Acceleration of Gas Bubble-Free Surface Interaction Computation Using Basis Preconditioners

Transcription

1 Acceleraton of Gas Bubble-Free Surface Interacton Computaton Usng Bass Precondtoners K. L. Tan, B. C. Khoo and J. K. Whte ABSTRACT The computaton of gas bubble-free surface nteracton entals a tme-steppng algorthm whereby a lnear system s solved at each tme-teraton. In our nvestgaton, the lnear systems are derved from a desngularzed boundary ntegral formulaton and are poorly condtoned. Ths leads to poor convergence rates when Krylov subspace methods are used to solve these systems. The convergence rates may however be mproved wth proper precondtonng. We lmt our nvestgaton to gas bubbles ntated at depths suffcently small such that a spke forms on the free surface durng the later stages of evoluton. Bubble dynamcs dctate that for gas bubbles ntated at such depths, the stages through whch the gas bubble and free surface evolve are smlar. Based on ths fact, we propose to perform one computaton run for a gas bubble ntated at one partcular depth, obtan a judcous set of a pror bass precondtoners from ths run and thereafter, use ths set of precondtoners on computaton runs for gas bubble ntated at dfferent depths. The computaton tme taken by the proposed method s, n general, 50% and 0% of the tme taken by the present method (wthout precondtonng) wth termnatng crtera of 1.0e-5 and 1.0e-7 n the nfnty-norm respectvely usng the B-conjugate Gradent Stablzed solver. The present method further enables computaton to an nfnty-norm termnatng crteron of 1.0e-10 n a shorter tme compared to the present method wth a crteron of 1.0e-5. K. L. Tan s a student wth the Sngapore-MIT Allance (SMA), Natonal Unversty of Sngapore, 10 Kent Rdge Cresent, Sngapore E-mal: smap9055@nus.edu.sg. B. C. Khoo and J. K. Whte are SMA fellows. INTRODUCTION Bubbles surrounded by a lqud can be found n many applcatons. Examples nclude bubble jet prnters, bolng water and underwater explosons. The chemcal ndustry uses bubble columns to enhance chemcal reactons and mxng. The study of bubbles can be dvded nto two man areas. The frst area, bubbles that do not undergo much change n volume, s studed extensvely. A seemngly smple problem of ths knd, the termnal rse velocty of a sngle bubble n water, s stll not yet fully understood. The second area entals bubble dynamcs pertanng to oscllatng bubbles. Examples of these nclude cavtaton bubbles and underwater exploson bubbles. Underwater exploson bubbles s an actve area of defense research. An exploson or gas bubble s formed when an underwater exploson takes place. Due to nerta, ths bubble wll overexpand and thereafter collapse. If ths collapse takes place near a sold surface, a hgh-speed jet wll be formed, drected towards the sold surface. On the other hand, f t takes place near a free water surface, the hghspeed jet wll then be drected away from the free surface and a water plume or spke wll be formed on the surface. All these problems have been studed usng the boundary ntegral method. Ths method necesstates the meshng of the surface only and not the entre doman thereby greatly reducng the sze of matrces to be solved. In ths paper, we seek to reduce the computaton tme taken to evolve a sngle gas bubble nteractng wth a free surface through use of precondtonng. Lmtng our nvestgaton to gas bubbles ntated at depths such that a spke forms on the free surface, the precondtoners are

