Chapter 2 Difficulties associated with corners

Transcription

1 Chapter Difficlties associated with corners This chapter is aimed at resolving the problems revealed in Chapter, which are cased b corners and/or discontinos bondar conditions. The first section introdces the cases of sch problems. The second section presents a mltiple node method and a program is implemented sing this method to overcome the problem. All problems in Chapter are sed to test this program. The third section presents another approach to the mltiple node method. The last section gives conclding remars.. Introdction In the traditional BEM, as eplained in the previos chapter, the bondar of the domain is discretised into bondar elements. The main process is to constrct a sstem of algebraic eqations to solve for vales of fnction and normal derivative at the bondar points and then se these vales to compte fnction vales at internal points in the domain. The bondar of the domain often contains corners and points with discontinos bondar conditions, which leads to certain difficlties. The difficlt with problems whose domain contains corners is cased b the ambigit in the normal derivative at a corner. The difficlt with problems with discontinos bondar conditions is cased b the fact that the shape fnctions maintain the continit of the bondar approimations. This has been the sbject of considerable research over the past decade and several methods have been developed to overcome these difficlties. These are divided into two categories namel, non-conforming element methods (Manolis and Benerjee, 986) and mltiple node methods (Mitra and Ingber, 993). The non-conforming methods inclde the se of discontinos elements and semidiscontinos elements. In these methods, nodes are moved awa from corners so that fles need not be evalated at corners. The mltiple node method introdces

2 additional nnowns into the formlation, and hence additional eqations are reqired to close the sstem of eqations. The relative merits of the two methods are discssed b Brebbia and Ni (988). A comparison between the two methods has been made (Sbia et al., 995). However, in this wor we shall focs on the mltiple node method onl, becase of its accrac and also the ease of programming. There are also several inds of mltiple node methods. This wor considers the mltiple node method with ailiar bondar collocation (Mitra and Ingber, 993) and the gradient approach (Alarcón et al., 979, París and Cañas, 997) This chapter gives the mathematical bacgrond of the mltiple node method and its implementation. In section. we first discss the algorithm for the mltiple node method with ailiar bondar collocation. We net eplain how to implement the MULBEM code based on the mltiple node method. Finall, we discss the efficienc of the program on several classic test problems compared with the BETIS program (París and Cañas, 997) which ses speciall constrcted singlar elements to tae accont of discontinities in geometr or in the nnowns. In particlar the well-nown Motz problem (Motz, 946) is introdced as a test for the MULBEM program. In section.3 we present the gradient approach (París et al., 98) which is a variation of the mltiple node method. GRABEM is an implementation of this method. The comptational reslts from GRABEM are compared with those from LINBEM and MULBEM. In section.4 we give some conclding remars.. The Mltiple node method and The MULBEM program. The method presented in this section is based on the mltiple node method with ailiar bondar collocation (Sbia et al., 995). The mathematical concept of the method is presented in section... We implement a program sing the method in section... The comptation reslts are discssed at the end of the section. 3

3 .. Formlation of the mltiple node method In the previos chapter we smmarised the mathematical bacgrond needed for the standard collocation form of the BEM sing linear continos elements. The sstem of eqations is epressed in matri form HU GQ (.) which H and G are N N matrices and U and Q are N colmn vectors In this eqation it is assmed that all elements of Q are single-valed. However, this condition is violated at corners and points with discontinos bondar conditions as shown in Figre... X Z X [] Y [] Z [] [] Y q q (b) X [] Y [] Z (a) q q (c) q q Figre.. q Y is the vale of q at the node Y. It is not singlevaled. q Y in element [] is not the same as q Y in element []. At corners, since, q where n is the otward normal vector to the element, q is n not single-valed. The same is tre at points with discontinos bondar conditions. One method to resolve this ambigit is to pt mltiple nodes at these points. In the case of M mltiple nodes, the dimension of the vectors U and Q becomes ( N M). Conseqentl, we need to add etra collocation points to obtain a sqare sstem of linear eqations. Before discssing how to allocate sch points we detail the main strctre of the algorithm. 4

