Yang, “Error estimation for boundary element m Engineering Analysis with Boundary Elements, Vol. 23, pp. 257-265, 1999.
[22]. Z. C. Li, T. T. Lu, H. S. Tsai, A. H. D. Chen eigenvalue problems”, Vol. 30, pp. 292-308, 2006.
[23]. Z. C. Li, “Error analysis of the Trefftz method for solving Laplace’s eigenvalue problem
2007.
[24]. Z. C. Li, “The Trefftz method for the Helmholtz eq Numerical Mathematics, Vol. 58, pp. 131-159, 2008.
[25]. Z. C. Li, L. J. Young, H. T. Huang, Y. P. Liu, A. D. Cheng, “Comparisons of fundamental solutions and particular solutions for Trefftz
Boundary Elements, Vol. 34, pp. 248-258, 2010.
[26]. H. Ma, Q. H. Qin, “Solving potential problems by a boundary-type meshless method-the boundary point method base
Vol. 31, pp. 749-761, 2007.
[27]. S. Suleau, A. Deraemaeker, P. Bouillard, “Dispersion and pollution of meshless solutions for the Helmholtz equation”, Com
Vol. 190, pp. 639-657, 2000.
[28]. T. Tsangaris, Y. S. Smyrlis, A. Karageorghis, “A Matrix Decomposition MFS Algorithm
in Annular Domains”, Computers, Materials & Continua, vol.1, od
puter Methods in Applied Mechanics and Engineering,
arbitrary domain”, Journal of Computational Physics, Vol. 209, pp.
s”, Computer Modeling in Engineering & Sciences, Vol. 19, No. 3 , pp. 197-221,
quation”, Computers
coustics ”, The Journal of the Acoustical Society of America, Vol.
International Journal
rnational Journal for
ry-value problems”, Advances in Engineering Software, Vol. 24, pp. 133-145, elements”, Engineering Analysis with Boundary Elements, Vol. 23, pp. 793-803, 1999.
for Biharmonic Problems no.3, pp.245-258, 2004.
[29]. M. Vable, “Controlling errors in the process of automating boundary elements meth analysis”, Engineering Analysis with Boundary Elements, Vol. 26, pp. 405-415, 2002.
[30]. J. G. Wang, G. R. Liu, “On the optimal shape parameters of radial basis functions used for 2-D meshless methods”, Com
Vol. 191, pp. 2611-2630, 2002.
[31]. D. L. Young, K. H. Chen, C. W. Lee, “Novel meshless method for solving the potential problems with
290-321, 2005.
[32]. D. L. Young, K. H. Chen and J. T. Chen, “A Modified method of fundamental solutions with source on the boundary for solving Laplace equation with circular and arbitrary domain
2007.
[33]. D. L. Young, C. C. Tsai, C. W. Chen, C. M. Fan, “The method of fundamental solutions and condition number analysis for inverse problems of Laplace e
and Mathematics with Applications, Vol. 55, pp. 1189-1200, 2008.
[34]. D. L. Young, K. H. Chen, C. W. Lee , “Singular meshless method using double layer potentials for exterior a
119, pp. 96-107,2006.
[35]. J. M. Zhang, Z. H. Yao, H. Li, “A hybrid boundary node method”, for Numerical Methods in Engineering, Vol. 53, pp. 751-763, 2002.
[36]. J. M. Zhang, M. Tanaka, T. Matsumoto, “Meshless analysis of potential problems in three dimensions with the hybrid boundary node method”, Inte
Numerical Methods in Engineering, Vol. 59, pp. 1147-1160, 2004.
[37]. X. Zheng, Z. H. Yao, “Some applications of the Trefftz method in linear elliptic bounda
1995.
[38]. Z. Zhao, X. Wang, “Error estimation and h adaptive boundary
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