中華民國振動與噪音工程學會第 8 屆學術研討會, 明志科技大學, 台灣台北, 年 6 月 日 The 8 h aional Conference on Sound and Vibraion (CSSV), Taipei, Taiwan, June,. I-54 An Applicaion of ew Error Esimaion Technique o he Boundary Elemen ehod Jun-Ting Chen, Cheng-Tzong Chen, Kue-Hong Chen, Deparmen of Civil Engineering, aion Ilan Universiy Deparmen of Hydraulic and Ocean Engineering, aional Cheng Kung Universiy khc677@niu.edu.w (Corresponding auhor: Kue-Hong Chen) (SC Proec umber: SC-98--E-97-3) ABSTRACT In his sudy, we develop a novel esimaion echnique o obain he opimal number of elemens in he boundary elemen mehod (BE) wihou having analyical soluion. By using he complee Treffz se as he analyical soluion, namely quasi-analyical soluion, he new error esimaor is presened in he paper. The error curve versus differen number of elemens can be derived in he proposed echniques by comparing numerical soluion wih he quasi-analyical soluion. By observing he error curve, we can obain he opimal number of elemens in BE. One numerical example is aken o demonsrae he accuracy and efficiency of he proposed esimaion echnique. Keywords: Treffz complee se, boundary elemen mehod, esimaion echnique, quasi-analyical soluion. difference beween he exac soluion and he numerical resul of he governing equaion, bu he exac soluion of engineering problems is difficul o find from mahemaical formulaion. Furhermore, in he boundary elemen analysis, number of degrees of freedom depends solely on an analys s experience and his/her inuiion. Someime we can ge he accurae numerical soluion, and someimes we can ge he poor resuls wihou having exac soluion when we choose he differen number of elemens. Obviously, he choice of number of elemens is a very obecive and ime-consuming process, and here is no guaranee ha he final soluion is sufficienly accurae. Obaining a reliable error esimaor is very imporan in order o guaranee a cerain level of accuracy of he numerical resul, and is a imporan ingredien of he sabiliy analysis in numerical mehods. Thus, esimaion of he discreizaion error in he Boundary Elemen ehod. ITRODUCTIO Discreizaion of he boundary inegral equaion is an imporan sage of he Boundary Elemen ehod (BE) in solving engineering problems [,, 4, 6, 7, 8,, ], he discreizaion process, which ransforms a coninuous sysem ino a discree sysem wih finie number of degrees of freedom, resuls in errors. Because of he fac ha he reliabiliy of he boundary elemen approximaion is direcly relaed o he discree boundary elemen model, in which a proper mesh should be used o represen accuraely he original problem boh in is geomery and condiion. In general, he discreizaion error is generaed from he (BE) is worhy of sudy. Differen inegral equaions can be used o find he residual of discreizaion [, 4]. A large number of sudies applied he hypersingular equaion o find he residual as error esimaor [, 4]. Boh he singular inegral equaion UT and hypersingular inegral equaion L in he dual BE can independenly deermine he unknown boundary daa for he problems wihou a degenerae boundary []. The residuals obained from hese wo equaions can be used as indexes of error esimaion. This provides a guide for remeshing wihou he problem of mismach of he collocaion poins on he boundary in he sample poin error mehod. However, i
中華民國振動與噪音工程學會第 8 屆學術研討會, 明志科技大學, 台灣台北, 年 6 月 日 The 8 h aional Conference on Sound and Vibraion (CSSV), Taipei, Taiwan, June,. I-54 canno be compared in he oal error quaniy in differen number of mesh since i is poinwise error which depends on he number of elemens. In his paper, we wan o find a way of obecive crierion o compare he error quaniies in differen number of mesh. Therefore, we develop he novel error esimaor o obain he opimal number of elemens of he BE wihou having analyical soluion. The convergen numerical soluions of he BE can be obained afer adoping he opimal number of elemens in unavailable analyic soluion condiion. This sudy has presened a way of calculaing he oal error quaniy as an asympoically exac error esimaor by implemening he new esimaor in BE based on complee Treffz se [3, 5, 8, 9,, 3] in solving poenial problem. A quasi-analyical