UNIVERSITY OF CINCINNATI

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1 UNIVERSITY OF CINCINNATI 5 August 006 Date: I,, hereby submit this work as part of the requirements for the degree of: in: Miind Shrikant Bapat Master of Science Department of Mechanica Engineering It is entited: Fast Mutipoe Boundary Eement Method For Soving Two-Dimensiona Acoustic Wave Probems This work and its defense approved by: Chair: Dr. Yiun Liu Dr. ay Kim Dr. Kumar Vemaganti

2 Fast Mutipoe Boundary Eement Method For Soving Two-Dimensiona Acoustic Wave Probems A thesis submitted to the Graduate Schoo of the University of Cincinnati in partia fufiment of the requirements for the degree of Master of Science In the Department of Mechanica, Industria and Nucear Engineering of the Coege of Engineering in December 006 by Miind Bapat B.E. Mechanica Engineering Maharashtra Institute of Technoogy, Pune, India. August 00. Committee Chair: Dr. Yiun Liu

3 ABSTRACT The boundary eement method BEM is a numerica method for soving boundary vaue probems. The boundary eement method has a cear advantage over other techniques ike finite eement method FEM in probems invoving infinite domains. Hence the boundary eement method has found many appications in the fied of acoustics which often exist in infinite domains. The traditiona approach for finding soutions to acoustic probems ug the boundary eement method has a computationa compexity of the order ON. This makes the computation very sow as the number of nodes increase. A new technique caed fast mutipoe method FMM has emerged in the ast decade. Repacing the norma matrix-vector mutipication with the fast mutipoe method reduces the computationa time to order ON. In this thesis the fast mutipoe method has been used to acceerate the boundary eement method for -D acoustic wave probems. The reevant formuae are derived. It is shown that the computationa time is of the order ON for this formuation. It is aso observed that the memory required is much esser and hence arger modes can be soved. The formuation is a very basic one and gives good resuts as shown by the numerica exampes. Use of higher-order eements and hyperguar formuation wi resut in a very capabe and robust sover in the future.

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5 Acknowedgements First and foremost, I woud ike to thank Dr. Yiun Liu for his abe guidance and support in this work. This work coud be competed ony through his suggestions and hep. I woud ike to thank Dr. ay Kim for his hep in topics reated to the fied of acoustics. I aso thank Dr. Kumar Vemaganti for serving on the committee and his suggestions which have made the report better. I woud aso ike to express my gratitude to my coeagues at the CAE Laboratory, Liang Shen and ian Sun for their insights into the topic of fast mutipoe methods. I woud aso ike to thank Niesh Biade, a coeague and friend for his hep in presenting this thesis. I aso thank my other coeagues Manas Phadke, Shardoo Chirputkar and Nimish agtap for their hep. I woud ike to express my appreciation for the necessary infrastructure provided by the CAE aboratory, OCC aboratories and the ibraries. I am gratefu to my friends, Kapak Gatne, Aniruddha Pasue, Ait Pharne, Rahu Patki, Amit Kukarni and Koustubh Ashtekar, without whose support this work woud not have been competed. Last but not the east; I thank my parents and my sister who have stood by me in this venture. I aso appreciate each and every contribution made to this thesis by everyone whose names I coud not mention here.

6 Tabe of Contents List of Figures... List of Tabes...5 Chapter 1 Introduction Introduction The Boundary Eement Method BEM BEM for the Acoustics Probem Fast Mutipoe BEM for Acoustics Probems Structure of This Thesis... 9 Chapter Conventiona Boundary Eement Method for the Acoustics Probem Introduction Hemhotz equation and BIE Formuation for -D Acoustics BEM Formuation with Constant Eement Soution to the System of Equations... Chapter 3 Fast Mutipoe Method Introduction Fast Matrix-Vector Mutipication Iterative Sovers and Fast Matrix-Vector Mutipication GMRES Method Fast Mutipoe Method Mathematica Formuation for FMM... 9 a. Decomposition for G kerne

7 b. MM Transation c. ML Transation... 3 d. LL Transation e. Transations for the F kerne The Fast Mutipoe Method Agorithm Appication to the BEM Probem Chapter Code Deveopment C Sover for Conventiona BEM C GMRES Sover Tree traversa in FMM BEM Sover a. Agorithm for creation of the tree... 0 b. Agorithm for upward pass... 1 c. Agorithm for downward pass Brief overview of the main casses and their functions for FMM BEM... 1 Chapter 5 Resuts Resuts for -D Acoustics probems... 3 a. -D Acoustics Interior Domain Probems... 3 b. -D Acoustics Exterior Domain Probems... 6 c. -D Acoustics Scattering Probems Comparison with the conventiona BEM code... 5 Chapter 6 Discussions and Future Work...5 References...56

8 Appendix A. Derivation of the Mutipoe Coefficients and Functions...6 Appendix B. Code isting for the FMBEM sover

9 List of Figures Figure 1 Point P inside the boundary Figure Point P on the boundary S surrounded by a sma semicirce Figure 3 A boundary S divided into m constant eements Figure A constant eement... 1 Figure 5 Mutipe node to node interaction... 8 Figure 6 Singe ce to ce interaction... 8 Figure 7 Decomposition for G kerne Figure 8 MM Transation Figure 9 ML Transation... 3 Figure 10 LL Transation Figure 11 Domain ened in a box Figure 1 Division of any ce... 3 Figure 13 Tree structure for the domain Figure 1 Interior Acoustics Probem... Figure 15 Resuts for an interior acoustics probem... 5 Figure 16 Exterior Acoustics Probem... 6 Figure 17 Exterior acoustics probem... 6 Figure 18 Pressure Pot for Exterior Acoustic case k Figure 19 Resuts for an Exterior acoustics probem... 9 Figure 0 Resuts for an acoustic scattering probem Figure 1 Computationa time for the FMM sover... 51

