UNIVERSITY OF HERTFORDSHIRE DEPARTMENT OF MATHEMATICS

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1 UNIVERSITY OF HERTFORDSHIRE DEPARTMENT OF MATHEMATICS A multiple node method for corners and discontinuous boundary problems in the boundary element method Wattana Toutip Technical Report April 999 Abstract The boundary element method (BEM) is now recognised as a powerful tool for solving problems in applied science and engineering. In the main scheme of this method, the boundary of the domain is divided into boundary elements and shape functions are used to approximate source densities. The use of linear continuous elements and shape functions leads to certain difficulties for problems where domains contain corners and for problems with discontinuous boundary conditions. The resolution of the difficulties has been the subect of considerable research over the past few years and several methods have been suggested. A multiple node method is one of the most efficient approaches and is simpler compared with the other methods from the programming point of view. In this report, we detail the multiple node method which is introduced to remove ambiguities in the normal derivative at corners and points with discontinuous boundary conditions. Consequently, errors caused by such problems can either be eliminated or substantially reduced.

2 Introduction The boundary element method has become an important tool for solving problems in applied science and engineering. Problems may be posed on complex domains. In the traditional BEM, the boundary of the domain will be discretized into boundary elements. The main process is to construct a system of algebraic equations to obtain the value of functions and normal derivatives at the boundary points and then use these values to compute function values at internal points in the domain. The boundary of the domain often contains corners and points with discontinuous boundary conditions, which leads to certain difficulties. The difficulty for problems whose domains contain edges and/or corners is caused by the ambiguity in the normal derivative along an edge or at a corner. The difficulty for problems with discontinuous boundary conditions is caused by the fact that the shape functions maintain the continuity of the boundary approximations. This has been the subect of considerable research over recent years and several methods have been developed to overcome these difficulties. These are divided in two categories namely, non-conforming element methods (Manolis and Beneree, 986) and multiple node methods (Mitra and Ingber, 993). The relative merits of the two methods is discussed by Brebbia and Niku (988). A comparison between the two methods has been identified (Subia and Ingber, 995). However, in this paper we shall focus on the multiple node method only, because of its accuracy and also the ease of programming. It is instructive to review the functions of the nodes within boundary elements. Typically, the nodes within boundary elements serve three purposes. Those that define the shape of the boundary elements which are called geometric nodes, those that are used to specify the source densities are named functional nodes, and those used as collocation points are called collocation nodes. In many boundary element formulations, all nodes fulfil all three roles so that there is no need to consider more than one set of nodes. However, more flexibility is available by identifying three separate sets of nodes; and the approach for handling corners and discontinuous boundary conditions is dictated by these choices. This report is intended mainly for readers who want to understand the mathematical background of the BEM, especially, the multiple node method. Hence it is divided into four main parts. The first part gives the basic mathematics for the BEM applied to the Laplace problem using linear continuous elements. This is the mathematical background of the LINBEM code (Mushtaq, 995) based on the code in Brebbia (978). In the second part we give details of the multiple node method, as implemented in the MULBEM program. In the third part we demonstrate the accuracy of the MULBEM program compared with LINBEM and known exact solutions. Furthermore, we discuss the efficiency of the program in several classic test problems compared with the BETIS program (Paris and Cañas, 997). In particular the well-known Motz problem (Motz, 946) is introduced as a test for the MULBEM program. Finally, in the last section we summarise the main features of the multiple node method compared with other approaches and draw some conclusions about the efficiency of the method.