soluion is simulaed o subsiue for real analyical soluion by employing he aid of he Treffz se. The convergence analysis of BE versus differen number of elemens can be derived in he proposed echniques by comparing wih he quasi-analyical soluion. By observing he error curve versus differen number of mesh, we can obain he opimal number of elemens in BE. We develop a sysemaic error esimaion scheme o search for he opimal number of elemens. B = B B = D problem, denoes he whole boundary of he domain D, in which of B is he essenial boundary (Dirichle boundary) in which he poenial is prescribed, B is he naural boundary (eumann boundary) where he normal derivaive of he poenial in he n direcion is specified. x. BE formulaion The boundary inegral equaion for he domain poin can be derived from Green's second ideniy as: πu( = T( s, u( db( U ( s x U ( s, ( db(, x D (4) where, ) is he fundamenal soluion which saisfies: U ( s, = δ ( x (5) in which U( s, = lnr and δ ( x is he Dirac-dela T ( x s funcion, and, ) is defined by U( s, T( s, = (6) n s in which n is he ou-normal direcion a he boundary s poin s. Discreizing he boundary B ino boundary elemens in Eq.(4) as follows:. PROBLE STATEET AD ETHOD OF SOLUTIO. Problem saemen We consider he behavior of he medium governed by he Laplace equaion wih he mixed-ype boundary condiions as: u( =, x D () u( = u (, x B () u( ( x x B = = ( ), (3) nx where is he operaor wih problem, u( is he poenial, D is he compuaional domain of he πu( = CPV RPV B B T ( s, u( db( U ( s, ( db(, x B (7) where CPV is he Cauchy principal value and RPV is he Riemann principal value. The boundary inegral equaion is discreized by using number of consan boundary elemens, hen he resuling algebraic sysem (UT mehod: convenional BE of singular inegral formulaion) can be obained as: [ U ]{ } = [ T ]{ u}. (8) For he problem wih mixed-ype boundary condiions, Eq.(8) can be decomposed ino u u [ ULU R ] = [ TL TR ]. (9) By collecing he given and unknown ses, we rearrange he influence marices ino
中華民國振動與噪音工程學會第 8 屆學術研討會, 明志科技大學, 台灣台北, 年 6 月 日 The 8 h aional Conference on Sound and Vibraion (CSSV), Taipei, Taiwan, June,. I-54 u [ U T ] = [ T U ] L R L R Eq.(8) can be simplified o [ A] x = b in which % % A = U L TR, x = % u [ ] [ ] u b = TL U R % [ ]. u () () () (3) We can derive he unknown vecor, x, by uilizing he linear algebra solver. ex, we can collocae he poins in he ineresed domain and calculae he poenial by using he Eq.(8). 3. ovel error esimaion echnique The derivaion in formulaing he analyical soluion in he realisic engineering problem is no obained easily. To overcome he drawback, an alernaive problem is defined o be solved by implemening BE. The domain shape and boundary condiion ype in he new problem are he same wih he original problem. Furhermore, he alernaive analyical soluion in he new problem is similar wih he real analyical soluion in he original problem, namely quasi-analyical soluion. Afer solving he new problem by BE and comparing wih he quasi-analyical soluion, we develop a novel error esimaion echnique in his sudy. The derivaion in he novel error esimaion echnique is presened as: 3. Definiion of new problem. Quasi-analyical soluion In his sudy, a new boundary-value problem are derived based on he Treffz concep, his geomery conour and boundary condiion ype in he new problem is he same wih original problem, and also saisfies he same differenial equaion (DE) operaor. The 3 poenial, u (, in he new problem a arbirary poin x in he domain is he known by funcion, he linear combinaion of he T-complee se funcions as follows: q u ( = = ν φ (, x D (4) φ( can be chose by he T-complee se funcions which saisfies he governing Eq.(), is he oal number of he T-complee funcions andν denoes he undeermined coefficien. Each of he funcions of T-complee se funcions saisfies he governing Laplace equaion in Eq.