ISSUES RELATED TO THE NUMERICAL IMPLEMENTATION OF A SPARSE METHOD FOR THE SOLUTION OF VOLUME INTEGRAL EQUATIONS AT LOW FREQUENCIES

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1 University f Kentucky UKnwledge University f Kentucky Master's Theses Graduate Schl 2010 ISSUES RELATED TO THE NUMERICAL IMPLEMENTATION OF A SPARSE METHOD FOR THE SOLUTION OF VOLUME INTEGRAL EQUATIONS AT LOW FREQUENCIES Kiran Arct University f Kentucky, kkarc2@uky.edu Click here t let us knw hw access t this dcument benefits yu. Recmmended Citatin Arct, Kiran, "ISSUES RELATED TO THE NUMERICAL IMPLEMENTATION OF A SPARSE METHOD FOR THE SOLUTION OF VOLUME INTEGRAL EQUATIONS AT LOW FREQUENCIES" (2010). University f Kentucky Master's Theses This Thesis is brught t yu fr free and pen access by the Graduate Schl at UKnwledge. It has been accepted fr inclusin in University f Kentucky Master's Theses by an authrized administratr f UKnwledge. Fr mre infrmatin, please cntact UKnwledge@lsv.uky.edu.

2 ABSTRACT OF THESIS ISSUES RELATED TO THE NUMERICAL IMPLEMENTATION OF A SPARSE METHOD FOR THE SOLUTION OF VOLUME INTEGRAL EQUATIONS AT LOW FREQUENCIES Cmputatinal electrmagnetic mdeling invlves generating system matrices by discretizing integral equatins and slving the resulting system f linear equatins. Many methds f slving the system f linear equatins exist and ne such methd is the factrizatin f the matrix using the s called lcalglbal slutin (LOGOS) mdes. Cmputer cdes t perfrm the discretizatin f the integral equatins, filling f the matrix, and the subsequent LOGOS factrizatin have previusly been develped by thers. Hwever, these cdes are limited t cmplex duble precisin arithmetic nly. This thesis extends and expands the existing cmputer by creating a mre general implementatin that is able t analyze a prblem nt nly in cmplex duble precisin but als in real duble precisin and bth cmplex and real single precisin. The existing cde is expanded using templates in Frtran 90 and the resulting generic cde is used test the perfrmance f the LOGOS (bth OL- and NL-LOGOS) factrizatin n matrices generated by discretizatin f the vlume integral equatin. As part f this effrt, we demnstrate fr the first time that the LOGOS factrizatin prvides an O(N lg N) cmplexity slutin t the vlume integral equatin frmulatin f lw-frequency electrmagnetic prblems. KEYWORDS: Cmputatinal Electrmagnetics, Vlume Integral Equatins, Templates in Frtran 90, Lcal-Glbal Slutin Mdes, Matrix Slver Kiran Arct 08/31/2010

3 ISSUES RELATED TO THE NUMERICAL IMPLEMENTATION OF A SPARSE METHOD FOR THE SOLUTION OF VOLUME INTEGRAL EQUATIONS AT LOW FREQUENCIES By Kiran Arct Dr. Rbert J. Adams Directr f Thesis Dr. Stephen D. Gedney Directr f Graduate Studies 08/31/2010

4 RULES FOR THE USE OF THESES Unpublished theses submitted fr the Master s degree and depsited in the University f Kentucky Library are as a rule pen fr inspectin, but are t be used nly with due regards t the rights f the authrs. Bibligraphical references may be nted, but qutatins and summaries f parts may be published nly with the permissin f the authr and with the usual schlarly acknwledgements. Extensive cpying r publicatin f the thesis in whle r in part als requires the cnsent f the Dean f the Graduate Schl f the University f Kentucky. A library that brrws this thesis fr use by its patrns is expected t secure the signature f each user. Name Date

5 THESIS Kiran Arct The Graduate Schl University f Kentucky 2010

6 ISSUES RELATED TO THE NUMERICAL IMPLEMENTATION OF A SPARSE METHOD FOR THE SOLUTION OF VOLUME INTEGRAL EQUATIONS AT LOW FREQUENCIES THESIS A thesis submitted in partial fulfillment f the requirements fr the degree f Master f Science in Electrical Engineering in the Cllege f Engineering at the University f Kentucky By Kiran Arct Lexingtn, Kentucky Directr: Dr. Rbert J. Adams, Prfessr f Electrical and Cmputer Engineering Lexingtn, Kentucky 2010 Cpyright Kiran Arct 2010

7 T my family

8 ACKNOWLEDGEMENTS I wuld like t thank Dr. Rbert J. Adams fr prviding me an pprtunity t wrk in his grup and cntribute t his research. He has been a patient and understanding advisr affrding me the freedm t understand the ideas and the cde in this field at my wn pace. The same freedm has als let me explre and expand my skills and knwledge in ther areas as well. He was never verly critical f the mistakes I made during my research and always pinted me in the right directin t remedy my mistakes. Fr his excellent tutelage I am ever thankful. I wuld als like t thank the remaining members f my thesis defense cmmittee Dr. William T. Smith and Dr. Cai-Cheng Lu. I wuld als like t thank my friends and clleagues Dr. Xin Xu and Dr. Zhying Zeng. I wuld especially like t thank Xin fr his invaluable advice and excellent directin he prvided me when I required it the mst. I can gladly say that he has been like a c-advisr and I will be ever thankful fr everything he has dne. Finally, I wuld like t thank my whle family: Amma, Dad, Deepi and Pavan. Withut their lve and supprt I wuld never have achieved everything that I have achieved s far in my life. I lve yu all and thank yu. iii

9 Table f Cntents ACKNOWLEDGEMENTS... iii List f Tables... v List f Figures...vi 1 Intrductin Basic Review Mtivatin Objective and Scpe Backgrund Maxwell s Equatins General Scattering Prblem [1] Vlume Equivalence Principle [1, 10] Scattering Prblem Slutin [1] Vectr Ptentials [1, 10] Vectr Ptentials Slutins [1] Vlume Integral Equatins [1] Discretizatin f the Vlume Integral Equatin Methd f Mments Lcally Crrected Nyström Methd Sparse Representatin f the System Matrix [14] Gemetrical Decmpsitin Using Oct-tree [14] ACA Methd [14, 15] Multi-Level Simply Sparse Methd [6, 7, 8, 14] LOGOS Factrizatin LOGOS Mdes [14, 19] NL-LOGOS Factrizatin OL-LOGOS Factrizatin [19] Existing Cde Structure and Cde Mdificatin Current Cde Setup Cde Mdificatin Numerical Verificatin f the Generic Linear Algebras Library Perl Scripting Perfrmance f OL-LOGOS and NL-LOGOS Factrizatin Numerical Results: Factrizatin Cmplexities Numerical Results: Bistatic RCS Numerical Results: Single Precisin vs Duble Precisin Cnclusins Summary f the Prject Future Wrk References Vita iv

10 List f Tables Table 3.1: Relative RMS errr cmparisn ( time) fr square matrices f rder Operatins perfrmed n cmplex duble precisin data...47 Table 4.1: Summary f peak memry scaling Table 4.2: Summary f factrizatin time scaling...59 v

11 List f Figures Figure 2.1: An bject illuminated by an incident electrmagnetic field in free space... 9 Figure 2.2: A dielectric rectangular cubid fit in t a 4-level ct-tree ^ ^ H Figure 2.3: MLSSM representatin f the system matrix (a) Z 4 (b) U 4 Z3 V4 (c) H H 4 U 3 Z 2 V 3 V 4 U...26 Figure 3.1: Cde structure and dependencies...34 Figure 3.2: Wall clck time cmparisn f the rutines in the existing library and the new generic library. (a) Cmparisn f the SVD, matrix-matrix multiplicatin, matrixmatrix additin and matrix cpy. Cmplex duble precisin rutines are cmpared. (b) Same as (a) but zmed in fr better reslutin. All peratins perfrmed n square matrices f rder Figure 3.3: Wall clck times f the peratins using the new generic linear algebras library. (a) Times fr SVD, matrix-matrix multiplicatin, matrix-matrix additin and matrix cpy peratins in cmplex duble (CD), cmplex single (CS), real duble (RD) and real single (RS) precisins. (b) Same as (a) but zmed in fr better reslutin. All peratins perfrmed n square matrices f rder Figure 3.4: Mdified cde structure and dependencies...51 Figure 4.1: Meshed dielectric shells...54 Figure 4.2: Peak memry used by the OL-LOGOS factrizatin and NL-LOGOS factrizatin fr tw different tlerances when ε r = Figure 4.3: Ttal time fr OL-LOGOS factrizatin and NL-LOGOS factrizatin fr tw different tlerances when ε r = Figure 4.4: Peak memry used by the OL-LOGOS factrizatin and NL-LOGOS factrizatin fr tw different tlerances when ε r = Figure 4.5: Ttal time fr OL-LOGOS factrizatin and NL-LOGOS factrizatin fr tw different tlerances when ε r = Figure 4.6: Peak memry used by the OL-LOGOS factrizatin and NL-LOGOS factrizatin fr tw different tlerances when ε r = Figure 4.7: Ttal time fr OL-LOGOS factrizatin and NL-LOGOS factrizatin fr tw different tlerances when ε r = Figure 4.8: Relative RMS errr fr the OL-LOGOS and the NL-LOGOS factrizatin when ε r = Figure 4.9: Relative RMS errr fr the OL-LOGOS and the NL-LOGOS factrizatin when ε r = Figure 4.10: Relative RMS errr fr the OL-LOGOS and the NL-LOGOS factrizatin when ε r = Figure 4.11: Relative errr f OL-LOGOS factrizatin. Simulatin carried ut in duble precisin fr ε r = Figure 4.12: (a) Bi-static RCS calculated with OL-LOGOS factrizatin and the NL- LOGOS factrizatin fr tw different tlerances when ε r = 64. The number f DOF is (b) RCS zmed in fr better reslutin Figure 4.13: (a) Bi-static RCS calculated with OL-LOGOS factrizatin and the NL- LOGOS factrizatin fr tw different tlerances when ε r = 16. The number f DOF is (b) Zmed in fr better reslutin Figure 4.14: Bi-static RCS calculated with OL-LOGOS factrizatin and the NL-LOGOS factrizatin fr tw different tlerances when ε r = 4. The number f DOF is (b) Zmed in fr better reslutin vi

