A Shooting Method for A Node Generation Algorithm

Transcription

1 A Shooing Mehod for A Node Generaion Algorihm Hiroaki Nishikawa W.M.Keck Foundaion Laboraory for Compuaional Fluid Dynamics Deparmen of Aerospace Engineering, Universiy of Michigan, Ann Arbor, Michigan 489 January, 2 Absrac In his paper, a shooing mehod is proposed for a node generaion algorihm. The node generaion algorihm generaes nodes succssesively equidisribuing a local average error, bu canno erminae a a specified poin. A shooing mehod is developed for his purpose which adjuss he equidisribuion quaniy by he Secan mehod. Numerical resuls are given ha shows he effeciveness of he mehod. Inroducion In his sudy, a shooing mehod is sudied o capure he endpoin by adjusing C. The idea is aken from [2] where a shooing mehod for he consrucion of an L opimum node disribuion is sudied. In [2], he nodes are generaed successively saring from a pair of nodes o minimize L error, and herefore he posiion of he las node depends upon he choice of he second node. The second node is hen be found by he minimizaion echnique of Bren s, which guaranees ha he las node is placed a he specified endpoin. In his sudy, we need o find he value of C such ha 2 Node Generaion Algorihm We consider generaing nodes for inerpolaion of a funcion in I = [, ] ha will equidisribue he leading erm of he local average error C E [] C E = E f 2 E Ψ2 E. (2.) where f() = dx/d, E = j+ j, f E = f( j+ ) f( j ), Ψ E = x E f( m ), m = j+ + j, (2.2) E 2 and E denoes he elemen defined by wo consecuive nodes, j and j+. The nodes can be generaed successively saring from he iniial poin (, ) = (, x()) once we specify he value of C E. Le C denoe his desired value, hen given a node j, we mus find j+ such ha Docoral Candidae, Aerospace Engineering and Scienific Compuing C E ( j+ ) C =. (2.3)

2 CFD Noes by Hiroaki Nishikawa The ieraion formula proposed in [] is k+ j+ = j + [ C C E ] p ( k j+ j ). (2.4) he superscrip k indicaes he number of ieraions, p is a posiive real number. The role of p is o damp an excessivelly large change, i.e. an exremely small fe k which will occur when he ieraion eners a region ha is near an inflecion poin or where he curve is locally linear. We erminae he ieraion when he equidisribuion is achieved wihm 3% error. The node generaion algorihm is summarized as follows.. compue C by C = 2 E 2(I) for a desired error E 2(I) ; 2. se j+ = j + ( j j ) ( j+ =. for j = ); 3. compue a new locaion j+ by (2.4); 4. if C E /C >.E-3, go o 3; 5. if j+ <., go o 2 (Nex node). Le N denoe he oal number of nodes generaed by he above algorihm, including he iniial node. Then noe ha we always have N >.. 3 Shooing Mehod Given a se of N nodes generaed by he node generaion algorihm, we consider solving he following nonlinear equaion for C. N (C). =. (3.) Noe ha his has many soluions unless N is fixed. For a convex funcion, his equaion has a unique soluion for a fixed N since N is a monoone funcion of C [2]. However i is no guaraneed for nonconvex funcions, and herefore we allow N o change, relaxing he requiremen for he soluion. Bu he soluion we seek is he one wih he oal number of nodes close o he original number N. Since he explici funcional relaionship beween N and C is no known, we use he Secan mehod o solve he problem. Given anoher se of nodes wih C (2) =.C and he iniial one C () = C, we herefore compue C by C (n) = C (n ) N (C (n ) ) [ N (C (n ) ) N (C (n 2) ) ] / [ C (n ) C (n 2)] (3.2) where n is he ieraion index greaer han wo, and when generae anoher se of nodes and also a each ieraion, we modify he sep 5 such ha he node generaion erminaes if he number of nodes reaches N +. The ieraion will be erminaed when he error in capuring he endpoin becomes less han 5% of he size of he las elemen, N (C (n) ). <.5 (3.3) N (C (n) ) N (C (n) ) The algorihm is summarized as follows.. compue N for C () = C; 2. compue N2 for C (2) =.C wih N 2 N +, se n = 3; 3. compue C (n) by (3.2); 4. compue Nn for C (n) wih N n N + ; 5. if (3.3) is no saisfied, se n = n + and go o 3; 6. se Nn =.. c 2 by Hiroaki Nishikawa 2