10 List of Tabes Tabe 1 Comparison of the vaues aong the centerine of the duct..1 Tabe Convergence of the pressure vaue at the center..1 Tabe 3 Comparison of the vaues aong a radia ine of the cyinder. 5

11 Chapter 1 Introduction 1.1 Introduction In many engineering probems, the soutions to partia differentia equations PDEs are sought. The boundary conditions are specified on the physica boundary of the domain in case of some probems. These are caed the boundary vaue probems BVPs. The process of finding the soutions anayticay to BVPs becomes increagy difficut with the increag compexity of the given geometries and boundary conditions. Many numerica techniques have been devised to sove these probems. Finite eement method FEM and Finite Difference Method are two we-known methods. Boundary eement method BEM, based on the boundary integra equation BIE is another numerica method for soving BVPs. Engineering probems in acoustics, soi mechanics, and fracture mechanics can be soved ug the boundary eement method. The method is very eegant and gives very accurate resuts. The conventiona way of soving the BEM probems takes considerabe computationa time. The introduction of fast mutipoe method FMM, for soving the BEM probems has acceerated the cacuation time. The fast mutipoe boundary eement method FMBEM is very robust in nature and can sove very arge modes. The method aso consumes esser memory and has an accuracy simiar to the conventiona approach. 1. The Boundary Eement Method The boundary eement method consists of soving an integra equation that is mathematicay equivaent to the origina partia differentia equation PDE. This integra equation is termed the boundary integra equation BIE. BIE is defined on the spatia 6

12 boundary of the domain aone. To obtain the soution to the integra equation for a compex geometry, the boundary is discretized into eements. On expresg the integra equation for each eement a inear system of equations is obtained. Some basic work of integra equation techniques was done by Fredhom [1]. Mushkeishivie [] studied the soutions to guar integra equations. Mikhin [3] studied the appication of integra equations to probems in mechanics. Kupradze [] discussed the soution to potentia probems ug the integra equations which used potentia theory by Keogg [5]. aswon [6], aso independenty appied integra equations in the potentia theory. Cruse and Rizzo [7] appied this method to eastostatics. This approach is simiar to the current approach or sometimes caed Direct method. The boundary eement method emerged as a computationay viabe technique during the 1970 s and has been deveoped substantiay in the foowing years. Since, ony the spatia boundaries are invoved for soving, a N-1 dimension mode of the spatia/physica boundary needs to be created for soving the same probem, where an N dimension mode is created for FEM. Aso it can be appied to infinite domains. This is the main advantage of BEM. Today the BEM has found appications in many fieds of computationa mechanics, such as the wave propagation, heat transfer, diffusion and convection, fuid fow, fracture mechanics, eectric probems, geomechanics, pates and shes, ineastic probems, contact probems, design sensitivity and optimization probems. 1.3 BEM for the Acoustic Probem The governing equation for the acoustic probems is the Hemhotz equation. The eariest attempts to sove acoustic probems ug BEM were by Chen [8]. He discussed the 7

13 radiation from an arbitrary body in an infinite medium. Chertock [9] discussed the method for predicting the fied generated by a body, on whose boundary the pressure is known. Copey [10] discussed the radiation probem for vibrating bodies. The above formuations do not correcty predict the fied at certain frequencies caed the eigen frequencies. Schenk [11] improved this formuation for soving the probem at eigen frequencies. This approach is widey accepted as the CHIEF method. Shaw [1] appied the deveoped method to some ocean engineering probems. Burton and Mier [13] proposed a hyperguar formuation for the probem of eigen frequencies. Shaw and Tai [1] made a study of the modes of these eigen frequencies. Beyond this point a ot of research has been carried out and the scope of these appications has been expanded. 1. Fast Mutipoe BEM for Acoustic Probems FMM finds its roots in tree-based agorithms. Barnes-Hut agorithm [15] is one such agorithm and is very ey reated to the FMM. These tree-based agorithms were originay deveoped to dea with mutibody probems in astrophysics or simuations in moecuar dynamics etc. Rokhin [16] first introduced FMM to sove the Lapace equation. Rokhin used a binary tree structure. Greengard [17] appied this method to mutibody probems. He aso introduced the use of quad-trees for soving -D probems. The use of a quad-tree for -D probems can be compared to the Barnes-Hut agorithm. Mammoi & Ingber [18] appied fast mutipoe BEM ug an indirect BEM for soving Stokes fow around cyinders. Yoshida [19] appied the FMM to sove crack propagation probems. FMM has been appied to a variety of probems. A ot of different aspects of the fast mutipoe method have aso been studied so far. Rokhin [0] first appied the FMM 8

14 method for soving the Hemhotz equation. Fukui [1] aso soved the Hemhotz equation ug FMM. Fukui and Rokhin discussed the formuation for 3-D acoustic probems. Simiar attempts were made by Zhao and Chew [] to sove the -D acoustic probem. Nishimura [3] reviewed the works ti then and discussed these methods. Chen [] used a hyperguar formuation in his FMBEM code to sove -D acoustic probem. 1.5 Structure of This Thesis The remaining chapters of this thesis are as foows: In Chapter, the boundary integra equation for the -D Acoustic probem is formuated. The BEM formuation for the probem is provided ug constant eements. Briefy the iterative methods for soving the probem are discussed. In Chapter 3, the fundamentas of the fast mutipoe method are discussed. The use of fast mutipoe method in association with iterative sovers is expained. The fast mutipoe method formuation for the -D acoustic probem is deveoped. In Chapter, a C code for the fast mutipoe method is discussed. In Chapter 5, the numerica resuts obtained by ug the code are presented. In Chapter 6, the thesis concudes with discussions and the scope for future work. 9

