Transactions on Modelling and Simulation vol 20, 1998 WIT Press, ISSN X

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1 r-adaptive boundary element method Eisuke Kita, Kenichi Higuchi & Norio Kamiya Department of Mechano-Informatics and Systems, Nagoya University, Nagoya , Japan Abstract This paper presents a r-adaptive mesh refinement scheme for boundary element methods. Sample point error estimation scheme are applied for the error estimation of the boundary element method. Then, the extended and the global error indicators are defined from the computational errors and the fundamental solutions. Mesh refinement is carried out so that, in the refined mesh, the error indicator distributes uniformly on all boundary elements. The present method is applied to the two-dimensional elastic problem in order to confirm the validity of the present scheme. Key Words: Boundary Element Method, Adaptive Scheme, Sample Point Error Analysis, Two-Dimensional Elastic Problem. 1 Introduction The computational accuracy in boundary element analysis is strongly dependent on the boundary element discretization of the object under consideration. Therefore, the error estimation and the adaptive mesh refinement schemes have been studied widely[1, 2]. In the primary studies, the schemes for thefiniteelement methods were directly extended to the boundary element methods. However, the adaptive schemes for the boundary element methods have the different difficulties from the finite element methods. Firstly, the boundary element

2 270 Boundary Elements methods are mainly formulated by means of the collocation method and thus, the error estimation schemes for thefiniteelement method are not adequate for the collocation-type boundary element methods. Secondly, the error on a boundary element affects ones on the other elements due to the global property of the fundamental solutions. For overcoming these difficulties, Kamiya et al. [3] presented the sample point error estimation scheme and the global error indicator. Then, they employed them to the h-adaptive mesh refinement schemes. On the other hands, in this paper, we apply them to the r-adaptive mesh refinement schemes. In the present scheme, the error estimation is carried out by the sample point error estimation scheme. Then, the extended and the global error indicators are calculated from the computational errors and the fundamental solutions. The boundary elements are redistributed so that, in the refined mesh, the indicator distributes uniformly on all boundary elements. The present scheme is applied to the two-dimensional elastic problem in order to confirm its validity. 2 BEM and Error Estimation 2.1 Boundary Element Analysis We shall consider the two-dimensional elastic problem. The governing equation and the boundary conditions are given as: and <ryj = 0 (in ft) (1) Uk = Uk (on r*j, tk = fk (on I\) (2) where u^ and f& (k x,y) denote the displacement and the traction components in the ^-direction and the upper bar the specified boundary value. ^2, P^ and F^ are the domain occupied by the object and its displacement- and the traction-specified boundaries, respectively. Taking the Kelvin's solutions as the weight functions, we have the following boundary integral equation from Eq.(l)[4]: where pi and PJ denote the source and the integral points. Besides, CM is the parameter depending on where the point pi is placed. (3)

3 Boundary Elements 271 Discretizing the Eq.(3) by TV conforming linear boundary elements, we have N (4) where T/, HI and t/ denote the displacement component in /-direction on the node i and the displacement and the traction components in /-direction on the element jf, respectively. Uji and tji are approximated by the interpolating functions (j)\ and (f>2 as follows: Substituting the Eqs.(5) to the Eq.(4), we have * i <*> Hu = Gt (6) where u and t denote the nodal displacement and traction vectors and H and G the coefficient matrices. The boundary conditions are substituted into Eq.(6), which is solved for the unknowns. 2.2 Sample Point Error Estimation Scheme The boundary integral equation (3) is not satisfied on the other boundary points p( than pi. Therefore, in order that the Eq.(3) are held on the sample point p\, the errors e^i and en are added to the numerical solutions u and </, respectively. So we have = 0 (7) The residual of the boundary integral equation r&(p^) is defined as (8) Finally, from Eqs.(7) and (8), we have the relationship between the residual and the errors: (9)

4 272 Boundary Elements where e^i 0 on the boundary F^ and en 0 on the boundary The error distribution on the element is approximated as e<ul = euljv, en - eufi (10) *=!- * (11) where ( denotes the local coordinate taken on the element, which is defined as ± 1 on the both ends of the element and = 0 on its center. Discretizing the Eq.(9), we have Pi ^/ *' (12) Pi 6w, / flwr (13) JTui where N^i and NU (N = N^i -f NU) are the total number of elements on the boundaries F^ and F*,, respectively. Finally, we have This is solved for the error vector e. r = Be (14) 3 Error Indicators and Adaptive Process 3.1 Extended Error Indicator From Eq.(14), we have The extended error indicator \\r^\\ is defined as

