Development of 3D-CFD code for Heat Conduction Process using CUDA. Dissertation

Transcription

1 Developmet of 3D-CFD code for Heat Coductio Process usig CUDA Dissertatio Submitted i partial fulfilmet of Evaluatio of Master of Techology Computer Egieerig By Yogesh Bhadke MIS No: Uder the guidace of Dr. Vadaa Iamdar ad Dr. Vikas Kumar ( C-DAC ) Dr. Supriyo Paul ( C-DAC ) Departmet of Computer Egieerig ad Iformatio Techology College of Egieerig, Pue Shivajiagar, Pue Jue-2014

2 DEPARTMENT OF COMPUTER ENGINEERING AND INFORMATION TECHNOLOGY, COLLEGE OF ENGINEERING, PUNE CERTIFICATE This is to certify that the dissertatio titled Developmet of 3D CFD Code for Heat Coductio Process Usig CUDA has bee successfully completed By Yogesh D. Bhadke MIS No: ad is approved for the partial fulfillmet of the requiremets for the degree of Master of Techology i Computer Egieerig Dr. Vadaa Iamdar Project Guide, Departmet of Computer Egieerig ad Iformatio Techology, COEP, Pue Dr. J. V. Aghav Head, Departmet of Computer Egieerig ad Iformatio Techology, COEP, Pue Dr. Vikas Kumar Project Guide, C-DAC, Pue Jue 2014 Dr. Supriyo Paul Project Guide, C-DAC, Pue

3 Ackowledgemet I would like to express my gratitude to all those who have provided timely guidace ad helpig had i makig my project a success. I m thakful to Ceter for Developmet of Advaced Computig (C-DAC) Pue for selectig me for project. I m thakful to Dr.Vikas Kumar for givig me opportuity to work for the project. I m deeply idebted to project guide Dr.Supriyo Paul whose guidace, stimulatig suggestios ad ecouragemet helped me i all the time of durig project discussios & codig phase. I wat to thak the Departmet of Computer egieerig, College of Egieerig, Pue for givig me opportuity to do this project as part of our fourth semester, to do the ecessary research work ad to use departmetal resources. I exted my sicere thaks to project guide Dr. Vadaa Iamdar who ecouraged me for this project. From this project work my guides have brought the best out of me. I eve wat to thak my departmet colleagues for all their help, support, iterest ad valuable hits. ii

4 Abstract Heat coductio is atural pheomeo which is govered by three dimesioal, trasiet partial differetial equatio. The partial differetial equatio ca be solved by may umerical method such as fiite differece method, fiite volume method, fiite elemet method etc. These methods require heavy computatio to solve the system of algebraic equatios. Graphics processig uit (GPU) ca be used to hadle the computatio of CPU as a coprocessor so the GPU will save a lot of time for computatio. The domiat proprietary framework for GPU computig is CUDA, provided by NVidia. It ca be used to solve computatio itesive task. The objective of this work is to develop a heat coductio code o CUDA platform, which will solve the system of algebraic equatios usig GPU framework. iii

5 List of Figures Fig. 1.1 Moder GPU architecture 5 Fig. 4.1 Computatioal Mesh where width ad height of cell is show by x ad y 14 Fig. 5.1 Flow Diagram for ADI ad Douglus method 17 Fig. 5.2 Physical domai ad physical domai after grid geeratio 21 Fig. 5.3 Set of XY-plae of physical domai chopped across Z- directio. 21 Fig. 5.4 Computatio of Tridiagoal system for row is assiged to 1 thread for X-Sweep. 23 Fig. 5.5 Computatio of Tridiagoal system for colum is assiged to 1 thread for Y-Sweep 23 Fig. 6.1 Time compariso betwee ADI, Douglus ad ADI CUDA 25 Fig. 6.2 Time compariso for Grid size 25 Fig. 6.3 Speedup 26 Fig. 6.4 Heat map for iitial stage 26 Fig. 6.5 Heat map for itermediate stage 27 Fig. 6.6 Heat map at covergece state 27 Fig.6.7 Validatio of ADI ad Douglus serial code 28 Fig. 6.8 Validatio of ADI-CUDA code. 28 iv

6 List of Tables Table 6.1 System specificatio 24 Table 6.2 Test data 24 Table 6.3 Executio time, No of iteratio for covergece i ADI, Douglus ad ADI-CUDA method 24 v

7 Cotets Ackowledgemet Abstract List of Figures List of Tables ii iii iv v 1 Itroductio Heat Trasfer Heat Coductio Heat Coductio Equatio Steady ad usteady heat trasfer Multi-dimesioal teat trasfer Boudary ad Iitial Coditios GPU GPU Computig GPU Architecture CUDA Memory hierarchy CUDA threads, block, grid Kerel 7 2 Literature Survey Literature review 8 3 Problem Statemet Research Gap Problem Statemet Motivatio Objectives 10 4 Methodology Geeral Steps Partial Differetial Equatio Numerical method Fiite differece method 13 vi

8 5 Implemetatio Alteratig Directio Method Douglus Method Thomas Algorithm ADI-CUDA Mappig strategy 21 6 Results 24 7 Coclusio ad Further work Coclusio Further work 29 Refereces vii