12 Figure 4.15: Cmparisn f (a) memry and (b) time used by the OL-LOGOS factrizatin and the NL-LOGOS factrizatin in single and duble precisin fr a unknwn spherical shell with ε r = 64 and a tlerance f Figure 4.16: (a) Bi-static RCS calculated with the OL-LOGOS factrizatin and the NL- LOGOS factrizatin in single and duble precisin fr unknwn prblem with ε r = 64 with a tlerance f 10e -3. (b) Zmed in fr better reslutin vii

13 1 Intrductin 1.1 Basic Review Frequency dmain cmputatinal electrmagnetic mdeling (CEM) usually invlves slving a linear system f equatins in the frm i Zx F (1.1) where Z is knwn as the system matrix, x is knwn as slutin vectr and i F is knwn as the excitatin vectr. The NxN system matrix Z can be generated using (a) Integral Equatin based methds (IE) r (b) Finite Element based methds (FEM). The unknwn vectr x is either field cefficients r the current cefficients and the excitatin vectr i F is determined by sampling an incident surce ver sme spatial pints. System matrices generated using IE methds are dense whereas thse generated using FEM are sparse [1]. Slving (1.1) t btain the x vectr usually requires emplying either direct matrix slvers t btain the inverse f the system matrix r iterative matrix slvers t estimate the slutin vectr x. The strategy f direct slvers given (1.1), invlves btaining the inverse f the system matrix, The prduct f 1 Z, and subsequently the slutin vectr by using Z 1 1 i Zx Z F (1.2) 1 Z and Z is an identity matrix and it fllws that x 1 i Z F (1.3) 1

14 One f the mst ppular methds t btain 1 Z is t decmpse the system matrix Z in t lwer and upper triangular matrices, L and U, respectively, where 1 1 L 1 Z U (1.4) Freely available sftware libraries such as LAPACK (Linear Algebra PACKage) [2], SuperLU [3] and MUMPS (MUltifrntal Massively Parallel sparse direct Slver) [4] btain the inverse f an NxN matrix efficiently and quickly. Sparse, direct matrix slvers like MUMPS use prperties such as symmetry, sparsity and definiteness f the matrix t cntrl the fill-in during the factrizatin t achieve high cmputatinal and memry efficiency. In general, direct slvers that d nt make use f any special prperties f the matrices have a CPU cmplexity f 3 2 O( N ) and a memry cmplexity f O ( N ) [1]. Sftware libraries that d take advantage f the special prperties achieve significantly lwer cmplexities. The system matrices that are generated using FEM based methds are always sparse and the slvers such as MUMPS are designed t take advantage f this sparseness when slving (1.1). Hwever, the same sparse slvers prvide n cmputatinal advantage fr slving (1.1) when the system matrix is generated using IE based methds as a cnsequence f the matrix being dense. The strategy f iterative slvers given (1.1) typically invlves seeking an estimate f the slutin vectr in the frm f x n x 1 p (1.5) n n 2

15 where x n1 is the previus estimate and p n is a crrectin vectr. In brad terms, all iterative slvers begin with an initial guess f x 0, and calculate the residual vectr r n i Zxn F (1.6) by perfrming the Zx n matrix-vectr multiplicatin [5]. Based n r n a new slutin vectr x n1 is estimated and the matrix-vectr multiplicatin is carried ut again. This prcess cntinues until the residual errr falls belw a required threshld. A few f the iterative algrithms that estimate the slutin vectr are cnjugate gradient, bicnjugate gradient and generalized minimal residual (GMRES). The efficiency f the iterative methds is dependent n the efficiency f the matrixvectr multiplicatin. If the matrix vectr prduct is very efficient and the matrix is well cnditined then the slutin cnverges very quickly. But iterative slvers can suffer frm slw cnvergence r n cnvergence at all fr prly cnditined matrices [5]. This lack f rbustness n the part f iterative slvers makes direct slvers mre appealing in slving (1.1). But that is nt t say iterative slvers d nt have a place in slving linear equatins. Direct slvers may be used as precnditiners t imprve the cnditin number f the system matrix and then an iterative slver may be used t btain a rapidly cnverging slutin vectr. 3

16 1.2 Mtivatin The system matrix generated by using IE methds is always a dense matrix and therefre sparse direct slvers that cmpute 1 Z using LU factrizatin are either t slw r t memry intensive t be directly applied in slving (1.1). Hwever, it has been shwn that there exist sparse apprximatins f the dense matrices generated by IE methds by Canning and Rgvin [6]. In this thesis, the sparse representatin f the system matrix is knwn as Multi Level Simply Sparse Matrix (MLSSM) [6-8]. Sparse direct slvers based n the sparse representatin f the dense matrices can and have been develped by Adams et. al. in [9]. The direct slver uses the cncept f lcal-glbal slutin (LOGOS) mdes t factr the MLSSM representatin f the system matrix. The LOGOS mdes are derived frm the MLSSM data structure and can be classified in t tw categries (a) verlapping and (b) nn-verlapping based n whether the surce mdes have verlapping r nn-verlapping supprt. The LOGOS mdes are als classified as lcalizing r nn-lcalizing based n whether the mdes lcalize the assciated scattered fields t regins utside a specified regin. The system matrix generated using the vlume integral equatin is a prime candidate t test the perfrmance f the LOGOS factrizatin. The system matrix can be apprximated using the MLSSM and thereafter LOGOS factrizatin can be applied. Bth verlapping and nn-verlapping LOGOS factrizatins can be tested t verify that the verlapping factrizatin prvides an asympttically better cmplexity cmpared t nn-verlapping factrizatin. 4

17 1.3 Objective and Scpe The bjective f this wrk is t extend the existing single data type cde t a mre generic cde and cmpare the perfrmance f verlapping and nnverlapping factrizatins f the matrices generated using the vlume integral equatin. As part f this thesis, the existing cde is extended t wrk with any f the fur Frtran 90 intrinsic data types viz., Cmplex Duble (CD), Cmplex Single (CS), Real Duble (RD) and Real Single (RS). After the extensin f the cde the factrizatin perfrmances are tested in real single precisin. Given belw is an utline f the thesis. Chapter tw reviews the general backgrund that is required in understanding CEM prblems. In this chapter a brief summary f Maxwell s equatins and the vlumetric equivalence principle are given. Additinally, the chapter als reviews the Nyström methd fr discretizing vlume integral equatins. Als discussed are the structure f the MLSSM and the LOGOS factrizatin. Chapter three discusses the current cde structure and describes the rigidity in the cde. Tw ptins, Frtran 2003 features and templating in Frtran90, are described fr the extensin f the existing cde in making the cde less rigid and mre generic. Furthermre, reasns fr chsing templates fr the extensin f the cde are discussed. Cde snippets are given t explain hw the cde was mdified using the elegant slutin f templates. Chapter fur prvides numerical results in cmparing verlapping and nn-verlapping LOGOS factrizatin in real single precisin fr vlume integral 5

18 equatins at lw frequencies. T be specific, memry and time cmplexities f the verlapping and the nn-verlapping factrizatins are cmpared. Bi-static RCS plts cmputed using the tw factrizatin methds are cmpared against the analytical slutin fr thin dielectric shells. Chapter five summarizes the purpse f this prject and the limitatins f the LOGOS factrizatin. Finally, remarks are made abut future wrk. 6

19 2 Backgrund 2.1 Maxwell s Equatins Maxwell s equatins are the pillars n which all f the macrscpic electrmagnetic thery is built. They are a set f equatins that dictate the interactins amng the electric and magnetic fields, the charge and current distributins and the cnstitutive material prperties. These partial differential equatins with space and time variables can describe the field vectrs and their relatinships with charge and current distributins at anyplace and anytime. Any material discntinuities in the regin where the prblem is defined give rise t discntinuities in the charge and current distributins, which in turn dictate the behavir f the fields. The relatinships between the fields, material parameters and the charge and current distributins are usually knwn as the bundary cnditins. These equatins alng with bundary cnditins are used t slve electrmagnetic bundary value prblems. The set f fur Maxwell s equatins in time harmnic frm fr a linear medium are given by and the cntinuity equatins are given by E j u H M (2.1) H j E J (2.2) e E (2.3) m H u (2.4) 7