4 Traditionall, Laplace problems are classified into three categories depending on the bondar conditions. A Dirichlet condition is that we now the fnction vale, a Nemann condition is that we now the normal derivative, a Robin condition is that we now a linear combination of the two. All the featres of the BEM are illstrated sing onl the Dirichlet and Nemann conditions. The Robin condition is easil incorporated and since it adds nothing to the nderstanding of the BEM we shall not consider sch problems. For details see París and Cañas (997). We shall consider the mied problem in which we have a Dirichlet condition on a part of the bondar together with a Nemann condition on the remainder. First of all, we define the technical terms for two elements which have a mltiple node in common as shown in Figre... We note that we se a conter-clocwise integration process. This integration direction leads to the terminolog for the elements associated with node Y: the previos element and the adjacent element, see Figre... [previos] Y [adjacent] Y Y Figre.. Mltiple node Y between a previos element and an adjacent one To implement the program we identif the mltiple node b means of the bondar conditions in the two elements it belongs to in the following wa: 5

5 Code Previos element Adjacent element Dirichlet Dirichlet Dirichlet Nemann 3 Nemann Dirichlet 4 Nemann Nemann We are now read to contine discssing the mltiple node approach. Sppose that we partition the bondar into N elements with the necessit of M mltiple nodes. The elegance of the method lies in the fact that the additional eqations are obtained from the framewor of collocation in the bondar integral eqation method and withot resorting to other laws, theorems, differentiation or finite differencing (Manolis and Benerjee, 986, París and Cañas, 997). We emphasise that for qantities, namel two potentials and two fles, are associated with each mltiple node. Among these for qantities, two are given as bondar conditions and two are nnown. Consider the sitations in Figre..(a), in which a corner is shown, and in Figres..(b), (c), in which discontinos bondar conditions are shown. In all these sitations, an ambigit in q ma eist at the geometric node Y. In order to resolve the ambigit, two fnctional nodes, Y and Y, are placed at the geometric node Y. After the addition of the mltiple nodes we rearrange the geometric nodes into fnctional nodes and collocation nodes. Collocation at N nodes will not provide a sfficient nmber of eqations for the soltion. To derive a sfficient nmber of eqations we proceed as follows: Code Referring to Figre..(a), we place the first mltiple collocation node in the previos element. We cannot place the second one at the same position as the fnctional node becase collocation eqation (.0) at this point will be linearl dependent on the previos one so that the soltion of the sstem of eqations is not niqe. To resolve this problem, we collocate the eqation at an bondar point on 6

6 the adjacent element and the vale of, as given b the Dirichlet condition, can be inserted in the left hand side of eqation (.0). Code Referring to Figre..(b), we place the first mltiple node in the previos element as sal. Frthermore, we can collocate the second one at the same fnctional node becase no linear dependence occrs. Consider the mltiple fnctional nodes that are nmbered i and j, where the conditions are Dirichlet and Nemann respectivel. Then from eqation (.0), the diagonal element on the ith row of the coefficient matri is c i, and the diagonal element on the jth row of the coefficient matri is c j. The elements in the jth colmn of the ith row and on the ith colmn of the jth row are zero. This different placement of the zeros maes the two eqations linearl independent. The rest of this scheme ses the continit of to close the sstem of eqations b the eqation (adjacent) = (previos), where (previos) is nown. Code 3 This scheme is similar to Code, the difference being the closing of sstem of eqations with the eqation (previos) = (adjacent), where (adjacent) is nown Code 4 This scheme is similar to the previos two schemes, ecept that we close the sstem with the eqation (previos) - (adjacent) = 0, where (previos) and (adjacent) are nnown. However, there is a major difference between this scheme and the others in the case of the Nemann problem i.e. the problem which has onl Nemann conditions. The soltion of the Nemann problem is niqe onl p to an arbitrar constant. Conseqentl, the nmerical soltion of sch problems is liel to ehibit this attribte. 7