15 Chapter Boundary Eement Method for Acoustic Probem.1 Introduction The phenomenon of the sound wave propagation has been studied in depth for many engineering appications. The starting point for most discussions is the Hemhotz equation. In the foowing section, a brief derivation of the Hemhotz equation is provided. A sound wave pasg through any point in a compressibe fuid creates a change of pressure at that point given by: φ T φ 0 φ 1 φ T is the tota pressure φ 0 is the ambient vaue of pressure φ is the acoustic pressure i.e. change produced in the pressure If c is the veocity of the wave, V the specific voume, ρ the density of the compressibe fuid, and ρ 0 the density of the fuid at that point, then the continuity equation is given by φ ρ c t V 0 The Euer s equation is given by 0. V ρ0 φ. 3 t Combining equations and 3 resuts in 10

16 1 1 φ ρ 0 φ 0 ρ0 c t Assuming a homogenous media, equation can be written as 1 φ φ 0 5 c t Considering that time harmonic waves are being soved, the pressure can be expressed as φ ~ φe ω i t 6 where ~ φ is the ampitude of the wave a function of space and frequency On substituting equation 6 into equation 5, the foowing resut is obtained ~ ~ φ k φ 0 7a where 7b k ω is the wave number. c This is the Hemhotz equation. Here k is the wave number and φ ~ the pressure at any point. In the foowing discussions, φ ~ wi be repaced by φ.the equation 6 can be appied to find the pressure at any given frequency.. Hemhotz Equation and BIE Formuation for -D Acoustic It was shown in the previous section that for a homogenous medium the mathematica formuation of the acoustic probem in frequency domain is given by the Hemhotz equation φ k φ 0 7a where φ is the acoustic pressure. 11

17 Now anaogous to the potentia probem, we can say that the fundamenta soution G wi be given by the equation G k G δ P, P 8 0 In this case the point P 0 is a guar point source and δ is the Dirac-deta function. G P, P is a function of two points and represents the pressure fied at point P due to a 0 source at point P 0. In a very basic formuation we wi consider a symmetrica radia soution to this equation i.e. G varies ony with respect to r in poar co-ordinate system in -D Rewriting the equation 8 for the entire pane except P 0 in poar co-ordinate system d G 1 dg k G 0 dr r dr 9 Substitute χ kr, we obtain the standard Besse equation of order zero d G dg χ χ χ G 0 10 dχ dχ The genera soution for the above equation, in terms of the first kind and second kind 1 Hanke functions of the nth order H and H, can be written as n 1 G AH kr BH kr 11 o o n Wu [6] has discussed the method to find the constants A and B. Integrating the Hemhotz equation over a tiny circuar area Ω bounding P 0 and appying divergence theorem, the equation 8 can be written as 1

18 ψ imω 0 dγ 1 n Ω 1 Wu [6] has shown that A0 and B-i/ Hence the fundamenta soution is given by i G H 0 kr 13 The norma derivative of the fundamenta soution is given by G n ik r H1 kr 1 n Ug these fundamenta soutions, the boundary integra equation for the probem is derived. The Green s second identity as shown beow wi be appied: [ u v v u] dv V S v u u v ds n n 15 The equation 8 for fundamenta soution can aso be written as G k G δ P, P Rearranging the terms, G k G δ P, P 17 0 Rearranging the terms of equation 7, φ k φ 18 Substituting u φ and v G in equation 15, [ φ G G φ] dv V S G φ φ G ds n n 19 13

19 Substituting the equations from 17 and 19, [ φ k G δ P, P0 G k φ ] dv V S G φ φ G ds n n 0 Hence, [ φ k G δ P, P0 Gk φ] dv V G φ φ G ds 1 n n Hence, [ φk G φδ P, P0 Gk φ] dv V Therefore, [ φδ P, P ] dv V S G φ φ G ds n n S G φ 0 φ G ds 3 n n S By the definition of Dirac-Deta function φ P G P, P0 φ P 0 G P, P0 φ P ds P n n S This can be written in a simpified form as [ Gq Fφ] φ P 0 ds 5 where S q φ and n F G n The equation 5 has converted a the terms in the equation to boundary vaue terms. In a we posed boundary vaue probem ony one of q or φ or a reating equation is given. Hence this equation cannot be used directy. To find the boundary variabes first we 1

20 consider a point P 0. Now we sha consider the imiting case as the point is moved to the boundary Fig. 1. Figure 1. Point P inside the boundary In other words, the point P 0 can be assumed to be ying on the boundary and a sma semicirce S ε is assumed Fig.. Figure. Point P on the boundary S surrounded by a sma semicirce The imit is found as ε tends to 0 Fig.. Appying to the equation 5 we have P S S [ Gq Fφ] imp S φ P im ds Evauating the first term on the right hand side first, 15

21 P S P, P0 q P ds S im G im G P, P0 q P ds 0 ε 7 0 S Separating the integra into two domains imε P, P0 q P ds imε 0 G P, P0 S Sε Sε S Sε and S ε 0 G q P ds 8 Substituting the equations 13 and 1, we have i 0 G P, P0 q P ds imε 0 H 0 kr q P rdθ 9 imε S Sε Sε π i 0 0 H 0 kε q ε S Sε 0 imε G P, P0 q P ds imε εdθ 30 But im εh kε 0 31 Therefore, ε 0 0 P S P, P0 q P ds S im G im G P, P0 q P ds 3 0 ε 0 S Sε No additiona term is added for the domain S ε i.e. when we reach the boundary. That is, there is no ump term here. Consider the second term on the right hand side P S P, P0 P ds S im F φ imε F P, P0 P ds 33 0 S 0 φ Again separating the integra into two domains imε P, P0 φ P ds imε 0 F P, P0 S Sε Sε S Sε and S ε 0 F φ P ds 3 Substituting the equations 13 and 1, 16

