FOR many years the authors of this paper have worked on

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1 The Fast Parametri Integra Equations System for Poygona D Potentia Probems Andrzej Kużeewski and Eugeniusz Zieniuk Abstrat Appiation of tehniques for modeing of boundary vaue probems impies three onfiting requirements: obtaining high auray of the resuts, high speed of the soution and ow oupany of omputers memory (RAM) It is very diffiut to satisfy suh requirements partiuary in proess of soving arge-sae engineering probems Numeria soution of these probems require omputations on arge matries Aurate resuts an be obtained ony by using appropriate modes and agorithms In the previous papers the authors appied the parametri integra equations system (PIES) in modeing and soving boundary vaue probems The first requirement was satisfied - the resuts were obtained with very high auray This paper fufis other requirements by nove approah to aeerate the PIES For this purpose the fast mutipoe method (FMM) is inuded into onventiona PIES, therefore the fast PIES method is obtained Index Terms parametri integra equations system, boundary vaue probems, fast mutipoe method I INTRODCTION FOR many years the authors of this paper have worked on deveopment and appiation of the parametri integra equations system (PIES) to sove boundary vaue probems The PIES has aready been used to sove probems modeed by D and 3D partia differentia equations, suh as: Lapae ], ], Hemhotz 3] or Navier-Lamé 4], 5], 6] The remarkabe advantage of the PIES, ompared to traditiona boundary integra equation (BIE), is diret inusion in its mathematia formaism a shape of the boundary of the onsidered probem 7] The shape of the boundary is generay defined using partiuar funtions For this purpose, the urves (eg Bézier, B-spine, et) or surfae pathes (suh as Coons, Bézier and others) widey used in omputer graphis, were appied in the PIES The PIES is an anaytia modifiation of traditiona BIE The above mentioned urves and surfae pathes are appied in modeing the shape of the boundary, instead of the ontour integra as in the ase of BIE Therefore in pratie, a sma number of ontro points is required to define any shape of the boundary It is definitey muh easier than in ase of eement methods (suh as boundary eement method BEM 8] or finite eement method FEM 9]) Moreover, the auray of soutions an be effiienty improved without interferene in modeing the shape of the boundary The former studies foused on auray and effiieny of the resuts obtained using the PIES in omparison with the we-known agorithms suh as the FEM or the BEM, as we as anaytia methods These studies onfirmed the effetiveness and auray of the PIES in soving D ], 4] and 3D ], 5], 7] boundary vaue probems The authors Manusript reeived Marh 4, 7; revised Apri 4, 7 A Kużeewski and E Zieniuk are with the Fauty of Mathematis and Informatis, niversity of Biaystok, 5-45 Białystok, Ciołkowskiego M, Poand, e-mais: akuze@iiuwbedup, ezieniuk@iiuwbedup aso proposed some extensions of the PIES method, ie to sove unertainy defined probems (interva PIES ], ]) or to sove transient probems ] The former studies show, that modeing of the boundary in the PIES is definitey more simpe and effiient ompared to the FEM or the BEM However effiient soving of arge-sae engineering probems by the PIES is imited (simiary to onventiona BEM) due to the fat, that the PIES in genera produes dense and non-symmetria matries That matries are definitey smaer in sizes than in the BEM, athough to ompute their oeffiients we need O(N ) operations and another O(N 3 ) operations to sove obtained system using diret sovers (where N - the number of equations of the agebrai system of inear equations) In the paper 3] the authors proposed parae version of the PIES obtained using OpenMP, whie in 4], 5], 6] they aeerated numeria omputing of oeffiients and soving of the system of inear equations using