3 Laplace Problems The Laplace equation satisfied by the potential u in a domain with a boundary as shown in Figure, can be converted into the well known integral equation (Brebbia, 978) cu uq d qu d () u where and we assume that u u on and q q on. n n denotes the unit outward normal to, and u is the fundamental solution of the Laplace equation and c is a real number which is described in the following paragraphs. n u u u 0 u q q n Figure : Laplace equation posed on the domain with the boundary Let P be a point in the domain, assume that P is an internal point, equation () becomes (Paris and Cañas, 997) u( P) qu d uq d () u where u is a fundamental solution and q. n On the other hand, in the case when P is a boundary point, equation () is of the form (Paris and Cañas, 997) where c( P) u( P) uq d qu d (3) ( P) c( P), (P) being the internal angle at the boundary point P. 3

4 For a two-dimensional domain, the fundamental solution is given by (Gilbert and Howard, 990) u ln( ) r (4) On the other hand, in a three-dimensional domain (Gilbert and Howard, 990) u 4r where r is the distance between the source point and the field point, see Figure. Field point r Source point Figure : Distance r between the source point and the field point The boundary integral equation (3) is discretized by partitioning the boundary into N elements. The element lies between node and node + as shown in Figure, equivalent to replacing the boundary curve by a polygon N Target element N + [] N r i 3 Base node Figure 3 : Discretization of the boundary into N elements 4

5 We choose a suitable set of basis function w ( s) :,,..., N where s is the distance around the boundary,, and consider the boundary element approximation N u ~ ( s) w ( s) U N q ~ ( s) w ( s) Q where U and Q are the, approximate, values of u and q at node. Setting u u~ and q q~ in equation (3) we collocate at the N nodal points to obtain a system of algebraic equation (Davies and Crann, 996) c u i i N N ( w s) U q ds w ( s) Q u ds (5) which we may write as c u i i N i N Hˆ U G Q (6) i where Hˆ G i i w ( s) q ds w ( s) q ds (7.) N w ( s) u ds w ( s) u ds (7.) N We shall consider linear elements in which w (s) is the usual hat function based at node as shown in Figure 4. 5

6 w (s) Figure 4 : Hat function based on node Consider an arbitrary segment such as the one shown in Figure 5, Nodal value of u and q = - = Node + Node Figure 5 : Relation between local co-ordinate and dimensionless co-ordinate, ( s s ) l s s l Local co-ordinate s s l With the basis function w (s) being the hat function shown in Figure 4, we see that the values of u and q at any point of an element can be defined in terms of their nodal values and the linear interpolation functions and such that u( ) U q( ) Q U Q U U Q Q 6

7 where is the dimensionless co-ordinate and and are the usual Lagrange interpolation polynomials ( ( ) ) The integral along the element in equation (5) becomes, for the left hand side, N U U w q ds q ds h h U (8) U [ ] where for each element we have two components, h q ds [ ] h q ds [ ] Similarly, for the right hand side we obtain Q Q w u ds q ds g g (9) Q Q N [ ] where for each element we have two components, g u ds [ ] g u ds [ ] So that, from equation (6), for each collocation node i we have H ˆ h h (0.) i ( ) and G i g g( ) (0.) 7

8 In this development of the boundary element method, equation c U i i N i N Hˆ U G Q () i has been obtained by collocating at the nodes defining the function values i.e. the collocation and functional nodes are the same. In equation (), defines a functional node and i denotes a collocation node. Let us define H ˆ where i i H i H i H ˆ c where i i i then we can write equation () as N N H iu G i Q () The system of linear equations may be written in matrix form HU = GQ (3) Applying the boundary conditions to identify the Dirichlet and Neumann boundary regions we can partition the system (3) in the form U U Q H H G G Q where U and Q comprise known boundary values and U and Q comprise unknown boundary values. The equations are then rearranged in the form AX = F Where A G H, Q X, and F [ GQ HU] U Once the values of U and Q on the whole boundary are known we can calculate the value of u at any internal point P k by using equation (). We obtain u k N G k Q N Hˆ U (4) k 8

9 The LINBEM code is implemented by using this mathematical algorithm and works well with smooth domain boundaries because there are no corners in such boundaries. However, we see in Example that the normal derivative values have errors in a domain which contains corners. Example : The corner problem Consider the potential problem u 0 in a square plate 0 x 6, 0 y 6 with solution u x. We partition the boundary into elements without multiple nodes. The boundary values of function and normal derivative are shown in Figure 6. (0,6) y q = 0 (6,6) u = 300 u = 0 (0,0) (6,0) q = 0 x Figure 6 : Boundary conditions for Problem We compute the solution this problem by the LINBEM program without special treatment at the corners and we see that errors in the normal derivative at the four corners are large, as shown in Table. 9