3 CFD Noes by Hiroaki Nishikawa E 2(I) N Ieraion N x u L2(I).E E-3.E E-5.E E-7 Table 3.: Example (a): p = 2. E 2(I) N Ieraion N x u L2 (I).E E-3.E E-5.E E-6 Table 3.3: Example (c): p = 8. E 2(I) N Ieraion N x u L2(I).E E-3.E E-5.E E-6 Table 3.2: Example (b): p = 3. E 2(I) N Ieraion N x u L2 (I).E E-2.E E-4.E E-6 Table 3.4: Example (d): p = 5. 4 Resuls Numerical ess were performed for he following four differen funcions which are aken from []. (a) a + ( a) { e ( /ϵ)} / { e ( /ϵ)} wih ϵ =.4 and a =.6 (b) a(.5) 2 a 4 ( + 8ϵ) + ( + 2ϵa) { e ( /ϵ)} / { e ( /ϵ)} wih ϵ =. and a = 3.5 (c) anh{2(.5)} (d) e ( ) + 2/{ + 4(.7) 2 } (a) is a sricly convex funcion, (b) and (c) have a single inflecion poin, and (d) has wo inflecion poins. All he compuaions were done wih double precision. The value of p was chosen such ha he node generaion algorihm converges for all he nodes. The resuls are summarized in Table 3. o 3.4 where E 2(I) is he specified L 2 error, N is he iniial number of nodes, N is he final number of nodes, and x u L2(I) is he acual L 2 error compued by he five poin Gaussian quadraure formula. Some plos are given in Figures 4. o 4.4. In amos all cases, he shooing mehod converges quickly a less han ieraions. For nonconvex funcions, his is due o he opion ha i is allowed o increase he oal number of nodes by. Wihou i, i would ake exremely long or even fail o converge. However here sill can be seen a pahological case. In he las example, he mehod ook very long o converge for E 2(I) =.E 3. One way o make i converge faser is o change he inpu value C by a very small amoun. We found ha E 2(I) =.E 3 insead of.e 3 made i o converge a he firs ieraion. However i should be remembered ha here can be in principle wo bad senarios in he mehod. Firs, C (n) can go negaive during he ieraion. Second he final number of nodes can be significanly less han he saring value. These problems would arise when he mehod is applied o funcions wih more inflecion poins. This robusness issue remains as a fuure sudy. 5 Concluding Remarks A shooing mehod was proposed o make he node generaion algorihm o end a a specified poin. I was shown ha he mehod is very effecive alhough no perfecly robus, and ha he mehod is much faser han he relaxaion algorihm proposed in []. However fuure work mus be done on he robusness of he mehod. Also, i would be necessary o invesigae he possibiliy ha he mehod is used o generae an equidisribuion grid for a given number of nodes which seems o be a more pracical problem. This is currenly possible only by he relaxaion algorihm. c 2 by Hiroaki Nishikawa 3

4 CFD Noes by Hiroaki Nishikawa Fig. 4.: Example (a). E 2(I) =.E-4. Circles indicae nodes, and sars are heir projecion ono -axis Fig. 4.2: Example (b). E 2(I) =.E Fig. 4.3: Example (c). E 2(I) =.E-2 Fig. 4.4: Example (d). E 2(I) =.E- c 2 by Hiroaki Nishikawa 4

5 CFD Noes by Hiroaki Nishikawa References [] Nishikawa, H., Accurae Piecewise Linear Approximaions o D Submanifolds: Error Esimaes and Algorihms. [2] Kezscher, R. and Forh, S., A Shooing Mehod for he Generaion of he bes L piecewise linear inerpolaion, AMOR 99/3, Cranfield Universiy c 2 by Hiroaki Nishikawa 5