CDA 7] These approahes redue quite signifianty the time of omputations, however the probem of imited resoures of RAM in PC sti exists Therefore, it not aows for onvenient and effiient soving of arge-sae engineering probems In the mid of 98s of XX entury Rokhin and Greengard proposed the fast mutipoe method (FMM) 8], 9], ], whih aows to redue the CP time in the FMM aeerated methods to O(N), as we as definitey redue oupany of RAM However, appiation of the FMM has inreased the ompexity of impementation of the PIES It requires a new approah for omputing oeffiients and soving the system of inear equations The main goa of this paper is to present possibiity of aeeration of the PIES for numeria soving of boundary vaue probems using the FMM The main onept of the FMM is adopted from the FMM-BEM method ] However we think, that inusion of the FMM into the PIES shoud be more effetive than into the BEM It is strity onneted with the different way of defining of BRC in the PIES and the BEM To verify this onept, we need to inude the FMM into onventiona PIES Additionay, we must modify the way of soving of the inear system of equations and appy iterative sover Therefore, we obtain the new fast PIES method In our preiminary studies we try to onfirm high effiieny of the fast PIES on the exampe of poygona D potentia boundary probem II THE PIES FOR TWO-DIMENSIONAL POTENTIAL BONDARY PROBLEM The PIES for two-dimensiona Lapaes equation was obtained as the resut of anaytia modifiation of boundary integra equations (BIE), where the boundary is defined

2 using boundary integras The main goa of modifiation was inusion of the shape of boundary in mathematia formaism of BIE The shape of boundary is defined by parametria inear (or non-inear) funtions The PIES for poygona domains is presented by the foowing formua 7]: u (s k ) = n j= { s j s j j (s k,s)p j (s)ds P s j j (s k,s)u j (s) J j, where: =,,,n, s s k s, s j s s j,j j is the Jaobian, n is the number of parametri segments that reates poygona boundary of domain in D In PIES defined in the parametri referene system,s ands j orrespond to the beginning of -th an-th segment, whie s and s j to their ends Integrands j(s k,s) and P j(s k,s) in () are presented in the foowing form: j (s k,s) = π n, S S P j (s k,s) = S n (j) S n (j) π S, S where:s = S () (s k ) S () j (s) ands = S () (s k ) S () j (s) ExpressionsS n (i) (s) (n = j,, i =,) are parametri inear funtions S n (i) (s) = a(i) n sb(i) n, (3) whih desribe partiuar segments of poygona domain Coeffiients a (i) n and b (i) n are obtained in an easy way by simpe agorithm utiizing orner points of poygon Simiary to the previous researhes ], 3], 6], the pseudospetra method ] was appied to numeria soving of the PIES () Boundary funtions are approximated by the foowing series: u j (s) = p j (s) = N N u (k) j L (k) j (s), p (k) j L (k) j (s), where u (k) j and p (k) j are unknown oeffiients on segment j, N - is the number of terms in approximating series (4), L (k) j (s) - the base funtions (Lagrange poynomias) on segment j Finay, an agebrai version of PIES () is transformed into system of agebrai equationsax = b To sove the system, Gaussian eimination with partia pivoting and iterative refinement is used Soutions on the boundary are obtained after soving the system III THE FAST PIES FORMLATION The FMM is appied to aeerate the soving of equation () The main idea of the FMM is to transform auating interations between segments into interations between the es that reate the hierarhia struture (tree) with the () () (4) smaest es (aed eaves) ontaining a number of segments Beause the PIES for D probems is defined on parametri ine s (Fig ), the tree struture is reated on the basis of that ine, unike in the FMM-BEM ], where whoe pane is used Exampe of the tree struture for the fast PIES is presented in Fig Fig system x () P n P =P n P s = j= P k d n P n- d n- P k dk d d P s = P s s e j= d k