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Evaluation of convergence behavior of metamodeling techniques for bridging scales in multi-scale multimaterial simulation. (English) Zbl 1349.76857

Summary: The effectiveness of several metamodeling techniques, viz. the Polynomial Stochastic Collocation method, Adaptive Stochastic Collocation method, a Radial Basis Function Neural Network, a Kriging Method and a Dynamic Kriging Method is evaluated. This is done with the express purpose of using metamodels to bridge scales between micro- and macro-scale models in a multi-scale multimaterial simulation. The rate of convergence of the error when used to reconstruct hypersurfaces of known functions is studied. For sufficiently large number of training points, Stochastic Collocation methods generally converge faster than the other metamodeling techniques, while the DKG method converges faster when the number of input points is less than 100 in a two-dimensional parameter space. Because the input points correspond to computationally expensive micro/meso-scale computations, the DKG is favored for bridging scales in a multi-scale solver.

MSC:

76T15 Dusty-gas two-phase flows
76F65 Direct numerical and large eddy simulation of turbulence

Software:

Matlab; DACE
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Full Text: DOI

References:

[1] Fedorov, A. V.; Kharlamova, Y. V.; Khmel, T. A., Reflection of a shock wave in a dusty cloud, Combust. Explos. Shock Waves, 43, 1, 104-113 (2007)
[2] Hambli, R., Numerical procedure for multiscale bone adaptation prediction based on neural networks and finite element simulation, Finite Elem. Anal. Des., 47, 7, 835-842 (2011)
[3] Hambli, R., Multiscale prediction of crack density and crack length accumulation in trabecular bone based on neural networks and finite element simulation, Int. J. Numer. Methods Biomed. Eng., 27, 461-475 (2011) · Zbl 1215.92004
[4] Hambli, R., Apparent damage accumulation in cancellous bone using neural networks, J. Mech. Behav. Biomed. Mater., 4, 868-878 (2011)
[5] Hambli, R.; Katerchi, H.; Benhamou, C.-L., Multiscale methodology for bone remodelling simulation using coupled finite element and neural network computation, Biomech. Model. Mechanobiol., 10, 133-145 (2011)
[6] Unger, J. F.; Könke, C., Coupling of scales in a multiscale simulation using neural networks, Comput. Struct., 86, 21-22, 1994-2003 (2008)
[7] Unger, J. F.; Eckardt, S., Multiscale modeling of concrete, Arch. Comput. Methods Eng., 18, 3, 341-393 (2011) · Zbl 1284.74105
[8] Bdzil, J. B.; Menikoff, R.; Son, S. F.; Kapila, A. K.; Stewart, D. S., Two-phase modeling of deflagration-to-detonation transition in granular materials: a critical examination of modeling issues, Phys. Fluids, 11, 2, 378 (1999) · Zbl 1147.76317
[9] Kapila, A. K.; Menikoff, R.; Bdzil, J. B.; Son, S. F.; Stewart, D. S., Two-phase modeling of deflagration-to-detonation transition in granular materials: reduced equations, Phys. Fluids, 13, 10, 3002 (2001) · Zbl 1184.76268
[10] Menikoff, R., Hot spot formation from shock reflections, Shock Waves, 21, 2, 141-148 (2011)
[11] Kapahi, A.; Udaykumar, H. S., Dynamics of void collapse in shocked energetic materials: physics of void-void interactions, Shock Waves, 23, 6, 537-558 (2013)
[12] Abdelhamid, Y.; El Shamy, U., Multiscale modeling of flood-induced scour in a particle bed, Bridges, 10, 740-749 (2014)