22 ik r 0 F P, P0 φ P ds imε 0 H1 kr φ P rdθ 35 n imε S Sε Sε im π ik ε 0 F P, P0 φ P ds imε 0 H1 kε φ P S Sε 0 εdθ 36 But i imε 0 εh1 kε 37 kπ Putting this vaue in equation 36 we have im π ik i ε 0 F P, P0 φ P ds imε 0 φ P kπ S Sε 0 π φ P π S Sε 0 imε F P, P φ P ds imε dθ dθ imε 0 F P, P0 φ P ds φ P0 0 S Sε This is the ump term. The integra is not continuous at P 0 when P0 is on the boundary. On substituting equations 3 and 0 in equation 5 the resut is [ G P, P0 q P F P, P P ] C P φ P ds P φ S Here 1 C P 0 if P 0 is on the boundary C P 0 1 if P 0 is inside the domain. This equation is appicabe for both interior and exterior probems. In the case of scattering probems an additiona term for the incident wave is introduced. 17

23 [ G P, P q P F P, P φ P ] I C P0 φ P0 0 0 ds P φ P0 S I where φ is the incident wave..3 BEM Formuation with Constant Eements Consider a boundary S Fig.3. Let it be divided into m constant eements. Constant eements are defined as those on which both the potentia function φ and its norma derivative q is constant over the entire eement []. Figure 3. A boundary S divided into m constant eements Assuming S to be divided into m constant eements On any eement S i, φ φ i a constant q q i a constant The boundary integra equation 1 is used to derive the formuation. For every source point on every eement S i, we have 18

24 [ Giq Fiφ ] C φ ds 3 i i S First term on the right hand side is evauated as: S G qds i m 1 S G qds i GiqdS q S S G qds i m 1 m 1 q S g G ds i i 5 6 where g G ds 7 i S i Simiary the second term can be evauated as S S S m Fφ ds FφdS 8 i 1 S m i φ Fφ ds F ds 9 i 1 m 1 S i Fφ ds φ fˆ 50 i where i fˆ F ds 51 i S i Thus equation 3 can be written as 19

25 m 1 m c φ q g φ fˆ 5 i i i 1 The equation 5 is modified ug the term i f i c δ fˆ 53 i i ii The fina equation becomes m 1 q g i m 1 φ f 5 i Putting in terms of matrices, we obtain: g g : g 11 1 m1 g g g 1 : m.... :.. g g g 1m m : mm q q : q 1 m f f : f 11 1 m1 f f f 1 : m.... :.. f f f 1m m : mm φ1 φ : φm 55 This equation is used for soving the unknown boundary vaues at the nodes. Evauation of f i and g i cannot be done anayticay and is done ug Gaussian quadrature. On the other hand, f ii and g ii are guar in nature and their vaues need to be found out. Consider an eement of ength R i Fig.. The Cauchy Principa vaue CPV is taken as the vaue of the integra. 0

26 R i Eement S i Source point Figure. A constant eement Now, g G ds 56 ii S i To find the CPV, i gii ε 0 i Si Sε im G ds 57 i g ii imε 0 H 0 kr ds 58 Si Sε g ii R i imε ε i 0 H 0 kr ds 59 [ S kr H kr S kr H kr ] R H kr Ri g ii π i 1 i 1 i 0 i 0 i 0 i 60 where S n denotes the Struve function of the nth order. This vaue is used in the BEM formuation. Simiary for finding f ii, 1

27 fˆ F ds 61 ii S i i ˆ F ds 6 fii imε 0 i Si Sε ik r f 0 kr ds 63 n ˆ ii imε H1 Si Sε f ˆ ii 0 6 f ii c fˆ 65 i ii f ii. Soution to the System of Equations The equation 55 is used for soving the matrix equation g g : g 11 1 m1 g g g 1 : m.... :.. g g g 1m m : mm q q : q 1 m f f : f 11 1 m1 f f f 1 : m.... :.. f f f 1m m : mm φ1 φ : φm In a we-posed boundary vaue probem, one of the vaues of q or φ is known for every boundary node. In the soution for such a BVP, a the unknown vaues are shifted to the eft and the known vaues to the right. This converts the above system into a matrix system Ax b where x is the vector of a the unknowns. A is unsymmetrica and aso dense. The soution to x can be found ug various sovers. Direct sovers ike Gaussian eimination give high accuracy. These sovers take very arge time though ON 3. Iterative sovers can aso be utiized to sove the system. The accuracy of these sovers is controed by a toerance. These sovers are much faster and have an efficiency of ON.

28 Chapter 3 Fast Mutipoe Method 3.1 Introduction The BEM formuation of the -D acoustic probem was discussed in the earier chapter. The output of the numerica formuation is a matrix equation of the form Ax b. Soving this system ug techniques ike Gaussian eimination, the time taken for computing the soution varies directy with the cube of the number of degrees of freedom i.e. ON 3. The memory requirement is aso directy proportiona to the square of the number of degrees of freedom ON. One can reduce the time taken by the sover by ug iterative schemes ike Gauss-Siede, BiCG, and GMRES etc. By ug iterative schemes the time taken is reduced to the order ON. But the memory requirement variation stays of the order O N. The memory requirement for each scheme changes as the overheads are different for each scheme. The fast mutipoe method, used to carry out matrix vector mutipication amidst the usage of the iterative sover reduces the soution time to the order of ON. The memory requirement is aso smaer ON.. 3. Fast Matrix-Vector Mutipication Matrix-vector mutipication is an essentia component of any iterative scheme which finds soution to the system Ax b. Setting up the matrix-vector mutipication in the simpest form, an agorithm can be written ike the one beow: Consider a matrix A, which has to be mutipied by a vector x Let v be the resutant vector. 3