2 chosen and computed based on some a pror knowledge of evoluton of such bubbles ntated at dfferent depths. MATHEMATICAL FORMULATION Mathematcal Formulaton Assumng the flud doman bounded by the gas bubble and free surface to be nvscd, ncompressble and rrotatonal, the flow wthn the doman Ω may then be descrbed by a potental feld φ ( x,y,z), where the flud velocty s gven by φ, that satsfes Laplace s equaton φ = 0. The knematc and dynamc boundary condtons governng the moton of the gas bubble wth no buoyancy force, and free surface are Dp = φ, Dt Tme Dscretzaton The Euler Forward tme dscretzaton scheme s used. The tme dscretzed equatons would thus be of the form 1 k k p = p + φ t for the update of the poston of the feld ponts, and and λ 1 k 1 k V0 φ = φ φ ε t k V 1 k 1 k φ = φ φ t for Ω S, for Ω Σ for the update of potental at the feld ponts. smplfcaton, we shall wrte the above two equatons as ( ) 1 k k k φ = φ + F φ,v t For and λ V0 = 1+ φ ε Dφ 1 Dt V for Ω S, Dφ 1 = 1 + φ for Ω Σ Dt where = ( x,y,z) p are the spatal coordnates of the feld pont n 3-D, t s the dmensonless tme varable, γ the dmensonless depth of ntaton, ε the dmensonless strength parameter, V the volume of the gas bubble, V 0 the ntal volume of the gas bubble, λ the rato of specfc heats, Ω the doman boundary, S the porton of Ω defned by the gas bubble and Σ the porton of Ω defned by the free surface (Best [1] and Wang et al. []). The far feld boundary condton s prescrbed as ( ) 0 φ p as p = x + y + z. The value of parameters used n our nvestgaton s ε = 100, for exploson bubble, and γ = 1. 4, for datomc gas (Best [1]). wth F representng the approprate functon for Ω S or Ω Σ. k φ s updated by solvng Laplace s equaton wth boundary condtons gven by φ k. In our code, φ s computed drectly through the ntegral equaton by dfferentatng the Greens functon n the three prncpal drectons. An alternatve method for computng φ would be to obtan the normal component of φ usng the ntegral equaton and the tangental components usng fnte dfferences (Wang et al. []). Soluton of Laplace s Equaton The Desngularsed Indrect Boundary Integral Method (DIBIM) s used to solve Laplace s equaton (Zhang et al. [3]) n order to derve the updates to φ. Based on the above method, we have ( p) φ = σ ( q ) p q where σ s the unknown source strength, = ( x,y,z) q are the spatal coordnates of the source pont, and p q s the

3 Eucldean dstance between the feld pont and the source pont. Intal Condtons The ntal bubble radus for the chosen value of ε s , obtaned by solvng the Raylegh equaton modfed for gas bubble (Best [1]). Assumng an ntal quescent free surface and notng the ntal velocty of the gas bubble surface to be neglgble, the ntal condton of φ = 0 may be prescrbed for the boundary. DISCRETIZATION The ntal geometry of the gas bubble s a dscretzed sphere of 49 nodes wth radus located at ( x,y,z ) = ( 00,,γ ). The ntal quescent free surface s dscretzed as a flat rectangular surface comprsng 79 nodes spannng ( x,y,z ) = ( 5:5, 5:5, 0). The applcaton of a truncated dscretzed free surface mesh s an approxmaton made possble by the far feld boundary condton prescrbed above. A pont collaton scheme s used wth the DIBIM descrbed above. The collocaton s performed on the nodes, 49 for the gas bubble and 79 for the free surface, defned by the dscretzaton. In order to have the number of unknowns equal to the number of known equatons, we set the number of source ponts to be equal to the number of dscretzed nodes, wth 49 ponts assocated wth the gas bubble and 79 ponts assocated wth the free surface. Ths results n a system wth N = 11 degrees-of-freedom. The source ponts are postoned n accordance wth Zhang et al. [3]. As mentoned above, the Euler Forward tme-steppng scheme s used to evolve the gas bubble and free surface. The tme-step s chosen such that the bound on the maxmum change n potental φ for any node s restrcted to 0.04 at each tme-teraton, φ 1 V0 max 1 + φ + ε V t = mn φ 1 max 1 + φ j λ, The value of 0.04 s typcal of the range of φ chosen for such computatons (Best [1] and Wang et al. []). The expected nodal potental values would range from 1 to 3. PRESENT IMPLEMENTATION φ Let = ( x, y, z) p be the poston of feld pont j, j N! be the vector of potentals at the N feld ponts, σ ponts and K be the number of tme-teratons to be taken. N! be the vector of source strengths at the N source The present mplementaton s as follows: 0 Set ntal condtons: p = p j = 1: N, φ = 0 where p j j s the ntal poston of the feld pont j. Do k = 1: K 1. Form matrx G where G j = p q. Solve Gσ = φ for σ usng teratve solver wth set termnatng crteron 3. Compute materal dervatve φ =Hσ where H = j p q j 4. Compute tme-step t 5. Update poston of feld ponts ( ) = + t j = 1: N j j j 1 k p p φ 6. Update potental of feld ponts ( ) φ = φ + F φ, z, V t j = 1: N 1 k j j j where F s the knematc boundary condton functon assocated wth node j 7. Compute new volume V End The teratve solver used s the B-conjugate Gradent Stablzed (BICGSTAB) solver. The G matrces generated j 0 j

4 based on the DIBIM above have been found to be poorly condtoned. A termnatng crteron of r < 1.0e 5 N where r! s the vector of resduals, s used. Ths lberal termnatng crteron s chosen because of the poor convergence characterstcs assocated wth usng a Krylov Subspace solver on ll-condtoned matrces. The DIBIM s also lmted n that t starts to fal as opposte sectons of the bubble surface approach each other. In our nvestgaton, we set K = 150 so as to end our computaton before ths scenaro arses. (c) The output based on the above mplementaton for the evoluton of free surface nteracton wth a sngle gas bubble ntated at γ = 1.0 s shown n Fgure 1. (d) (a) (e) (b) Fgure 1: Evoluton of Free Surface Interacton wth Gas Bubble Intated at 1.0 γ = PROPOSED METHOD The bound on convergence rate of a Krylov Subspace solver for the soluton of a general lnear system Ax = b