[13] van der Hoef, M.; van Sint Annaland, M.; Deen, N.; Kuipers, J., Numerical simulation of dense gas-solid fluidized beds: a multiscale modeling strategy, Annu. Rev. Fluid Mech., 40, 1, 47-70 (2008) · Zbl 1231.76327
[14] Pan, W.; Fedosov, D. A.; Caswell, B.; Karniadakis, G. E., Predicting dynamics and rheology of blood flow: a comparative study of multiscale and low-dimensional models of red blood cells, Microvasc. Res., 82 (2011)
[15] Sun, C. T.; Vaidya, R. S., Prediction of composite properties from a representative volume element, Compos. Sci. Technol., 56, 171-179 (1996)
[16] E, W.; Engquist, B.; Huang, Z., Heterogeneous multiscale method: a general methodology for multiscale modeling, Phys. Rev. B, 67, 9, 1-4 (2003)
[17] Jacobs, G. B.; Don, W.-S., A high-order WENO-Z finite difference based particle-source-in-cell method for computation of particle-laden flows with shocks, J. Comput. Phys., 228, 5, 1365-1379 (2009) · Zbl 1409.76113
[18] Shotorban, B.; Jacobs, G. B.; Ortiz, O.; Truong, Q., An eulerian model for particles nonisothermally carried by a compressible fluid, Int. J. Heat Mass Transf., 65, 0, 845-854 (2013)
[19] Davis, S.; Dittman, T.; Jacobs, G. B.; Don, W. S., High-fidelity Eulerian-Lagrangian methods for simulation of three dimensional, unsteady, high-speed, two-phase flows in high-speed combustors, (47th AIAA/ASME/SAE/ASEE Joint Propulsion Conference (2011))
[20] Davis, S.; Dittmann, T.; Jacobs, G.; Don, W., Dispersion of a cloud of particles by a moving shock: effects of the shape, angle of rotation, and aspect ratio, J. Appl. Mech. Tech. Phys., 54, 6, 900-912 (2013)
[21] Stokes, G. G., On the effect of the internal friction of fluids on the motion of pendulums, Transactions of the Cambridge Philosophical Society, 9 (1850)
[22] Clift, R.; Grace, J. R.; Weber, M. E., Bubbles, Drops, and Particles (1978), Academic Press
[23] Boiko, V. M.; Poplavskii, S. V., Drag of nonspherical particles in a flow behind a shock wave, Combust. Explos. Shock Waves, 41, 1, 71-77 (2005)
[24] Loth, E., Compressibility and rarefaction effects on drag of a spherical particle, AIAA J., 46, 9, 2219-2228 (2008)
[25] Tedeschi, G.; Gouin, H.; Elena, M., Motion of tracer particles in supersonic flows, Exp. Fluids, 26, 4, 288-296 (1999)
[26] Tong, X.; Remotigue, M.; Luke, E.; Kang, J., Multiphase simulations of blast-soil interactions, (Proceedings of the ASME 2013 Fluids Engineering Division Summer Meeting (2013), ASME), No. FEDSM2013-16549
[27] Feng, Z.-G.; Michaelides, E. E., Drag coefficients of viscous spheres at intermediate and high Reynolds numbers, J. Fluids Eng., 123, 4, 841-849 (2001)
[28] Lu, C., Artificial neural network for behavior learning from meso-scale simulations, applications to multi-scale multimaterial flows (2010), The University of Iowa, Ph.D. thesis
[29] Lu, C.; Sambasivan, S.; Kapahi, A.; Udaykumar, H. S., Multi-scale modeling of shock interaction with a cloud of particles using an artificial neural network for model representation, Proc. IUTAM, 3, 25-52 (2012)
[30] Davis, S.; Sen, O.; Jacobs, G.; Udaykumar, H., Coupling of micro-scale and macro-scale Eulerian-Lagrangian models for the computation of shocked particle-laden flows, (ASME 2013 International Mechanical Engineering Congress and Exposition, vol. 7A (2013), ASME)
[31] Kleijnen, J. P., Statistical Tools for Simulation Practitioners (1986), Marcel Dekker, Inc. · Zbl 0629.62004
[32] Simpson, T. W.; Peplinski, J. D.; Koch, P. N.; Allen, J. K., Metamodels for computer-based engineering design: survey and recommendations, Eng. Comput., 17, 129-150 (2001) · Zbl 0985.68599
