Meccanica 32: 79 86, 997. c 997 Kluwe Academic Publishes. Pined in he Nehelands. Applicaion of Opeaional Quadaue Mehods in Time Domain Bounday Elemen Mehods M. SCHANZ and H. ANTES Technical Univesiy of Baunschweig, Insiue of Applied Mechanics, Spielmannsaße, D-3806 Baunschweig, Gemany (Received: 7 Januay 997) Absac. The usual ime domain Bounday Elemen Mehod (BEM) conains fundamenal soluions which ae convolued wih ime-dependen bounday daa and inegaed ove he bounday suface. Hee, a new appoach fo he evaluaion of he convoluion inegals, he so-called Opeaional Quadaue Mehods developed by Lubich, is pesened. In his fomulaion, he convoluion inegal is numeically appoximaed by a quadaue fomula whose weighs ae deemined using he Laplace ansfom of he fundamenal soluion and a linea mulisep mehod. To sudy he behaviou of he mehod, he numeical convoluion of a fundamenal soluion wih a uni sep funcion is compaed wih he analyical esul. Then, a ime domain Bounday Elemen fomulaion applying he Opeaional Quadaue Mehods is deived. Fo his fomulaion only he fundamenal soluions in Laplace domain ae necessay. The popeies of he new fomulaion ae sudied wih a numeical example. Sommaio. L usuale meodo agli elemeni di conono (BEM) nel dominio del empo coniene soluzioni fondamenali che sono convolue con dai al conono dipendeni dal empo e inegai supeficie di conono. Nel pesene aicolo viene pesenao un nuovo appoccio pe la valuazione degli inegali di convoluzione svilluppao da Lubich, i cosiddei meodi opeazionali di quadaua. In quesa fomulazione, l inegale di convoluzione viene appossimao numeicamene con una fomula di quadaua i cui pesi sono deeminai usando la asfomaa di Laplace della soluzione fondamenale e un meodo lineae a più passi. Pe sudiae il compoameno del meodo, la convoluzione numeica di una soluzione fondamenale con una funzione di passo uniaio viene compaaa con i isulai analiici. Infine, una fomulazione agli elemeni di conono nel dominio del empo viene deivaa applicando i meodi opeazionali di quadaua. Pe quesa fomulazione sono necessaie solo le soluzioni fondamenali nel dominio di Laplace. Le popieà di una nuova fomulazione sono sudiae con un esempio numeico. Key wods: BEM, Time domain, Mulisep mehod, Tansfom mehods, Solid mechanics. Inoducion The Bounday Elemen Mehod (BEM) has become a widely used numeical mehod. In he case of ansien elasodynamic poblems, he BEM is mosly used in fequency o Laplace domain followed by an invese ansfomaion, e.g., []. Mansu [] developed one of he fis Bounday Elemen fomulaions diecly in ime domain. I was fomulaed fo he scala wave equaion, and lae on exended by Anes [2] o elasodynamics. This fomulaion was, e.g., applied o hee dimensional conac poblems by Seinfeld [4] and exended o viscoelasiciy by Schanz [2]. All fomulaions in ime domain, howeve, equie an adequae choice of he ime sep size. An impope chosen ime sep size leads o insabiliies o numeical damping. A fis impovemen of his behaviou is shown by Jäge [6] fo acousics and by Schanz e al. [3] fo elasodynamics. Anohe disadvanage of he fomulaion in ime domain is, howeve, ha no fo all physical poblems ime-dependen fundamenal soluions ae known in an explici analyical fom, e.g., in pooelaso dynamics [5].