9 Chapter 1 INTRODUCTION 1.1 Heat Trasfer Heat is the form of eergy that ca be trasferred from oe system to aother as a result of temperature differece. A thermodyamic aalysis is cocered with the amout of heat trasfer from oe equilibrium state to aother equilibrium as a system udergoes a process. The Heat trasfer is a stream of thermal egieerig that cocers the creatio, use, coversio, ad trasactios of thermal eergy ad heat amid physical systems. Heat trasfer is categorized ito assorted mechaisms such as thermal coductio, thermal covectio, thermal radiatio ad trasfer of eergy by phase chage [12]. As these mechaisms have differet characteristics, they occur simultaeously i the same system. The basic coditio for the heat trasfer is temperature differece. There ca be o et heat trasfer amid two mediums that are at the alike temperature. The rate of heat trasfer i a particular directio depeds o the magitude of the temperature gradiet (the temperature differece per uit legth or the rate of chage of temperature) i that directio. The larger the temperature gradiet, the higher the rate of heat trasfer [12]. 1.2 Heat Coductio Heat trasfer by coductio is the flow of thermal eergy iside solids ad o-flowig fluids, drive by thermal o-equilibrium usually measured as a heat flux (vector), i.e. the heat flow per uit time at a surface [12]. Coductio is the trasfer of eergy from the more eergetic particles of a substace to the adjacet less eergetic oes as a result of iteractios betwee the particles. Coductio ca occur i solids, liquids, or gases. I gases ad liquids, coductio occurs due to the diffusio of the molecules across their radom motio. I solids, it is due to the combiatio of vibratios of the molecules i a lattice ad the eergy trasferred by free electros. The rate of heat coductio across a medium depeds o the geometry of the medium, its thickess, ad the physical of the medium, as well as the temperature differece across the medium [12]. 1

10 1.3 Heat Coductio Equatio Heat coductio equatio is the parabolic partial differetial equatio (PDE). Heat trasfer have directio as well as magitude. The rate of heat coductio i a particular directio is proportioal to the temperature gradiet. Heat coductio i a medium is three-dimesioal ad depeds o time. That is, u=u(x, y, z, t) here u is the temperature fuctio over space variable x, y, z ad time t. The temperature i a medium varies alog positio as well as time [12] Steady ad usteady heat trasfer Heat trasfer problems are usually categorized as beig steady or trasiet. Steady state coductio is a form of coductio that happes whe the temperature differece is costat [12], so that after a equilibrium the temperatures i the object does ot chage ay further. I steady state coductio, the heat goig i the object is equal to heat comig out. I steady state coductio, temperature is idepedet of time ad ca be represeted i the form like u=u(x, y, z). Trasiet coductio occurs as the temperature iside a object chages as a fuctio of time. I trasiet coductio temperature fuctio ca be represeted like u=u(x, y, z, t) Multidimesioal heat trasfer Heat trasfer problems are additioally categorized as beig oe-dimesioal, twodimesioal, or three-dimesioal, depedig o the relative magitudes of heat trasfer rates i various directio ad the level of accuracy required [13]. The temperature dissipatio across the medium at a particular time ca be delieated by a set of three coordiates x, y, ad z i the rectagular or Cartesia co-ordiate system [12]. Where α = k ρc p u α 2 u = 0 t x (1.1) 2 2

11 k is the thermal coductivity ρ is the desity of the material ad c p is the specific heat capacity The equatio i two-dimesios takes the form: u u t α( u x 2 y Ad i three dimesios: u u t α( u x 2 2) = 0 (1.2) + 2 u y 2 z 1.4 Boudary Coditios ad Iitial Coditio 2) = 0 (1.3) The differetial equatios do ot iclude ay iformatio regardig the coditios o the surfaces such as the exteral temperature or a specified heat flux. Yet we uderstad that the heat flux ad the temperature dissipatio i a medium deped o the coditios at the surfaces ad the descriptio of a heat trasfer problem i a medium is ot fiished uless a maximum descriptio of the thermal coditios at the boudig surfaces of the medium is specified. The mathematical expressios for the thermal coditio at the borders are called as Boudary coditios [12]. To delieate a heat trasfer problem completely, two boudary coditios have to be give for every sigle directio of the coordiate system where the heat trasfer is sigificat. Therefore, we have to give two boudary coditios for oe-dimesioal problem, four boudary coditios for two-dimesioal problem, ad six boudary coditios for three-dimesioal problems [13]. For example, for a three dimesioal box of size L3 the boudary coditios ca described as: T(x=0, y, z, t) = Tx1 T(x, y=0, z, t) = Ty1 T(x, y, z=0, t) = Tz1 T(x=L, y, z, t) = Tx2 T(x, y=l, z, t) = Ty2 T(x, y, z=l, t) = Tz2 Iitial coditio is a mathematical expressio for the temperature dissipatio of the medium at the begiig of the simulatio. Note that we demad merely oe iitial coditio for a heat coductio problem as the coductio equatio is first order i time. 3