7 .. The MULBEM program The LINBEM code is modified in order to handle problems cased b corners and discontinos bondar conditions b appling the mltiple node method. We call the new code MULBEM. The program is an implementation of the algorithm described in the previos section. The program is tested with all problems in Chapter. The first eample is the sqare problem. The comptational reslt is compared with that from LINBEM. The second eample is the Motz problem. We compare the comptational reslt with that from LINBEM, BETIS and other researchers (Smm, 973, Whiteman and Papamichael, 97, Lefeber, 989). The final eample is the L-shaped region mied bondar problem. The comptational reslt is compared with that from LINBEM and BETIS. Eample.. The sqare problem We se the sqare problem in Eample.4.. The internal soltions sing MULBEM are compared with those sing LINBEM and shown in Table... Table.. The potential at the internal points for the sqare problem Point LINBEM MULBEM Eact soltion We see from Table.. that the MULBEM reslt is the better compared with that from LINBEM. The normal derivative at the for corners is shown in Table... We can see from Table.. that the normal derivative at the for corners sing MULBEM is mch better than that from LINBEM 8

8 Table.. The normal derivative at the for corners Point LINBEM MULBEM Eact soltion Eample.. The Motz problem We se the Motz problem as in Eample.4.. In this eample we discretise the bondar into 56 elements. We see that the domain of this problem contains 4 corners and discontinos bondar point, so that we inclde 5 mltiple nodes at sch points as shown in Figre A O 5 5, B Figre..3 Discretisation of the bondar in to 56 elements with 5 mltiple nodes inclding a singlar point O We also compare the reslt with a variet of reference soltions (Lefeber, 989, Smm, 973, Whiteman and Papamichael, 97) as shown in Table..3. We can see from Table..3 that the MULBEM reslt agrees well with the BETIS reslt. Both reslts are better than those from the LINBEM. For the points when the other references have soltions, MULBEM reslts are in good agreement. 9

9 The approimate vale of the integral of normal derivative along the bondar sing MULBEM is which is an improvement on the vale from LINBEM. Table..3 The soltion of the Motz problem Distance Method Reference along OB MULBEM LINBEM BETIS Whiteman Smm Lefeber ******* ******* ******* ******* ******* ******* ******* ******* ******* ******* ******* ******* ******* ******* ******* ******* ******* ******* ******* ******* ******* The smbol "*****" in the table means there is no soltion from that reference The soltion is indistingishable from that compted sing the BETIS code as shown in Figre..4. Relative difference compared with the reference soltions Relative difference MULBEM LINBEM BETIS r Figre..4 Relative difference compared with the reference soltions 30

10 Frthermore, we also test the accrac of the program sing the soltions near the point (7,0) as shown in Table..4 to approimate the coefficients 0 and as in Eample.4.. Table..4 Soltions of the points near the singlar point O where r is the distance from O along OB (see Figre..3) r The coefficients 0 and sing MULBEM are compared with those from LINBEM and the eact vales as shown in Table..5. Table..5 The coefficients 0 and Coefficient LINBEM MULBEM Eact We can see from Table..5 that the coefficient sing MULBEM is better than that sing LINBEM. 3

11 Eample..3 The L-shaped region mied bondar problem We again se the problem in Eample.4.3. Consider the potential problem defined in the L-shaped region when the bondar is discretised as shown in Figre.4.5. We note that the domain of this problem contains 6 corners and one of those is re-entrant. We are interested particlarl in the accrac of soltions near this point. We now approimate the vale of as in Eample.4.3 and the order of the singlarit sing the soltion near the singlar point O in Table..6. Appling the least sqares method to fit the fnction log q log log r to this data we obtain the order of singlarit and the coefficient and these are compared with those from LINBEM and BETIS in Table..7. Table..6 The normal derivative at the points near the re-entrant O, r being the distance from O along OB r q Table..7 Approimation of the order of the singlarit and the coefficient Method BETIS LINBEM MULBEM Eact No.of elements ***** The smbol " ****** " in the table means there is no soltion in the item 3

