In this paper, a new methodology for predicting fluid free surface shape using Model Order Reduction (MOR) is presented. Proper Orthogonal Decomposition combined with a linear interpolation procedure for its coefficient is applied to a problem involving bubble dynamics near to a free surface. A model is developed to accurately and efficiently capture the variation of the free surface shape with different bubble parameters. In addition, a systematic approach is developed within the MOR framework to find the best initial locations and pressures for a set of bubbles beneath the quiescent free surface such that the resultant free surface attained is close to a desired shape. Predictions of the free surface in two-dimensions and three-dimensions are presented.Singapore-MIT Alliance (SMA

Khoo, Boo Cheong

Lim, K. M.

My-Ha, D.

Willcox, Karen E.

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    Abstract—In this paper, a new methodology for predicting fluid free surface shape using Model Order Reduction (MOR) is presented. Proper Orthogonal Decomposition (POD) combined with a linear interpolation procedure for its coefficient is applied to a problem involving bubble dynamics near to a free surface. A model is developed to accurately and efficiently capture the variation of the free surface shape with different bubble parameters. In addition, a systematic approach is developed within the MOR framework to find the best initial locations and pressures for a set of bubbles beneath the quiescent free surface such that the resultant free surface attained is close to a desired shape. Predictions of the free surface in two-dimensions and three-dimensions are presented.  Index Terms— Bubble Dynamics, Model Order  Reduction, Proper Orthogonal Decomposition  I. INTRODUCTION set of underwater gas bubbles created by explosions can lead to the formation of a water plume in the sea surface. It is often desired that the water plume attain a particular shape to serve certain functionality, such as obstructing objects skimming above the water surface. This shape can be achieved by solving an optimization problem which minimizes the deviation of the simulated free surface from the desired shape with the decision variable set being the bubble parameters such as lateral positions, depths and strengths. Since it is not efficient and cost effective to couple a full model of bubble and free surface in an optimization problem, the Proper Orthogonal Decomposition (POD) is used to build a low-cost, low-order approximation model for the current problem. A series of linear POD basis functions are obtained from a set of solution snapshots of the bubble and free surface interaction problem. These basis functions and their POD coefficients are then used to simplify the optimization of  Manuscript received by December 15, 2004.  My-Ha D. was with the High Performance Computation for Engineered Systems, Singapore-MIT Alliance, National University of Singapore (e-mail: g0200755@nus.edu.sg).  Lim K. M. is with the Department of Mechanical Engineering, National University of Singapore. (e-mail: limkm@nus.edu.sg). Khoo B. C. is with the Department of Mechanical Engineering, National University of Singapore. (e-mail: mpekbc@nus.edu.sg). Willcox K. is with the Department of Aeronautics and Astronautics, Massachusetts Institute of Technology. (e-mail: kwillcox@mit.edu). bubble parameters to attain the desired water plume.   II. PROPER ORTHOGONAL DECOMPOSITION Given a set of snapshots ( ){ }xu  which are solutions of the system collected at different instants in time, the optimal POD basis functions ( ){ }∞=1jj xφ  are chosen to minimize the truncation error due to the construction of the snapshots ( )xu  using M  basis functions ( )( )( )( )φφφψψψψ ,,,,max22 uu =  (1)  The POD basis functions ( ){ }∞=1jj xφ  satisfy  ( ) ( ) ( )xxdxxxK λφφ =′′′∫Ω ,  (2) where the kernel ( )xxK ′,  is the autocorrelation function with the dimension of NN ×  ( N  is dimension of u , usually in order of 410 ), (Newman, 1996). In the Method of Snapshots (Sirovich, 1987), the POD basis vectors are calculated as the linear combination of the snapshots ( ) ( )∑==Miijij xubx1φ , Mj ,...,1=    (3) where jb  satisfies  bRb λ=  (4) Here R  is the modified correlation matrix,  ( )jiij uuMR ,1=  (5) The approximation is given by the linear combination of the basis functions ( ) ( )∑=≈qjjj xaxu1φ  (6) where q is chosen to satisfy a desired level of accuracy ( Mq ≤  and Nq << ); the POD coefficient ja  is determined as a function of time. The POD procedure can also be applied to parameter-dependent problems whose snapshots are collected at a series of varying parameters (Bui-Thanh et al., 2003). The coefficient ja  is then a function of these varied parameters. Let { }iuδ  be the snapshot taken corresponding to the Model Order Reduction for Determining Bubble Parameters to Attain a Desired Fluid Surface Shape My-Ha D.1, Lim K.M.1, Khoo B.C.1, Willcox K.21 National University of Singapore, 2 Massachusetts Institute of Technology   A   parameter value Mii ,...,1, =η . Basic POD is applied on the set of snapshots { }Miiu 1=δ  to obtain the orthonormal basis { }Mjj 1=φ . The coefficient ijaδ  is given by the projection  ( )ii ua jj δδ φ ,=  (7) The POD coefficients δja  for intermediate values of η  which is in the range of interest but not included in the ensemble can be found by interpolation among the ijaδ . The prediction of δu  is given by ∑==qjjjau1φδδ  (8) In the next section, POD in parametric space coupled with bubble dynamics will be used in an optimization context in the problem of water barrier construction.  III. WATER BARRIER SIMULATION The free surface formed by 0N  number of bubbles can be approximated by the linear superposition of the individual free surfaces, Our objective is to minimize the difference between the constructed surface ∑== oNkkSS1 and the desired surface 0S . It is difficult to obtain directly the bubble parameters that give the desired shape, so we divided the problem into two stages. In the first stage, the lateral position { } oNkkr 10 =  and the POD coefficient { }qjNkkja ,..,1;,..1 0 == for the bubbles are solved: BPS1: qjNkar okjk ...1,...1,,0min ==  ( )[ ]∑= −oMiiSS120             s.t. ∑ ∑= =⎟⎟⎠⎞⎜⎜⎝⎛ += oNkqjkjkjk aS1 1φφ  ( )kk r0φφ =   oNk ...1=      ( )kjkj r0φφ =   oNk ...1= , qj ...1=      drrr kk ≥− 21 00   oNkkkk ..1,; 2121 =≠      maxmin jkjj aaa ≤≤  oNk ...1= , qj ...1=   Here, kφ and kjφ  are the snapshot mean and the thj  POD mode, respectively, for the thk  bubble. The minimum lateral distance between centers of two arbitrary bubbles is 0.2=dr  to ensure the validity of the superposition. The range ja  is determined by the range of interest of the bubble strength and depth, [ ]maxmin ,εε  and [ ]maxmin ,γγ . The first-stage problem can be solve approximately using the Approximation Function (AF) algorithm which uses exponential functions to approximate the first POD mode, k1φ , and the mean, kφ . The functions have the form of and  ( ) 221 rCeCrf −=   (9) The coefficients 1C  and 2C  are specified by solving the optimization problem  AF:  21 ,minCC ( ) 20 iff −        s.t.  ( ) 221 rCeCrf −=  where 0f  is the mean or the first POD mode. Using these functions, the free surface corresponding to the thk  bubble can be represented by  ( ) ( ) ( )2012202 1111 kkm rrCkrrCmk eCaeCrS −−−− +=  (10) BPS1b: okk Nkar ...1,, 10min=  ( )[ ]∑= −oMiiSS120             s.t. ( ) ( ) ( )( )∑=−−−− += o kkmNkrrCkrrCm eCaeCrS111112012202  drrr kk ≥− 21 00   oNkkkk ...1,, 2121 =≠  max11min1 aaak ≤≤  oNk ...1=  The lateral positions of the bubbles, { } oNkkr 10 = , are obtained by solving this problem. Then, we substitute these lateral positions back to the BPS1 to obtain the complete set of POD coefficients  ( ){ }qjNka okj ,...,1,,...,1,* ==  (11) In the second-stage, the strength ε  and depth γ  of each bubble are obtained from the POD coefficients by interpolation of a piecewise linear function ( ) ( ){ }1,...,1,1,...,1,,,,, 11, −=−=≤≤≤≤= ++ mtnsaa sstttsjj γγγεεεγεγε (12) The second-stage is written as BPS2:         kk γε ,min   ( )[ ]∑= −qjjkj aa12*          s.t. ( ) ( )kkstjkkj aa γεγε ,, =  1+≤≤ tkt εεε  1+≤≤ sks γγγ  1,...,1;1,...,1 −=−= mtns ; qj ...1=  The problem BPS2 involves minimizing of a piecewise function and this is given by the global minimum of the solutions of its piecewise function elements. The bubble problem is fully solved when the lateral position, the depth and the strength of the bubbles are determined.  