20 J j (2.5) e M j (2.6) The - symbl n tp f the variables in (2.1)-(2.6), and als in equatins frm here n, indicates that the variable is a vectr quantity. E, H, m M, J, e, m, and u are electric field (V/m), magnetic field (A/m), magnetic current density (V/m 2 ), electric current density (A/m 2 ), electric charge density (C/m 3 ), magnetic charge density (Wb/m 3 ), electric permittivity (F/m) and magnetic permeability (H/m), respectively. The general bundary cnditins at the interface f tw regins, regin 1 and regin 2, with different material parameters are E 2 E M s H 2 H J s D 2 D es B 2 B ms ^ n 1 ^ n 1 ^ n 1 ^ n 1 (2.7) (2.8) (2.9) (2.10) Where M S, J S, are the magnetic and electric surface current densities, respectively. es, ms are the electric and the magnetic surface charge densities, respectively, and D 2 is the electric flux density (Clumbs/m 2 ) in regin 2 and D 1 is the electric flux density in regin 1. Similarly, B s are the magnetic flux densities (Webers/m 2 ) in regins 1 and 2. Finally, ^ n is the unit nrmal vectr pinting frm regin 1 in t regin 2. 8

21 2.2 General Scattering Prblem [1] Cnsider a hmgeneus r an inhmgenus (scatterer) bject with permittivity ( r ) and permeability ( r ) different frm the free space permittivity and permeability illuminated by an electrmagnetic field as shwn in Figure 2.1. In the figure ^ k is knwn as the wavenumber f the medium (free space in this case). inc E, inc H are the electric and the magnetic fields f the incident wave, which are prduced by surces lcated far away frm the scatterer. These fields are the nes that wuld exist in the absence f the scattering bject. s E and H s are the scattered electric and magnetic fields due t induced currents n the surface r in the vlume f the scattering bject. Let E and H be the ttal fields that are present due t the presence f the scatterer in free space. The ttals fields can then be dented as inc s E E E (2.11) inc s H H H (2.12) These ttals fields are typically the quantities that are f interest in a general scattering prblem. E s inc E, ^ k r r H s H inc Figure 2.1: An bject illuminated by an incident electrmagnetic field in free space 9

22 2.3 Vlume Equivalence Principle [1, 10] T slve the scattering prblem ne can use the vlume equivalence principle which in general terms can be stated as: a scatterer can be replaced by equivalent induced vlume currents that radiate in free space [1]. Equatins (2.1)-(2.4) can be rewritten as E j u H M (2.13) H j E J (2.14) where E e (2.15) u m H (2.16) M j ( 1 H (2.17a) r ) ( r 1) M E r (2.17b) J j ( r 1) E (2.18a) ( r 1) J H (2.18b) r 1 e r E r 1 m r H r (2.19) (2.20) The vlume currents J and M nw radiate in free space and slving fr fields radiated in free space is much simpler that slving fr fields in inhmgeneus media. The currents are still unknwn at this pint and the intrductin f these currents has nt slved the riginal prblem. 10

23 2.4 Scattering Prblem Slutin [1] At this pint we knw we can replace the scatterer with equivalent currents. We als knw that inc E and must satisfy the vectr Helmhltz equatins inc H, when away frm the riginal surces, 2 2 inc inc 2 E k E 0 (2.20) inc inc 2 H k H 0 (2.21) and the scattered fields equatins given by s E and s H, are the slutins t the vectr wave 2 2 E H s s k k 2 2 E H s s J j J j M j M j M J (2.22) (2.23) where J and M are the equivalent vlume surces frm (2.17) and (2.18) Vectr Ptentials [1, 10] One methd t slve fr the scattered fields in (2.22) and (2.23) is using vectr ptentials. Vectr ptentials ( A, F ) are merely mathematical tls that aid in the slutin prcess f btaining the fields. In a surce free regin the magnetic flux density is always slenidal (divergence is zer), and thus can be represented as a curl f a vectr quantity as B A s s H A (2.24) A where A is the magnetic vectr ptential. 11

24 The scattered magnetic field can be represented as H A S A (2.25) Substituting (2.25) in (2.13) assuming surce free regin results in r Using the identity, we set E s A j A (2.26) A j A 0 (2.27) E S 0 (2.28) e E s A ja (2.29) e where e is knwn as the scalar electrical ptential. Next, we take the curl f (2.25) t btain Using the vectr identity H A S A (2.30) in (2.30) and equating the result t (2.14) gives 2 A A A (2.31) s A 2 A A J j E (2.32) Substituting (2.29) in (2.32) and rearranging a few terms leads t A ( j J 2 2 A k A ) e (2.33) where 2 k = 2. Rearranging terms in (2.33) results in 12

25 2 2 A k A ( A j ) J e (2.34) We are free t define the divergence f A and here we define it as A j (2.35) e This is knwn as the Lrentz gauge. Frm (2.35) Substituting (2.36) in (2.34) leads t A e (2.36) j 2 A k 2 A J (2.37) Using (2.36) in (2.29) results in E s A j A j A (2.38) Frm (2.25) and (2.38), it is clear that we nw have the scattered fields nly in terms f the magnetic vectr ptential. Starting with the electric flux density as slenidal and fllwing a prcedure similar t the ne described abve, we can derive the scattered fields in terms f the electric vectr ptential ( F ). The ttal scattered fields are then btained frm superpsitin f the fields due t the magnetic vectr ptential ( A ) and fields due t the electrical vectr ptential ( F ). The fields resulting frm the electric vectr ptential are given by and E s F F (2.39) 13

26 H s F j F jf (2.40) Finally, the equatin fr the electric vectr ptential is given by 2 F k 2 F M (2.41) The ttal scattered electric field is the superpsitin f (2.38) and (2.39) and the ttal scattered magnetic field is the superpsitin f (2.25) and (2.40). The ttal fields can then be written as and E H s s s s j A F E A EF j A (2.42) s s j F A H A HF jf (2.43) Vectr Ptentials Slutins [1] It seems cunterprductive t intrduce additinal unknwn vectr ptentials t slve a scattering prblem. Frtunately, slutins t (2.37) and (2.41) that satisfy the radiatin cnditin fr the scattered fields maybe written as A J G (2.44) F M G (2.45) where G is the Green s functin and the * symbl indicates three dimensinal cnvlutin. The well-knwn Green s functin is given by G e jkr (2.46) 4 r 14

27 and the three-dimensinal cnvlutin fr the magnetic vectr ptential may be written as r ' jkr r' e A( r) J dr' (2.47) 4 r r' and the cnvlutin fr the electric vectr ptential may be written as ' jkr r' e F( r) M r dr' (2.48) 4 r r' where r is the bservatin crdinate and r is the surce crdinate. Therefre, (2.47) and (2.48) can be substituted in (2.42) and (2.43) t make the equatins relatively simpler and in terms f the equivalent currents J and M Vlume Integral Equatins [1] Integral equatins, mre precisely integr-differential equatins, are cnstructed t describe the interactins between electrmagnetic fields and the scaterrers that are cmpsed f dielectric r magnetic materials. We can rearrange terms in equatins (2.11) and (2.12) t btain E H inc inc s E E (2.49) s H H (2.50) Substituting (2.42) and (2.43) in (2.49) and (2.50), respectively, results in E inc H j A r E r j A inc F (2.51) j F r H r jf A (2.52) 15

28 These equatins are knwn as the vlume integral equatins. T reiterate, ( A, F ) are the vectr ptentials and are given by (2.47) and (2.48). In equatins (2.51) and (2.52) we nt nly have the vectr ptentials that are in terms f the unknwn equivalent currents, but als have the unknwn ttal fields. As an alternative, ne culd pse the prblem nly in terms f the unknwn ttal fields by using equatins (2.17)-(2.18). Ding s will make it pssible fr (2.51) t be expressed entirely in terms f the unknwn E field and (2.52) t be expressed entirely in terms f the unknwn H field. If the scattering bjects are cmpsed f nly dielectric material, then the last term f the (2.51) may be drpped because there will be n magnetic currents induced in the vlume f the bject. Similarly, if the scattering bject is cmpsed f nly magnetic material, then the last term in (2.52) maybe drpped because there will be n electric currents induced in the vlume f the bject. 2.5 Discretizatin f the Vlume Integral Equatin In the previus sectins, vlume integral equatins were derived by using Maxwell s equatins and the vlume equivalence principle. Slving (2.51) and (2.52) fr either the ttal fields r the vlume currents is nn-trivial. There exist nly a few scatterer gemetries such as dielectric slid spheres and dielectric spherical shells, fr which the vlume currents maybe cmputed analytically. Fr any ther gemetries, (2.51) and (2.52) have t be slved using numerical techniques. T that end, matrix equatins are generated by discretizing the vlume integral equatin and the subsequent system f linear equatins is 16

29 slved using matrix slvers such as LU factrizatin, LOGOS mdes etc. One such methd that discretizes integral equatins is the Methd f Weighted Residuals r the Methd f Mments (MM), and it is briefly explained belw Methd f Mments Cnsider a scatterer cmpsed f nly dielectric material, and expand (2.51) cmpletely and in terms f the electric currents J t give E inc r j r ' J j 1 r V j J r ' V J r ' jk rr' e dv' 4 r r' jk rr' e dv' 4 r r' (2.53) In this equatin the LHS is knwn and the equatin needs t be slved fr J. Using the MM, currents inside the scatterer are apprximated in the scatterer. The general steps in slving (2.53) fr the currents using the MM are (a) Discretizatin Determine a discrete representatin f the gemetry (b) Basis Functins Chse a set f functins t represent the unknwn quantities n the discretized gemetry (c) Testing Functins Impse the underlying equatin (VIE) with respect t a discrete set f testing functins (d) Matrix Equatin The result f this prcess is a matrix equatin fr the unknwn cefficients J. Let s first write the integral equatin in a simplified frm as [11] f x (2.54) where is the integral peratr, f is the knwn frcing functin (excitatin vectr) and x is the unknwn functin (current density) distributed thrughut the 17