10 Table : Percentage errors of normal derivatives at corners Corner Solution Error(%) Exact LINBEM (0,0) (6,0) (6,6) (0,6) In the next section, we discuss some problems caused by corners and also discontinuous boundary conditions and how to implement the MULBEM program to resolve such problems. 0

11 Multiple node method In the previous section we have summarised the basic background of the BEM using linear continuous elements. There are a number of source codes to implement the BEM algorithm. In this paper, however, we referred to the LINBEM program which uses only the framework of the BEM. The following paragraphs will indicate some errors caused by corners and discontinuous boundary conditions. A multiple node method (Mitra and Ingber, 993) is one of the tools developed to overcome these errors. This section will explain how to implement the MULBEM program, based on the LINBEM code, by using the multiple node method. The BEM system equations are given by (3) as HU = GQ which H and G are N N matrices and U and Q are N column vectors In this equation it is assumed that all elements of Q are single-valued. However, this condition is violated at corners and points with discontinuous boundary conditions as shown in Figure 7. X Z X [] Y [] Z [] [] Y u u u u u u q q (b) X [] Y [] Z (a) q q (c) q q Figure 7 : q Y is the value of q at node Y. It is not single-valued. q in element [] is not the same as q in element []. Y Y u At the corners, by the definition, q where n is the outward normal vector to n the element, q is not single-valued. The same is true at points with discontinuous boundary conditions. One method to resolve this ambiguity is to put multiple nodes at these points. In the case of M multiple nodes, the dimension of the vectors U and Q become ( N M ). Consequently, we need to add extra collocation points to obtain a square system of linear equations. Before discussing how to allocate such points we detail the main structure of the algorithm.

12 Traditionally, Laplace problems are classified into three categories depending on the boundary conditions. A Dirichlet condition is that we know the function value, a Neumann condition is that we know the normal derivative, a Robin condition is that we know a linear combination of the two. All the features of the BEM are illustrated using only the Dirichlet and Neumann conditions. The Robin condition is easily incorporated. Since it adds nothing to the understanding of the BEM we shall not consider such problems. For details see Paris and Cañas (997). We shall consider the mixed problem as shown in Figure in which we have a Dirichlet condition on a part, of the boundary, together with a Neumann condition on the remainder,. First of all, we define the technical terms of two elements which have a multiple node in common as shown in Figure 8. [previous] Y [adacent] Y Y Figure 8 : Multiple node Y between a previous element and an adacent one To implement the program we identify the multiple node by means of the boundary conditions in the two adacent elements in the following ways: Code Previous element Adacent element Dirichlet Dirichlet Dirichlet Neumann 3 Neumann Dirichlet 4 Neumann Neumann We are now ready to continue discussing the multiple node method and the implementation of the MULBEM program. Suppose that we partition the boundary into N elements with the necessity of M multiple nodes. The elegance of the method lies in the fact that the additional equations are obtained from the framework of collocation in the boundary integral equation method and without resorting to other laws, theorems, differentiation or finite differencing (Manolis and Beneree, 986, Paris and Cañas, 997). We emphasise that four quantities, namely, two potentials and two fluxes, are associated with each multiple node. Among these four quantities, two are given as boundary conditions and two are unknown.