12 We see from Table..7 that the MULBEM program wors as well as the BETIS code. Both programs wor better than LINBEM. The approimate vale of the integral of normal derivative along the bondar sing MULBEM is which is an improvement on the vale from LINBEM. In the net section, we introdce a variant of the mltiple node method, the gradient approach..3 The gradient approach and the GRABEM program The mltiple node method is one of the efficient methods to resolve corner problems in the BEM. There are several other methods similar to this approach (Sbia and Ingber, 995). The gradient approach (Alarcón, et al, 979, París and Cañas, 997) is a frther variation of the mltiple node method which we are going to discss in section.3.. Althogh it is more complicated than the previos mltiple node method from a programming point of view, it is sitable to appl with the dal reciprocit method in Chapter3. In section.3. we present GRABEM, which implements a program sing the method, and discss comptational reslts at the end of the section..3. Formlation of the gradient approach The gradient approach is based on relating two nnown fles associated with a corner, etending to the domain close to the corner the hpothesis of linear interpolation along the elements adjacent to the corner. This wa there will be three variables at each node: one potential and two fles. The continit of potential and bondar conditions are sitable to redce the variables to have one nnown for all cases of bondar conditions ecept the Dirichlet condition. We discss onl the case when the Dirichlet conditions appl at consective elements. The approach reqires the nowledge of the potential at ever node. Sppose that we have potentials, and at node -, and + respectivel, as shown in Figre.3.. Then we can obtain a linear approimation to, from the nodal vale, over the triangle defined b the three nodes. We can then find grad over the triangle 33

13 and nowing the nit normal vectors to each of the elements we can obtain the fles at node in each of sch elements. + ( 3 3, ) (, ) (, ) - Figre.3. Linear approimation of the potential srronding node De to the fact that the potential is nown at nodes,, and, the eqation of the plane represented in Figre.3. can be easil generated as (.) The vale of can then be epressed as ( )( ) ( )( ) ( ) 3 ( )( 3 ) ( )( 3 ) ( 3)( ) ( )( ) ( ) ( )( 3 ) ( )( 3 ) (.3) Using this epression, the components of the gradient at point, i.e. node, can be generated. 34

14 35 Let s define as ) )( ( ) )( ( 3 3 (.4) Then the components of the gradient are ) ( ) ( 3 3 (.5) ) ( ) ( 3 3 (.6) In general, if the gradient is nown at the node, then an fl along a certain direction can be epressed projecting the gradient onto this direction. Let v be the nit vector along the gradient direction as shown in Figre.3.. The vector R, n denotes the nit normal vector to the right element. The vector L, n denotes the nit normal vector to the left element. The nit normal vector is v =, (.7) + - ) ( L, n R, n v R q, Figre.3. The gradient approach at the node L q,

15 Conseqentl, the fles on each side can be epressed in the form q L, ( L, q ) cos( v, n ) (.8) R, ( R, ) cos( v, n ) (.9) If the right-hand sides of eqations (.8) and (.9) are nown, then the can be sed to calclate the two fles at the node. In this wa these eqations epress both fles as fnctions of a single nnown (the modls of the gradient). The advantage of this approach is that we need not introdce new variables in the sstem of eqations. N.B. The gradient method introdces a linear approimation to (, ) and hence the fles (.8) and (.9) are also approimations. An alternative approach at a right angle corner wold be to se the tangential derivatives on one side to obtain the normal derivative on the other. Then to se a finite difference approimation to obtain the normal derivatives in terms of nodal potential vales. While this is possible it is more difficlt to deal with non-right angle corners whereas the gradient approach covers all cases..3. The GRABEM program The GRABEM program implements the idea described above. The gradient approach is added in the process of appling the bondar condition to the final sstem of eqations to be solved. It mst be pointed ot that the nmber of variables associated with a node is initiall 3, as shown in Figre.3.3. q L, q R, + [] [-] Figre.3.3 Three variables at node - 36