29 for i 1:1:n for 1:1:n v i v i A i,x; end end Thus there are two nested oops. The number of operations performed in this cacuation is proportiona to n ON. But if there is knowedge about the matrix the cacuation can be done ug agorithms which invove fewer cacuations. For exampe if the matrix is a diagona matrix the agorithm can be written as: for i 1:1:n v i A i,i x i; end Simiary if the matrix A can be given by A i, y i -z where y and z are two vectors. Then, A i,x y i -z x y i x - y i z x z x Therefore, v i x yi z x y i z ϕ vi Pyi Qyi R Thus every vi can be written as a function of yi and three constants P, Q and R. The agorithm can be written as :

30 for i 1:1:n PPxi; QQzixi; RRzi xi; end for i 1:1:n vi Pyi -QyiR; end There are two oops here but they are not nested. The number of operations performed in this cacuation is proportiona to n ON. Note that in the cacuation Ai, is not used and hence the evauation of the entire matrix is not required. The vectors x and y are used instead of the matrix A. Fast mutipoe method is one agorithm to do the fast matrix-vector mutipication. The above exampe of fast matrix-vector mutipication has some key concepts. In genera they can be summarized as : 1 The matrix A is not present in the cacuation. A is expressed as a function of two vectors y and z. In genera, the matrix can be written as a function of one or more vectors. 3 In the resutant vector v, every vi can be written as a function of yi aone and does not depend on any other term from y. 5

31 3.3 Iterative Sovers and Fast Matrix-Vector Mutipication Anaytica or exact methods for finding soution to the system Axb require a ong time to get resuts ON 3. Iterative or numerica methods to sove the same system were devised to reduce this arge computationa time. Many iterative schemes have been deveoped. A the schemes can be expained by the foowing steps: 1 Start with an initia guess for the soution vector x say x 0 Compute the residua vector r 0 Ax 0 -b. 3 Create another vector δx fr. The computation for the vector δx is based on the iterative scheme. A common way of doing this is Mδx r. Here M is a matrix, caed the preconditioner matrix. If M is identica to A, a ge iteration is required to sove the probem. In genera, the preconditioner matrix M is simiar to A. Update the soution vector x n x n-1 δx 5 Find the residua vector r n Ax n -b 6 If r n ies within the given toerance, x n is the soution, ese go back to step 3. In the above method at every step the computation Ax n is performed. The mutipication of matrix and vector generay takes ON operations. A the other computations are of the nature ON. Hence if the matrix-vector mutipication matrix can be done by ug an agorithm which takes ON steps, the entire operation of soving the system becomes an ON process. 3. GMRES Method In the code for soving the -D Acoustic probem, the iterative scheme caed Generaized Minima Residua Method GMRES has been empoyed. Saad and Schutz [7] deveoped 6

32 this method for soving non-symmetric inear systems. The GMRES method is based on the Kryov subspaces. The residuas r 0, Ar 0, A r 0 form an orthogona space span. In the code deveoped in this research, the GMRES agorithm which has been used is based on the agorithm deveoped by Barrett et a [9]. A arge amount of memory is required as it invoves storing the residua at each step. The GMRES agorithm uses restarts for this reason. The number of iterations required for convergence is not very arge for systems which are positive definite or neary positive definite. The agorithm for GMRES is as foows 1 Start with an initia guess x 0. Compute the residua vector Mr 0 Ax 0 -b 3 The basis v is formed expicity ug Gram-Schmidt orthogonaization. This is aso caed Arnodi process. Vector x is created ug this basis. A east square technique is appied to update the vector x. Check for convergence and continue. 3.5 Fast Mutipoe Method The fast mutipoe method was deveoped by Greengard and Rokhin [] to do fast matrixvector mutipication method. The agorithm fas into the wide category of tree-based agorithms. Agorithms ike Barnes-Hut were deveoped to sove muti-body gravity probems. Agorithms ike fast Fourier transform etc. were deveoped to sove probems in vibrations etc. A of them invove creating a tree and doing common cacuations higher up in the tree structure. Fast mutipoe method aso is based on a simiar concept. The points nodes are stored in a tree structure in a method which is based on the spatia ocation. A 7

33 number of points are stored in one ce. Every point to point interaction is no onger necessary here. Ce to ce interactions and ce to point interactions are performed. Consider n points Fig. 5. The tota number of interactions in this case is nn-1/. Figure 5. Mutipe node to node interaction Now if they are divided spatiay into sets with n1 and n points respectivey, there are n1 ce to point interactions in the first ce and n ce to point interactions in the second ce. The number of ce to ce interactions is 1. The interactions thus reduce to nn11. This is the ge eve formuation Fig. 6. Figure 6. Singe ce to ce interaction 8

34 On further ening mutipe ces into arger ces hierarchicay, the number of interactions gets reduced even more significanty. This is the muti eve formuation. To use the fast mutipoe method, the node to node interaction must be decomposed into equivaent combination of ce to ce interaction and point to ce interactions. This depends on the system i.e. nature of the matrix A. The important mathematica requirements are as beow: 1 Far fied expansion shoud exist. Near fied expansion shoud exist Duraiswami [8] has isted the other requirements for doing the fast mutipoe method mutipication. 3.6 Mathematica Formuation for FMM As discussed in Chapter, the numerica formuation of the probem eads to a matrix equation GqFφ. Based on the prescribed boundary condition the system is converted to the form Axb. a. Decomposition for G kerne The G kerne for the -D Hemhotz probem is given by i G H 0 kr. The G kerne can be spit ug Graf s equation Refer Appendix A. [ O r I r O r I r ] i G