5 s a functon of the condton number of the matrx A, κ ( A ). In general, the better the condton of A, the hgher s the convergence rate and vce-versa. Block Dagonal Precondtonng The use of precondtonng to acheve better convergence propertes for teratve solvers s not new. There are also many forms of precondtoners (Saad [5]). In our nvestgaton, we choose to use block dagonal precondtoners. The ratonale for such a choce s explaned below. As stated n Step 1 of the mplementaton shown before, the entres n the G matrx are G j = p q. Ths means that the magntude of entry G s nversely j proportonal to the dstance between feld pont j and source pont. Let the 49 nodes assocated wth the gas bubble be ndexed j = 1: 49 and the 79 nodes assocated wth the free surface be ndexed j = 493 :11. The resultng G matrx generated would be of the form G1 G3 G = G4 G where G1! s the sub-matrx assocated wth the gas bubble nodes only, G! s the sub-matrx assocated wth the free surface nodes only, and G3! and G4! are the cross-coupled terms. j ponts n accordance wth Zhang et al. [3], t s expected that the value of all entres n matrces G3 and G4 to be much smaller n magntude as compared to the value of the entres n matrces G1 and G. As the gas bubble evolves, t s observed that only the top secton of the bubble surface comes nto close proxmty wth the free surface. Ths mples that most of the couplng occurs wthn matrces G1 and G, and not wthn G3 and G4. Based on the above observaton, f we are able to construct a precondtoner M for the G generated at every tme-teraton such that G1 0 M = 0 G then (rght-)precondtonng the matrx G wth M would yeld I G3 G GM =, G4 G1 I resultng n a better condtoned matrx. It would however be computatonally expensve to nvert sub-matrces G1 and G at every tme-teraton for use as precondtoners. It s observed from Fgure 1a that durng the ntal stages of evoluton, the dstance between two bubble nodes s much smaller compared to the dstance between a bubble node and a free surface node. Because of the postonng of the source

6 The need to compute precondtoners at every tmeteraton can be elmnated f we relax the requrement that the block precondtoner sub-matrces be exact nverses of sub-matrces G1 and G. In fact, our am s just to precondton the G matrx such that t has better convergence propertes and not to convert the dagonal submatrces nto dentty sub-matrces. Fgures a, b and c show the gas bubble and free surface for γ = 1.0 at tmeteratons 60, 65 and 70 respectvely. It s observed from Fgure that the change n geometry over several tmeteratons s lmted. Ths mples that the change n value of entres n the G matrx from one tme-teraton to the next s also lmted. In ths case, we can compute the precondtoner for the G matrx generated at a partcular tme-teraton and make use of ths precondtoner to precondton the G matrces generated at tme-teratons about that partcular tme-teraton from whch the precondtoner s obtaned. In ths case, what we obtan s (a) G1 0 M = 0 G and after (rght-)precondtonng (b) (c) Fgure : Gas Bubble and Free Surface Geometry at Tmeteratons 60 (a), 65 (b) and 70 (c) for 1.0 γ =. I G3 G GM = G4 G1 I whch would stll result n a relatvely well-condtoned matrx. By judcously choosng a handful of G matrces throughout the entre computaton run to generate the precondtoners from, effectve precondtonng may be acheved for the entre computaton run. The above method may mprove the condton of the matrx and therefore reduce the tme taken to solve the lnear system at each tme-teraton. However, ths method would requre a pror knowledge of the G matrces so as to compute the precondtoners. Ths would mply that, n order to accelerate the computaton, we would have to perform the sad computaton frst, whch s somewhat neffcent. An alternatve method s to decde a pror at whch tme-

