Nested dissection method is an elimination method for a set of linear algebraic equations with minimum fillins. Physically it divides a domain into four subdomains, and each subdomain is again divided into four. This procedure is repeated till all nodes are included in some subdomains. Using this characteristic, the authors examine the efficiency of the method on the transputer

Favier, J.

Pinelli, A.

Piomelli, U.

Revell, A.

Skillen, A.

Taniguchi, Takeo

Irie, Hirotsugu

Okayama University Digital Information Repository

Nested Dissection Method on Transputer

Okayama University Scientific Achievement Repository

Memoirs of the Faculty of Engineering,Okayama University,VoI.25, No.2, pp.1l5-122, March 1991Nested Dissection Method on TransputerTakeo TANIGUCHI* & Hirotsugu IRIE**(Received February 8 ,1991)SYNOPSISNested dissection method is an elimination method fora set of linear algebraic equations with minimum fill-ins. Physically it divides a domain into four subdomains,and each subdomain is again divided into four. This pro-cedure is repeated till all nodes are included in somesubdomains. Using this characteristic, the authors ex-amine the efficiency of the method on the transputer.1. INTRODUCTIONAccording to the development of the ability of computers, the sizeof problems treated in engineering fields has been growing. Then,faster solvers have been always required, and a number of effectivesolvers were proposed and applied for the actual problems. For thefinite element users solvers for a large sparse set of linear equa-tions have been required. Using the characteristics of the coefficientmatrices, the sparse matrix technique is effectively introduced in thesolvers and at present we find, for example, the band matrix method,the skyline method, and so on.The efficiency of solvers can be judged by the ammount of necessarymemories and also the CPU time required for the solution. Especially,the restriction of the memory size becomes important in rec~nt years,when very large problems are treated.* Department of Engineering Science** Department of Civil Engineering115116 Takeo TANIGUCHI and Hirotsugu IRIEFig, 1 Hardware SystemPC-9801RX I , Hard lJisk Unit IIIRoot Processor Subprocessor(T-800) (T-800)lHB Hemory lHB HemoryIHemory Board(6HB)Any method based on the elimination necessarily requires additionalmemory for the storage of new nonzero entries called fill-in appearingduring the forward elimination process. At present it is known thatthe nested dissection method generates the least number of fill-ins[I,2].The elimination process used in the nested dissection method sug-gests us another important aspect, i.e. the parallel computation. Atusing the nested dissection method, physical domain is divided intosubdomains, and the elimination of these subdomains can be treated in-dependently. This characteristic obviously fits to the property ofparallel machine.In'this paper the authors investigate the efficiency of the nesteddissection method on a parallel machine. For this purpose they firstlyexplain the elimination procedure of the nested dissection method,discuss on the parallel coding of the method, and show the results oftest problems. The results are compared with ones by other solvers.2. HARDWARE AND SOFTWARE SYSTEMThe computer used for this investigation is a parallel machinecalled transputer with 2 processors, T-800. One of these processor iscalled the root processor, which governs another processor called sub-processor. Each processor has 1MB memory, and auxiliary 6MB memory isadded to the root processor. The structure of this system prepares 8MBmemory as a whole, but the subprocess or can store, at most, 1MBmemory.A microcomputer, NEC PC-9801RX,is used as I/O machine of thetransputer, and a hard disk unitof 40 MB is attached to it.Operation system is MS-DOS. So,the user can treat whole systemonly using MS-DOS. For the compu-tation, Parallel Fortran is used.Total hardware system is illust-rated in Fig.l.For the efficient use of anyparallel machine, the tasks load-ed on each processor should beNested Dissection Method on TmnsjJuter 117equated. In above hardware system we can use two processors with 7MBand 1 MB, and the divided task for the machine is restricted by thesubprocessor. That is, two processors can fully work when each proces-sor subjects to the load less than 1MB.3. ELIMINATION PROCESS FOR SPARSE SYSTEMLetAx b (1)be a set of linear algebraic equations obtained by the application ofthe finite element method or the finite difference method.Now, we consider on the elimination process for (1). Let a(i,j) bean entry of the matrix A. Then, the elimination of the k-th row of Areplaces a(i,j) to a'(i,j).a'(i,j) a(i,j) - a(i,k)*a(k~j)/a(k,k) (2)This relation shows that a(i,j) must be modified, if both of a(i,k)and a(k,j) are nonzeros. Then, the zero entry at a(i,j) becomes non-zero if the product a(i,k)*a(k,j) is nonzero. New nonzero entrya'(i,j) is called "fill-in".The application of the theory of graph can clarify the generationof fill-in during the elimination ordering. At the presentation of thecoefficient matrix A of (1) by a graph, we set firstly n nodes for ex-pressing a matrix A(n*n), and indicate the existence of a nonzeroentry a(i,j) of A by a line connecting a pair pf nodes labelled i andj. Then, we can express the appearance of a fill-in also by a newline. Fig.2-a shows an example of a coefficient matirx, Fig.2-bpresents its graph, and Fig.2-c shows the graph after the eliminationof a node. located at the centre. Dotted lines in Fig.2-c indicate the~ppearance of fill-ins, and we can remark that the sub graph consistedby nodes which are connected to the eliminated node is replaced by acomplete graph after the elimination. That is, all nodes which areconnected to the eliminated node are directly connected each otherafter the elimination.In case of the band matrix method it is known that fill-ins appearonly for entries inside of the bandwidth. For example, if the matrix118 Takeo TANIGUCHI and Hirotsugu IRIE2 3 4 5 6 7 8 91 X X X2 X X X X3 X X XtI X X X X 05 X X X X X ( b)6 X X X X7 X X XA 18 X X X X ... ;'" : ............. ",'" : " ...9 X X X ~~-----~ --- - ~:::-', -1......... ' ... '"... ,..... : ", ..... '"(a ) <; ' .. I ...........(c)Fig. 2 Hatl'ix and Its Graphof Fig.2-a is eliminated according to the ordering of the matrix, allfill-ins appear within the halfbandwidth 4 including the maindiagonal.3. NESTED DISSECTION METHODLet R be a rectangular area, and subdivide the domain into N'~Nsmall rectangles by placing N nodes on each edge of the domain. Then,a grillage with (N+l)*(N+l) nodes are placed there.Now, we consider on the elimination of nodes set in the area. Apart of the grillage is shown as M(O) in Fig.3. The effect of theelimination of a node is restricted only within the subarea consistingof nodes which are connected to the eliminated one. The state afterthe first elimination is shown as M(l) in the figure, and bigger rec-tangle with 8 nodes show a complete graph. M(2) shows new state afterthe elimination of 2 nodes. At this elimination we can remark that nofill-ins appear for nodes which locate on the common edge of these twocomplete graphs, because fill-ins are already generated at the firstelimination. The figures in Fig.3 clarify that fill-ins once generatedare effectively used for the successive eliminations.Nested dissection method uses the property of the elimination pro-cedure mentioned above effectively. Assume a grillage with (N+l)*(N+l)nodes is placed in a rectangular area. The rectangular area is dividedinto four smaller rectangles, and call nodes on the bisecting lines asNested Dissection Method on Transputer11THeO)119M(l) M(2) M(3)m@sH(4) M(S) M(6)000H(?) M(8) M(9)Fig.3 Nested Dissection MethodG(X). Each rectangle is divided into four again, and nodes newlyselected are classified into G(X-l). This subdivision is repeated un-till all nodes are included in some groups G( j). The final group isset in G(l). Then, "X" of G(X) indicates the number of groups. Theelimination is started from nodes in the final group, and continued tonodes in successive group till all nodes are eliminated. The elimina-t ion 0 r d e r i n g 0 f nod e sin the sameg r 0 u pis arb i t r a r y , but theelimination of another group must be done after the elimination ofprevious one. The different ordering of the elimination for nodes inthe same group does not give any difference of tne number of fill-ins.The elimination ordering of the nested dissection method candecrease total number of fill-ins, since(1) the number of nodes connected to a node to be eliminated is120 Takeo TANIGUCHI and Hirotsugu IRIErestricted to be small, and(2) nonzero entries once generated during the elimination aremodified as many times as possible not to create new fill-ins.4. PROBLEMS AND RESULTS ON TRANSPUTERFor nodes on the grillage the elimination process can be easilyparallelized. Since the elimination of a node modifies only the valuesof entries as shown in Fig.3, another node which is not influenced bythe elimination can be eliminated at the same time.This property of the nested dissection method suggests the