12 For example u(x, y, z, t=0) = 0 Aalysis of multidimesioal heat coductio PDE is complicated ad requires umerical aalysis by high ed computer with great computatio power. So ow a days there is hurdle i icrease i computatio power (GHz) of workstatio so oe thik of usig the Graphics Processig Uit to offload the heavy computatio task [13]. 1.5 GPU GPU stads for Graphics Processig Uit (GPU) a chip with computig capabilities hosted o a video card mostly used for 3D acceleratio i games or codec acceleratio i movie editig. NVidia pioeered GPU computig i Now a days GPU are power eergy-efficiet datacetres i uiversities, govermet labs, uiversities, eterprises ad small ad medium busiess aroud the world GPU computig GPU-computig is the adoptio of the GPU plus CPU to speed-up egieerig, eterprise ad scietific applicatios. GPU speed-up the applicatio by offloadig the heavy computatio task o the GPU whereas rest part of the executio is hadled by the CPU. If we compare CPU ad GPU based o the umber of cores the Itel core i7 CPU have 6 cores ad 1.17 billio trasistors ad NVidia GTX 580 SC GPU have 512 cores ad 3 billio trasistors. A CPU cosists of the few core optimised for the sequetial processig while GPU is built from thousads of smaller ad efficiet cores desiged for processig multiple tasks parallelly. As compared to CPU, GPU s are optimised for data-parallel throughput computatio, architecture of GPU is tolerat of memory latecy ad more trasistors are dedicated for computatio [17]. 4

13 1.5.2 GPU architecture Fig. 1.1 Moder GPU architecture GPUs are developed as array of threaded streamig multiprocessors. Every streamig multiprocessor cotais umber of streamig processors that share cotrol logic ad istructio cache. Now, every sigle GPU comes alogside graphics double data rate (GDDR) DRAM, that is called as globe memory as show i Fig. 1.1 Moder GPU architecture. This global memory differs from the oe i CPU. I 2006, NVIDIA idustrialized parallel computig framework CUDA to use GPU effectively for geeral purpose computatios [17]. 1.6 CUDA CUDA is a scalable parallel programmig model ad software eviromet for parallel computig. CUDA stads for Compute Uified Device Architecture. It is miimal 5

14 extesio to the C/C++ eviromet. I CUDA cotext, GPU is addressed as the Device CPU as host ad Kerel is the fuctio that rus o the device. Parallel portios of a applicatio are executed o the device as kerels, oe kerel is executed at a time ad may threads execute each kerel. CUDA threads are extremely lightweight have very little creatio overhead. I CUDA programmig eviromet we ca use thousads of thread to achieve maximum efficiecy. I CUDA parallel computig, may threads execute i parallel ad all execute the same istructio at the same time [17] Memory hierarchy Every CUDA eabled GPU provides several differet types of memory. These differet types of memory have differet properties such as access latecy, address space, scope ad lifetime. Registers are the fastest memory accessible without ay latecy o each clock cycle just as o a regular CPU.A thread's register caot be shared with other kerels. Shared Memory is comparable to L1 cache memory o a regular CPU. It resides close to the multiprocessor ad very little access time. It is shared amog threads from the same block. Global Memory available o the device but off-chip from the multiprocessors so that access to the global memory ca be 100 times greater tha shared Memory. All threads from all blocks have access to the Global memory ad used for iter-block commuicatio betwee thread. Local Memory is thread specific private memory stored where global memory is stored. Arrays declared iside a thread are stored i local memory. Costat memory resides off-chip from multiprocessors ad is mostly readoly. The host code writes to the device s costat memory before lauchig the kerel ad the kerel may the read this memory. All thread have access to the shared memory. Costat memory access is cached so that subsequet reads from costat memory ca be very fast. Texture memory is aother variety of read-oly memory that ca improve performace ad reduce memory traffic whe reads have certai access patters. 6

15 1.6.2 CUDA threads, blocks, grid Threads i CUDA are extremely lightweight ad they have their ow registers ad program couter [16]. All threads share a memory address space called Global Memory ad threads withi the same block share access to a very fast shared memory that is more limited i size while threads across the block uses very slow global memory for commuicatio. Withi the same block thread share the istructio stream ad execute istructio i parallel. Whe the thread executio diverges the the differet braches of executio are ru serially util the diverget sectio is completed the at that poit all threads i the block ca resume their executio i parallel agai. CUDA devices ru may threads simultaeously. For example, NVidia Tesla C2075 has 14 multiprocessors, each of which have 32 cores so that 448 threads may be ruig simultaeously. Threads ruig uder CUDA must be grouped uder blocks, a block ca hold at most 512 or 1024 threads. Now a days GPU based o kepler architecture ca support 2048 threads also. Grid is the blocks of threads which may be oe, two or three dimesioal as programmer prefers Kerel Kerel is the mai uit of work that the mai program ruig o the host computer offloads to the GPU for the computatio o the device. I CUDA, lauchig a kerel requires specifyig three thigs The dimesios of the grid The dimesio of the Blocks The kerel fuctios to ru o that device. Kerel fuctio is specified by declarig global i the code ad a special sytax is used i the code to lauch these fuctios o the GPU with specificatio of the block ad grid dimesios. These kerel fuctios acts as the etry poit for the GPU computatios ad kerel fuctio ca call aother fuctio called as device fuctio declared usig device keyword. Both fuctio have the retur type as the void. 7