d 3 P3 s 3 = P k P 4 P 3 d 4 P 5 x () parametri referene system s j= s s j Pn s n = j= Mapping of the shape of the boundary into parametri referene On the basis of the tree the foowing steps of the FMM proedure are performed: mutipoe expansion (auation of moments), moment-to-moment transation, moment-tooa transation and oa-to-oa transation During the omputations, the ompex notation is introdued, due to onvienient desribing of points on the pane They are redued to the form of P = P () ip () (Fig ), where - ompex, () - oordinate in diretion x (), () - oordinate in diretion x (), i = Fig e eaf Exampe of the tree struture for the fast PIES Leve ) Mutipoe expansion: At first, we onsider the integra onneted with the kerne j(s k,s) Transformations for the kerne P j (s k,s) are presented in the ast subsetion Considering the ompex notation in the kerne (), it an be 3 4

3 noted, that: j (s k,s) = π n S S = = { π R ns S j = R { j (s k,s), where S = S () is (), S j = S () j is () j and j (s k,s) = S π n S j,r-rea part of ompex number The ooation point s k is oated in the segment S, and observation point s j in the segment S j Assuming that introdued point s (oated in the segment S ), whih is the key eement of the FMM, is ose to the points j (Fig ), the kerne an be expanded abouts using the Tayor series expansion: j (s k,s) = π k= { n S (k )! S S S S In order to simpify the auations we shoud hange the base of integration s into S simiary to (3): S = a j sb j, where a j = P P j j and b j = P j (P P j j )sj, j =,,,n Therefore, new imits of integration are P j (ower) and P j (upper) We aso need to hange ds by ds: ds ds = a j => ds = ds a j Substituting the kerne j (s k,s) into integra () and = a j, we obtain the foowing expression: s j = j (s k,s)p j (s)ds = P P j k= = π π p j(s) S (k )! S { n S S k (S,S )M k (S ), where M k (S ) are aed moments about S : M k (S ) = P P j S S ds = p j (S) ds They are independent of the ooation point s k and shoud be omputed one ony Expressions k (S,S ) have the foowing form: ns k (S,S S for k = ) = (k )! S S where s are mid-point of eaves for k (5) (6) (7) ) Moment-to-moment transation: If we want to move the point s to a new oation s (when we hange the eve of the tree during omputations), we an use the foowing expression to find new moments about s : M k (S ) = P P j Taking into aount, that: S S = S S and appying the binomia formua: (ab) n = pj(s) ds (8) ] (S )(S S k ) n ( n m m= at ast we obtain moments in the point s : M k (S ) = k m= ) a m b n m (9) ] S S (k m) M m (S ) () (k m)! using a finite number of term in the transation 3) Moment-to-oa transation: Assuming, that the point s e is ose to the ooation point s k (see Fig ), the equation (7) an be expanded abouts e (the segment, where the point s e is oated) using the Tayor series expansion: s j where: and j (s k,s)p j (s)ds = = π = L (S,S ) = n L (S,S ) = ( ) L (S e,s ) k= S e S ] S e! M (S ) (k )!M k (S S e ) () (k )!M k (S ) for S e Pointss e, simiary tos, are mid-points of eaves Desribed proedure is aso aed oa expansion 4) Loa-to-oa transation: Simiary to moment-tomoment transation, the point s e an be moved to new oation s e (when we hange the eve of the tree during omputations) It is performed using the binomia formua (9) and the foowing transformation: = m= = m= =m

4 At ast we obtain the foowing oa-to-oa transation: s j { j (s k,s)p j (s)ds = ( ) π m= = (k m )!M k (S ) m S e S e S e (m )! ] m S ] S e! () 5) Transations for the kerne P j (s k,s): The kerne P j(s k,s) in the ompex notation an be omputed on the basis of the foowing expression: j (s k,s) = j (s k,s) P n = n j (s k,s), (3) S where n = n in - norma vetor to segment j in the ompex notation Hene { P j (s k,s) = R P j (s k,s) = n R j (s k,s) S n I j (s k,s) S (4), where R,I - rea and imagine part of ompex number Finay, after auating the derivative (3) we obtain: P s j where j (s k,s)u j (s)ds = N k (S ) = = π