30 scatterer. Step (a) requires that the vlume f the scatterer be apprximated by using smaller blcks such as hexahedral r tetrahedral cells. Step (b) calls fr the expansin f the unknwn x as a set f knwn pn-functins unknwn cefficients n, giving n weighted by x N n1 (2.55) n n where n spans a functin space f linearly independent functins that have supprt n vlume V and interplate x t sme plynmial rder p. These functins are knwn as basis functins. Expanding (2.55) in (2.54) results in f pn n1 (2.56) n n Step (c) requires the intrductin f anther set f N-testing functins, n, that are linearly independent and have supprt n vlume V. Next, define an inner prduct r g r dv f, g f (2.57) V Perfrm the inner prduct f (2.54) with each f the N testing functins pn, f, K (2.58) m This leads t a pn x pn linear system f matrix equatins represented as n1 n m n f Z (2.59) Finally, step (d) requires slving (2.59) t btain the unknwn cefficients. This can be dne by using LU factrizatin r the LOGOS factrizatin. 18

31 2.5.2 Lcally Crrected Nyström Methd The vlume integral equatin f (2.53) is discretized using the s called Lcally Crrected Nyström (LCN) methd t btain the system matrix [12, 13]. The integral equatin f (2.53) is singular when the surce and bservatin crdinates cincide. When perfrming numerical quadrature, the LCN methd handles such singularities by lcally crrecting them. The general idea f such lcal crrectins is described here. Cnsider an integral equatin f the frm [13] where and G r' r Gr r' J r ' dv' J is the unknwn current density, r (2.60) V is the knwn frcing functin is the kernel. Equatin (2.60) may be apprximated by numerical quadrature as N r Gr r ' J r ' (2.61) i1 n and sampling (2.61) at N discrete pints leads t square matrix f rder N and the m-th rw f the matrix is given by N r Gr r ' J r ' m i1 n m n n n (2.62) n At vanishing distances between r m and ' n r, the kernel G will becme singular. T handle this, define the exact kernel as G m, n Gm, n Lm, n (2.63) 19

32 where L m, n is a lcal crrectin matrix and m n G, is defined as G m 0, m n, n (2.64) Gr m rn ', m n Assume that the current density can be expanded with a set f knwn basis functins define where functins f k r that are distributed thrughut the vlume V. Then frm (2.63) N r Gr r' f r ' dv G f r N nlm, n f k n m k ' n m, n k n (2.65) n1 V n1 nm L m, n is the m-th rw f the lcal crrectin matrix. Ding this fr K basis f k r leads t linear system f equatins. This system can be slved fr m-th rw f the L m, n using LU factrizatin. Once the crrectin matrix is cmputed, a linear system f equatins may be cnstructed as r G L J r ' (2.66) m m, n m, n n which can be slved fr the current density. Fr detailed explanatin refer t [12]. Fr this thesis the Nyström methd is used generate the matrix equatins and the LOGOS factrizatin is used t slve fr the currents. 2.6 Sparse Representatin f the System Matrix [14] Up t this pint we have discussed the generatin f the system matrix fr the vlume integral equatin. Since the matrix is generated using an integral equatin and because its kernel cntains a nn-lcal peratr, the matrix is 20

33 dense. The system matrix basically represents the cupling between the surce and field pints in the vlume (vlume integral equatins) r n the surface (surface integral equatins) f the scatterer. The system matrices that are generated by the discretizatin f the integral equatins have sub-blcks that are weakly cupled [15]. The weak cupling crrespnds t interactins between surce and field pints that are sufficiently away frm each ther. Thus, Adaptive Crss Apprximatin (ACA) [15] can be used t fill the far interactin blcks. Further cmpressin f the system matrix is achieved by string the system matrix using the Multi Level Simply Sparse Methd (MLSSM) (discussed belw). This thesis uses the ACA methd t fill the far interactin blcks and the MLSSM t represent the system matrix in a cmpressed frm Gemetrical Decmpsitin Using Oct-tree [14] The first step in btaining a sparse representatin f the system matrix is the recursive decmpsitin f the underlying scatterer gemetry using a nested ct-tree structure. Starting with level l = 1, the spatial samples f the meshed gemetry are decmpsed in t a multi-level ct-tree with L levels, by cntinuus sub-divisin f the grups at level l. The rt level is l = 1 and has nly ne grup, which cntains all the spatial samples. Let the ttal number f nn-empty grups - grups that cntain spatial samples at each level be M(l). The number f levels, L, is selected s that the spatial grups at that level L cntain at least 20 DOF. At level l, the i th grup is dented as i(l). The grups that share bundaries with the i th grup are knwn as the near-neighbrs f the i th grup. In additin, the i th grup is als a near neighbr f itself. The remaining grups at 21

34 level l are knwn as the nn-near grups r far grups. Furthermre, the ntatin z i(l) is used dente the sub-matrix blck f a level l matrix Z l, assciated with the surce grup i(l) ACA Methd [14, 15] As mentined previusly, ACA can be used t fill the far interactin blcks f the system matrix because f the weak cupling that exists between the surce and field grups that are placed far apart. After the gemetry has been decmpsed via the ct-tree, the system matrix Z can be expressed as where L near Z Z (2.67) l2 l near Zl is the near-neighbr interactin blcks f the system matrix at level-l. Aside frm near Z L the near interactin blcks at level-l can be thught f as the nn-near-neighbr blcks at level-(l+1), in ther wrds near nnnear L Zl1 Z. Therefre, (2.67) may be written as near L L l3 nnnear l Z Z Z (2.68) Write Z nnnear l as nnnear nnnear nnnear nnnear l Z1 ( l ),..., Z i ( l ),..., Z M ( l ) Z (2.69) Using the prcedure described in [15] an uter prduct representatin fr each f the nn-near blcks in (2.68) is calculated. Each f the blcks may be written as 22

35 ~ ( K) ~ ~ H nnnear i ( l ) Zi( l) Ui( l) Vi(l) Z u v (2.70) K k1 k H k where K is the effective rank f the nnnear l Z matrix. U i( l) and ~ ~ H i(l) V are full rectangular matrices with clumns and rws given by uk and H v k. The ACA prcedure is used t btain the uk and v adaptively fr k 1,2,... K. Clumns H k and rws, u k and H v k, respectively, are cntinuusly cmputed until the cnvergence criteria is met. The cnvergence is determined by checking hw gd the apprximatin in (2.68) is with the additin f an extra clumn in U i( l) ~ and an extra rw in ~ H i(l) V. Fr detailed descriptin f the algrithm and the cnvergence criteria refer t [14] Multi-Level Simply Sparse Methd [6, 7, 8, 14] The MLSSM representatin f the system matrix is mre efficient than the ACA representatin f the system matrix. T describe the structure f the MLSSM Figure 2.2 will be used. Cnsider the meshed dielectric rectangular cubid in Figure 2.2 which depicts an ct-tree with fur levels built n tp f the meshed gemetry. In this case a fur level ct-tree cntains very few grups at the furth level and indeed the grups at the finest level will nly be the eight hexahedra that make up the cubid. Levels fur, three, tw and ne cntain eight, fur, tw and ne grups, respectively as indicated in Figure 2.2. The structure f the MLSSM fllws a recursive relatinship given by 23

36 where ^ H Z l Z l U l Z l1 V l, l 2,3.. L (2.71) ^ Z l cntains all the near-neighbr interactins at level-l f the ct-tree that were nt represented at a finer level f the ct-tree. The far interactins are cmpressed and represented by the rectangular, rthnrmal, blck diagnal 1 Level 1 2 Level Level 3 Level Figure 2.2: A dielectric rectangular cubid fit in t a 4-level ct-tree. matrices U l and H V l. The riginal matrix may be recvered by setting l = L in (2,71) and stpping the recursin at l = 2 where because the matrices U 2 and ^ 2 Z2 Z (2.72) H V 2 are defined at neither level l =2 nr level l =1. Perfrming the recursin n the ct-tree represented in Figure 2.2 leads t 24

37 ^ Z4 ^ 4 Z3 H 4 Z U V U U Z V V (2.73) H 3 H 4 ^ Beginning with Z 4, the three terms n the RHS f (2.73) represent the near neighbr interactin blcks at levels 4, 3 and 2 that have nt been represented at finer levels. The blcks that crrespnd t each f the RHS terms in (2.73) are shwn in Figure 2.3 (a), (b) and (c), respectively. Referring back t the ct-tree, it is evident that grups (1,2), (2,3), (3,4) and s n are neighbrs and therefre ^ frm the near neighbr interactins f the MLSSM at level 4 ( Z 4 ) which is apparent in Figure 2.3 (a). Again frm Figure 2.2 it is bserved that grups (1, 2) and (3, 4) at level-3 are neighbrs and thus frm a near neighbr interactin blck at level-3. Figure 2.3 (b) shws the matrix blcks that make up the near neighbr interactin blcks at level-3 with the exceptin f the near neighbr blcks that have already been represented at level 4. And these are blcks f the system matrix that crrespnds t the 2 nd term f the RHS in (2.73). Figure 2.3 (c) can be understd fllwing a similar reasning. Finally, als ntice that the shaded parts in Figure 2.3 (a), (b) and (c) crrespnd t different chunks f the system matrix assciated with ne surce grup at level-4. 25