16 Chapter 2 LITERATURE SURVEY 2.1 Literature Review Federio E. Teruel ad Rizwa-uddi implemeted a parallel umerical code for the simulatio of the heat coductio betwee packed spheres [9]. The complex geometry motivates the authors to build the tool because modellig of such geometry is computatioally expesive. This tool helps to compute the temperature distributio of ay kid of arragemet or cofiguratio of packed spheres which are costraied to differet boudary coditio. The authors used the fiite volume method for discretizatio. The authors created a parallel versio of the code because umber of sphere to be simulated ca become very large icreasig the computatioal load. Each sphere is hadled by a sigle processor. The authors observed that the parallel code was givig good performace whe the umber of spheres was greater tha 50. Yuzhi Su ad Idrek Wichma i a techical ote [11] preseted the theoretical solutio to the heat coductio i oe-dimesioal three-layer composite slab. Eige fuctio expasio ad fiite differece method was used to fid out ad compare the solutio of the above metioed heat coductio problems. Xiaohua Meg et al [1] desiged a CPU based heat Coductio algorithm by leveragig CUDA. The algorithm implemets the calculatio of displacemet ad velocity of each particle o GPU i parallel. To simulate process of oe-dimesioal heat coductio o the oe-dimesioal particles structure system they implemeted CPU based serial algorithm of the Ruge-kutta method. I parallel versio of the algorithm, they used the cellular elemet method to orgaize the data i parallel fashio. This method divides the computatioal domai ito a series of thread grids, each grid cotais multiple particles ad each particle's state is oly relevat with two adjacet particle's state. Their parallel algorithm process divided ito three parts: 1.Iitialize ad read each particles state's iformatio, which is doe o the CPU. 2. Establish the mappig from particles to thread grid. This part is also completed i CPU. 3. State of each particle is calculated. This step is performed o GPU. The third part is very time cosumig (aroud 80% of the total time). 8

17 Cohe ad Molemaker [3] implemeted a secod order fiite volume code for CFD simulatios o CUDA. They studied buoyacy drive flow with this implemetatio ad observe over 8x performace gai o a sigle GPU over a 8-core CPU based system. Jocobse et. al. [4] studied a mixed MPI-CUDA implemetatio of icompressible flow computatio o a GPU cluster. They implemeted a dual-level parallelism to solve Navier-Stokes equatio for simulatig buoyacy drive flow. CUDA is used for fie grai data-parallelism executig o each GPU ad MPI for coarse grai parallelism across the whole cluster. They observe over 130x performace gai usig 128 GPUs o 64 odes over a CPU based implemetatio. Tomasz P. Stefaski, Timothy D. Drysdale [7] implemeted the 3D ADI FDTD method usig graphics processig uit to accelerate the performace. They used parallel cyclic reductio algorithm to solve the tridiagoal system of equatios. The performace obtaied from parallel implemetatio is 8 times that of the serial implemetatio of the 3D ADI-FDTD method. 9

18 Chapter 3 Problem Statemet 3.1 Research Gap Accordig to literature review may authors have tried to implemet the heat coductio equatio i oe-dimesioal ad i two-dimesioal they also got good results. Implemetatio i the three-dimesioal ad i multidimesioal is much complicated as compared to the oe or two dimesioal heat coductio equatio. Oe author headed for the three-dimesioal heat coductio equatio ad implemeted the parallel algorithm o the GPU, he used the parallel cyclic reductio algorithm to solve the tridiagoal system of equatios ad the performace received by him was 8X. As I m also headig towards the implemetatio of three-dimesioal heat coductio equatio my goal is to desig a effective parallel program strategy which ca best utilise the memory hierarchy i CUDA, fie-graied (thread) ad coarse graied (block) parallelism. I eed to select the best tridiagoal solver accordig to my parallel program strategy so that my performace should be at least greater tha 8X. 3.2 Problem Statemet To develop a parallel CFD code for three-dimesioal trasiet Heat coductio process o the CUDA platform. 3.3 Motivatio Basically I was curious about parallel computig ad wat to lear the same so I selected the NVidia s CUDA framework for parallel computig because NVidia is the pioeer i the utilisig the GPU for geeral purpose parallel computig. Oe of the reaso for selectig Computatioal Fluid Dyamics (CFD) domai is that CUDA has major applicatio i the CFD domai. Problems i the CFD ivolves huge computatio task so these problem ca be well efficietly solved by the combiatio of the CPU ad GPU. 3.4 Objectives To examie the best algorithm to solve partial differetial equatio. May fiite differece method are available to solve the partial differetial equatio like explicit method, implicit method, semi implicit method. Choose the best amog them to solve the equatio. 10

19 To develop a sequetial code for three-dimesioal trasiet heat coductio equatio i C laguage. To modify the sequetial code ito parallel code to work o CUDA platform. Develop a ew parallel program strategy to implemet the parallel program i cotext of the CUDA framework to yield the best performace. To validate the above code with a stadard CFD solver. Accuracy of the results obtaied from the serial ad parallel code must be validated with the referece results obtaied from the stadard CFD solver. 11

20 Chapter 4 METHODOLOGY 4.1 Geeral Steps To solve ay CFD problem followig geeral steps are used. 1. The Geometry (physical bouds) of the problem is defied. 2. The problem domai is divided ito discrete cells called as odal etwork or mesh. The mesh may be uiform or o-uiform. 3. At boudary grid poits, apply the boudary coditios. Ay derivatives i boudary coditios are replaced by oe sided fiite differece approximatios ivolvig values at boudary ad iterior odes. 4. Collect the algebraic coditios at all the gird poit iterior as well as boudary to obtai a system of algebraic equatios i terms of ukow values of the variable at these odes. 5. Solve the resultig equatios i terms of ukow values of the variable at these odes. The solutio obtaied at the grid poits ca be iterpolated ad processed to obtai the desired physical quatities i the so-called post processig step of the simulatio. 4.2 Partial Differetial Equatio Partial Differetial Equatio (PDE) is a differetial equatio that cotais the umber of ukow multivariable fuctios. A partial differetial equatio is the equatio which cotais the partial derivative show i the Eq. (1.1) i which u is treated as the fuctio of x ad t. Partial differetial equatio are ofte used to costruct model of the most basic theories uderlyig physics ad egieerig [14]. Partial differetial equatio are used to model the multidimesioal systems. The Heat equatio is a parabolic partial differetial equatio which represets the variatio of temperature with respect to time over a give regio. 12