16 In Figre.3.3, is a potential variable and q L,, R node. There are five sitations which can occr at each node: q, are two fl variables at the. Nemann-Nemann condition. Nemann-Dirichlet condition 3. Dirichlet-Nemann condition 4. Dirichlet-Dirichlet condition at corner 5. Dirichlet-Dirichlet condition with smooth bondar We recall from chapter that we mst appl the bondar conditions to the sstem of eqations (.9) HU GQ (.0) in order to rearrange them into a final sstem A (.) where consists of nnown bondar vales. We se the nmber before each condition to be the code of the bondar condition in the program. The scheme of appling these conditions is shown in Table.3.. Table.3. The scheme of appling the variables at node. Code q L, q R, Unnown() Known Known Known Unnown() Known 3 Known Known Unnown(3) 4 Known Unnown(4a) Unnown(4b) 5 Known Unnown(5) Unnown(5) 37

17 We see from the scheme that each condition in the first three codes has onl one nnown. Hence we can appl to eqation (.0) directl. Code 4 is the most difficlt case to appl the bondar condition. The last case is a conseqence of case 4 sing the fact that the two fles are eqal. We give details for all cases in the rest of this section. Code Nemann-Nemann condition This implies that the otward normal derivative of the potential is nown at both sides of node and conseqentl, we obtain eqation as follows. Ai H i... G, ql, G, qr,... Code Nemann-Dirichlet condition This condition implies that the potential is nown along the element [] and the normal derivative is nown along the element [-]. The application of the bondar condition is then A i G, q R,... Hi G, ql,,... Code 3 Dirichlet-Nemann condition This condition implies that the potential is nown along the element [-] and the normal derivative is nown along the element []. The application of the bondar condition is then A i G, 38

18 q L,... Hi G, qr,,... Code 4 Dirichlet-Dirichet condition at a corner This condition implies that the potential is nown along the element [-] and element []. Before appling the bondar condition, let s define new variables in eqation (.8) and (.9) as D () (.) C cos( v, n L, ) (.3) L, C cos( v, n R, ) (.4) R, Hence, we can rewrite these eqations as q L DCL,, (.5) q R DCR,, (.6) We can see from eqations (.5) and (.6) that the two fles are fnctions of one variable, namel, follows. D. Therefore, we can now appl the bondar condition as A i G, CL, G, CR, D... H,... i Code 5 Dirichlet-Dirichlet condition with smooth bondar 39

19 This condition implies that the potential is nown along the element [-] and element []. According to the smoothness of the bondar, the angle between the normal vector and the gradient vector is zero, hence C L, CR, We can now se the algorithm for Code 4 and we obtain the sstem of eqations b sing A i G, G, D... H,... i After the sstem of eqations has been solved, the soltions of most cases are directl obtained from the variables of their codes ecept those of code 4. The soltion for case 4 is the modls of the gradients as in eqation (.). To obtain the two normal derivatives at the node, we sbstitte that vale in eqation (.5) and (.6). We now discss the performance of GRABEM on all problems presented in section.4. Frther applications will be considered in Chapter 3. Eample.3. The sqare problem We se the problem in Eample.4.. The internal soltions obtained with GRABEM are compared with those sing LINBEM and MULBEM and shown in Table.3.. Table.3. The potential at the internal points Point LINBEM MULBEM GRABEM Eact soltion

20 We see from Table.3. that the reslt from GRABEM and MULBEM is better than that from LINBEM. The normal derivatives on the bondar nodes are shown in Table.3.3. We also see from Table.3.3 that the normal derivative sing GRABEM is the best of the nmerical reslts. Table.3.3 The normal derivative at the bondar nodes Point LINBEM MULBEM GRABEM Eact soltion Eample.3. The Motz problem We se the Motz problem as in Eample.4.. In this eample we discretise the bondar into 56 elements as in Figre.4.4. We compare the comptational reslts with those from LINBEM, MULBEM and other references in Table.3.4. Table.3.4 The soltion of the Motz problem Distance Method Reference from O to B LINBEM MULBEM GRABEM BETIS Smm ******* ******* ******* ******* ******* ******* The smbol " ****** " in the table means there is no soltion from that reference 4

