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International Journal for Uncertainty Quantification
Fator do impacto: 4.911 FI de cinco anos: 3.179 SJR: 1.008 SNIP: 0.983 CiteScore™: 5.2

ISSN Imprimir: 2152-5080
ISSN On-line: 2152-5099

Open Access

International Journal for Uncertainty Quantification

DOI: 10.1615/Int.J.UncertaintyQuantification.2020030630
pages 83-100


Xiao Liang
School of Mathematics and System Science, Shandong University of Science and Technology, Qingdao, Shandong, China and Sony Astani Department of Civil and Environmental Engineering, University of Southern California, Los Angeles, CA
R. Wang
Institute of Applied Physics and Computational Mathematics, Beijing, China
Roger Ghanem
Sony Astani Department of Aerospace and Mechanical Engineering, University of Southern California, 210 KAP Hall, Los Angeles, California 90089, USA


Mathematical models used to describe detonation consist usually of coupled nonlinear partial differential equations, with phenomena occurring at a multitude of scales. While numerical solutions of these problems require significant computational resources, the evolution of the physics along multiple spatial and temporal scales makes the associated predictions sensitive to fluctuations that are beyond normal experimental control. Modeling, characterizing, and propagating uncertainties in predictions of detonation dynamics exacerbates both the mathematical, algorithmic, and computational challenges. These challenges are addressed in the present paper by using basis adaptation in the context of polynomial chaos expansions. The multivariate Rosenblatt transformation is used to first map all the random variables to independent Gaussian variables, following which a rotation is affected on these Gaussians that is adapted to any specified quantity of interest. Thus, accurate estimates of statistical moments and even probability density functions are obtained at specified Lagrangian reference points.


  1. Wang, R. and Jiang, S., Mathematical Methods for Uncertainty Quantification of the Nonlinear Partial Differential Equation and Numerical Solution, Sci. China Math., 45:1-18, 2015.

  2. Liang, X. and Wang, R., Sensitivity Analysis and Validation of Detonation CFD Model, Acta Phys. Sin., 109:114-121,2017. (in Chinese).

  3. Wang, R. and Liang, X., Research on Validation Experiment Hierarchy of Validation for Physical Modeling in Numerical Simulation of Detonation, China Meas. Test, 42:13-20,2016.

  4. Fickett, W. and Davis, W., Detonation Theory and Experiment, Boston, MA: Dover Publications, 1979.

  5. Oberkampf, W. and Roy, C., Verification and Validation in Scientific Computing, Boston, MA: Cambridge University Press, 2010.

  6. Aeschliman, D. and Oberkampf, W., Experimental Methodology for Computational Fluid Dynamics Code Validation, AIAA J., 36:733-741,2015.

  7. Kozmenkov, Y., Kliem, S., and Rohde, U., Validation and Verification of the Coupled Neutron Kinetic/Thermal Hydraulic System Code DYN3D/ATHLET, Ann. Nucl. Energy, 84:153-165,2015.

  8. Scovel, C. and Menikoff, R., High Explosive Verification and Validation: Systematic and Methodical Approach, J. Clin. Invest, 96:2661-2666,2011.

  9. Ghanem, R. and Spanos, P., Stochastic Finite Elements: A Spectral Approach, New York: Springer-Verlag, 1989.

  10. Soize, C. and Ghanem, R., Physical Systems with Random Uncertainties: Chaos Representations with Arbitrary Probability Measure, SIAMJ. Sci. Comput., 26:395-410,2004.

  11. Ghanem, R. and Red-Horse, J., Polynomial Chaos: Modeling, Estimation, and Approximation, in Handbook on Uncertainty Quantification, R. Ghanem, D. Higdon, and H. Owhadi, Eds., New York: Springer, 2017.

  12. Doostan, A., Ghanem, R., and Red-Horse, J., Stochastic Model Reduction for Chaos Representation, Comput. Methods Appl. Mech. Eng., 196:3951-3966,2007.

  13. Tipireddy, R. and Ghanem, R., Basis Adaptation in Homogeneous Chaos Spaces, J. Comput. Phys., 259:304-317, 2014.

  14. Tsilifis, P. and Ghanem, R., Reduced Wiener Chaos Representation of Random Fields via Basis Adaptation and Projection, J. Comput. Phys, 341:102-120, 2017.