80 M. Schanz and H. Anes Theefoe, hee, a BE fomulaion in ime domain is pesened which is based on he Opeaional Quadaue Mehods published by Lubich [7]. This mehod is a quadaue fomula which appoximaes he convoluion inegal in he ime domain bounday inegal equaion. The quadaue weighs ae deemined fom he fundamenal soluions in Laplace domain. In Secion 2, his quadaue mehod is summaized. In Secion 3, he bounday elemen fomulaion fo elasodynamics is developed using his quadaue mehod. Afe ha, numeical esuls of a 3-d ba ae pesened. 2. Convoluion Quadaue In he following, he convoluion in he ime domain bounday inegal equaion is appoximaed by he so-called Opeaional Quadaue Mehod developed by Lubich [7]: a convoluion inegal of he fom Z y() =f()g()= f( )g()d () 0 can be appoximaed by he Opeaional Quadaue Mehod using he Laplace ansfomed funcion ^f (s). Subsiuing f () by he invese Laplace ansfomaion of ^f (s) in he convoluion inegal () and exchanging he inegals leads o Z Z c+ir f ( )g( ) d = lim 0 2i R! c ir Z ^f (s) e s( 0 ) g( ) d {z } x() ds; (2) wih a eal consan c. The inne inegal, abbeviaed wih x(), is a soluion of he diffeenial equaion of fis ode d x() =sx() +g() wih x(0) =0; (3) d wih vanishing iniial condiions. Theefoe, x() can be appoximaed by a linea mulisep mehod x() kx j=0 j x n j = kx j=0 j (sx n j + g((n j))); (4) wih equal ime seps ; = n and he saing values x k = = x =0. Unfounaely, his epesenaion of he mulisep mehod does no allow o exac he discee values x n which shall be used P o appoximae Equaion P (2). Taking a epesenaion wih (fomal) powe seies fo x() = x n z n and g() = g(n)z n he mulisep mehod becomes X X s x n z n = g(n)z n ;= 0 ++ k z k 0 ++ k zk: (5) The used mulisep mehod, chaaceized by he quoien of he geneaing polynomials, should be A()-sable wih posiive angle, sable in a neighbouhood of infiniy, songly zeo-sable and consisen of ode p, (see [0]). Well known examples of possible geneaing
Opeaional Quadaue Mehods in BEM 8 polynomials ae he backwad diffeeniaion fomulas of ode p 6 6, e.g., of ode 2 given by = 3 2 2z+ 2 z2. Wih he epesenaion of x() in Equaion (5) he convoluion () is appoximaed by X y(n)z n = 2i lim R! Z c+ir c ir ^f (s) X ds s g(n)z n : (6) The inegaion along he cuve c ir o c + ir is changed o a closed conou, by adding a half cicle a is ends. If he funcion ^f (s) saisfies he assumpion j ^f(s)j!0 fo R(s) > c and jsj!; (7) he inegal in Equaion (6) can be deemined by he value of he inegand a he singula poin s = = (Cauchy s inegal fomula) Z c+ir 2i lim R! c ir ^f (s) ds = ^f s Repesening he funcion ^f by a powe seies ^f = X wih he coefficiens Z! n () = 2i : (8)! n ()z n ; (9) jzj= ^f z n dz (0) and being he adius of a cicle in he domain of analyiciy of ^f, Equaion (8) can be simplified by Cauchy s poduc of wo seies X X! n ()z n g(n)z n = X nx j=0! n j ()g(j)z n : () Taking now he nh coefficien of he powe seies (), he final quadaue fomula eads y(n) = nx k=0! n k ()g(k); n = 0; ;:::;N: (2) The inegaion weighs! n ae deemined by Equaion (0). Afe a pola coodinae ansfomaion his inegal is appoximaed by a apezoidal ule wih L equal seps 2=L ([8])! n () = n L LX `=0 ^f (e i` 2 L )! e in` 2 L : (3) Now, using he echnique of he Fas Fouie Tansfomaion (FFT), he weighs! n can be calculaed vey fas fo all n = 0; ;:::;N. If one assumes ha he values of ^f in Equaion (3) ae compued wih an eo bounded by, hen he choice L = N and N = p yields an eo in! n of size O( p ), (see [8]).