21 4.2.1 Numerical methods Two ways are available to solve partial differetial equatios oe is by aalytical way ad other is by umerical Method. Heat coductio problems with simple physical bouds are solved i aalytical way but i reality the problem ivolves the complex geometry with complex boudary coditios which caot be solved by aalytical method. I that case high performace computers are used to get accurate approximate solutio by usig umerical method. I aalytical method, goverig differetial equatio are solved with boudary coditios which yields the temperature fuctios at every poit i the medium. I the umerical method, differetial equatio are substituted by the algebraic equatio ad for the ukow temperature values i the medium, algebraic equatios are solved simultaeously. Numerical formulatio for the heat coductio problem ca be obtaied by the fiite differece method, fiite volume method ad fiite elemet method. Each method have its ow pros ad cos ad each used i practice Fiite differece method I every umerical method, cotiuous partial differetial equatios are replaced by the discrete approximatios [14]. Here discrete coveys the meaig that solutio is kow oly for defiite umber of poits i the physical domai. These umber of defiite poits i the domai is selected by the user or programmer of the method. Icrease i the umber of discrete poits i the domai icreases the accuracy ad resolutio of the umerical solutio. The set of discrete poits i the physical domai where discrete solutio is computed is called as Mesh.These poits are called as the odes ad if adjacet poit are coected by the lies the the resultig structure looks similar to the et or mesh. Δx, Δt are the key parameters i the mesh which are local distace betwee adjacet poits i the space ad local distace betwee the adjacet time steps. 13

22 Fig. 4.1 Computatioal Mesh where width ad height of cell is show by x ad y. I Fig. 4.1 two-dimesioal plae is divided across x ad y directios with width ad height x ad yfor cell (i, j) respectively. The temperature at the cell (i, j) is deoted as the ui, j. I Fiite differece method, each derivative of partial differetial equatio is substituted by its trucated Taylor series expasio while equatio is discretized term by term. Time derivative is liear u = u(t+δt) u(t) t Δt (4.1) For Spatial order derivative at least secod order developmet is eeded to approach secod-order derivatives 2 u = u(x+δx) 2u(x)+u(x Δx) t 2 (Δx) 2 (4.2) First order derivatives are usually approximated by the cetral differece to avoid bias whe selectig betwee forward differece ad backward differece. u = u(x+δx)+u(x Δx) t 2(Δx) (4.3) 14

23 Whe regular mesh is used, fiite differece method produces the highly structured systems of equatios. Major advatage of fiite differece method is that the simple formulatio of method. Disadvatage is that the method demads the simple geometry with a structured grid i.e. method becomes complicated i a o-rectagular geometries. Fiite differece method ca be implemeted i explicit or implicit maer, explicit method is relatively simple to set up ad implemet tha implicit method. I explicit method there is stability costraits over time step which i some cases results i selectig the very small time step which results i loger computatio time. I case of implicit approach, stability costraits over time step is maitaied over large values of time step which results i lesser computatio time. Implicit method more complicated to program ad the set up. Fiite differece method ca also be implemeted from Crak-Nicolso method which is semi-implicit method because it takes the average of the explicit ad implicit method. Whe the Crak-Nicolso method used for multidimesioal heat equatio, the umber of ukow i differece equatio teds to icrease with the dimesio so the differece equatio becomes more complicated ad time cosumig to solve. So solutio of the problem is to use splittig method called ADI which produces the tridiagoal system of equatio i each split which ca be efficietly solved by Thomas algorithm. 15

24 Chapter 5 IMPLEMENTATION 5.1 Alteratig Directio Implicit Method I this sectio we show how ADI method is used to solve a three-dimesioal heat coductio equatio. We also itroduce TDMA algorithm which is uderlyig basic block of ADI method. For the three-dimesioal heat equatio (Eq. (1.3)) the differece equatio is give as +1 u i,j,k ui,j,k t = u +1 i+1,j,k u i,j,k +ui 1,j,k ( x) 2 + u +1 i,j+1,k u i,j,k +ui,j 1,k + ( y) 2 +1 u i,j,k u i,j,k +ui,j,k 1 (5.1) ( z) 2 Whe the set of simultaeous equatio is solved usig Crak-Nicolso method for two dimesioal differece equatio (Eq. (1.2)) which results i coefficiet matrix which is peta-diagoal. The solutio for peta-diagoal system of equatios is very time cosumig. I case of three-dimesioal heat equatio (Eq. (1.3)), correspodig solutio of differece equatio results i coefficiet matrix which is hepta-diagoal i ature whe solved usig Crak-Nicolso method which is also very time cosumig. Oe way to overcome this is to use the splittig method. This method is kow as the Alteratig Directio Method or ADI. I ADI maily the computatio is split ito the two steps i case of two-dimesio case ad i three steps i case of three-dimesioal case. I case of three-dimesioal case, i first step implicit method is applied i X-directio ad explicit method i Y-directio ad Z directio producig a itermediate solutio. I secod step, implicit method is applied i Y-directio ad explicit method i X-directio ad Z-directio. I third step, implicit method is applied i Z-directio ad explicit method is applied i X-directio ad Y-directio. Flow diagram for ADI is show i Fig