ap-plicability of it to the parallel computation. Obviously, the elimina-tion of nodes included in the group, G(X), must be done in one proces-sor, because they form a complete graph at the stage. That is, theelimination procedure of the nested dissection method can be parallel-ized except the treatment for nodes in G(X).For the comparison of the efficiency of the nested dissectionmethod, we choose following two solvers:1. Gauss Elimination Method2. Band Matrix MethodIn following discussion GEM, BMM and NDM are used for Gauss Elimina-tion Method, Band Matrix Method, and Nested Dissection Method, respec-tively.These three solvers are tested using single processor and also twoprocessors. The parallel coding for GEM and BMM is as following: Oddrows are solved by the root processor, and even rows are by the sub-processor. The parallel coding for NDM is due to the method mentionedabove.Now, we explain the problems used for the test. Two problems areprepared for our tests, in whichTest 1 and 2 include 49 and 225Fig's 1 2 1 3 12 1nodes, respectively. 4 and2 2 2 3 2 2 25 show the groups of nodes of Test 1 2 1 3 1 2 11 and Test 2, respectively. In the 3 3 3 3 3 3 3figures, numbers from 1 to 4 indi- 1 2 1 3 1 2 1cate the nodes clasified into G(l) , 2 2 2 3 2 2 2G(2) , G(3) and G(4), respectively. 1 2 1 3 1 2 1Test 1 is almost one fourth of Test2, which is enclosed by nodes classi- Fig. 4 Groups of Nod!Js of Test 1Nested Dissectian Method an Transputer1 2 1 ~ 1 2 1 4 1 2 1 3 1 2 1"2 2 2 3 2 2 2 4 2 2 2 3 2 2 21 2 1 3 1 2 1 4 1 2 1 3 1 2 13 3 3 3 3 3 3 4 3 3 3 3 3 3 31 2 1 3 1 2 1 4 1 2 1 3 1 2 12 2 2 3 2 2 2 4 2 2 2 3 2 2 21 2 1 3 1 2 1 4 1 2 1 3 1 2 14 4 4 4 4 4 4 4 4 4 4 4 4 4 41 2 1 3 1 2 1 4 1 2 1 3 1 2 12 2 2 3 2 2 2 4 2 2 2 3 2 2 21 2 1 3 1 2 1 4 1 2 1 3 1 2 13 3 3 3 3 3 3 4 3 3 3 3 3 3 31 2 1 3 1 2 1 4 1 2 1 3 1 2 12 2 2 3 2 2 2 4 2 2 2 3 2 2 21 2 1 3 1 2 1 4 1 2 1 3 1 2 1Fig. J Groups of Nodes of Test 2Table 1 Results of Test 1 and Test 2(msec)Test 1 Test 2------------------------------ -----------------------------------Single Pa ra Single Pa ra.GEH 321.98 275.33 31690.05 24149.63BHH 17.' 98 20. 74 288.96 272.19NOH 17. 73 17.02 253.89 233.22121fied in G(4).For NDM the elimination is started from nodes in G(l), and succes-sively continued to nodes in G(2), G(3) and G(4). For BMM the elimian-tion ordering is determined so that the half bandwidth is minimised.In these tests the half bandwidth is 7 for Test 1 and 15 for Test 2,and all data are stored in 2-dimensioanl array. Same elimination or-derings are introduced in GEM, but the elimination is applied for allentries of the coefficient matrix A.The results of Test 1 and 2 are summarized in Table 1. From thetable we obtain following conclusions.(1) Nested dissection method is the fastest solver among them for thesingle and also parallel computings.122 Takeo TANIGUCHI and Hirotsugu IRIE(2) The efficiency of the parallel computaion for the nested dissec-tion method appears already in small-size problem like Test 1. On theother hand, the band matrix method requires longer execution time inthis case, and it can show the efficiency of the parallel computationfor larger problem like Test 2. This difference is caused by the I/Otime for the parallel procedure of these solvers.The results of this table suggest that the nested dissection methodbecomes more effective for larger problems.5.CONCLUDING REMARKSThe efficiency of the nested dissection method is surveyed using aparallel computer, and the comparison of CPU time showed that it isfaster than the Gauss elimination method and the band matrix method.Moreover, the results suggest that it will be more effective forlarger problems.Te s t pr 0 b lems t r ea ted in thi s in vest iga ti on are sel ec ted so tha tthe nested dissection method can be easily applied. If more generalproblems are chosen for the tests, the use of the nested dissectionmethod becomes difficult. In this case, the one-way dissection methodshould be introduced instead of the nested dissection method.[3]ACKNOWLEDGEMENTThis work was financially supported by the scientific research fundof the Ministry of Education of Japan.REFERENCES[1] A. George," Nested dissection of a regular finite element mesh",SIAM J. Numer. Anal., Vol.10, No.2 (1973) pp.345-363[2] T. Taniguchi & K. Numata,"Fundamental study on the fill-in mini-mization problem", Memoirs of School of Eng., Okayama Univ., Vol.15-1(1980) pp.101-110[3] A. George,"An automatic one-way dissection algorithm for irregularfinite element problems", Lecture Notes in Mathematics 630 (ed.G.A.Watson) (1977) pp.76-89