82 M. Schanz and H. Anes Figue. Numeical and exac evaluaion of he convoluion U0() H(). To check he pocedue, he convoluion beween he fundamenal soluion of he displacemen ( = p i i wih i = x i y i ) U ij (x; y; ( 4% 2 i j 3 " c 2 )= 3i j 3 ij c H c 2 2 c H # + ij c 2 c 2 2 + c 2 ) c 2 (4) and he uni sep funcion H() ae calculaed. Figue shows he numeical and exac evaluaion of he convoluion inegal U 0 ()H(). Obviously, he ageemen of he analyical and numeical soluion is good, wih he excepion of he neighbouhood of he jumps. Thee, an oveshooing is obseved depending on he choice of he mulisep mehod and he ime sep size. The paamees in his es ae chosen as suggesed above wih = 0 0.The used mulisep mehod is a backwad diffeenial fomula of second ode. 3. BE Fomulaion Fo consisency, he bounday inegal equaion fo elasodynamics in ime domain is ecalled. The field equaions of a homogeneous elasic domain (Young s Modulus E, densiy % and Poisson s aio )aegivenby (c 2 c 2 2 )u i;ij + c 2 2 u j;ii + b j % =u j (5) wih displacemens u j and wave speeds c 2 E( = ) %( 2)( + ) ; c2 2 = E %2( + ) : (6) In he above equaions, () ;i denoes he deivaive wih espec o he spaial coodinae x i, and u j is he acceleaion. On he bounday = u [ of he domain, he bounday condiions i (x;)= ij n j = p i (x;)x 2 ; u i (x;)=q i (x;)x 2 u (7)
Opeaional Quadaue Mehods in BEM 83 ae given. ij is he sess enso and n j he ouwad nomal on he bounday. Fo a complee iniial bounday value poblem also he iniial condiions u i (x; 0) and _u i (x; 0); x 2,haveo be pescibed. Hee, hey ae assumed o be zeo u i (x; 0) =0; _u i (x;0)=0 x 2: (8) Assuming also vanishing volume foces, he dynamic exension of Bei s ecipocal wok heoem leads o he inegal equaion (see [2]) c ij (y)u j (y;)= Z x [U ij (x; y;) j (x;) T ij (x; y;)u j (x;)] d x ; (9) whee U ij and T ij ae he ime-dependen fundamenal soluions of he displacemens and acions, especively. The inegal fee em c ij (y) is he same as in elasosaics [4], e.g., c ij (y) = ij =2 fo a smooh bounday. As shown by Bonne [3], he fis inegal in Equaion (9) is weakly and he second songly singula. Theefoe, he second inegal can only be defined in he sense of a Cauchy Pincipal Value. Fo he numeical soluion of he bounday inegal Equaion (9) in an abiay domain, a disceizaion mus be inoduced. Theefoe, he bounday is divided in E iso-paameic elemens e whee F polynomial shape funcions Ne f (x) fo he spaial vaiable ae defined. This pocedue yields EX FX EX u j (x;)= Ne f (x)uef j (); j(x;)= Ne f (x)ef j (); (20) e=f= e=f= wih he ime dependen nodal values u ef j () and ef j (). Inseing hese ansaz funcions in Equaion (9) leads o c ij (y)u j (y;) = EX FX Z e=f= I e e FX U ij (x; y;)ne f (x)d e ef j () T ij (x; y;)ne f (x)d eu ef j () : (2) If he ime is disceized in N equal ime seps, he convoluion beween he fundamenal soluions U ij o T ij and he nodal values ef j () o uef j (), especively, is appoximaed by he Convoluion Quadaue fomula (2). This esuls in he new bounday elemen fomulaion in ime domain EX FX nx c ij (y)u j (y;n) = f! ef ef n k ( ^U;y;)j (k) e=f= k=0! ef n fo n = 0; ;:::;N, wih he weighs! n ef ( ^U;y;)= n LX L `=0 Z k ef ( ^T;y;)uj (k)g (22) e ^U x;y; (`) n Ne f (x)d e` ; (23) fo he displacemens (` = e i` 2 L ) and a simila expession fo he weighs of he acions.