P P j k= P k (S,S )N k (S ), S S (k )! u j (S)n ds (5) Expressions N k (S ), simiary to M k (S ) (7), are aed moments about S and they are independent of the ooation points k and shoud be omputed one ony Expressions P k (S,S ) have the foowing form: P k (S,S ) = (k )! for k, S where s are mid-point of eaves It an be noted, that a transations desribed in previous subsetions remain exaty the same for the kerne P j (s k,s), exept for the fat that N (S ) Therefore, we an direty appied them for the moments N k (S ) 6) The fast PIES agorithm: The fast PIES agorithm runs in severa steps The first step is to determine the struture of the tree On the basis of D probems mapped into parametri referene system (Fig ) the struture of the tree is reated Leve e overs the whoe probem It is the straight ine where the PIES is defined The ength of this ine is equa to the sum of a segments, whih reated the boundary of the probem Leve parent e is divided into two equa hid es of eve Then, dividing is ontinued in the same way for eah eve parent es The division is arried out unti the number of segments in a e is ess than or equa to the predefined maximum number of segments in a e Eah e, whih has no hidren, is aed a eaf The division is ompeted if at the highest eve we obtained eaves ony or predefined maximum number of eves is reahed The next step is aed upward pass Starting from the highest number eve, moments in a eaves are omputed (up to k terms in the Tayor series expansion) Then, traing the tree struture upward and using moment-to-moment transation, a moments are omputed in eah parent e, up to the eve (Fig 3) Fig 3 s' e s e s'' e s'' e s s' e oa expansion moment-to-oa transation oa-to-oa transations Transations in the fast PIES s s' s s'' s' moment-to-moment transations Leve The next step is aed downward pass To eary desribed this pass, we shoud define a few terms onneted with es neighbourhood Two es are adjaent at eve i if they have ommon end Two es are we separated at eve i if they are not adjaent at eve i, but their parent es are adjaent (at eve i ) Then the interation ist of e K is reated using the ist of we-separated es from a eve i e K Two es are far es if their parent es are not adjaent Coeffiients of oa expansion are omputed starting from the eve and traing the tree struture downward to a eaves (Fig 3) Coeffiients of oa expansion at e K are the sum of the ontributions from the es in the interation ist of e K (omputed using moment-to-oa transation) and from a the far es (omputed using oato-oa transation) For a e K at eve, ony moment-tooa transation is used to ompute oeffiients of the oa expansion At the highest number eve, ontributions from eaf K and its adjaent es are omputed direty, as in the onventiona PIES Finay, we obtain a vetor, whih is the resut of mutipiation matrix A by a vetor x and there is no need to store the entire matrix A in the omputer s memory This approah requires the use of iterative sovers for system of equations (eg GMRES) IV TESTING EXAMPLE The foowing exampe shows the auray and effiieny of the fast PIES in soving D potentia probems Inte Core i5-459s (4 ores, 4 threads, 3 GHz, 6MB ahe memory) with 8 GB RAM was used during tests Both the fast and onventiona PIES was ompied using g

5 with -O optimization Numeria tests were arried out on 64-bit buntu Linux operation system (kerne 44) The testing exampe onerns the probem desribed by Lapaes equation The shape of the boundary is shown in Fig 4 It is simpe boundary probem, however in order to inrease its ompexity, we assumed that the edge ooks ike a gear with 56 teeth We used 5 inear segments to mode the probem using both version of the PIES Boundary onditions are identia on eah tooth and they are presented in Fig 4 (where p Neumanns and u Dirihets boundary onditions) On eah segment we have defined the same number of ooation points (from to 8) and finay have soved the system of 4 to 496 agebrai