The stall-delaying properties of the humpback whale flipper have been observed and quantified in recent years, through both experimental and numerical studies. In the present work, numerical simulations of an infinite-span wing with an idealized representation of this geometry are reported at a Reynolds number of 1.2×105. Using large-eddy simulation, an adequate spatial resolution is first established before also examining the spanwise extent of the domain. These results are then analyzed to provide an explanation of the conditions that drive the lift observed beyond the conventional stall angle. The undulating leading-edge geometry gives rise to a spanwise pressure gradient that drives a secondary flow toward the regions of minimum chord. In turn, this leads to the entrainment of higher-momentum fluid into the region behind the maximum chord, which energizes the boundary layer and delays stall. Aside from demonstrating a significant poststall lift, the undulations also have the added benefit of substantially reducing lift fluctuations

The University of Manchester - Institutional Repository

Flow over a Wing with Leading-Edge Undulations

International audienceThe stall-delaying properties of the humpback whale flipper have been observed and quantified in recent years, through both experimental and numerical studies. In the present work we report numerical simulations of an infinite span wing with an idealised representation of this geometry, at a Reynolds number of 1.2 × 105 . Us- ing Large Eddy Simulation, we first establish an adequate spatial resolution before also examining the spanwise extent of the domain. We then proceed to analyse these results to provide an explanation of the conditions that drive the lift observed be- yond the conventional stall angle. The undulating leading-edge geometry gives rise to a span-wise pressure gradient that drives a secondary flow towards the regions of minimum chord. In turn, this leads to the entrainment of higher-momentum fluid into the region behind the maximum chord, which energises the boundary layer and delays stall. Aside from demonstrating a significant post-stall lift, the undulations also have the added benefit of substantially reducing lift fluctuations

Skillen, Alex

Revell, Alistair

Pinelli, Alfredo

Piomelli, Ugo

Favier, Julien

HAL AMU

The stall-delaying properties of the humpback whale flipper have been observed and quantified in recent years, through both experimental and numerical studies. In the present work, numerical simulations of an infinite-span wing with an idealized representation of this geometry are reported at a Reynolds number of 1.2×10(to the power of 5). Using large-eddy simulation, an adequate spatial resolution is first established before also examining the spanwise extent of the domain. These results are then analyzed to provide an explanation of the conditions that drive the lift observed beyond the conventional stall angle. The undulating leading-edge geometry gives rise to a spanwise pressure gradient that drives a secondary flow toward the regions of minimum chord. In turn, this leads to the entrainment of higher-momentum fluid into the region behind the maximum chord, which energizes the boundary layer and delays stall. Aside from demonstrating a significant poststall lift, the undulations also have the added benefit of substantially reducing lift fluctuations

Nested Dissection Method on Transputer

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