5 Boundary Elements 273 where \\rj^\\ (k = x,y) is defined as 3.2 Global Error Indicator N?ll - E 11*2411 (17) i=x,y The extended error indicators %,%, -,77^ mean the effect from the error on the element i to the accuracy of the elements 1, 2, - -, N. On the other hands, the indicators 77%, %j,,7/7^ mean the effect from the elements 1, 2,, TV to the element i. Therefore, in order to simultaneously measure the effects from and to the element i, we define the global error indicator Hi as 3.3 Convergence Property of Error Assuming that there is no error at the nodes of the element, the relationship between the error on the element and the length of the element is given as lk-^l = ^ + o(^) (19) where && and /%& indicate the numerical solution and the length of the element at fc-th iteration, respectively. In the same manner, the following relationship is held at (fc-f l)-th iteration. 11% - 6&+i = #/4Li + o (/^Li) (20) Substituting the global error indicator Eq.(18) into the error norms on the left-hand sides of Eqs.(19) and (20), we have By eliminating K, we have H* ~ h,

6 274 Boundary Elements 3.4 Mesh Redistribution Process The whole boundary of the object under consideration is divided into boundary segments at the geometric corner points and at the boundary points where the boundary conditions vary. The segments are considered as the macro-elements. The process of the present scheme is as follows: 1. Specify the data of the macro elements (the coordinates of the end-points, the line-type of the segment, etc.). 2. Specify the total number of the boundary elements N. 3. Discretize the whole boundary by N equal-length boundary elements. 4. Perform boundary element analysis. 5. Perform error estimation. 6. Check the convergence criterion. If the convergence criterion is satisfied, the process is terminated. If not so, the process goes to the next step. (a) Calculating the global error indicator Hi and its average value (b) Calculate the number of the boundary elements on the macro element j\ where the summation Hi is taken for all boundary elements on the macro element j. (c) Discretize the macro element j by Nj boundary elements so that, in the refined mesh, the errors in all boundary elements are almost equal to (d) Go to step 4.

7 Boundary Elements 275 ty=1.0(kgf/cm2) G=1000(kgf/cm2) v=0.3 1 Figure 1: Object under consideration H h Initial 5th iteration H I I I I I I I h -4* Figure 2: Mesh partitions 4 Numerical Example A numerical example is shown in Fig.l. An initial mesh is constructed by the 40 equal-length elements. Figure 2 shows the boundary element meshes at initial and 5th (final) iterations. At the 5th iteration, the elements move to the stress concentration points A and B. The function distributions at both meshes are shown in Fig.3. We notice

8 276 Boundary Elements CO O Q_ X (a) Initial f I I t *x ly (b) 5th iteration Figure 3: Function distributions

9 200 Boundary Elements 111 O X (0 O T3 C LU I I I I iteration Figure 4: Convergence property of error indicators that the distributions at the 5th iteration are improved fairly since the elements move to the points A and B. Figure 4 shows the convergence property of the error indicators. Tf^ax and Tf^in means the maximum and the minimum values of H^ respectively. The maximum value and the difference between the maximum and the minimum values decreases as the process goes. 5 Conclusion This paper presented the r-adaptive schemes for the boundary element method. In the present schemes, the computational error is estimated by the sample point error estimation scheme. Then, according to the relationship between the error and the length of the element, the elements are redistributed on the whole boundary. The present scheme was applied to the two-dimensional elastic problem. As the process goes, the elements move to the stress concentration points and then, the error indicators are improved monotonously.

10 278 Boundary Elements References [1] N. Kamiya, editor. Special Issue on Error Estimate and Adaptive Meshes for FEM/BEM, Advances in Engineering Software, Vol. 15, pp [2] E. Kita and N. Kamiya. Recent studies on adaptive boundary element methods. Advances in Engineering Software, Vol. 19, No. 1, pp , [3] K. Kawaguchi and N. Kamiya, An adaptive BEM by sample point error analysis. Advances in Engineering Software, Vol. 9, pp , [4] C. A. Brebbia. The Boundary Element Method for Engineers. Pentech Press, 1978.

A mesh refinement technique for the boundary element method based on local error analysis

A mesh refinement technique for the boundary element method based on local error analysis J. J. Rodriguez