84 M. Schanz and H. Anes Figue 2. Disceizaion, loading and bounday condiions of he ba. Figue 3. Longiudinal displacemen a poin P vesus ime fo diffeen values of. Noe ha he calculaion of he inegaion weighs (23) is only based on he Laplace ansfomed fundamenal soluion. Theefoe, fo he spaial inegaion in Equaion (23), he echniques well known fom he elasodynamic fequency domain fomulaion ae used. Finally, Equaion (22) is solved wih he collocaion mehod and a diec equaion solve. 4. Numeical Example As a fis applicaion, a ba (geomey: 3 m m m, maeial: E = N m 2 ; =0;% = kg m 3 ) is consideed (see Figue 2). The ba is aken o be fixed on one end, and is loaded wih a uni sep funcion in ime on he ohe fee end. The emaining sufaces ae acion fee. The ba is disceized wih 56 iangles, and linea shape funcions ae used. The paamee and L ae chosen as suggesed in Secion 2: L = N and N = p, wih he eo bound = 0 0. Smalle values of, e.g., below = 0 30, lead o compleely unsable esuls. The spaial inegaion is done wih sandad Gauß quadaue fomulas. The weakly singula inegals in Equaion (22) ae egulaized wih a coodinae ansfomaion and he songly singula inegals wih he mehod suggesed by [5]. Resuls fo he longiudinal displacemen a he poin P vesus ime ae ploed fo diffeen ime sep sizes in Figue 3. Thee, a backwad diffeenial fomula of second ode is applied fo he undelying mulisep mehod. These esuls ae compaed wih he -d soluion [4], which is denoed wih exac. Obviously, hee exiss a ciical value of he ime sep size, below
Opeaional Quadaue Mehods in BEM 85 Figue 4. Compaison of diffeen mulisep mehods. which he esuls ae unsable ( < 0:5). This is in accodance wih he invesigaions fo he bounday inegal equaion of he wave equaion by [9]. Fo lage ime sep sizes ( = 0:9), a kind of phase shifing is obseved because he ime sep size is oo lage o appoximae he peaks coecly. Since he mehod suggesed by Anes [2] shows an unsable behaviou using small ime seps and numeical damping fo lage ime seps, he ime sep size has hee be esiced o he ineval [0.7,.0]. In compaison wih his fomulaion, hee, a smalle lowe ciical value and no numeical damping is obseved. Howeve, his behaviou depends heavily on he undelying mulisep mehod. Theefoe, finally he influence of he undelying mulisep mehod is sudied in Figue 4. Thee, he esuls wih a backwad diffeeniaion fomula of he fis ode (BDF ) and hose ofhesecondode (BDF2) aecompaed. Anopimalchoiceofheimesepsize is used in boh calculaions. The esuls of he BDF 2 ae close o he -d soluion han he esuls of BDF, bu wih he BDF pocedue much smalle ime sep sizes ae possible. 5. Conclusions The pesen pape descibes a bounday elemen fomulaion diecly in ime domain whee only he fundamenal soluions in Laplace domain ae used. This fomulaion is based on he Opeaional Quadaue Mehods developed by Lubich [7]. Applying hese quadaue fomulas o he convoluion inegal in he bounday inegal equaion, a numeical inegaion fomula is obained whee he weighs depend only on he Laplace ansfomed fundamenal soluions. Wih hese fomulas, a new ime domain bounday elemen fomulaion is deived. A numeical example shows ha a ciical ime sep size exiss, below which he mehod becomes unsable. This ciical value depends on he undelying mulisep mehod. Compaed wih he diec ime domain based fomulaion suggesed by [2], he ciical ime sep size is much smalle. Fuhemoe, all advanages of he Laplace domain bounday elemen fomulaion can be used. Theefoe, his mehod seems o be suiable also in he case of he hypesingula acion bounday inegal equaion, and fo all cases whee he ime-dependen fundamenal soluion is no known, e.g., in viscoelasiciy o pooelaso dynamics.
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