25 Start Setup the Grid Iitialise ad apply the boudary coditio for the Grid X-sweep Y-sweep TDMA Z-sweep No Covergece? Yes Exit Fig.5.1 Flow Diagram for ADI ad Douglus method The fiite differece equatio of model equatio i the ADI formulatio are +1/3 u i,j,k ui,j,k t 3 +2/3 +1/3 u i,j,k ui,j,k t /3 u i,j,k ui,j,k t 3 = [ δ x 2 +1/3 u i,j,k ( x) 2 = [ δ x 2 +1/3 u i,j,k ( x) 2 = [ δ x 2 +2/3 u i,j,k ( x) 2 + δ y 2 u i,j,k ( y) 2 + δ y 2 +2/3 u i,j,k ( y) 2 + δ y 2 +2/3 u i,j,k ( y) δ z 2 u i,j,k ] ( z) (5.2) 2 + δ z 2 +1/3 u i,j,k ] ( z) 2 (5.3) + δ z 2 +1 u i,j,k ] ( z) (5.4) 2

26 Where δ 2 x u i,j,k δ 2 y u i,j,k δ 2 z u i,j,k = u i+1,j,k = u i,j+1,k = u i,j,k+1 2 u i,j,k 2 u i,j,k 2 u i,j,k + u i 1,j,k + u i,j 1,k + u i,j,k 1 Boudary coditios are applied o the domai boudaries ad for ay typical ode (i, j, k) the algebraic equatio is give as d1 u +1/3 i+1,j,k + (1 + 2d1)u +1/3 i,j,k d1u +1/3 i 1,j,k = (1 2d2 2d3)u i,j,k + d2( u i,j+1,k + u i,j 1,k ) + d3( u i,j,k+1 + u i,j,k 1 ) (5.5) d2 u +2/3 i,j+1,k + (1 + 2d2)u +2/3 i,j,k d2u +2/3 i,j 1,k = (1 2d1 2d3)u +1/3 i,j,k + d1( u +1/3 i+1,j,k + u +1/3 i+1,j,k ) + d3( u +1/3 i,j,k+1 + u +1/3 i,j,k 1 ) (5.6) +1 d3 u i,j,k+1 + (1 + 2d3)u i,j,k d3u i,j,k 1 = (1 2d1 2d2)u +2/3 i,j,k + d1( u +2/3 i+1,j,k + u +2/3 i+1,j,k ) + d2( u +2/3 i,j+1,k + u +2/3 i,j 1,k ) (5.7) Where d1 = α. t 3( x 2 ) d2 = α. t 3( y 2 ) d3 = α. t 3( z 2 ) ADI is the implicit approach where the ukow must be obtaied by meas of a simultaeous solutio of the differece equatio applied at all grid poit at a give time level. If we write these system of equatio i matrix form the it will look like b1 c u1 d1 a2 b2 c2 0 0 u2 d2 0 a3 b3 c3 0 u3 = d3 [ a5 b5] [ u5] [ d5 ] (5.8) This coefficiet matrix i Eq. (5.8) is a tridiagoal matrix, defied as havig ozero elemets oly alog the three diagoals. The solutio of the system of equatios deoted by coefficiet matrix ivolves maipulatio of tridiagoal arragemet; such a solutio ca be obtaied by usig Thomas algorithm which becomes almost the stadard for the treatmet of tridiagoal systems of equatios. Drawback of ADI method is that method is ucoditioally stable (meas irrespective of how large the time step ( t) is, method produces the stable results) for two-dimesioal case but for geeralisatio to three dimesioal case it is ot ucoditioally stable.so there is a alterative to ADI method is available which is Douglus Method. 18