35 Figure 7. Decomposition for G kerne Here, 1 poar ange of z 0 with respect to z c poar ange of z with respect to z c r 1 distance between z 0 and z c r distance between z and z c O r H kr O r H kr I r kr 69 I r kr 70 But each term of the matrix is GqdS So Therefore, the mutipoe expansion is So i GqdS O r1 M kr O r1 M kr 71 where, 30

36 M r kr qds 7 So M r kr qds 73 So These two are the two formuae for the two sets of moments. The points z 0 and z are separated by the introduction of the point z c. The moments in the equations 7 and 73, need to be computed ony once. This is simiar to the cacuation of P, Q and R in the exampe in section 3.. b. MM Transation Now consider that the coocation point has changed from z c to z c'. Figure 8. MM Transation The formuae for MM transations are given by Refer Appendix A: 31

37 M M kr M r kr M r r 1 7 kr M r kr M r r 1 where, poar ange of z 0 with respect to r distance between z 0 and z c' 75 z c' c. ML Transation Consider a point z which is e to the source point. Figure 9. ML Transation The formuae for ML transation are as derived in Appendix A L L H kr M r H kr M r r H kr M r H kr M r r

38 where, 6 poar ange of z 0 with respect to z 5 poar ange of z with respect to z c r 6 distance between z 0 and z r 5 distance between z and z c The ML transation converts the expression of a matrix vector product as far fied expansion to that as a near fied expansion. The near fied formuation can be used to compute the product. d. LL Transation Now consider a point z ' which is e to the source point than z. The formuae for LL transations are: Figure 10. LL Transation 33

39 L L kr L r kr L r r kr L r kr L r r where, 7 poar ange of z ' with respect to z 8 poar ange of z 0 with respect to r 7 distance between z ' and z r 8 distance between z ' and z 0 79 z ' Both, MM and LL transations are done to utiize the hierarchica tree structure. The use of tree structure is a key aspect of the FMM. e. Transations for the F kerne The formuation for the transations for the F kerne is simiar to that of G kerne. The detaied formuae for the F kerne are derived in Appendix A. 3.7 The Fast Mutipoe Method Agorithm The fast mutipoe method agorithm can be described as foows: 3

40 1. Creation of the tree: The entire domain is ened in a square box. Fig. 11 Figure 11. Domain ened in a box This is aso the root eve ce or eve 0 ce. This ce is subdivided into four more equa ces Fig. 1. Figure 1. Division of any ce These are the eve 1 ces. Every subdivided ce is said to be a chid ce of the origina ce. The origina ce is said to be the parent ce of these chid ces. A the ces of eve 1 which contain eements are subdivided into ces at eve. A ce is said to 35

41 contain an eement if the midpoint of the eement ies within the ce. Every domain has to be subdivided ti eve because ML interactions cannot be performed on eve 1 ces. Every ce which contains eements is continued to be subdivided into four ces unti each one has fewer eements than the preset maximum number of eements per ce Fig. 13. A ce is said to be a eaf ce if the ce has no chid ces. Figure 13. Tree structure for the domain. Upward pass: In the next step the upward pass is performed. The moments are cacuated for every eaf ug the direct formua for M equations 7 and 73. The moments of a ge eaf is the sum of the corresponding moments of a the eements beow it. The moments for the other ces are carried out by doing the MM transation 36

42 equations 7 and 75. For every ce the moment is the summation of the MM transations of its chid ces. These moments are found for a ces whose eve. 3. Downward Pass: The actua product of Ax is cacuated in this step. The ces of eve which share at east a common vertex are said to be adacent ces at eve. ML transations cannot be performed on adacent ces. Two ces whose parents are adacent but the ces themseves are not adacent are caed we separated ces. The ist of a we separated ces of a particuar ce is caed the interaction ist of that ce. ML interactions can be carried out with each ce of the interaction ist. At eve, the downward moment is computed by adding the corresponding moments formed from a of the ces in the interaction ist of that ce. From eve 3 onwards, the LL interaction with its parent is added to this sum to get the downward moment at that point. This process is carried out ti one reaches the eaf.. Evauation of the integra: The near fied formua is used to evauate the integra inside the eaf. The direct formua for computation of the product Fq or Gφ is used for eements in the eaf and eements in the adacent ces. The sum is obtained by adding the near fied expansion to the summation from the direct formua. 3.8 Appication to the BEM Probem The process of finding the matrix-vector product is carried out for both the F and G kernes. The resutant vector is obtained as the difference v Gq-Fφ. The known quantities are input in the first case and the approximation x is used in the next case. This vector v is passed back to the GMRES sover. The resutant vector v is used to find the next approximation to the soution vector for the next iteration. Use of preconditioner acceerates the process of finding the soution to the equation system. In the code 37

43 deveoped, the use of a bock acobi preconditioner is empoyed. Eements in one eaf are used to form a ge bock in the bock matrix. The vector v is cacuated ce by ce. The use of preconditioner can be done immediatey after finding the vaue of the components of v for each ce eve. 38

44 Chapter Code Deveopment.1 C Sover for Conventiona BEM The code for both the methods, conventiona BEM and FMBEM, has been written ug the C anguage. C is a midde eve anguage and is usefu for writing code invoving data structures and mathematica cacuations. Most of the mathematica cacuation routines are standard ibrary functions. Evauation of Besse functions has been done ug standard ibrary code avaiabe on The input is provided to the sover in the form of a fie input.dat. The points inside the domain, for which the pressure is to be found out, are input as points.dat. The sover writes the output in a fie output.dat.. C GMRES Sover A critica part of the sover is the iterative sover. The iterative scheme used is the GMRES. The GMRES sover used in this particuar code has been downoaded from Some of the routines have been modified to match the overa fow of the sover deveoped in this work. This is possibe because the GMRES sover code uses the paradigm of cpp tempates. The above sover was created at the Oak Ridge Nationa Laboratory at Tennessee..3 Tree Traversa in FMM BEM Sover 39