7 teraton to compute the precondtoners. Ths would however necesstate the nverson of matrces durng the computaton run thus makng the computaton run expensve. In fact, there may not even be any savngs n computaton tme n usng ths method over the present method. A Pror Computed Bass Precondtoners Gven n Fgure 3 s the evoluton of free surface nteracton wth a sngle gas bubble ntated at γ = 1.1. Comparng Fgures 3 and 1, t s observed that the gas bubble n both cases, though ntated at dfferent depths, goes through the same sequence of evoluton wth small dfferences n geometry. (d) (a) (b) (e) Fgure 3: Evoluton of Free Surface Interacton wth Gas Bubble Intated at 1.1 γ = As mentoned before, our am of precondtonng s not to generate dentty sub-matrces but smply to mprove the condton of the G matrces generated such that they have better convergence propertes. It s therefore plausble to make use of precondtoners generated from the computaton run of γ = 1.0 for the computaton run of γ = 1.1. Ths clam s substantated by the observaton above that for a sngle gas bubble ntated at varous depths such that a spke forms on the free surface, the gas bubble and free surface would go through the same sequence of evoluton wth margnal dfferences n geometry In ths case, the need to have a pror knowledge of the G matrces for each new computaton run of a dfferent γ s elmnated. In fact, we need only perform one computaton run of one value of γ usng the present method, compute the precondtoners from the G matrces obtaned at judcously chosen tme-teratons and then make use of these precondtoners for other computaton runs of any value of γ that results n spke formaton on the free surface. (c)

8 Proposed Implementaton The proposed mplementaton s gven below. Perform only once to obtan the bass precondtoners: Judcously select the tme-teratons to compute the bass precondtoners from. Let the number of bass precondtoners selected be n#. For a selected value of γ = γ#, perform computaton run usng the present method. If tme-teraton k = one of the n# pre-selected tme-teratons above, 1. Extract sub-matrces G1 and G, and compute and G. Form M 3. Wrte M to fle G1 At the end of the computaton run, there should be n# a pror computed bass precondtoners. To obtan the soluton to any value of γ : Set ntal condtons. Do k = 1: K 1. Form matrx G. Precondton G wth the most approprate M from the set of n# bass precondtoners 3. Solve Gσ = φ for σ usng teratve solver wth set termnatng crteron 4. Compute materal dervatve φ =Hσ 5. Compute tme-step t 6. Update poston of feld pont ( ) = + t j = 1: N j j j 1 k p p φ 7. Update potental of feld ponts 1 k φ φ φ ( ) = + F, V t j = 1: N j j where F s the knematc boundary condton functon assocated wth node j 8. Compute new volume V End In Step above, the most approprate M would be the precondtoner whose geometry whch t s computed from, most closely matches the geometry at tme-teraton k, whch matrx G s computed from. Rght-precondtonng s used snce ths preserves the resdual of the orgnal lnear system. Selecton of Precondtoners In the selecton of precondtoners, we would not want to select too large a number snce ths would degrade the performance of the proposed method through excessve readng of fles. On the other hand, we would not want to select too low a number of precondtoners snce ths would not effcently precondton the G matrces and also degrade the performance of the proposed method. For the computaton of the bass precondtoners, we use γ = 1.0. Ths value of γ s used because t s approxmately the mean of the range of γ values that we are nterested n nvestgatng. The selecton crteron we choose s based on the amount of change n geometry wthn a range of tme-teratons, rather than smply a unform dstrbuton based on the number of tme-teratons or geometry of bubble. The ratonale for the above crteron s that f the geometry changes mnmally over a wde range of tme-teratons, there s no need for many precondtoners over that range of tme-teratons on the bass of the lmted change n geometry. On the other hand, f the change n geometry s tremendous over a lmted range of tme-teratons, there s no need for many precondtoners over that range of geometry on the bass of the lmted number of tme-teratons taken to evolve over that range of geometry. Snce the volume of the gas bubble s solely a functon of the geometry of the bubble, we can replace geometry n the above crteron wth volume of the gas bubble. We can also use the volume to establsh the most approprate precondtoner to use. The alternatve vew s that we can determne the range of volumes that we use a partcular precondtoner for. Based on the above crteron, we look to the volumeteraton plot gven n Fgure 4 to select the volume, and thus the tme-teraton to compute the precondtoners from.