() as: [ φ() ( ] =, [ φ() ( ] =, L, [ φ ( ] =, [ φ ( ] = ( ) ( ) (5) Because of he linear propery of differenial equaion operaor in G.E., he poenial, u q (, saisfies he G.E. as: [ q ] [ ] u ( = v φ() ( + v [ φ() ( ] + L + v [ φ ( ] + v [ φ ( ] = ( ) ( ) (6) By collocaing number of collocaion poins o mach original B.C. in original problem, he undeermined coefficien, ν, can be deermined. Therefore, he quasi-analyical soluion is similar o real analyical soluion. The wo problems have he same boundary conour and boundary condiion ype, and he boundary condiions of he new problem are given as: = ν φ (, x (7) = u ( B and is derivaive in he normal direcion (flu as follows: u ( ( = = n = = x = φ ( ν n ν ω (, x B where u ( and ( x (8) are he known poenial and is derivaive in he normal direcion (flu. The error analysis beween he new defined problem and he original problem are formulaed on he nex secion as:
中華民國振動與噪音工程學會第 8 屆學術研討會, 明志科技大學, 台灣台北, 年 6 月 日 The 8 h aional Conference on Sound and Vibraion (CSSV), Taipei, Taiwan, June,. I-54. Error analysis beween he new defined problem and he original problem The relaionship beween he real analyical soluion and quassi-analyical soluion is shown as: e q u ( = u ( + R ( (9) where n = + n R ( = v φ ( The remainder funcion R ( saisfies he G. E. and i is exponenial convergence as R ( = O( r ), < r < n Therefore, he difference in he wo solver of space is derived as: e q u ( u ( = R ( C( r ), % % where C is bounded consan. u = u( R ) u( R )]/ ln( R / ) () [ R where R and R denoe he inner and ouer radii, respecively. The nodes disribuion is ploed in Fig. (b), he field soluion in Eq. () is depiced in Fig.3 (a). The RS resuls wih differen number of erms of Treffz complee se funcion comparing wih quasi analyical soluion and real analyical soluion, respecively, are shown in Fig.4. By observing he error curves, he opimal number of elemens is 4. The poenial is ploed by using he 4 number of elemens (opimal elemen in Fig.3 (b) and 8 elemens (less opimal elemen in Fig.3 (c), respecively. The field soluion along he radius r=.43 by using he differen elemens are show in Fig.5, and he poenial a he poin u(x=.48, y=.39) by using he differen elemens are shown in Fig.6. 3. R..S comparing wih quasi-analyical soluion The error quaniy of numerical soluion adops he roo mean squared (r.m. error by solving he new problem and comparing wih quasi-analyical soluion, which is defined as follows: rel = ( u ~ u) u () i= i= is he number of field poins, u ~ ( x ) where is he numerical soluion of he new problem by he BE. The flowchar of he formulaion in implemening he novel errormaor is shown in Fig.. 4. UERICAL EXAPLE To show he performance of he error esimaion scheme, we consider a raher sandard problem, subeced o he mixed-ype boundary condiion as shown in Fig. (a) which have been solved by CHE ec [5]. Our error esimaion scheme can be applied o more general case wih loss of generaliy hrough he case. The analyical soluion of he radial emperaure disribuion is given by u r) = u( R ) + u ln( r / ) () ( R in which 4 5. COCLUSIO In his paper, a new esimaion echnique is developed o obain he opimal number of elemens for he BE, we successfully applied he sysemaic error esimaion scheme o solve -D poenial problems wihou having analyical soluion. The numerical examinaion verifies he validiy of he error esimaor echnique. The echnique plays a role in deermining he opimal number of elemens which can be seen as a obecive way o obain he relaive errors in differen number of elemens wihou having analyical soluion, and we can obain he numerical soluion efficienly. The perplexing number of elemens in he BE can ge. The convergen resul is found from he convergen sudy in he case. umerical resuls agreed very well wih he analyical soluions. 6. ACKOWLEDGEET Financial suppor from he aional Science Council under Gran no.sc-98--e-97-3 o he firs auhor for aion Ilan Universiy is graefully acknowledged. 7. REFERECES