13 Consider the situations in Figure 7(a), in which a corner is shown, and in Figures 7(b),(c), in which discontinuous boundary conditions are shown, In all these situations, an ambiguity in q may exist at the geometric node Y. In order to resolve the ambiguity, two functional nodes, Y and Y, are placed at the geometric node Y. After the addition of the multiple nodes we rearrange the geometric nodes into functional nodes and collocation nodes. The collocation at N nodes will not provide a sufficient number of equations for the solution. To derive a sufficient number of equations we proceed as follows: Code : Referring to Figure 7(a), we place the first multiple collocation node in the previous element. We cannot place the second one at the same position as the functional node because collocation equation () at this point will be linearly dependent on the previous one so that the solution of the system of equations is not unique. To resolve this problem, we collocate the equation at any boundary point on the adacent element and the value of u, as given by the Dirichlet condition, can be inserted in the left hand side of equation (). Code : Referring to Figure 7(b), we place the first multiple node in the previous element as usual. Furthermore, we can collocate the second one at the same functional node because no linear dependence occurs as. Consider the multiple functional nodes that are numbered i and, where the conditions are Dirichlet and Neumann respectively. Then from equation (), the diagonal element on the ith row of the coefficient matrix is c i, and the diagonal element on the th row of the coefficient matrix is c. The elements in the th column of the ith row and on the ith column of the th row are zero. This different placement of the zeros makes the two equations linearly independent. The rest of this scheme uses the continuity of u to close the system of equations by the equation u(adacent) = u(previous), where u(previous) is known. Code 3 : This scheme is similar to Code, the difference being the closing of system of equations of with the equation u(previous) = u(adacent), where u(adacent) is known Code 4 : This scheme is similar to the previous two schemes, except that we close the system with the equation u(previous) - u(adacent) = 0, where u(previous) and u(adacent) are unknown. However, there is a maor difference between this scheme and the others in case of the Neumann problem i.e. the problem which has only Neumann conditions. The solution of the Neumann problem is unique only up to an arbitrary constant. Consequently, the numerical solution of such problems is likely to exhibit this attribute. To make the solution unique we have to set a Dirichlet condition at one or more nodes. In the next section we demonstrate the use of the program and test the accuracy with exact solutions and some classic problems. 3

14 Computational results To illustrate the accuracy of the multiple node method, we implemented the MULBEM program modified from the LINBEM code by using the multiple node method as discussed in the previous sections. We also add a subroutine to approximate the integral based on the Gaussian quadrature up to ten points. Furthermore, in the case of collocation away from functional nodes, we add a subroutine to collocate at any positions on the adacent elements. In order to cover all the boundary conditions and singular point problems we will consider 5 examples. The first one is a potential problem in a square which concern a heat conduction problem. In this problem we discuss all three boundary problems namely, Dirichlet problem, Neumann problem and mixed boundary problem. The second one is a steady state problem in the unit square. This problem we examined the analytic solution by separation of variables method compared with the numerical results of the program. The third one concern the cylindrical shape problem to examine the boundary solutions on the part of the cylindrical boundary. We compare the result with the method using a spreadsheet for the standard linear elements (Davies and Crann, 996). The fourth one is the Motz problem (Motz, 946) which is usually introduced to test a new method. We compare the computational result with the BETIS program (Paris and Cañas, 997) and other researchers (Symm, 973, Whiteman and Papamichael, 97, Lefeber, 989). The last example is defined on an L-shaped region with a re-entrant corner. Example : Potential problem in a square In this problem we continue discussing problem of Example in the first section. We shall investigate all conditions namely, the mixed boundary condition, the Dirichlet condition and the Neumann condition, respectively. Mixed boundary condition problem We also use the boundary conditions as shown in Figure 6 of Example. Initially the boundary is modelled with boundary elements and 4 multiple nodes at the corners. The calculation of the potential at five internal points is required at the points shown in Figure 9. 4

15 : Internal point : Multiplel node : Functional point Figure 9 : Discretization of the boundary into elements with 4 multiple nodes at the corners and 5 internal points The boundary solution of this problem is shown in Figure 0. u x q 0 (0,6) (6,6) (0,6) (6,6) u 300 u 0 q 50 q -50 (0,0) u x (6,0) (0,0) q 0 (6,0) Figure 0 : The boundary solution u x of Example The internal solutions obtained using the MULBEM program are compared with those of the LINBEM code and the exact solution in Table. Table : Internal values for the mixed boundary condition problem Internal point (,) (,4) (3,3) (4,) (4,4) MULBEM LINBEM Exact