equations We assumed vaue of GMRES onvergene riterion (GMRES toerane) equa to e-8 and the number of terms in Tayor series is 5 mm 5 mm TABLE II ACCRACY OF THE FAST PIES SOLTIONS VS VALE OF GMRES CONVERGENCE CRITERION No No of iterations CP time s] Re error norm %] of for GMRES to for GMRES to for GMRES to eq e-8 e-6 e-8 e-6 e-8 e e-5 84e e-4 96e e e-4 48 than 5%), however the number of iterations is redued, as we as omputation time CP time s] onventiona PIES fast PIES Number of equations Fig 5 Comparison of the CP time used by the fast PIES and onventiona PIES Fig 4 The shape of modeed probem u= TABLE I THE FAST PIES VS CONVENTIONAL PIES: CP TIME AND RAM OCCPANCY No of CP time s] RAM oupany MB] Speed-up equations fast PIES PIES fast PIES PIES Tabe I presents obtained resuts of aeerating auations in the fast PIES ompared to onventiona PIES Growing number of equations resuts in sma inrease of omputation time in the fast PIES, ontrary to onventiona PIES The fast PIES aso needs more than 3 times ess RAM during omputations Reative error norm L between soutions obtained by the fast and onventiona PIES is presented in Tabe II The vaue of L for GMRES toerane equa e-8 do not exeed % in a ases Dereasing toerane to e-6 resuts in grow of soutions errors (in the worst ase L norm is ess p= Fig5 presents omparison of used CP time between onventiona PIES and fast PIES It shoud be noted, that the shape of potted ines is onsistent with theoretia onsiderations Conventiona PIES needs O(N ) operations to ompute their oeffiients and another O(N 3 ) operations to sove the system by the diret sover Appiation of the fast PIES is definitey more effiient Additionay, we sove the exampe using a bit od appiation of the FMM-BEM, whih has been written in fortran by the authors of ] We want to find soutions omparabe with the fast PIES, therefore the mesh in the FMM-BEM is omposed of 4, 48, 37 or 496 inear eements (eah teeth is desribe by 4, 8, or 6 eements) The exampe of disretization (for 4 eements mesh) of a few teeth of the gear is presented in Fig 6 Toerane of GMRES is e-8 and the number of terms in Tayor series is 5 TABLE III THE FAST PIES VS THE FMM-BEM: CP TIME AND RAM OCCPANCY No of CP time s] RAM oupany MB] equations fast PIES FMM-BEM fast PIES FMM-BEM Tabe III presents obtained resuts of aeerating auations in the fast PIES ompared to the FMM-BEM Growing number of equations resuts in sma inrease of omputation time in the fast PIES ontrary to the FMM-BEM However,

6 The authors woud ike to thank prof Naoshi Nishimura from Department of Appied Anaysis and Compex Dynamia Systems Graduate Shoo of Informatis, Kyoto niversity, for sharing soure ode of the FMM-BEM appiation Fig 6 u= boundary eement p= Exampe of the FMM-BEM disretization for 4 eements mesh the probem of the FMM-BEM is rather onneted with too big size of aoated RAM ompared to reay used Therefore, we an use maximum 7 eements in a e Fig 7 BEM CP time s] FMM BEM fast PIES Number of equations Comparison of the CP time used by the fast PIES and the FMM- Fig7 presents omparison of used CP time between the fast PIES and the FMM-BEM Appiation of the fast PIES is more effiient than the FMM-BEM However, we shoud highight the fat, that the FMM-BEM program is a bit od Additionay, reative error norm L omputed between the FMM-BEM and the fast PIES is smaer than % V CONCLSION The paper presents possibiity of aeeration of omputation and redution RAM oupany for numeria soving of boundary vaue probems using the PIES Verifiation of this onept required inusion of the FMM into onventiona PIES The numeria exampe shows signifiant redution of omputation time of the fast PIES The speed-up in omparison to onventiona PIES inreases with the size of the onsidered probem We noted amost no differene between auray of soutions obtained by the fast PIES and onventiona one The fast PIES is aso sighty faster than the FMM-BEM, whie auray of obtained