38 ` 4 (a) 1 2 (b) 1 2 (c) ^ Figure 2.3: MLSSM representatin f the system matrix (a) Z 4 (b) 2.7 LOGOS Factrizatin U H H 4 U 3 Z 2 V 3 V 4 ^ H U 4 Z3 V4 (c) T slve the cmpressed representatin f the system matrix btained using integral equatin methds, ne can use iterative slvers [1] r use fast, direct slutin methds [9, 16, 17, 18]. A LOGOS based direct slver is ne such methd and is used in this thesis t slve system matrix generated using the vlume integral equatin. The use f a LOGOS-based slver assumes the availability f the sparse representatin f the system matrix (MLSSM). Tw different flavrs f LOGOS factrizatins, Nn-Ovelapped Lcalizing (NL- LOGOS) and Over-Lapped, Lcalizing (OL-LOGOS) are discussed belw in 26

39 brief. The perfrmance f these tw factrizatins fr the vlume integral equatin is cmpared in the later chapters. The details f these factrizatin algrithms may be fund in [19] LOGOS Mdes [14, 19] A single LOGOS mde is knwn as an excitatin/slutin pair. Let the simulatin dmain S, be divided in t tw nn-verlapping regins S 1 and S 2 as S S 1 S 2 (2.74) Additinally, let S 1 and S 2 be dented as Regin 1 and Regin 2, respectively. As a result f this decmpsitin, the system equatin may be rewritten as Z Z Z Z x x 1,m 2,m F F i 1,m i 2,m (2.75) where x1, m and 2, m x are parts f the slutin vectr x m assciated with Regins 1 and 2 respectively. Let Z 11 be the part f the impedance matrix that crrespnds t interactins between surces and bserver in Regin 1 and Z 21 be the part that crrespnds t fields excited in Regin 2 as a result f surces in Regin 1. Similar definitins apply t Z 22 and Z 12. LOGOS mdes are frmed by each excitatin/slutin pairing f ( F i m, x m ). T determine the LOGOS mdes that have supprt nly in Regin 1 ( x 0, x 0), the lcal cnditin is 1,m 2,m Z x 1, F i 11 m 1,m (2.76) with the glbal cnditin 27

40 Z x 1, F i 21 m 2,m (2.77) Substituting x 1, m frm (2.76) in t (2.77) leads t the lcal-glbal cnditin -1 i Z21 Z11x1,m F 2,m (2.78) satisfied by all LOGOS mdes. Utilizing (2.78) LOGOS mdes that are cnfined t Regin 1 can be btained t O. At this juncture we can intrduce tw classificatins f LOGOS mdes: lcalizing vs nn-lcalizing and verlapping vs nn-verlapping. Lcalizing LOGOS mdes are btained by letting F i 2,m 0, implying that the surces in Regin 1 d nt radiate any fields in t Regin 2. The mdes btained with i F 2,m 0, are knwn as the nn-lcalizing mdes. The secnd classificatin results frm the chice f the supprt f the surces. If the supprt fr the surces in Regin 1 ( x 1, m ) extends the bundary f Regin 1, then there is present an verlap with the surces defined in ther spatial regins. Such a chice f supprt fr the surces will lead t verlapping LOGOS mdes. Alternatively, if the surces in Regin 1 d nt have supprt beynd the bundary f Regin 1 then the LOGOS mdes are referred t as nn-verlapping mdes. Using these tw classificatins, fur types f LOGOS mdes can be btained: 1. NN-LOGOS mdes: Nn-verlapping nn-lcalizing mdes. 2. NL-LOGOS mdes: Nn-verlapping lcalizing mdes. 3. ON-LOGOS mdes: Overlapping nn-lcalizing mdes. 4. OL-LOGOS mdes: Overlapping and lcalizing mdes. 28

41 As mentined previusly, this thesis cmpares the perfrmance f NL-LOGOS vs OL-LOGOS factrizatin in Frtran90 s real single precisin. T this end, a brief descriptin f these tw factrizatins is given here NL-LOGOS Factrizatin The fllwing derivatin and ntatin f the NL-LOGOS factrizatin can be fund in [14] and are reprduced here fr cnvenience. NL-LOGOS mdes are calculated by impsing F i 2,m 0 t rder-ε in (2.78). T cmpute the lcalizing mdes satisfying this cnditin, the matrix blck assciated with surces in regin is decmpsed using QR factrizatin t btain Z Z Q R 1 Z 21 Q 21 (2.79) where Q 11 ( Q 21) is the same size f Z 11( Z 21) and R 1 is a square upper triangular matrix. Perfrming a singular value decmpsitin (SVD) n Q 11 results in Q (2.80) H 11 u1s1v1 where s1 is a set f n singular values srted in descending rder. The lcalizing mdes are btained by selecting thse singular vectrs that crrespnd t singular values clse t unity. If there are N L singular values that are clse t unity (t rder-ε), then let the crrespnding N L right singular vectrs be 29

42 dented by ( L) v 1 and the rest f the vectrs by (L) (N) (L) ( N) v v and s s 1 1 v1 1 1 s1 v ( N) 1. Using this ntatin we have. This leads t (2.79) being rewritten as Z 1 u1s1 H v1 R1 Q 21v (2.80) 1 Right multiplying bth sides by -1 R 1 and then by v 1 leads t (L) (N) u - 1s 1 1 u 1s1 u1s1 Z 1R1 v1 (L) (N) (2.81) Q 21v1 Q 21v1 Q 21v1 The value f N L is determined such that the apprximatin can be made and (2.81) is written as (L) Q v 0 (2.82) 21 1 (L) (N) (L) (N) (L) -1 (N) u 1s1 u1s1 v v Z Λ R v -1 Z 1R (N) (2.83) 0 Q 21v1 In (2.83) Λ cntains the lcalizing LOGOS surce mdes (L) -1 (L) 1 R1 v1 crrespnding t Regin 1. Fllwing similar prcedure the lcalizing surce mdes fr Regin 2 are als calculated. Therefre, Z Z Z Z Λ1 0 (L) Λ 0 (L) 2 Λ (N) 1 0 Λ 0 (N) 2 u1s 0 (L) L u 0 s (L) 2 2 Z Z (N) 11 (N) 21 Z Z (N) 12 (N) 22 (2.84) where ( N) Λ 1 and ( N) Λ 2 are the rthnrmal cmplements f the lcalizing surce mdes f Regin 1 and Regin 2, respectively. Let P ( L) 1 u1 (N) and P2 u 2 span the lcalized field space, then the prjectin matrix is cmprised f ( L) P 1 and ( L) P 2 30

43 and their rthnrmal cmpnents factrizatin is given by ( N) P 1 and P ( N) 2. The cmplete ne level P P ( L) ( N) H H Z Z Z Z Λ ( L) Λ ( N ) ~ I 0 Z Z ( LN ) ( NN) (2.85) where (L) (L) Λ 1 0 Λ (L) and similar definitin apply t 0 Λ 2 (N) Λ, (L) P and (N) P. The identity matrix ( L s1 0 0 ) I ~ ( L) s 2 cntains apprximately unity diagnal elements. The factrizatin is written as Z Z ~ 11 Z12 21 Z 22 ( LN ) ( L) ( N) ( L) ( N ) P P I Z Λ Λ 1 0 Z ( NN) (2.86) In a multi-level decmpsitin f the gemetry the grups in each level define the Regins 1 and 2. At the finest level L f the tree the factrizatin is Z ( NN) L1 ( LN) ( L) ( N) ( L) ( N) P P I Z L Λ Λ 1 L L ~ 0 Z ( NN) L L L (2.86) The cmplete factrizatin is carried ut recursively by starting at the finest level l = L and stpping the factrizatin at level l = 2. Further details can be fund in [14]. 31

44 OL-LOGOS Factrizatin [19] As indicated previusly OL-LOGOS mdes are mdes fr which the surces generally have supprt extending beynd Regin 1 but have the scattered fields lcalized nly t Regin 1 [20]. The general factrizatin step at each level is f the frm 1 1 ) ( ) ( ) ( l l NN l LN l l LN l l ( N) l ( L) l ( NN) l ( LN) l ( N) l ( L) l ( NN) l Λ Λ Z 0 Z I P 0 Z I P Λ Λ Z 0 Z I P P Z ~ ~ ~ (2.86) The additinal terms 1 ) ( ) ( l NN l LN l l Λ Z 0 Z I P ~ cme frm the analysis f (NN) l Z using a shifted tree t btain s called intermediate mdes. The OL-LOGOS mdes are thereafter fund within the intermediate mdes. The details f the OL- LOGOS factrizatin can be fund in [19].