21 We can see from Table.3.4 that the GRABEM reslt agrees ver well with that from the MULBEM and BETIS. The reslt is better than that from LINBEM. For the points at which Smm qotes a soltion, GRABEM is in good agreement. Frthermore, we also test the accrac of the program sing the soltions near the point (7,0) as shown in Table.3.5 to approimate the coefficients 0 and as in Eample.4.. In this case we also refine the mesh near the point (7, 0) and the whole bondar contains 98 elements. Table.3.5 Soltions of the points near the singlar point O where r is distance from O along OB (see Figre.4.3) r The coefficients 0 and sing GRABEM are compared with those from LINBEM, MULBEM and the eact vales as shown in Table.3.6. Table.3.6 The coefficients 0 and Coefficient LINBEM MULBEM GRABEM Eact

22 We can see from Table.3.6 that the coefficient 0 from GRABEM is nearl the same as that from MULBEM. On the other hand the coefficient is slightl better. However, both coefficients are an improvement on those sing LINBEM. The approimate vale of the integral of normal derivative along the bondar sing GRABEM with 98 elements is compared with that from the previos program in Table.3.7. Table.3.7 The approimation of integral of derivative along the bondar LINBEM MULBEM GRABEM We see from Table.3.7 that the approimation from MULBEM and GRABEM is almost the same and it is better than that from LINBEM. Eample.3.3 The L-shaped region mied bondar problem We again se the problem in Eample.4.3. Consider the potential problem defined in the L-shaped region with the bondar dicretised as shown in Figre.4.5. We now approimate the vale of as in Eample.4.3 and the order of the singlarit sing the soltion near the singlar point O in Table.3.8. The approimation is compared with that from LINBEM and MULBEM in Table.3.9. We see from Table.3.9 that the GRABEM reslt is nearl the same as that from MULBEM. Both programs wor better than LINBEM. 43

23 Table.3.8 The normal derivative at the points near the re-entrant O, r being the distance from O along OB r q Table.3.9 Approimation of the order of the singlarit and the coefficient Method LINBEM MULBEM GRABEM Eact No.of elements ***** The approimate vale of the integral of normal derivative along the bondar sing GRABEM is compared with that from the previos program in Table.3.0. Table.3.0 The approimation of the integral of the derivative along the bondar LINBEM MULBEM GRABEM We see from Table.3.0 that in this eample the normal derivative given b GRABEM is mch better than that from LINBEM or MULBEM. 44

24 .4 Conclding remars There are two main methods for overcoming problems cased b geometric corners and/or discontinos bondar conditions in the bondar element method. In the first method, sall nown as the non-conforming method, fnctional and collocation nodes are moved off geometric corners or points of discontinos bondar conditions. In the second method, the mltiple node method, mltiple nodes are placed at corners and points with discontinos bondar conditions. Varios athors claim one method is better than the other (Brebbia and Ni, 988, Manolis and Benerjee, 988). However, Sbia and Ingber (995) sggest that both methods are eqall reliable. From the programming point of view, we believe that the mltiple node approach (in MULBEM) is simpler. The elegance of this method is that the additional eqations from the new collocation nodes at the mltiple nodes are obtained within the framewor of the bondar element method. We need not se other laws, theorems, differentiation or finite differencing (Sbia and Ingber, 995). We have shown that the mltiple node method wors well and is a significant improvement on the standard collocation method. The slight increase in comptational cost and compleit of inpt data is more than compensated b the increased accrac. The gradient approach sed in GRABEM is more complicated than the mltiple node with ailiar bondar collocation (from the programming point of view). The gradient approach needs more mathematical bacgrond as eplained in section.3. On the other hand the mltiple node method ses onl the framewor of BEM. However the gradient approach is sitable to appl for the dal reciprocit method to solve Poisson-tpe eqations for which the mltiple node sed in section. is nsitable. We will discss this capabilit together with frther problems in chapter 3. 45