resuts is amost the same This paper is our first attempt to use the fast PIES for soving D potentia boundary vaue probems The presented tehnique of aeeration of omputations shoud be extended to the probems modeed by non-inear segments Obtained resuts suggest that hosen diretion of studies shoud be ontinued REFERENCES ] E Zieniuk, Modeing and effetive modifiation of smooth boundary geometry in boundary probems using B-spine urves, Engineering with Computers, Vo 3, No, pp 39-48, 7 ] E Zieniuk and K Szerszeń, Trianguar Bézier surfae pathes in modeing shape of boundary geometry for potentia probems in 3D, Engineering with Computers, Vo 9, No 4, pp 57-57, 3 3] E Zieniuk and K Szerszeń, Trianguar Bézier pathes in modeing smooth boundary surfae in exterior Hemhotz probems soved by PIES, Arhives of Aoustis, Vo 34, No, pp 5-6, 9 4] E Zieniuk and A Bołtuć, Non-eement method of soving D boundary probems defined on poygona domains modeed by Navier equation, Internationa Journa of Soids and Strutures, Vo 43, No 5-6, pp , 6 5] E Zieniuk, A Bołtuć and K Szerszeń, Modeing ompex homogeneous regions using surfae pathes and reiabiity verifiation for Navier-Lamé boundary probems, Proeedings of The Internationa Conferene on Sientifi Computing WORLDCOMP, pp 66-7, 6] E Zieniuk, A Bołtuć and A Kużeewski, Agorithms of identifiation of muti-onneted boundary geometry and materia parameters in probems desribed by Navier-Lamé equation using the PIES, in Advanes in Information Proessing and Protetion: Internationa Advaned Computer Systems Conferene 6, pp 49-48, 6 7] E Zieniuk, Computationa method PIES for soving boundary vaue probems (in poish), PWN, Warszawa, 3 8] C A Brebbia, J C F Tees and L C Wrobe, Boundary eement tehniques, theory and appiations in engineering, Springer-Verag, New York, 984 9] O C Zienkiewiz The Finite Eement Methods, MGraw-Hi, London, 977 ] E Zieniuk and M Kapturzak and A Kużeewski, Conept of modeing unertainy defined shape of the boundary in two-dimensiona boundary vaue probems and verifiation of its reiabiity, Appied Mathematia Modeing, Vo 4, No 3-4, pp 74-85, 6 ] E Zieniuk, A Kużeewski and M Kapturzak, The infuene of interva arithmeti on the shape of unertainy defined domains modeed by osed urves, Computationa & Appied Mathematis, doi:7/s , to be pubished ] E Zieniuk, D Sawiki and A Bołtuć, Parametri integra equations systems in D transient heat ondution anaysis, Internationa Journa of Heat and Mass Transfer, Vo 78, pp , 4 3] A Kużeewski and E Zieniuk, OpenMP for 3D potentia boundary vaue probems soved by PIES, AIP Conferene Proeedings 6: 3th Internationa Conferene of Numeria Anaysis and Appied Mathematis ICNAAM 5, No 4898, 5 4] A Kużeewski and E Zieniuk, GP-based aeeration of omputations in eastiity probems soving by parametri integra equations system, Advanes in Engineering Software, Vo 79, pp 7-35, 5 5] A Kużeewski, E Zieniuk and M Kapturzak, Aeeration of integration in parametri integra equations system using CDA, Computers & Strutures, Vo 5, pp 3-4, 5 6] A Kużeewski, E Zieniuk and A Bołtuć, Appiation of CDA for Aeeration of Cauations in Boundary Vaue Probems Soving sing PIES,, Leture Notes in Computer Siene: Parae Proessing and Appied Mathematis PPAM 3, PT II, pp 3-33, 4 7] CDA C Programming Guide, aess date: 3 Marh 7] 8] V Rokhin, Rapid soution of integra equations of assia potentia theory, Journa of Computationa Physis, Vo 6, No, pp 87-7, 985 9] L F Greengard and V Rokhin, A fast agorithm for partie simuations, Journa of Computationa Physis, Vo 73, No, pp , 987 ] L F Greengard, The rapid evauation of potentia fieds in partie systems, The MIT Press, Cambridge, 988 ] Y J Liu and N Nishimura, The fast mutipoe boundary eement method for potentia probems: A tutoria, Engineering Anaysis with Boundary Eements, Vo 3, No 5, pp 37-38, 6 ] D Gottieb and S A Orszag, Numeria Anaysis of Spetra Methods: Theory and Appiations, SIAM, Phiadephia, 977 ACKNOWLEDGMENT