16 Errorr(%) Percentage errors at the internal points are shown in Figure. Percent of Errors at Internal points MULBEM LINBEM Figure : Percentage errors at the internal points (,), (,4),(3,3),(4,) and (4,4) Dirichlet problem Consider the Dirichlet problem as shown in Figure. u x u 300 u 0 u x Figure : Boundary conditions of the Dirichlet problem The internal solutions obtained using the MULBEM program are compared with those of the LINBEM code and the exact solution in Table 3. 6

17 Error(%) Table 3 : Internal solutions of the Dirichlet condition problem Internal point (,) (,4) (3,3) (4,) (4,4) MULBEM LINBEM Exact Percentage errors at the internal points are multiplied by 00 and shown in Figure 3. Percentage errors at Internal points MULBEM LINBEM Figure 3 : Percentage errors at the internal point (,),(,4),(3,3),(4,) and (4,4) Neumann problem The Neumann problem is described in Figure 4. q 0 q 50 q -50 q 0 Figure 4 : Boundary conditions of the Neumann problem 7

18 Temperature As we mentioned in the previous section, we set a Dirichlet condition at the point (6,0) to be zero to make the solution unique. The internal solutions obtained using the MULBEM program are compared with those of the LINBEM one in Figure 5. Internal solution of The Neumann problem MULBEM LINBEM Exact Figure 5 : Solution at the internal points (,),(,4), (3,3),(4,) and (4,4) We also use the problem of this example to investigate the effect of the position of the collocation point. Since we move the collocation point off the functional node only at the multiple node with Dirichlet condition, we examine the effect in the Dirichlet problem. We concentrated on the position closed to the multiple node and found that the nearer to the multiple node the more accuracy in computing. The calculation result of all internal points of the Dirichlet problem using percent of collocation as 0.0, 0.50,.00 and 5.00 is demonstrated in Table 4. Table 4 : The effect of internal solutions by the position of collocation Point Exact Percent collocation of multiple nodes solution 0.0% 0.50%.00% 5.00% (,) (,4) (3,3) (4,) (4,4)

19 Error(%) We conclude this example by examining the effect of the number of Gauss points in the numerical quadrature. A subroutine is introduced to execute by means of the Gaussian quadrature (Gerald and Wheatley, 994) up to ten points. We see as the number of Gauss points increase so does the accuracy of the solutions as shown in Figure 6. Percentage errors relative to Gauss points MULBEM Figure 6 : Percentage errors of potential at the point (,) Relative to the number of Gauss points (3 to 0) 9

20 Example 3 : Dirichlet problem Consider the Dirichlet problem in the unit square with boundary conditions as shown in Figure 7. y u = (0,) (,) u = 0 u = 0 (0,0) u = 0 (,0) x Figure 7 : Boundary condition of the Dirichlet problem on the unit square The analytic solution is u( x, y) n n ( ( ) )sin nx sinh ny nsinh n (5) To compare with this exact solution we discretize the boundary into 6, 3, 64, and 8 elements together with the 4 multiple nodes at the corners. The values at the internal points are shown in Table 5. Table 5 : Internal values of the Dirichlet problem Point MULBEM solution Analytic solution (0.5,0.75) (0.50,0.75) (0.75,0.75) (0..50,0.60) (0.50,0.50) (0.50,0.40) (0.50,0.5)

21 errors (%) errors(%) We can see that the MULBEM solutions converge to exact solutions as the number of elements increases as shown in Figure 8. Percentage errors of the Dirichlet problem (.5,.75) (.50,.75) (.75,.75) (.50,.60) (.50,.50) (.50,.40) (.50,.5) Figure 8 : Percentage errors of internal solutions for each partition For this problem, we concentrate on the internal points on the line x = 0.5 close to the edge y =. It is cleared from the 6-elements solution that the errors increase near the boundary as shown in Figure 9. Relation between errors and distance elements Figure 9 : Relation between errors and distance from the edge y = on x = 0.5 of the 6-elements partition

22 This effect is also true in the other partitions but least than the 6-elements partition as shown in Table 6. Table 6 : Relation between percentage errors of each partition and distance from the edge y = Distance Percentage errors E E E E