  15. Huan, X., Geraci, G., Safta, C., Eldred, M.S., Sargsyan, K., Vane, Z.P., Oefelein, J.C., andNajm, H.N., Multifidelity Statistical Analysis of Large Eddy Simulations in Scramjet Computations, in 2018 AIAA Non-Deterministic Approaches Conf., p. 1180, 2018.

  16. Doostan, A. and Owhadi, H., A Non-Adapted Sparse Approximation of PDEs with Stochastic Inputs, J. Comput. Phys, 230:3015-3034,2011.

  17. Huan, X., Safta, C., Sargsyan, K., Vane, Z.P., Lacaze, G., Oefelein, J.C., and Najm, H.N., Compressive Sensing with Cross Validation and Stop-Sampling for Sparse Polynomial Chaos Expansions, SIAM/ASA J. Uncertainty Quantif., 6:907-936,2018.

  18. Weibull, W., Spread around the Initiating Point of the Detonation Wave in High Explosives, Nature, 160:402-403, 1947.

  19. Metchell, D. and Paterson, S., Spread of Detonation in High Explosives, Nature, 161:438-439, 1947.

  20. Romick, R., Aslam, T., and Powers, J., Verified and Validated Calculation of Unsteady Dynamics of Viscous Hydrogen-Air Detonations, J. Fluid Mech, 769:154-181,2015.

  21. Bdzil, J. and Stewart, D., The Dynamics of Detonation in Explosive Systems, Ann. Rev. Fluid Mech., 39:263-292,2007.

  22. Schlesinger, S., Terminology for Model Credibility, Simulation, 27:103-104, 1979.

  23. Mader, C. and Forest, C., Two-Dimensional Homogeneous and Heterogeneous Detonation Wave Propagation, Chem. Explosive, 21:1-17, 1976.

  24. Mader, C., Numerical Modeling of Detonations, Livermore, CA: University of California Press, 1979.

  25. Lee, E. and Tarver, C., Phenomenological Model of Shock Initiation in Heterogeneous Explosives, Phys. Fluids, 23:2362.

  26. Long, Y. and Chen, J., Theoretical Study of the Reaction Kinetics and the Detonation Wave Profile for 1,3,5-Triamino-2,4,6-Trinitrobenzene, J. Appl. Phys, 120:185902, 2016.

  27. Hu, Y., Lu, W., Chen, M., Yan, P., and Zhang, Y., Numerical Simulation of the Complete Rock Blasting Response by SPH- DAM-CFEM Approach, Simul. ModelPract. Theory, 56:55-68, 2015.

  28. Castedo, R., Natale, M., Lopez, L., Sanchidrin, J., Santos, A.P., Navarro, J., and Segarra, P., Estimation of Jones-Wilkins-Lee Parameters of Emulsion Explosives Using Cylinder Tests and Their Numerical Validation, Int. J. Rock Mech. Mining Sci, 112:290-301,2018.

  29. Wilkins, M., The Equation of State of PBX 9404 and LXO4-01, Lawrence Radiation Laboratory, Livermore, CA, Rep. No. UCRL-7797, 1964.

  30. Jasak, H., Weller, H., and Gosman, A., High Resolution NVD Differencing Scheme for Arbitrarily Unstructured Meshes, Int. J. Numer. Methods Fluids, 31:431-449, 1999.

  31. Jasak, H. and Tukovic, Z., Automatic Mesh Motion for the Unstructured Finite Volume Method, Trans. Famena, 30:1-20, 2006.

  32. Bartlma, F. and Schroder, K., The Diffraction of a Plane Detonation Wave at a Convex Corner, Combust. Flame, 66:237-248, 1986.

  33. Yuan, X., Zhou, J., Lin, Z., and Cai, X., Numerical Study of Detonation Diffraction through 90-Degree Curved Channels to Expansion Area, Int. j. Hydrogen Energy, 42:7045-7059, 2017.

  34. Liang, X. and Wang, R., Verification and Validation of Detonation Modeling, Defence Technol, 15:398-408, 2019.

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