[33] Jin, R.; Du, X.; Chen, W., The use of metamodeling techniques for optimization under uncertainty, Struct. Multidiscip. Optim., 25, 2, 99-116 (2003)
[34] Jin, Y., A comprehensive survey of fitness approximation in evolutionary computation, Soft Comput., 9, 1, 3-12 (2005) · Zbl 1059.68089
[35] Wang, G. G.; Shan, S., Review of metamodeling techniques in support of engineering design optimization, J. Mech. Des., 129, 4, 370-380 (2007)
[36] Chen, V. C.; Tsui, K.-L.; Barton, R. R.; Meckesheimer, M., A review on design, modeling and applications of computer experiments, IIE Trans., 38, 4, 273-291 (2006)
[37] Jin, R.; Chen, W.; Simpson, T. W., Comparative studies of metamodelling techniques under multiple modelling criteria, Struct. Multidiscip. Optim., 23, 1, 1-13 (2001)
[38] Fang, H.; Horstemeyer, M. F., Global response approximation with radial basis functions, Eng. Optim., 38, 04, 407-424 (2006)
[39] Xiu, D.; Hesthaven, J. S., High-order collocation methods for differential equations with random inputs, SIAM J. Sci. Comput., 27, 3, 1118-1139 (2005) · Zbl 1091.65006
[40] Barthelmann, V.; Novak, E.; Ritter, K., High dimensional polynomial interpolation on sparse grids, Adv. Comput. Math., 12, 273-288 (2000) · Zbl 0944.41001
[41] Ma, X.; Zabaras, N., An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations, J. Comput. Phys., 228, 8, 3084-3113 (2009) · Zbl 1161.65006
[42] Chen, S.; Cowan, C. F.M.; Grant, P. M., Orthogonal least squares learning algorithm for radial basis function networks, IEEE Trans. Neural Netw., 2, 2, 302-309 (1991)
[43] Park, J.; Sandberg, I., Universal approximation using radial-basis-function networks, Neural Comput., 3, 246-257 (1991)
[44] Haykin, S., Neural Networks: A Comprehensive Foundation (1994), Mc Millan: Mc Millan New Jersey · Zbl 0828.68103
[45] Li, H.; Mulay, S. S., Meshless Methods and Their Numerical Properties (2013), CRC Press · Zbl 1266.65001
[46] Trochu, F., A contouring program based on dual Kriging interpolation, Eng. Comput., 9, 160-177 (1993)
[47] Lophaven, S. N.; Nielsen, H. B.; Søndergaard, J., Dace-a Matlab Kriging toolbox, version 2.0 (2002), Tech. rep.
[48] Song, H.; Choi, K. K.; Lamb, D., A study on improving the accuracy of Kriging models by using correlation model/mean structure selection and penalized log-likelihood function, (10th World Conference on Structural and Multidisciplinary Optimization (2013)), 1-10
[49] Smolyak, S. A., Quadrature and interpolation formulas for tensor products of certain classes of functions, Dokl. Akad. Nauk SSSR, 4, 240-243 (1963) · Zbl 0202.39901
[50] Sgarbi, M.; Colla, V.; Reyneri, L. M., A comparison between weighted radial basis functions and wavelet networks, (ESANN, Citeseer (1998)), 13-20
[51] Wendland, H., Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Adv. Comput. Math., 4, 1, 389-396 (1995) · Zbl 0838.41014
[52] Wu, Z., Compactly supported positive definite radial functions, Adv. Comput. Math., 4, 1, 283-292 (1995) · Zbl 0837.41016
[53] Morse, B. S.; Yoo, T. S.; Rheingans, P.; Chen, D. T.; Subramanian, K. R., Interpolating implicit surfaces from scattered surface data using compactly supported radial basis functions, (ACM SIGGRAPH 2005 Courses (2005), ACM), 78
[54] Ferrari, S.; Maggioni, M.; Borghese, N. A., Multiscale approximation with hierarchial radial basis function networks, IEEE Trans. Neural Netw., 15, 1, 178-188 (2004)