45 Fast mutipoe method is a tree based agorithm. The agorithm starts with creation of a tree. The upward and downward passes invove tree traversas. The code for the tree data structure is contained in the cass tree and cass treenode. Very simpe routines for creating, traverg and deeting the tree are incuded in the code for these two casses. a. Agorithm for Creation of the Tree The agorithm for creating the tree is based on the seria input of the eements to the tree creation ogic. This fact can be used for extending the tree if additiona eements are added. The agorithm starts with creation of a root ce. This ce corresponds to the bounding box of the entire given boundary. A quad-tree is used and hence every parent has four chid ces. Each chid ce corresponds to a eve 1 ce. The agorithm for fast mutipoe method requires at east eve two ces. The four chid ces are created for the root. Chid ces are created further for the eve 1 ces. The initia subdivision of the tree ces into chid ces stops at eve. The agorithm for every new eement is as beow: 1 The coordinates of the new eement are read. The eve ce for the eement is determined. Insert the eement in that ce s eement ist if it is a eaf. A eaf is a ce with no chid ces. If the ce has chid ces determine the chid ce for the eement. Repeat this procedure ti the eement is paced in a ist. 3 If the eements in the eaf exceed a preset maximum, the eaf is subdivided into four chid ces. A the eements are then distributed in these next eve chid ces. The parent ce s eement ist is emptied. This process is repeated ti every chid ce has fewer eements than the preset maximum. 0

46 b. Agorithm for Upward Pass The tree traversa agorithm is a simpe recursive agorithm. The cass momentinfo stores the upward moments for each ce. c. Agorithm for Downward Pass The tree traversa agorithm is aso a simpe recursive agorithm. The cass momentinfo stores the downward moments for each ce aso. The interaction ist of each ce is stored. This heps in expediting the cacuation. Simiary the ist of adacent ces for every ce is aso saved. The adacent ces may be at a different eve. The cacuation for such cases is generay engthy and storing this ist aso expedites the sover code.. Brief Overview of the Main Casses and Their Functions for the FMM BEM 1 Cass tree This cass is the container cass for the tree data structure. Agorithms for tree traversa use this cass. Cass treenode This cass contains the information about each ce ike its coordinates, its chidren etc. 3 Cass momentinfo- This cass is the data structure used for storing moments. The code for a the transations are functions of this cass. Cass eement This cass contains information about the constant eement used in each case. 5 Cass node This cass is the data structure for the BEM node. 1

47 6 Cass Preconditioner1 The cass has the code for preconditioning for the GMRES sover. 7 Cass Vector1 and Cass Operator1 These casses contain the impementation for the GMRES sover. 8 Besse1, Besse, HankeSecond, Gaustegrator and Struve are casses which do some mathematica operations. These casses have been created to have easier readabiity for the code. 9 The fie main.cpp has the routines for reading the input fie, writing the output fie. It aso maintains the overa fow for the program. 10 Gmres.h, yn.c, n.c are standard ibrary fies.

48 Chapter 5 Numerica Resuts 5.1 Resuts for -D Acoustic Probems a. -D Acoustic Interior Domain Probems To test the accuracy of the BIE formuation and the codes, a very simpe test case has been utiized. The test is that of a cross section of an acoustic duct. The geometry of the duct is square with side ength 1.00 Fig. 1. The boundaries parae to x axis are the was of the duct. In the input for the sover a mixed boundary condition is defined. Veocity boundary conditions are prescribed on the two was of the duct. Pressure boundary conditions are prescribed on the other two boundaries. 1.0 q0 1.0 φ 1000i φ 0 100i 0,0 q0 Figure 1. Interior Acoustics Probem The anaytica soution for this probem is a simpe 1-D soution of the nature: 3

49 ikx φ φ 0 e φ The numerica resuts for both the conventiona code and FMBEM code agree very we with this anaytica vaue. The vaues for pressure at 9 equay spaced points aong the centerine of the duct are found out. Here we compare the vaue of rea part of the pressure ug both the codes with the anaytica vaue, for the wave number k The error for these vaues is cacuated. The number of nodes used is 00. It is found that the L norm of the error for the conventiona code is whie the L norm of the error for the FMBEM code is The errors in both the cases are of the simiar order. Tabe 1 Comparison of the pressure vaues aong the centerine of the duct Conventiona Point Anaytica FMBEM BEM code % error Vaue code vaue x y vaue % error % % % % % % % % % % % % % % % % % % The study for convergence of the resuts has aso been conducted. The same probem has been modeed with different number of nodes. The vaue of the rea part of the pressure at

50 the center of the square is evauated. In the tabe beow, the % error with the increag number of nodes is shown. The accuracy increases with the increag number of nodes. Tabe Convergence of the pressure vaue at the center No. of nodes Conventiona FMBEM BEM code % error code vaue vaue % error % % % % % % The interior pressure distribution for a constant vaue of φ 100 and varying vaues of k1.5707, , 39.5 is potted Fig. 15. For the case of k , the resutant domain has a quarter waveength. This has been caed the Quarter Waveength Probem in the foowing discussion. a Interior Acoustic Probem for k

51 b Interior Acoustic Probem for k c Interior Acoustic Probem for k39.5 Figure 15. Resuts for an interior Acoustic probem b. -D Acoustic Exterior Domain Probems The next case considered is an exterior domain probem. The boundary conditions are specified and the vaues of pressure are found in the exterior domain. 6