23 Example 4 : Mixed problem This example is concerned with the potential flow past a cylinder between two parallel flat plates. By virtue of the symmetry of the problem, we consider the region shown in Figure 0 in which an artificial boundary is set at four radii from the centre of the cylinder. This problem has been tested by the method using spreadsheet (Davies and Crann, 996). To compare with their solutions, we discretize the boundary into 60 elements and use five multiple nodes at the corners. The partition is shown in the following figure. (0,4) (4,4) (0,0) 6 (4,0) Figure 0 : Discretization of the boundary into 60 elements with 5 multiple nodes at the corners The boundary value problem is subect to u 0 u = 0 on y = 0 u = 0 on ( x 4) y, y 0 q = 0 on x = 4, < y < u = on y = u = y on x = 0 We concentrate on the boundary solution, especially, on the cylindrical boundary. The normal derivatives at the points on such boundary are shown in Table 7. 3

24 Table 7 : The boundary solution on the cylindrical boundary Point Boudary solution MULBEM LINBEM (3.0,0.46) (3.0,0.60) (3.30,0.7) (3.40,0.80) (3.50,0.87) (3.60,0.9) (3.70,0.95) (3.80,0.98) (3.90,0.99) We see that from Table 7 that there is little difference between the two methods. The internal solution is compared with the LINBEM solution as shown in Table 8. Table 8 : The internal solution Point Internal solution MULBEM LINBEM (0.75,0.5) (0.75,.75) (.5,0.5) (.5,.00) (3.00,.75) Although we have not the exact solution, we see from Table 8 that the multiple node and the standard one are very similar. 4

25 Example 5 : The Motz problem This problem is often used as a test for a new method since the solution has a singularity. The geometry and boundary conditions of the problems are shown in Figure. q = 0 7 q = 0 A u = 500 O r q = 0 B u = Figure : Boundary conditions of the Motz problem on the 7 x4 rectangular domain with singular point O The analytic solution (Motz, 946) in the neighbourhood of the point (7,0) in polar form is u ( r, ) 0 r cos r cos 3r cos... (6) where r is the distance from (7,0) and is measured anti-clockwise from the line y = 0, x > 7. Whiteman and Papamichael (97) showed, using a conformal transformation method, that the first three i are given by 0 500, 5. 65, (7) To compare with the BETIS program which use a discontinuous linear element mehtod and applying a particular shape function at the singular element (Paris and Cañas, 997), we discretize the boundary into 56 elements. We see that the domain of this problem contains 4 corners and discontinuous boundary point, so that we include 5 multiple nodes at such points as shown in Figure. 5

26 A O 5 5, B Figure : Discretization of the boundary in to 56 elements with 5 multiple nodes including a singular point O We also compare the result with a variety of reference solutions (Lefeber, 989, Symm, 973, Whiteman and Papamichael, 97) as shown in Table 8. Table 8 : The solution of the Motz problem Distance Method Reference along OB MULBEM LINBEM BETIS Whiteman Symm Lefeber We investigate the solution in the neighbourhood of the point (7,0) and we see that the algorithm works well; the solution is indistinguishable from that computed using the BETIS code as shown in Figure 3. 6

27 Potential Percent difference compared with the reference solutions MULBEM LINBEM BETIS Figure 3 : Percentage difference compared with the reference solutions Furthermore, we test the accuracy of the program by using the solutions near the point (7,0) to approximate the coefficients 0 and in equation (6) and to compare with the reference value (7). We refine the mesh near the point (7,0) and the whole boundary contains 98 elements. We use the solution near the origin in Table 9 to approximate such values. Table 9 : Solutions of the points near the singular point O where r is distance from O along OB u r Applying the least squares method of data in Table 9, we obtain the coefficients which close to the coefficients of analytic solution, and in equation (7). Example 6 : The L-shaped region, mixed boundary problem 7