[55] Fornberg, B.; Piret, C., A stable algorithm for flat radial basis functions on a sphere, SIAM J. Sci. Comput., 30, 1, 60-80 (2007) · Zbl 1159.65307
[56] Fasshauer, G. E.; McCourt, M. J., Stable evaluation of gaussian radial basis function interpolants, SIAM J. Sci. Comput., 34, 2, A737-A762 (2012) · Zbl 1252.65028
[57] Fornberg, B.; Driscoll, T. A.; Wright, G.; Charles, R., Observations on the behavior of radial basis function approximations near boundaries, Comput. Math. Appl., 43, 35, 473-490 (2002) · Zbl 0999.65005
[58] Leonard, J.; Kramer, M.; Ungar, L., Using radial basis functions to approximate a function and its error bounds, IEEE Trans. Neural Netw., 3, 4, 624-627 (1992)
[59] Chen, S.; Grant, P. M.; Cowan, C. F.N., Orthogonal least-squares algorithm for training multioutput radial basis function networks, IEEE Proc. F, 139, 6, 378-384 (1992)
[60] Benoudjit, N.; Verleysen, M., On the kernel widths in radial-basis function networks, Neural Process. Lett., 18, 139-154 (2003)
[61] Dachapak, C.; Kanae, S.; Yang, Z.-J.; Wada, K., Orthogonal least squares for radial basis function network in reproducing Kernel Hilbert space, (IFAC Workshop on Adaptation and Learning in Control and Signal Processing and IFAC Workshop on Periodic Control Systems (2004)), 847-852
[62] Du, K.-L.; Swamy, M. N., Neural Networks in a Softcomputing Framework (2006), Springer · Zbl 1101.68757
[63] Huang, G.-B.; Saratchandran, P.; Sundararajan, N., A generalized growing and pruning RBF (GGAP-RBF) neural network for function approximation, IEEE Trans. Neural Netw., 16, 1, 57-67 (2005)
[64] Neruda, R.; Vidnerova, P., Learning errors by radial basis function neural networks and regularization networks, Int. J. Grid Distrib. Comput., 1, 2, 49-58 (2009)
[65] Tao, K. M., A closer look at the radial basis function (rbf) networks, (1993 Conference Record of The Twenty-Seventh Asilomar Conference on Signals, Systems and Computers (1993)), 401-405
[66] Moody, J.; Darken, C. J., Fast learning in networks of locally-tuned processing units, Neural Comput., 1, 281-294 (1989)
[67] Cressie, N., Statistics for spatial data, Terra Nova, 4, 5, 613-617 (1992)
[68] Matheron, G., Principles of geostatics, Econ. Geol., 58, 1246-1266 (1963)
[69] Zhao, L.; Choi, K.; Lee, I.; Gorsich, D., A metamodeling method using dynamic kriging and sequential sampling, (The 13th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference. The 13th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Fort Worth, TX (Sept. 2010)), 13-15
[70] Lophaven, S. N.; Nielsen, H. B.; Søndergaard, J., Aspects of the Matlab toolbox dace (2002), Informatics and Mathematical Modelling, Technical University of Denmark, DTU, Tech. rep.
[71] Stein, M., Large sample properties of simulations using Latin hypercube sampling, Technometrics, 29, 143-151 (1987) · Zbl 0627.62010
[72] Goel, T.; Haftka, R.; Shyy, W.; Watson, L., Pitfalls of using a single criterion for selecting experimental designs, Int. J. Numer. Methods Eng., 75, 127-155 (2008) · Zbl 1195.62129
[73] Burkardt, J.; Gunzburger, M.; Peterson, J.; Brannon, R., User manual and supporting information for library of codes for centroidal Voronoi placement and associated zeroth, first, and second moment determination (2002), Tech. rep.
[74] Boiko, V.; Kiselev, V. P.; Kiselev, S. P.; Papyrin, A.; Poplavsky, S.; Fomin, V., Shock wave interaction with a cloud of particles, Shock Waves, 7, 275-285 (1997) · Zbl 0900.76241
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