52 φφ 1000i 0,0.0 Figure 16. Exterior Acoustics Probem The numerica exampe uses a pusating cyinder of radius 1.0. The anaytica soution for the probem is φ φ0h 0 kr This soution is in poar coordinates. A constant pressure condition is prescribed in the probem Fig. 16. The numerica resuts agree with the anaytica resuts. In this case we use a pressure condition of φ 100 on the entire boundary. The vaue of pressure is cacuated at some points in the domain for the vaue of wave number k1. The pot is ony of a region of the infinite domain. Figure 17. Exterior acoustic probem: Cyinder in an infinite domain 7

53 The probem has been soved ug both conventiona BEM and FMBEM code. In the tabe beow the cyinder is modeed ug 100 nodes. A wave number of k1 has been used. Soving the case anayticay we obtain φ * i The comparison of the vaues is given in tabe 3. Tabe 3 Comparison of the vaues aong a radia ine of the cyinder Point Anaytica vaue FMBEM code vaue Conventiona BEM code vaue x y Rea part Imaginary Imaginary Imaginary Rea part Rea part part part part The pot for the above tabe is as beow: 8

54 Figure 18. Pressure Pot for Exterior Acoustic case k1 The -D pressure variation in the externa domain has been potted for different vaues of k, 5 and 10 respectivey Fig. 19. a Exterior Acoustic Probem k 9

55 bexterior Acoustic Probem k5 cexterior Acoustic Probem k10 Figure 19. Resuts for an Exterior Acoustic probem c. -D Acoustic Scattering Probems A scattering probem by a cyinder has been considered in this discussion. The cyinder serves as an obstace to the pasg pane wave. The veocity condition is prescribed on the cyinder boundary q 0. The incident pressure wave is a traveing wave given by the I ikx equationφ φ 0 e. The wave number k0.5 The use of conventiona BEM sover and FMBEM sover was done to find the soution. 50

56 The comparison has been done with the anaytica soution for the pressure at a fied point, 0. The anaytica vaue is given by [50]: 1 I n n ka H n kr φ φ 1 ε ni nφ n0 1 H n ka where a radius of the cyinder and ε n 1 for n0 otherwise. The vaue by the conventiona BEM sover and by the FMBEM sover compares we with the anaytica vaue of at the point,0. The pressure for the fied points for the same probem is potted. Refer Fig. 0. Figure 0. Resuts for an acoustic scattering probem 51

57 5. Comparison with the conventiona BEM code In this section, the CPU time for the computation wi be studied. The conventiona BEM code ug the same GMRES sover is taken as a guideine. Test cases of the quarter waveength probem discussed in section 5.1 with different number of nodes for the probem were created. The tests were run on a PC with 3.6 GHz CPU and,.8 GB RAM. The computationa times required ug both the sovers are potted Refer Fig. 1. Figure 1. Computationa time for the FMM sover The conventiona sover can ony sove cases with up to 000 nodes. The FMBEM sover can sove cases with a maximum of 100,000 nodes. The memory required for the conventiona BEM sover is of the order ON. The memory required for the FMM sover is of the order ON. Hence ony sma modes can be soved by the conventiona BEM 5

58 code. The sope for the FMBEM sover is whie sope of the conventiona sover was The sopes have been cacuated by ug a east square approximation for the curves. This is quite e to the expected sope of 1.0. The conventiona sover is faster in the cases with fewer numbers of nodes about 50. The FMM sover is faster for cases with more than about 50 nodes. 53

59 Chapter 6 Discussions and Future Work In the previous chapters, the fast mutipoe method for the -D acoustic probem was discussed. The fast mutipoe method gives accurate resuts for the different casses of Acoustic probems. The fast mutipoe method aso is much faster for arger sized modes. The graph of computationa time vs. number of nodes for the FMM sover has a sope of In other words, the computationa time is directy proportiona to N 1.1. This is very e to the theoretica variation of ON. There are some overheads invoved in the sover ike the formation of the tree structure. The convergence time aso varies due to the GMRES sover used. The GMRES sover used requires more steps for convergence as number of nodes increases. The conventiona sover is faster than the FMBEM sover for sma number of nodes. The cacuation of moments uses a p to p approximation. This resuts in p cacuations being performed at each step. Consequenty kp N cacuations are performed for finding the product where k depends on the number of eves in the tree structure. Hence the crossover point number of nodes for which both the sovers take same time depends argey on the vaue of p. The crossover occurs at about 50 nodes. The number of eements per eaf is a critica vaue. Increag the number of eements per eaf decreases the number of eves in the tree. This does not necessariy decrease the computationa time ce direct cacuation needs to be done for a the eements in a eaf and thus it increases the corresponding computationa time. The optimum number of eements per eaf was found to be 5 for the cases studied. 5

60 Exterior acoustic probems have better conditioned matrix than interior acoustic probems. Hence the number of iterations for convergence is much esser in the exterior case. Despite being better than the reguar BEM code, there is scope for further improvement of the FMBEM sover. The decomposition of the F and G kernes has been made ug two sets of moments. Since the two sets of moments are so ey reated, it must be possibe to use a ge set by the use of compex numbers. The current formuation is simpistic and does not consider the fictitious eigen frequencies. The hyperguar formuation for the same probem with the use of Burton-Mier BIE shoud sove the probem of fictitious frequencies for exterior acoustic probems. The formuation for higher-order eements shoud aso be possibe. The higher-order eements reduce the tota number of eements for the same accuracy. In a, this is the first step of the formuation of FMM BEM for -D Acoustics. It can be extended to encompass a arge number of other probems with the same or better efficiency. 55

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