中華民國振動與噪音工程學會第 8 屆學術研討會, 明志科技大學, 台灣台北, 年 6 月 日 The 8 h aional Conference on Sound and Vibraion (CSSV), Taipei, Taiwan, June,. I-54 [] K.H. Chen, J.T. Chen, C.R. Chou, C.Y. Yueh,, Dual boundary elemen analysis of oblique inciden wave passing a hin submerged breakwaer, Engineering Analysis wih Boundary Elemens, Vol.6, pp.97 98. [] H. Chen, J. Jin, P. Zhang, P. Lu uli-variable on-singular BE for -D Poenial Problems, TSIGHUA SCIECE AD TECHOLOGY, Vol., o.,pp. 43-5, 5. [3] K. H. Chen, C. T. Chen, J. F. Lee, Adapive error esimaion echnique of he Treffz mehod for solving he [] Z. C. Li, Young L.-J., Huang H. T., Liu Y.-P., A.H.D. Cheng, Comparisons of fundamenal soluions and paricular soluions for Treffz mehods, Engineering Analysis wih Boundary Elemens, Vol. 34, pp. 48 58,. []. Vable, Conrolling errors in he process of auomaing boundary elemens mehod analysis, Engineering Analysis wih Boundary Elemens, Vol. 6, pp. 45-45,. [] Z. Zhao, X. Wang, Error esimaion and h adapive boundary elemens, Vol. 3, pp. 793-83, 999. over-specified boundary value problem, Engineering [3] X. Zheng, Z. H. Yao, Some applicaions of he Treffz Analysis wih Boundary Elemens, Vol. 33, pp. 966-98, 9. mehod in linear ellipic boundary-value problems, Advances in Engineering Sofware, Vol. 4, pp. 33-45, 995. [4].A. Golberg, H. Bowman, Superconvergence and he use of he residual as an error esimaor in he BE. II: Collocaion, numerical inegraion and error indicaors, Vol. 5, pp. 5-5, 999. [5] S. C. Huang, R. P. Shaw, The Treffz mehod as an inegral equaion, Advances in Engineering Sofware, Vol. 4, pp. 57-63, 995. [6] A. B. Jorge, Gabriel O. Ribeiro, Timohy S. Fisher, Applicaion of new error esimaors based on gradien recovery and exernal domain approaches o D elasosaics problems, Engineering Analysis wih Boundary Elemens, Vol. 9, pp. 963-975, 5. [7].T. Liang, J.T. Chen, S.S. Yang, Error esimaion for boundary elemen mehod, Engineering Analysis wih Boundary Elemens, Vol. 3, pp. 57-65, 999. [8] E. Kia,. Kamiya, T. Iio, Applicaion of a direc Treffz mehod wih domain decomposiion o D poenial problems, Engineering Analysis wih Boundary Elemens, Vol. 3, pp. 539-548, 999. 應用在邊界元素法的新誤差評估技術 陳俊廷 陳誠宗 陳桂鴻 國立宜蘭大學土木工程學系 國立成功大學水利及海洋工程學系 摘要 本研究發展出新的誤差評估方法, 在不參照解析解的情況下, 獲得邊界元素法的最佳元素數目 藉由使用 Treffz 完全集合函數來創造新的解析解, 並比較此新的解析解, 我們可以得到邊界元素法在不同元素下的收斂行為分析 我們發展一套系統化的誤差評估技術來搜尋邊界元素法的最佳元素數目, 最後提供數值案例證明在無需解析解的情況下, 所提出的系統化誤差評估技術的有效性和準確性 關鍵詞 : 元素離散,Treffz 完全集合函數, 收斂行為分析, 系統化的誤差評估技術 [9] E. Kia,. Kamiya, Treffz mehod: an overview, Advances in Engineering Sofware, Vol. 4, pp. 3-, 995. 5
中華民國振動與噪音工程學會第 8 屆學術研討會, 明志科技大學, 台灣台北, 年 6 月 日 The 8 h aional Conference on Sound and Vibraion (CSSV), Taipei, Taiwan, June,. I-54 Sar Obain new problem, shape, boundary condiion ype Creae quasi-analyical soluion of new problem by Treffz concep Creae B.C. of new problem by quasi-analyical soluion Solving he new problem by BE Esimae R..S error comparing wih quasi-analyical soluion Obain opimal number of elemens by error curve versus differen number of elemens Solving he original poenial problem by BE End Fig.. Flowchar of he sysemaic error esimaion scheme. u = = u = u = = R = R = 4 (a). problem skech (b). elemen mesh Fig..(a) problem skech, (b) elemen mesh.. 6
中華民國振動與噪音工程學會第 8 屆學術研討會, 明志科技大學, 台灣台北, 年 6 月 日 The 8 h aional Conference on Sound and Vibraion (CSSV), Taipei, Taiwan, June,. I-54.5.5 -.5 - - -.5.5.5 (a). analyical soluion.5.5 -.5 - - -.5.5.5 (b). 4 elemens.5.5 -.5 - - -.5.5.5 (c). 8 elemens Fig. 3. Field soluions (a) analyical soluion, (b) 4 elemens (opimal), (c) 8 elemens. 7
中華民國振動與噪音工程學會第 8 屆學術研討會, 明志科技大學, 台灣台北, 年 6 月 日 The 8 h aional Conference on Sound and Vibraion (CSSV), Taipei, Taiwan, June,. I-54. Compared wih analyical soluion Compared wih similar analyical soluion(=3) Compared wih similar analyical soluion(=35) Compared wih similar analyical soluion(=39) Compared wih similar analyical soluion(=45). R..S.. E-5 E-6 4 8 6 umber of elemens Fig. 4. The error analysis for he field soluion wih he differen erms of Treffz basis. 68.6 Analyical solve Elemens=8 Elemens= Elemens=4 Elemens=8 Elemens=6 68.4 u(r=.43,θ) 68. 68..4.8..6 θ Fig. 5. The error analysis for he field soluion along he radius r=.43 wih he differen elemens. 8
中華民國振動與噪音工程學會第 8 屆學術研討會, 明志科技大學, 台灣台北, 年 6 月 日 The 8 h aional Conference on Sound and Vibraion (CSSV), Taipei, Taiwan, June,. I-54 68.8 Analyical soluion Original problem new problem (=3) new problem (=35) new problem (=45) 68.4 u(.48,.39) 68. 68.6 4 6 8 umber of elemens Fig. 6. u (.48,.39) versus number of elemens and wih he differen erms of Treffz basis. 9