45 3 Existing Cde Structure and Cde Mdificatin Chapter ne intrduced the cncept that mst frequency dmain CEM invlves slving a system f equatins. The system matrices generated using FEM techniques result in sparse matrices, whereas IE based techniques result in dense matrices. Chapter ne als tuched n direct sparse matrix slvers and iterative slvers. Chapter tw gave an verview f the steps invlved in the generatin f the system matrices in case f a vlume integral equatin frmulatin. These steps began with the Maxwell s equatins and the vlume equivalence principle. Using the vlume equivalence principle the vlume integral equatin was develped. The LCN methd was intrduced t discretize the vlume integral equatin. Then later in chapter tw, sparse representatins f the dense matrices were intrduced. Finally, the chapter ended with the descriptin f the LOGOS factrizatin algrithm t slve the sparse representatins f the IE system matrices. This chapter describes the cmputer cde that was develped t perfrm the matrix generatin, sparse representatin and the factrizatin described in the previus chapter and detailed descriptin f hw the cde was extended t a generic cde as part f this prject. 3.1 Current Cde Setup Starting with the prblem descriptin t slving the system utilizing LOGOS factrizatin is achieved here, at the University f Kentucky, by using the Material Scattering (MSCAT) cde in cnjunctin with the Mdular Fast Direct slver library (MFDlib). Bth the MSCAT and the MFDlib cdes are written in Frtran90/Frtran77. The cmplete cde set up and the cde dependencies are 33

46 shwn in Figure 3.1. The arrws indicate dependencies. Each f the blcks in the figure can be thught f as a separate blck ( a mdule in Frtran 90). Universals mdule is the base mdule and cntains all the cnstants, memry cunter rutines, timing rutines etc. The Linear Algebras mdule cntains all the linear algebra rutines such as matrix-matrix multiplicatin, matrix-vectr multiplicatin, matrix-matrix additin, matrix accumulatin etc. The MSCAT mdule reads in the input gemetry files, perfrms the quadrature fr integral equatin frmulatins (EFIE, CFIE, VIE) and the pst prcessing. Universals Linear Algebras MSCAT Tree MFD-MSCAT Cnnectr ACA MLSSM MFD Library Figure 3.1: Cde structure and dependencies 34

47 The Tree mdule is used t decmpse the underlying gemetry in t an ct-tree. The ACA mdule cntains cde t perfrm the adaptive crss apprximatin (ACA) mentined in chapter tw. The MLSSM mdule has all the rutines that build the system matrix in the sparse structure described in the Multi Level Simply Sparse Methd sectin in chapter tw. Finally, MFD Library cntains the cde t perfrm the NL-LOGOS and OL-LOGOS factrizatins. All the libraries/mdules can perfrm the simulatin nly in cmplex duble precisin. The Linear Algebras mdule has rutines that can perate nly n cmplex duble precisin data bjects. Fr example, the cmplex duble precisin derived data types have been defined as type, public :: Cmplex16Mat integer :: numrws = 0, numcls = 0 cmplex*16, pinter :: blk(:,:) => null() end type type, public :: Cmplex16Vec integer :: numelem = 0 cmplex*16, pinter :: vec(:) => null() end type where Cmplex16Mat is the name f the data structure, blk hlds the pinter t a cmplex*16 matrix blck, numrws and numcls indicate the number f rws and number f clumns in the matrix, respectively. Similar explanatin applies t Cmplex16Vec, hwever vec pints t an array f cmplex*16 values instead f a matrix. An example functin invlving Cmplex16Mat is the Cmplex16MatInstantiate whse functin definitin is functin Cmplex16MatInstantiate(numRws, numcls) result(a) implicit nne integer, intent(in) :: numrws, numcls type(cmplex16mat) :: A integer err 35

48 if((numrws >= 0).AND. (numcls >= 0)) then A%numRws = numrws A%numCls = numcls if ((numrws > 0).AND. (numcls > 0)) then allcate(a%blk(numrws, numcls), stat = err) call CheckState(err, 'In functin: Cmplex16MatInstantiate') A%blk = 0.d0 call CuntHeapMem(flat(numRws)*flat(numCls)*16*MB_Per_Byte) endif else print *, "numrws = ", numrws, " numcls = ", numcls call CaughtABug("Cmplex16MatInstantiate: Dimensin is negative.") end if end functin This functin accepts tw integer values (numrws and numcls) and allcates memry fr the Cmplex16Mat return variable A and cunts up a glbal memry cunter. Similarly, ther functins/rutines in the Linear Algebras mdule are designed t wrk with nly cmplex duble precisin data bjects. Since mst ther mdules in Figure 3.1 are dependent n the Linear Algebras mdule they have als been written t wrk with nly cmplex duble precisin matrices/vectrs. Fr instance, the MLSSM mdule has the data structure ssmlevel, given by type ssmlevel type(cmplex16mat), pinter :: u(:) => null() type(cmplex16mat), pinter :: vh(:) => null() type(cmplex16mat), pinter :: T(:,:) => null() end type where u and vh are arrays f matrices f the type Cmplex16Mat, and T is a tw dimensinal array f matrices f the type Cmplex16Mat. Hence, the whle cde is nt very flexible. One f the main bjectives f this thesis was t mdify the existing cde s that it is valid fr the fur intrinsic Frtran 90 data types: cmplex duble precisin (CD), cmplex single precisin (CS), real duble precisin (RD), real single precisin (RS). In cases fr which the simulatins culd be carried ut in 36

49 single precisin, the memry and time savings wuld be significant. Cnsequently, prblems with duble the number f unknwns culd be simulated while sacrificing a little accuracy. In an bject riented prgramming language such as C++ there are features, specifically templates, which make it easy t have a single algrithm that wrks fr different data types. Sample C++ cde fr using the templates is given belw. The GenericAdd functin adds tw numbers and returns the result. The explicit instantiatin tells the cmpiler what kinds f data types are t be assciated with the GenericAdd. In the example the functin is valid fr C++ int, flat and duble data types. A call t the functin is just the GenericAdd n matter which ne f the three C++ data type values need t be added. Cnsequently, in the eyes f a prgrammer nly ne functin exists that handles different data types. If the current cde were written in an bject riented prgramming language such as C++, cnversin t a generic cde wuld be relatively straightfrward. Hwever, a majrity f the MSCAT and MFD cde is written in Frtran 90 and unfrtunately n features such as templates are available in Frtran 90. // C++ Generic functin definitin template <class T> T GenericAdd(T a, T b) { T result; result = a + b; return result; } // Explicit instantiatin template int GenericAdd(int a, int b); template flat GenericAdd(flat a, flat b); template duble GenericAdd(duble a, duble b); 37

50 3.2 Cde Mdificatin Tw ptins were explred in rder t mdify the existing cde in t a generic cde. One was using Frtran 2003 features such as plymrphic entities which are btained using the keywrd CLASS instead f TYPE; hwever, there were tw majr drawbacks with this ptin. First, nt all Frtran 2003 features have been adpted by all cmpilers. Secnd and mre imprtantly, majr rewrite f the existing cde wuld have been required since the current cde uses a prcedural prgramming paradigm and the Frtran 2003 features fllw a mre bject riented apprach. Mrever, while researching the Frtran 2003 features it was bserved that the examples fr using the features were scant. It lead us t the cnclusin that Frtran 2003 features have nt been widely adpted yet. And it was thught unwise fr us t delve in t smething that des nt yet have wide spread use. The secnd ptin t achieve flexible cde was emulating templates in Frtran 90. The idea fr templates in Frtran 90 riginated frm [21]. The advantages in using this ptin were (a) n majr rewrite f the existing cde was required and (b) since the templates culd be implemented in Frtran 90 the prcedural prgramming paradigm f the existing cde culd be cntinued withut any prblems. The gist f [21] is explained here with an example. Cnsider the cde belw where there are tw mdules: single and duble. In each mdule there is an integer parameter prec, whse value is evaluated t 4 and 8 in the single and duble mdules, respectively, curtesy f the kind functin. Each mdule has a public subrutine name genericname and an 38

51 interface statement fr the rutine. The actual subrutine definitin resides in the functin definitin f RutineDef. Finally, bth mdules cntain include RutineDefinitin.f90 statement. The effect f the include statement is that the cmpiler replaces the include statement with the cntents f the RutineDefinitin.f90 file. mdule single integer, parameter :: prec = kind(0.0e0)! Evaluates t 4 fr single precisin public :: genericname interface genericname; mdule prcedure RutineDef; end interface cntains include "RutineDefinitin.f90" end mdule single mdule duble integer, parameter :: prec = kind(0.de0)! Evaluates t 8 fr single precisin public :: genericname interface genericname; mdule prcedure RutineDef; end interface cntains include "RutineDefinitin.f90" end mdule duble Nw, cnsider the RutineDefinitin.f90 file which has the actual definitin f the subrutine RutineDef given by subrutine RutineDef(a,b,c) implicit nne real(kind=prec), intent(in) :: a,b real(kind=prec), intent(inut) :: c : : end subrutine In the subrutine definitin the precisin f the arguments a, b, and c have been defined as prec. When the cmpiler replaces the include statement f the tw mdules with the actual subrutine definitin, prec has a different value in the 39

