Concluding remarks

In this thesis, a new estimation technique is developed. We successfully applied the estimation technique in the MFS to derive the optimal parameter without having analytical solution. The technique plays a key role to maintain the system characteristic of MFS due to its excellent performance and high convergence order due to its exponential error convergence rate. The main disadvantage of using MFS which yields the problem in perplexing fictitious boundary can be overcome by the obtained the optimal parameter. The convergent result is found from the convergent study in the cases. Numerical results agreed very well with the analytical solutions. Finally, the several numerical examinations successfully verify the validity of the error estimation technique. We successfully gain the good prediction capability of the error estimation scheme.

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Chapter 3

An Application of New Error Estimation Technique to the Boundary Element Method

Summary

In this chapter, we develop a novel estimation technique to obtain the optimal number of elements in the boundary element method (BEM) without having analytical solution. A new problem is defined to substitute for the original problem. The governing equation, domain shape and boundary condition type in the new problem are the same as the original problem.

By using the complete Trefftz set as the analytical solution, the analytical solution in the new problem is the known, namely quasi-analytical solution, which is similar with the real analytical solution in the original problem. By implementing the BEM to solve the new problem and comparing with the quasi-analytical solution, the novel error estimation scheme is developed. The error curve of the R.M.S error versus different number of elements can be plotted by the proposed technique. As a result, we can obtain the optimal number of elements in BEM. Several numerical examples are taken to demonstrate the accuracy and efficiency of the proposed estimation technique.

Keywords: Trefftz complete set, boundary element method, estimation technique, quasi-analytical solution

3.1 Introduction

Discretization of the boundary integral equation is an important stage of the BEM in solving engineering problems [2, 6, 11, 13, 17, 21, 29, 38], the discretization process, which transforms a continuous system into a discrete system with finite number of degrees of

freedom, results in errors. Because of the fact that the reliability of the boundary element approximation is directly related to the discrete boundary element model, in which a proper mesh should be used to represent accurately the original problem both in its geometry and condition. In general, the discretization error is generated from the difference between the exact solution and the numerical result of the governing equation, but the exact solution of engineering problems is difficult to find mathematical formulation out. Furthermore, in the boundary element analysis, number of degrees of freedom depends solely on an analyst’s experience and his/her intuition. Sometime we can get the accurate numerical solution, and sometimes we can get the poor results without having exact solution when we choose the different number of elements. Obviously, the choice of number of elements is a very objective and time-consuming process, and there is no guarantee that the final solution is sufficiently accurate. Obtaining a reliable error estimator is very important in order to guarantee a certain level of accuracy of the numerical result, and is a important ingredient of the stability analysis in numerical methods. Thus, estimation of the discretization error in the BEM is worthy of study.

Different integral equations can be used to find the residual of discretization [6, 13]. A large number of studies applied the hypersingular equation to find the residual as error estimator [6, 13]. Both the singular integral equation UT and hypersingular integral equation LM in the dual BEM can independently determine the unknown boundary data for the problems without a degenerate boundary [6]. The residuals obtained from these two equations can be used as indexes of error estimation. This provides a guide for remeshing without the problem of mismatch of the collocation points on the boundary in the sample point error method.

However, it cannot be compared in the total error quantity in different number of mesh since it is pointwise error which depends on the number of elements. Indication of error trend can only been known, but it does not give the error magnitude. In this thesis, we want to find a way of objective criterion to compare the error quantities in different number of mesh.

Therefore, we develop the novel error estimator to obtain the optimal number of elements of the BEM without having analytical solution. The convergent numerical solutions of the BEM can be obtained after adopting the optimal number of elements in unavailable analytic solution condition. This study has presented a way of calculating the total error quantity as an asymptotically exact error estimator by implementing the new estimator in BEM based on complete Trefftz set [8, 14, 16, 17, 25, 37] in solving potential problem. A quasi-analytical solution is simulated to substitute for the real analytical solution by employing the aid of the Trefftz set. The convergence analysis of BEM versus different number of elements can be derived in the proposed techniques by comparing with the quasi-analytical solution. By observing the error curve versus different number of mesh, we can obtain the optimal number of elements in BEM. We develop a systematic error estimation scheme to search for the optimal number of elements.