IV. RESULTS The parametric POD is applied to the ensemble that contains 312 snapshots corresponding to 39 values of   initial depth in the range [-1.25, -5.05] with interval step of 0.1, and 8 values of strength in the range [100, 800] with interval step of 100. The following figures and tables show the results for the water barrier problems using the AF algorithm implemented in LOQO. We can see that the AF algorithm is very efficient for solving 2D and reasonable-size 3D problems.         TABLE I COMPUTATION TIME FOR 2D PROBLEMS NO. OF BUBBLES COMPUTATION TIME 5 bubbles 16.0s 6 bubbles 19.7s 7 bubbles 23.2s 8 bubbles 28.6s Full simulation of 1 bubble 45.0s  Computation time for solving two-dimensional optimization problem in the domain of [0,14], 141 grid points (grid size 0.1) using AF algorithm. Solution is obtained by running a LOQO program on a Pentium 4 1.6MHz processor, RAM 256Mb 1 2 3 4 5 6 7 8 9 10-4-2024681012bubble No.depthstrength ε/100inception time 100*tFig. 5.  Optimal parameter values of the bubbles (depth, strength, and inception time). 024680246800.10.20.30.4xyS Fig. 4.  Constructed surface using 10 bubbles. 024680246800.10.20.30.4xyS Fig. 3.  Desired surface shape for 3D problem. 1 2 3 4 5 6 7 8-4-20246810121416bubble No.horizontal positiondepthstrength ε/100inception time 100*t Fig. 2.  Optimal parameter values of the bubbles (depth, strength, and inception time and position). 0 2 4 6 8 10 12 1400.050.10.150.20.250.30.350.4rS Fig. 1.  Solution to 2D convex sine-shape surface using 8 bubbles (dash line is desired surface, solid line is predicted surface)       V. CONCLUSION We presented an efficient method to determine the bubble parameters that will give a desired fluid surface shape for both 2D and 3D problems. The POD method using linear interpolation in parametric space, together with the AF algorithm, provides a good approximate solution to this optimization problem REFERENCES  [1] Wang Q.X., Yeo K.S., Khoo B.C. and Lam K.Y. Strong Interaction Between a Buoyancy Bubble and a Free Surface. Theoretical Computational Fluid Dynamics, Vol. 8, 73-88, 1996. [2] Rungsiyaphornrat S., Klaseboer E., Khoo B.C. and Yeo K.S. The Merging of Two Gaseous Bubbles with an Application to Underwater Explosion. Computer  & Fluid, Vol. 32, 1049-1074, 2003. [3] Holmes P., Lumley J.L. and Berkooz G. Turbulence Coherent structures, Dynamical Systems and Symmetry. Cambridge University Press, 1996. [4] Newman A.J. Model Reduction via the Karhunen-Lo `eve Expansion. Part I and II. T.R. 96-32, T.R. 96-33, 1996. [5] Sirovich L. Turbulence and the Dynamics of Coherent Structures, Part 1: Coherent Structures. Quarterly of Applied Mathematics, Vol. 45, No. 3, 561-571, October 1987. [6] Dowell E.H., Hall K.C., Thomas J.P., Florea R. and Epureanu B.I. Reduced-Order Models in Unsteady Aerodynamic. AIAA Paper, 99-0655, 1999. [7] Epureanu B.I., Dowell E.H. and Hall K.C. A Parametric Analysis of Reduced-Order Models of Potential Flows in Turbomachinery using Proper Orthogonal Decomposition. 2001-GT-0434, Proceedings of ASME TURBO EXPO 2001, New Orleans, Louisiana, June 2001. [8] Willcox K. and Peraire J. Balanced Model Reduction via the Proper Orthogonal Decomposition. AIAA paper, vol. 40, no. 11, 2323-2330, November 2002. [9] Hung V. Ly and Hien T. Tran. Modeling and Control of Physical Processes using Proper Orthogonal Decomposition. Journal of Mathematical and Computer Modeling. [10] Everson R. and Sirovich L. The Karhunen-Loeve for Gappy Data. J.Opt.Soc.Am.,12: 1657-1664, 1995. [11] Bui-Thanh T., Damodaran M. and Willcox K. Aerodynamic Data Reconstruction and Inverse Design using Proper Orthogonal  Decomposition. AIAA paper, 2003-4231.   TABLE II COMPUTATION TIME FOR 3D PROBLEMS NO.  BUBBLES 8x8 grid 15x15 grid 29x29 grid 6 bubbles 19.9s 24.7s 44.2s 8 bubbles 27.7s 39.3s 84.7s 10 bubbles 43.7s 63.0s 136.0s 12 bubbles 64.2s 255.0s 598.0s 16 bubbles 340.1s 1310.6s 2216.6s Full simulation of 1 bubble (1717 nodes)    542.0s  Computation time for solving three-dimensional optimization problems  with different numbers of bubbles and grid sizes using AF algorithm. Solution is obtained by running a LOQO program on a Pentium 4 1.6MHz processor, RAM 256Mb -1 0 1 2 3 4 5 6 7 8 9-1012345678912345678910xyFig. 6. Lateral positions of the bubbles 

Model Order Reduction for Determining Bubble Parameters to Attain a Desired Fluid Surface Shape

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Model Order Reduction for Determining Bubble Parameters to Attain a Desired Fluid Surface Shape

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