27 5.2 Douglus Method Douglus method is ucoditioally stable ad method is i complete aalogy with ADI method so the steps for the Douglus method remais same as that of ADI steps. The fiite differece equatio of three-dimesioal heat equatio Eq.(3) i the Douglus method formulatio are +1/3 u i,j,k ui,j,k t = [ 1 2 δ 2 +1/3 x u i,j,k +δx 2 u i,j,k ( x) 2 + δ y 2 u i,j,k ( y) 2 + δ z 2 u i,j,k ] ( z) (5.9) 2 +2/3 u i,j,k ui,j,k t = [ 1 2 δ 2 +1/3 x u i,j,k +δx 2 u i,j,k ( x) δ 2 +2/3 y u i,j,k +δy 2 u i,j,k ( y) 2 + δ z 2 u i,j,k ] ( z) 2 (5.10) +1 u i,j,k ui,j,k t = [ δ 2 +1/3 x u i,j,k +δx 2 u i,j,k ( x) δ 2 +2/3 y u i,j,k +δy 2 u i,j,k + ( y) 2 δ 2 +1 z u i,j,k +δz 2 u i,j,k ] (5.11) ( z) 2 Where δ 2 x u i,j,k = u i+1,j,k 2 u i,j,k + u i 1,j,k δ 2 y u i,j,k = u i,j+1,k 2 u i,j,k + u i,j 1,k δ 2 z u i,j,k = u i,j,k+1 2 u i,j,k + u i,j,k 1 Boudary coditios are applied o the domai boudaries ad for ay typical ode (i, j, k), the algebraic equatio is give as d1 u i+1,j,k d2( u i,j+1,k (1 + 2d1)u 3 3 i,j,k d1u i 1,j,k = (1 2d1 2d2 2d3)u i,j,k + + u i,j 1,k ) + d3( u i,j,k+1 + u i,j,k 1 ) + d1( u i+1,j,k + u i+1,j,k ) (5.12) where d1 = α. t 2( x 2 ) d2 = α. t ( y 2 ) d3 = α. t ( z 2 ) d2 u +2/3 i,j+1,k + (1 + 2d2)u +2/3 i,j,k d2u +2/3 i,j 1,k = (1 2d1 2d2 2d3)u i,j,k + d1( u +1/3 i+1,j,k + u +1/3 i+1,j,k ) + d2( u i,j+1,k + u i,j 1,k ) + d3( u i,j,k+1 + u i,j,k 1 ) (5.13) Where d1 = α. t 2( x 2 ) d2 = α. t 2( y 2 ) d3 = α. t ( z 2 ) 19

28 +1 d3 u i,j,k+1 + (1 + 2d3)u i,j,k d3u i,j,k 1 = (1 2d1 2d2 2d3)u i,j,k + d1( u +1/3 i+1,j,k + u +1/3 i+1,j,k ) + d2( u +2/3 i,j+1,k + u +2/3 i,j 1,k ) + d3( u i,j,k+1 + u i,j,k 1 ) (5.14) Where d1 = α. t 2( x 2 ) d2 = α. t 2( y 2 ) d3 = α. t 2( z 2 ) 5.3 Thomas Algorithm Gaussia elimiatio is the stadard method for the solvig the system of liear, algebraic equatio. Thomas algorithm is essetially the result of applyig Gaussia elimiatio to the tridiagoal system of equatios. Thomas algorithm is a efficiet solver for Eq. (7) ad it has two sweeps forward elimiatio ad backward substitutio. Specifically we wish to elimiate the lower diagoal term (the a s) as follows. I the first sweep, the lower diagoal is elimiated by c 0 = c 0 b 0, c i = d 0 = d 0 b 0 c i b i c i a i i = 1,2,, M 1 (5.15), d i = d i d i 1 a i i = 1,2,, M 1 (5.16) b i c i a i The tridiagoal system i.e. a system of equatios with fiite coefficiet oly o the mai diagoal (the b s), the lower diagoal (the a s) ad the upper (the c s). The secod sweep solves ukows from the last poit of the domai to the first as u M 1 = d M 1, u i = d i c i u i+1 i = M 2, M 3,,0 (5.17) Thomas algorithm is serial i ature. I order to calculate c i, d i ad u i their immedi- ately precedig terms c i 1, d i 1 ad u i 1 have to calculate ahead. 5.4 ADI-CUDA I this sectio we are talk about strategy required to covert serial code ito parallel code i CUDA. For parallel implemetatio, as we are discussig about three dimesio we cosider oly symmetrical or o-symmetrical cubes. I CUDA we are talkig about the lots of threads so we have to arrage CUDA threads accordig to our eed. 20

29 5.2.1 Mappig strategy As show i Fig.5.2 the physical domai i.e. the cube is divided ito discrete cells ad the resultat cube after meshig is also show i the same Fig.5.2. Fig. 5.2 Physical domai ad physical domai after grid geeratio For Mappig CUDA threads we cosider the cube as the set of XY-plaes chopped across the Z-directio as show i Fig.5.3. For performig parallel implemetatio we have to cosider set of 1D blocks with set of 1D threads i it. We ca assume the block i the CUDA as the oe XY plae i the physical domai. Oe XY plae cosists of the umber of rows ad umber of colum ad set of XY-plaes ca be see across Z- directio. We are usig the set of 1D threads, oe thread ca hadle oe row or colum for calculatig the temperature values. Fig. 5.3 Set of XY-plae of physical domai chopped across Z-directio. 21

30 I CUDA eviromet, there is o effective way available for iter-block sychroisatio so we have to create every ew call to GPU whe we require the sychroisatio amog the threads from differet block. So i our desig we have created GPU call for X-sweep, Y-Sweep, ad Z-Sweep ad also for the checkig covergece of the solutio ad for copy temperature values betwee successive time steps. BLOCK Tridiagoal matrix 1D thread XY plae with set of rows Fig. 5.4 Computatio of Tridiagoal system for row is assiged to 1 thread for X- Sweep. I X-Sweep, the umber of thread geerated for kerel lauch is equal to the umber of rows across every plae i the discretized domai as show i Fig.5.4. Oe thread ca hadle the oe row completely to geerate the coefficiet matrix of the tridiagoal system ad agai gives call to the tridiagoal system solver to solve the tridiagoal system ad write back the results from tridiagoal system solver. Each thread stores the coefficiet for the tridiagoal matrix i its private memory which is i tur stored i the local memory. I Y-Sweep, the umber of thread geerated for the kerel lauch is equal to the umber of colum across every plae i the discretized domai as show i Fig Here also oe thread ca hadle oe colum completely ad call tridiagoal solver ad write back results. I Z-Sweep, the umber of thread geerated for kerel lauch is equal to the umber of rows i oe plae oly. As i Z-Sweep, temperature values lies across blocks i the gird there is o use for geeratig the threads across the block. 22