52 single and duble mdules, specifically 4 and 8, signifying single and duble precisin, respectively. A call t the subrutine is made by using the public name f the rutine which in this case is genericname. In the main prgram (the calling rutine/prgram) belw, bth single and duble mdules are being use d. a, b and c are real single precisin variables and d, e, f are real duble precisin variables. It is pssible t use bth single and duble mdules at the same time because generic interfaces fr genericname extend each ther. Cnsequently, during cmpilatin the first call t genericname is assciated with the public name in the single mdule and the secnd call is assciated with the public name in the duble mdule. As a result the first call evaluates the subrutine in single precisin and the secnd call evaluates the subrutine in duble precisin. Thus, ne can use the same prcedure definitin fr different data types. prgram main use single use duble real*4::a,b,c real*8 ::d,e,f : : call genericname(a,b,c) call genericname(d,e,f) : : end prgram The same templates idea was extended fr derived types, and in ding s the Linear Algebras library f Figure 3.1 was mdified extensively t make it valid fr the fur intrinsic Frtran 90 data types viz., CD, CS, RD and RS. The extensin/rewrite f the cde will be illustrated by cnsidering the 40

53 Cmplex16MatInstantiate functin mentined earlier. Accrding t its functin definitin Cmplex16MatInstantiate allcates memry fr nly a Cmplex16Mat (read as cmplex16 matrix). T understand the steps in the mdificatin cnsider the cde belw. The mdule MatrixDefinitins cntains the type declaratins f Cmplex16Mat, Cmplex8Mat, Real8Mat and Real4Mat which are thught f as matrices f the fur intrisic data types (CD, CS, RD, RS). Belw MatrixDefinitins mdule are fur mdules named CmplexDublePrec, CmplexSinglePrec, RealDublePrec and RealSinglePrec, each f which has include Surce1.F90 statement. Surce1.F90 file cntains the definitin f the mdified Cmplex16MatInstantiate rutine. Ntice that in each f the CmplexDublePrec, CmplexSinglePrec, RealDublePrec and RealSinglePrec mdules the lcal name (MatrixType) is renamed as Cmplex16Mat, Cmplex8Mat, Real8Mat and Real4Mat, respectively. mdule MatrixDefinitins type Cmplex16mat integer :: numrws, numcls cmplex*16, pinter :: blk(:,:)=>null(); end type type Cmplex8Mat integer :: numrws, numcls cmplex*8, pinter :: blk(:,:)=>null(); end type type Real8Mat integer :: numrws, numcls real*8, pinter :: blk(:,:)=>null(); end type type Real4Mat integer :: numrws, numcls real*4, pinter :: blk(:,:)=>null(); end type end mdule MatrixDefinitins 41

54 !==================================================== mdule CmplexDublePrec use MatrixDefinitins, MatrixType => Cmplex16Mat private include "Surce1.F90" end mdule CmplexDublePrec! mdule CmplexSinglePrec use MatrixDefinitins, MatrixType => Cmplex8Mat private include "Surce1.F90" end mdule CmplexSinglePrec! mdule RealDublePrec use MatrixDefinitins, MatrixType => Real8Mat private include "Surce1.F90" end mdule RealDublePrec! mdule RealSinglePrec use MatrixDefinitins, MatrixType => Real4Mat private include "Surce1.F90" end mdule RealSinglePrec Next, cnsider the definitin f the mdified Cmplex16MatInstantiate functin in Surce1.F90 file, reprduced belw. The first three lines declare genericinstantiate t be the public name f the MatrixInstantiate private rutine whse definitin is given right belw the interface. This subrutine accepts numrws (integer), numcls (integer) and A (MatrixType) as arguments and allcates memry fr a numrws-by-numcls A matrix. When the cmpiler replaces the include statement in each f the CmplexDublePrec, CmplexSinglePrec, RealDublePrec and RealSinglePrec, MatrixType in the subutine definitin is implicitly, replaced with Cmplex16Mat, Cmplex8Mat, Real8Mat and Real4Mat, respectively. This rutine als perfrms dimensin checks and cunts up a glblal memry cunter based n glbal variables such as MB_Per_Byte and data_size. The rutine als has an allcatin errr checking rutine in CheckState and bug reprting rutine in CaughtABug. 42

55 public::genericinstantiate interface genericinstantiate; mdule prcedure MatrixInstantiate; end interface cntains subrutine MatrixInstantiate(numRws, numcls, A) implicit nne integer, intent(in) :: numrws, numcls type(matrixtype), intent(inut) :: A integer err if((numrws >= 0).AND. (numcls >= 0)) then A%numRws = numrws A%numCls = numcls if ((numrws > 0).AND. (numcls > 0)) then allcate(a%blk(numrws, numcls), stat = err) call CheckState(err, 'In subrutine:matrixinstantiate') A%blk = 0. call CuntHeapMem(flat(numRws)*flat(numCls)& *data_size*mb_per_byte) endif else print *, "numrws = ", numrws, " numcls = ", numcls call CaughtABug("MatrixInstantiate: Dimensin is negative.") end if end subrutine MatrixInstantiate Sample cde that tests memry allcatin fr the fur different matrix types is given belw. In the main prgram all five mdules described abve are being use d and A, B, C and D have been declared as matrices f the fur different types. In the next few lines the same subrutine (genericinstantiate) is being called t allcate memry fr Cmplex16Mat, Cmplex8Mat, Real8Mat and Real4Mat matrices. prgram main use MatrixDefinitins! Req d fr declaring A, B, C, D belw use CmplexSinglePrec! Req d fr call t Cmplex Single instantiate use CmplexDublePrec! Req d fr call t Cmplex Duble instantiate use RealSinglePrec! Req d fr call t Real Single instantiate use RealDublePrec! Req d fr call t Real duble instantiate implicit nne type(cmplex16mat) A; type(cmplex8mat) B; 43

56 type(real8mat) C; type(real4mat) D; integer rws, cls; rws = 10; cls = 10;! Allcate memry fr 10x10 matrices call genericinstantiate(rws, cls, A); call genericinstantiate(rws, cls, B); call genericinstantiate(rws, cls, C); call genericinstantiate(rws, cls, D); end prgram T explain hw this is pssible let us revisit the definitin f MatrixInstantiate. It shuld be bserved that what was earlier a functin (Cmplex16MatInstantiate) has been rewritten as a subrutine. This is s because, t be able t use CmplexDublePrec, CmplexSinglePrec, RealDublePrec and RealSinglePrec mdules all at the same time and have genericinstantiate extend ver the fur mdules requires the cmpiler t discriminate calls t genericinstantiate based n the argument types. Indeed, this is exactly what happens when subrutines are used. In the cde abve when the first call t genericinstantiate is encuntered by the cmpiler, the call has as its arguments int, int, and Cmplex16Mat. The cmpiler can then assciate this call with the rutine fund in the CmplexDublePrec mdule as a result f the argument match. Similarly, the next three calls are assciated with the crrect rutines in the ther mdules based n the third argument, which is unique in all fur calls t genericinstantiate. Hwever, if functins were used instead f subrutines then the cmpiler wuld nt be able t make the crrect assciatin because Cmplex16MatInstantiate, r rather a generic MatInstantiate functin wuld nly accept as its arguments tw integer values numrws and numcls and return a matrix. N matter what type f matrix is returned ne wuld have a 44

57 generic functin with the same name and the same type f arguments (tw integers) in fur different mdules. This wuld be tantamunt t ne call and fur functin definitins leading t ambiguity fr the cmpiler. Thus, subrutines need t be used instead f functins fr templates in Frtran 90. The existing Linear Algebras library cntained many such prcedures that were functins and were rewritten as subrutines. After rewriting functins as subrutines interfaces were generated fr every subrutine in a fashin similar t genericinstantiate described abve. Besides peratins n matrices the Linear Algebras library als has rutines that perate n blcks and arrays f matrices. Interfaces were als intrduced t these rutines, thereby cnverting the CD Linear Algebras library t a generic Linear Algebras library that wrks with CD, CS, RD and RS data types. In ttal 38 functins were rewritten as subrutines and interfaces fr 127 subrutines were generated. In rder t cnfirm that the generic Linear Algebras library did nt intrduce any unfreseen perfrmance degradatin, cmputatinal times f a few rutines were btained using the new library and cmpared against the times btained using the ld library. 3.3 Numerical Verificatin f the Generic Linear Algebras Library Figure 3.2 (a) depicts the wall clck times fr the SVD, matrix Multiply, matrix Additin and matrix Cpy peratins btained by averaging the individual times calculated in 20 trials. The times were btained n a machine with Intel Cre 2 CPU (1.86 GHz) with 8 GB memry. All peratins were perfrmed n square matrices f rder The cmparisn is nly in CD precisin because the ld library culd perate n nly CD precisin matrices. The red clred bars 45

58 shw the wall clck times fr the peratins using the generic Linear Algebras library and the blue bars depict the times fr the peratins using the ld library. Frm Figure 3.2 (a) it is bserved that the SVD peratin takes apprximately 52 secnds in bth the ld and the new libraries. Figure 3.2 (b) is the same as Figure 3.2 (a) but zmed in fr better reslutin f the Multiply, Add and Cpy times. Frm Figure 3.2 (b) it is nticed that the ld and the new multiply rutines take apprximately 5 secnds and the Add and Cpy rutines take less that 0.1 secnds t cmplete. Table 3.1 shws the relative RMS errr defined as Relative RMS N x x i, ldlib i, newlib i1 Errr (3.1) N 2 fr the SVD, Multiply, Add an Cpy peratins, where N is the number f trials (20) and x i is the time required fr the peratin in the th i trial. Frm Table 3.1 and Figure 3.2 it can be safely cncluded that there is n significant perfrmance degradatin in the newer library. Any discrepancies can be attributed t the uncertainty in the elapsed time calculatin rutine. 46