28 Consider the potential problem defined in the L-shaped region as shown in Figure 4. We note that the domain of this problem contains 6 corners and one of those is a reentrant which is the point to be interested. u = -4 0 A 93 q = 0 q = O 7 u = B 55 q = q = Figure 4 : Discretization and boundary conditions of the L-shaped region with 6 multiple nodes including the re-entrant O We concentrate on the normal derivative at points along OB in the neighbourhood of the re-entrant O. The analytic solution in polar form is (Paris and Cañas, 997) q 3 3 r 3r where r is the distance from O along OB. The dominant term close to the singular point is the first term and hence q in the neighbourhood of O. r Letting k = - be the order of the singularity, we see that the relation between log q and log r is a straight line whose slope is -k. We now approximate the value of and the order of the singularity k using the solution near the singular point O in Table Table 0 : The normal derivative at the points near the re-entrant O, r being the 8

29 distance from O along OB q r Applying the least squares method we obtain the order of singularity k and the coefficient and these are compared with those of the BETIS results as shown in Table. Table : Approximation of the order of the singularity k and the coefficient Method BETIS MULBEM Exact No.of element k We see from Table that the MULBEM program works as well as the BETIS code. 9

30 Conclusion There are two main methods to overcome problems caused by geometric corners and/or discontinuous boundary conditions in the boundary element method. In the first method, usually known as the non-conforming method, functional and collocation nodes are moved off geometric corners, edges, or lines of discontinuous boundary conditions. In the second method, the multiple node method, multiple nodes are placed at corners and points with discontinuous boundary conditions. Various authors claim one method better than the other (Brebbia and Niku, 988, Manolis and Beneree, 988). However, Subia and Ingber (995) suggest that both methods are reliable From the programming point of view, we believe that the multiple node approach is simpler. The elegance of this method is that the additional equations from the new collocation nodes at the multiple nodes are obtained within the framework of the boundary element method. We need not use other laws, theorems, differentiation or finite differencing (Subia and Ingber, 995). We have shown that the multiple node method works well and is a significant important on the standard collocation method. The slight increase in computational cost and complexity of input data is more than compensated by the increased accuracy. 30

31 References Becker, A.A. (99) The boundary element method in engineering, Berkshire: McGraw W-Hill. Beneree, P.K. (994) The boundary element methods in engineering, Cambridge: McGraw-Hill International(UK) Limited. Brebbia, C.A. (978) The boundary element method for engineers, Pentech press. Brebbia, C.A. and Niku, S.M. (988) Conforming versus non-conforming boundary elements in three-dimensional elastostatics: Letter to the editor. International ournal for numerical methods in engineering 6, Brown, J.W. and Churchill, R.V. (993) Fourier series and boundary value problems, 5 edn. Hightstown: McGraw-Hill. Ciskowski, R.D. and Brebbia, C.A. (99) The boundary element method in acoustics, Bath: Computational Mechanics publications. Davies, A.J. (980) The finite element method: A first approach, New York: Oxford University Press. Davies, A.J. and Crann, D. (996) Using a spreadsheet for the boundary element method: interior and exterior problems. University of Hertfordshire Division of Mathematics, Technical Report 4 Gerald, C.F. and Wheatley, P.O. (994) Applied numerical analysis, 5 edn. Addison-Wesley Publishing Company. Gibson, G.S. (987) Boundary element fundamentals: Basic concepts and recent developments in the Poisson equations, Southampton: Computational Mechanics Publications. Gilbert, R.P. and Howard, H.C. (990) Ordinary and partial differential equations with applications, West Sussex: Ellis Horwood Limited. Hall, W.S. (994) The boundary element method, Dordrecht: Kluwer Academic Publishers. Humi, M. and Miller, W.B. (99) Boundary value problems and partial differential equations, Massachusetts: PWA-KENT. Ingham, D.B. and Yuan, Y. (994) The boundary element method for solving improperly posed problems, Southampton: Computational Mechanics Publications. Kost, A. and Shen, J.X. (990) Treatment of singularities in the computation of magnetic fields with periodic boundary conditions by the boundary element method. IEEE Transactions on Magnetics 6,

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