31 1D Thread Block XY plae with set of colums Tridiagoal matrix Fig. 5.5 Computatio of Tridiagoal system for colum is assiged to 1 thread for Y- Sweep Covergece of the solutio is said to be achieved whe the maximum temperature differece betwee the each discrete poit of the domai is almost zero or costat i successive time steps 23

32 Chapter 6 RESULTS ADI serial ad Douglus code which is writte i C-laguage ad the ADI-CUDA which i writte i CUDA-C is tested o the system with followig specificatio Processor Itel Xeo E GHz X 4 RAM 16GB OS Ubutu LTS 64-bit GPU GeForce GTX 480 Compute capability 2 CUDA versio 5.5 Table 6.1 System Specificatio Followig are the boudary coditios for Grid Temperature at left ed ( c) 80 Temperature at Right ed ( c) 20 Temperature at Top ed ( c) 60 Temperature at Bottom ed ( c) 30 Temperature at Frot ed ( c) 70 Temperature at Back ed ( c) 20 Desity(ρ) (kg/m3) 7800 Sp. Heat(C) (J/K) 473 Coductivity(κ) (W/(m K) 43 Time step (secod) Table 6.2 Test data Followig results shows compariso betwee ADI, Douglus ad ADI-CUDA method wrt executio time ad No. iteratio take to coverge the solutio also the speed-up is show wrt ADI ad ADI-CUDA Grid size ADI Douglus ADI Parallel No of Iteratio No of it- Time Time eratio Time to executvercutverecute for co- to exe- for co- to exgecgece 24 No of iteratio for covergece Speedup 10X10X X20X X30X X40X X50X Table 6.3 Executio time, No of iteratio for covergece i ADI, Douglus ad ADI- CUDA method

33 Time i secod Time i secod From the above table oe ca easily ifer that irrespective of grid size umber of iteratio for particular method are costat. Followig graphs shows the results obtaied by executig the of ADI, Douglus, ADI- CUDA program o the system with system specificatio metioed i Table 6.1 ADI Douglus ADI-CUDA X10X10 20X20X20 30X30X30 40X40X40 50X50X50 Grid Size Fig. 6.1 Time compariso betwee ADI, Douglus ad ADI CUDA Graph from Fig.6.1 implies that for ay gird size the time take for executio is greatest for Douglus method, itermediate for ADI method ad miimum for ADI-CUDA method. ADI Douglus ADI-CUDA Grid Size Fig. 6.2 Time compariso for Grid size 25

34 Speedup Graph from Fig.6.2 implies that as the Grid size icreases the time take for executio also icreases expoetially Speedup X10X10 20X20X20 30X30X30 40X40X40 50X50X50 Grid Size Speedup Fig. 6.3 Speedup Graph from Fig.6.3 implies that as the grid size icreases there is icrease i speedup also. Heat Map for sample grid size 10X10X10 at iitial stage, at itermediate stage ad at covegece state resp. Fig. 6.4 Heat map for iitial stage 26

35 Heat map is a graphical represetaio of the temperature values cotaied i amatrix form as a color Fig. 6.5 Heat map for itermediate stage Fig.6.6 Heat map at covergece state 27

36 Validatio of results with the referece datum Fig. 6.7 Validatio of ADI ad Douglus serial code As show i the Fig. 6.7 the lie of the ADI ad Douglus method are overlappig which implies that the results are totally idetical because Douglus method is desiged i aalogy with ADI method. Although the lies of the ADI ad Douglus method are deviatig from the referece lie still the results are acceptable because the equilibrium temperature is same ad also system specificatio ad method used for computatio is ot kow for the referece results. Fig. 6.8 Validatio of ADI-CUDA code. Iterestigly from Fig. 6.8 the results of the parallel implemetatio are i complete aalogy with the referece lie ad the accuracy of the results i parallel ADI-CUDA method is more as compared to the serial ADI ad Douglus method. 28

37 Chapter 7 CONCLUSION ad FURTHER WORK 7.1 Coclusio Alteratig Directio Implicit method is a alterative to the Crak-Nicolso method which is computatioally expesive whe exteded to multidimesioal. ADI method shows the high level of data parallelism which ca exploited usig the high ed GPU. Speedup up to the 11X is obtaied i this project, which is icreasig with respect to the grid size. I parallel implemetatio local memory which is private memory of the CUDA thread is best utilised i this project. As there is o sigificat iter-block sychroisatio available i CUDA, we were forced to split out kerel call for X-sweep, Y-sweep, Z-sweep which is sigificat overhead i case of parallel implemetatio. 7.2 Further Work If i future ay sigificat iter-block sychroisatio available the agai it will be possible to improve the speedup with the same or aother strategy. 3D heat coductio PDE which is used i this project is almost similar to the some of PDE which are used i the computatioal fiace with some chage i this project oe ca easily exted this project for computatioal fiace applicatio. Speedup of the this project is possible to icrease if the memory layout of the data stored i the GPU is tued with the memory access durig X-sweep, Y-sweep ad Z-sweep i parallel implemetatio as the memory access i these sweeps is widely scattered. 29

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