59 60 6 S e c n d s S e c n d s SVD Multiply Add Cpy 0 Multiply Add Cpy Cmplex16 Old Cmplex16 New Cmplex16 Old Cmplex16 New (a) (b) Figure 3.2: Wall clck time cmparisn f the rutines in the existing library and the new generic library. (a) Cmparisn f the SVD, matrix-matrix multiplicatin, matrix-matrix additin and matrix cpy. Cmplex duble precisin rutines are cmpared. (b) Same as (a) but zmed in fr better reslutin. All peratins perfrmed n square matrices f rder Table 3.1: Relative RMS errr cmparisn ( time) fr square matrices f rder Operatins perfrmed n cmplex duble precisin data. Operatin Relative RMS Errr SVD Multiply Add Cpy Fr cmpleteness the wall clck times btained using the new generic Linear Algebras library fr SVD, matrix Multiply, matrix Add and matrix Cpy in all fur data types (CD, CS, RD, RS) are shwn in Figure 3.3 (a). The blue, dark red, light green and purple bars shw the wall clck times fr CD, CS, RD and RS precisins, respectively. All peratins were perfrmed n square matrices f rder As expected the cmputatinal times fr CD, CS, RD and RS have a descending trend, since each CD, CS, RD and RS variables are represented 47

60 using 16 ( 8 fr real part and 8 fr imaginary), 8 (4 fr real and 4 fr imaginary), 8 and 4 bytes, respectively. Figure 3.3 (b) is the same as (a) but zmed in fr better reslutin. S e c n d s CD CS RD RS S e c n d s Multiply Cpy Add CD CS RD RS (a) Figure 3.3: Wall clck times f the peratins using the new generic linear algebras library. (a) Times fr SVD, matrix-matrix multiplicatin, matrix-matrix additin and matrix cpy peratins in cmplex duble (CD), cmplex single (CS), real duble (RD) and real single (RS) precisins. (b) Same as (a) but zmed in fr better reslutin. All peratins perfrmed n square matrices f rder (b) Once Linear Algebras library was mdified in t a generic Linear Algebras library, ne culd fllw the same prcedure and cnvert the rest f the libraries in Figure 3.1 (which shws the cde structure) t generic libraries. This is pssible because the data structures in the ACA, the MLSSM and the MFD libraries are derived frm the fur (CD, CS, RD and RS) basic data structures f the generic Linear Algebras library. Therefre, every rutine in the ACA, the MLSSM and the MFD libraries wuld have fur mdules with interfaces t the fur basic data types. In ther wrds, the cmpiler wuld implicitly have fur versins f the entire cde. Hwever, it was nticed that the CD, CS, RD and RS versins f the entire algrithm need nt exist at the same time in the same 48

61 executable. Therefre, what was needed was a flexible way t cnvert the cde t the required data type and precisin. This meant that the generic Linear Algebras library wuld be a cnstant and the rest f the libraries that depended n the generic Linear Algebras library culd be made specific depending n the user s requirement. Perl scripts culd cnveniently d this jb. 3.4 Perl Scripting Using Perl and regular expressin (regex) matching apprpriate phrases can be replaced in the surce cde and cmpiled separately t prduce CS, RD r RS libraries. The assumptin in using Perl scripts is that the scripts will cnvert CD precisin libraries t thers. The new mdified cde structure and dependencies are shwn in Figure 3.4, where Perl Scripts affects the Tree, the ACA, the MLSSM and the MFD libraries by cnverting them in t any f the ther data type and precisin libraries. An example cde f hw Perl scripts perfrm regex match and replace is given belw. The variable $datatype is input frm the user that is used t decide hw t replace the lines. If the user decides that the Tree, the ACA, the MLSSM and the MFD libraries need t be cnverted t RS precisin, then $datatype matches real4 and all instances f Cmplex16Mat (scripts can cnvert nly CD t the ther types) n a single line are replaced with Real4Mat. Simliar lgic applies if the user selects the ther data type and precisins. Thus, every line f the surce cde in the ACA, the Tree, the MLSSM, and the MFD libraries can be searched and replaced with the apprpriate phrases. Similarly, ther phrases such as cmplex*16 can als be matched and replaced with any ne f 49

62 cmplex*8, real*8 and real*4. Once all the surce cde has been mdified, it can be cmpiled t generate CS, RD and RS libraries withut the need fr all fur cmbinatins f data type and precisin existing simultaneusly. if ($datatype eq 'real4') { $MdifiedLine =~ s/cmplex16mat/real4mat/gi; } elsif ($datatype eq 'real8') { $MdifiedLine =~ s/cmplex16mat/real8mat/gi; } elsif ($datatype eq 'cmplex8') { $MdifiedLine =~ s/cmplex16mat/cmplex8mat/gi; } T recap, the cmplex duble precisin nly Linear Algebras library was mdified by using templates in Frtran 90 t btain a flexible generic Linear Algebras library. Other libraries can be cnverted frm cmplex duble precisin t any f cmplex single, real duble r real single precisin just by replacing a few phrases using Perl scripts. The cde wuld still be valid since mst f the time the algrithms in thse ther libraries call the rutines in the generic Linear Algebras library. And because the generic Linear Algebras library can handle any f the fur types f matrices the cde des nt break. Thus, simple slutins were btained fr the prblem f making the entire cde flexible. 50

63 Universals Generic Linear Algebras Perl Scripts MSCAT Tree MFD-MSCAT Cnnectr ACA MLSSM MFD Library Figure 3.4: Mdified cde structure and dependencies 51

64 4 Perfrmance f OL-LOGOS and NL-LOGOS Factrizatin As mentined previusly, nce the system matrix is btained via the discretizatin f the integral equatins, a linear system f equatins needs t be slved. The slutin can be achieved via direct slvers such as LU factrizatin r the direct slvers that use the LOGOS mdes. Direct slvers typically have a 3 2 time cmplexity f O N and a memry cmplexity f O N 2 the NL-LOGOS based slver is bunded by N. The efficiency f O because f the bundaries separating the nn-verlapped surce mdes [21]. The efficiency f the OL- LOGOS based slver is bunded by O N [21]. The efficiencies mentined abve f the NL and OL LOGOS based slvers are expected fr prblems invlving lw frequencies. Lw frequency prblems are prblems where the maximum linear dimensin f the scatter is much less than the wavelength f the harmnic excitatin. The fllwing simulatins were carried ut using the new generic Linear Algebras library. 4.1 Numerical Results: Factrizatin Cmplexities Scattering by thin dielectric spherical shells frmulated using the vlume electric field integral equatin and discretized using the Nyström methd is presented here. The efficiency f the NL-LOGOS factrizatin is cmpared against the OL-LOGOS factrizatin fr different tlerances and different dielectric cnstants. The Radar Crss Sectin (RCS), defined belw, f the scatterers is als cmpared against the analytical slutin. Balanis in [10] defines RCS as the area intercepting the amunt f pwer that, when scattered 52

65 istrpically, prduces at the receiver a density that is equal t the density scattered by the actual target. If the transmitter and the receiver are lcated at the same pint then the RCS is knwn as mnstatic RCS and it is knwn as bistatic RCS if they are lcated at different pints. The RCS plts presented here are all bistatic RCS unless therwise specified. The simulatin frequency is fixed at 1 khz. Dielectric shells were chsen such that the uter radius f the shells is kept cnstant at 1 and the thickness f the shell is varied accrding t 0.13, 0.065, 0.033, , m. Cnsequently, with decreasing thickness but cnstant uter radius, the number f unknwns N increases. This als means that there is nly a single layer f meshed elements as shwn in Figure 4.1. In this figure, starting frm the tp left and mving clckwise the number f unknwns is 5184, 20736, 80640, and Bth NL-LOGOS and OL-LOGOS factrizatins were carried ut fr cnstant relative permittvities ε r = 4, 16 and 64. The factrizatins were carried ut in real single precisin fr factrizatin tlerances f 10e -2 and 10e- 3. The results were btained n a machine with Intel Xen CPU with six cres (3GHz) and 64 GB RAM. The peak memry usage during the factrizatin and the factrizatin time were extracted frm the internal cunters in the cde. 53

66 Figure 4.1: Meshed dielectric shells Figure 4.2 shws the peak memry used during the LOGOS factrizatin when the dielectric cnstant is 64. Fr a tlerance f 10-3 (10-2 ) the memry scaling fr NL-LOGOS factrizatin is apprximately ( N ) ( ( N ) ) and the OL-LOGOS factrizatin scaling is apprximately ( N ) ( ( N ) ). Figure 4.3 displays the ttal time scaling fr the NL-LOGOS and the OL-LOGOS factrizatins fr the same case as in Figure 4.2. Fr a tlerance f 10-3 (10-2 ) the NL-LOGOS factrizatin scales at apprximately ( N ) ( ( N ) ) and the OL- LOGOS factrizatin scales at apprximately ( N lg( N)) ( ( N lg( N)) ). It is nted that the time fr OL-LOGOS factrizatin becmes less than the time fr NL-LOGOS factrizatin as the number f unknwns increases. Therefre, we see an intersectin pint where the OL-LOGOS factrizatin time equals the NL- LOGOS factrizatin time. 54