9 It s observed from Fgure 4 that the ncrease n volume durng the frst 40 tme-teratons s mnmal. The rate of ncrease pcks up dramatcally at about tme-teraton 60, taperng off upon nearng maxmum volume. The rate of change n volume durng the collapse phase s slghtly less than that durng the rapd expanson phase. Based on the selecton crteron detaled above, the volumes and the correspondng tme-teratons we choose to compute the precondtoners for varous ranges of volume are gven n Table 1. volume tme-teraton Fgure 4: Gas Bubble Volume aganst Tme-teraton Plot for 1.0 γ =. Desred Volume Selected Tmeteraton Range of Volume Selected For Intal Table 1: Volumes Chosen for Computaton of Precondtoners RESULTS AND DISCUSSION The cpu-tme (CPU) and wall-clock (WC) tme taken to complete a computaton run of 150 tme-teratons for varous γ s s gven below. The teratve soluton method used s the BICGSTAB method wth the termnatng crteron beng the nfnty norm of the resdual as s wth the present method. Tables, 3 and 4 show the tmes taken wth termnatng crtera of 1.0e-5, 1.0e-7 and 1.0e-10 respectvely. Table 5 shows a comparson of the total number of teratons taken by the teratve solver durng the entre computaton run of 150 tme-teratons. It s observed from the results above that wth a termnatng crteron of 1.0e-5, the proposed method reduces the tme taken by almost 50% over the present method. Upon tghtenng the termnatng crteron to 1.0e-7, the proposed method reduces the tme taken by almost 80% over the present method. There s no tmng recorded for the present method wth a termnatng crteron of 1e-10. However, an alternatve way of nterpretng the results would be that wth the proposed method, we can acheve an mprovement of 5 orders n the nfnty norm of resdual over the present method whle takng less tme to complete the computaton run. Further evdence of the postve effect of the proposed precondtonng method on the convergence characterstcs of the G matrces formed may be observed n Table 5, whch shows the total number of teratons taken by the teratve solver for an entre computaton run of 150 tme-teratons at varous γ s. Fgure 5 shows the plots of number of teratons taken by the BICGSTAB solver at each tme-teraton for 0.8 γ = usng the present method and the proposed method wth a termnatng crteron of 1.0e-7. It s observed from Fgure 5 that the convergence rate s mproved throughout the entre computaton run. Havng determned the range of volume assocated wth each precondtoner, a new precondtoner s read nto memory from fle as and when necessary durng the computaton.

10 Proposed Method Present Method Rato CPU (s) WC (mn:s) CPU (s) WC (mn:s) CPU WC γ = : : γ = : : γ = : : γ = : : γ = : : Table : Tme taken for 150 Tme-teratons wth Iteratve Solver Termnatng Crteron of 1.0e-5 Proposed Method Present Method Rato CPU (s) WC (mn:s) CPU (s) WC (mn:s) CPU WC γ = : : γ = : : γ = : : γ = : : γ = : : Table 3: Tme taken for 150 Tme-teratons wth Iteratve Solver Termnatng Crteron of 1.0e-7 Proposed Method CPU (s) WC (mn:s) γ = :08 γ = : γ = :55 γ = :56 γ = :09 Table 4: Tme taken for 150 Tme-teratons wth Iteratve Solver Termnatng Crteron of 1.0e-10 Tolerance 1.0e-5 1.0e-7 1.0e-10 Method Proposed Present Rato Proposed Present Rato Proposed γ = γ = γ = γ = γ = Table 5: Total Number of Iteratons taken by Iteratve Solver for 150 Tme-teratons

11 number of BICGSTAB teratons present method proposed method 3. Zhang, Y. L., Yeo, K. S., Khoo, B. C. and Chong, W. K., 1999, Smulaton of Three-dmensonal Bubbles Usng Desngluarsed Boundary Integral Method, Internatonal Journal for Numercal Methods n Fluds Vol. 31, No. 8, pp Zhang, Y. L., Yeo, K. S., Khoo, B. C. and Chong, W. K., 1998, Three-dmensonal Computaton of Bubbles Near a Free Surface, Journal of Computatonal Physcs 146, pp Saad, Y., 1996, Iteratve Methods for Sparse Lnear Systems, PWS Publshng Company tme-teraton Fgure 5: Comparson of Number of Iteratons taken by Iteratve Solver at each Tme-teraton between Present Method and Proposed Method CONCLUSIONS Based on some a pror knowledge of the evoluton of free surface nteractng wth a sngle gas bubble ntated at the range of depths under nvestgaton, a set of bass precondtoners s computed from the computaton run made for one partcular depth. Ths set of bass precondtoners s then used to precondton the matrces generated at each tmeteraton for computaton runs made for dfferent depths. Usng ths proposed method, the tme taken to complete 150 tme-teratons of evoluton of free surface nteractng wth a sngle bubble s 50% and 0% of the tme taken by the present method wth termnatng crtera of 1.0e-5 and 1.0e-7 respectvely for the teratve solver used. Furthermore, the proposed method allows computaton to a tolerance of 1.0e- 10 n a tme less than that taken by the present method to compute to a tolerance of 1.0e-5. REFERENCES 1. Best, J. P., 1991, The Dynamcs of Underwater Explosons, Ph.D. Thess, The Unversty of Wollongong, Australa.. Wang, Q. X., Yeo, K. S., Khoo, B. C. and Lam, K. Y., 1996, Nonlnear Interacton Between Gas Bubble and Free Surface, Computers & Fluds Vol. 5, No. 7, pp