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3 - Modeling and computational simulation

from Part I - Fundamental concepts

Published online by Cambridge University Press:  05 March 2013

William L. Oberkampf  and
Christopher J. Roy
Christopher J. Roy
Affiliation:
Virginia Polytechnic Institute and State University
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Summary

The phrases modeling and simulation and computational simulation are becoming prevalent in a wide variety of technical, economic, governmental, and business activities (Schrage, 1999). Indeed, the phrases are becoming so common that one is even beginning to see them in the mass media. What do they mean? These phrases can have a wide variety of meanings depending on the field and the context. Here, we are concerned with the fields of the physical sciences and engineering. By examining the fundamentals of modeling and simulation (M&S) and scientific computing, our goal is to see the similarities in model formulation and computational issues across a wide range of physical systems. Our approach to scientific computing emphasizes the similarities that exist in mathematical form and structure of models in many technical disciplines. Then a framework, an overarching structure, is constructed for either attacking more detailed features of the system or for attacking more complex systems. Commonly, more complex systems involve coupling different types of physical phenomena, incorporation of additional elements from the system or the surroundings, and effects of human intervention in the system. Similarities in model formulation issues exist because many of the model properties are not determined by their physical nature, but by their mathematical structure, the interaction of the system with the surroundings, and the similarity in the nature of the system responses.

Many of the difficult issues that must be dealt with in verification, validation, and uncertainty quantification (VV&UQ) can be traced back to ambiguities and inconsistencies in the model formulation, the mapping of continuum mathematics models to discrete mathematics models, and vague or improper characterizations of uncertainties. This chapter deals with many of those issues by examining the fundamentals of M&S.

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Verification and Validation in Scientific Computing , pp. 83 - 144
Publisher: Cambridge University Press
Print publication year: 2010

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References

AEC (1966). AEC Atomic Weapon Safety Program. Memorandum No. 0560, Washington, DC, US Atomic Energy Commission.Google Scholar
Almond, R. G. (1995). Graphical Belief Modeling. 1st edn., London, Chapman & Hall.CrossRefGoogle Scholar
Andrews, J. D. and Moss, T. R. (2002). Reliability and Risk Assessment. 2nd edn., New York, NY, ASME Press.Google Scholar
Ang, A. H.-S. and Tang, W. H. (2007). Probability Concepts in Engineering: Emphasis on Applications to Civil and Environmental Engineering. 2nd edn., New York, John Wiley.Google Scholar
Aughenbaugh, J. M. and Paredis, C. J. J. (2006). The value of using imprecise probabilities in engineering design. Journal of Mechanical Design. 128, 969–979.CrossRefGoogle Scholar
Aven, T. (2005). Foundations of Risk Analysis: a Knowledge and Decision-Oriented Perspective, New York, John Wiley.Google Scholar
Ayyub, B. M. (1994). The nature of uncertainty in structural engineering. In Uncertainty Modelling and Analysis: Theory and Applications. Ayyub, B. M. and Gupta, M. M., eds. New York, Elsevier: 195–210.Google Scholar
Ayyub, B. M., ed. (1998). Uncertainty Modeling and Analysis in Civil Engineering. Boca Raton, FL, CRC Press.Google Scholar
Ayyub, B. M. and Klir, G. J. (2006). Uncertainty Modeling and Analysis in Engineering and the Sciences, Boca Raton, FL, Chapman & Hall.CrossRefGoogle Scholar
Bae, H.-R., Grandhi, R. V., and Canfield, R. A. (2006). Sensitivity analysis of structural response uncertainty propagation using evidence theory. Structural and Multidisciplinary Optimization. 31(4), 270–279.CrossRefGoogle Scholar
Bardossy, G. and Fodor, J. (2004). Evaluation of Uncertainties and Risks in Geology: New Mathematical Approaches for their Handling, Berlin, Springer-Verlag.CrossRefGoogle Scholar
Baudrit, C. and Dubois, D. (2006). Practical representations of incomplete probabilistic knowledge. Computational Statistics and Data Analysis. 51, 86–108.CrossRefGoogle Scholar
Beck, M. B. (1987). Water quality modeling: a review of the analysis of uncertainty. Water Resources Research. 23(8), 1393–1442.CrossRefGoogle Scholar
Bedford, T. and Cooke, R. (2001). Probabilistic Risk Analysis: Foundations and Methods, Cambridge, UK, Cambridge University Press.CrossRefGoogle Scholar
Ben-Haim, Y. (1999). Robust reliability of structures with severely uncertain loads. AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference and Exhibit, AIAA Paper 99-1605, St. Louis, MO, American Institute of Aeronautics and Astronautics, 3035–3039.CrossRef
Bogen, K. T. and Spear, R. C. (1987). Integrating uncertainty and interindividual variability in environmental risk assessment. Risk Analysis. 7(4), 427–436.CrossRefGoogle ScholarPubMed
Bossel, H. (1994). Modeling and Simulation. 1st edn., Wellesley, MA, A. K. Peters.CrossRefGoogle Scholar
Box, G. E. P. (1980). Sampling and Bayes’ inference in scientific modeling and robustness. Journal of the Royal Statistical Society: Series A. 143(A), 383–430.CrossRefGoogle Scholar
Box, G. E. P., Hunter, J. S., and Hunter, W. G. (2005). Statistics for Experimenters: Design, Innovation, and Discovery. 2nd edn., New York, John Wiley.Google Scholar
Breeding, R. J., Helton, J. C., Gorham, E. D., and Harper, F. T. (1992). Summary description of the methods used in the probabilistic risk assessments for NUREG-1150. Nuclear Engineering and Design. 135, 1–27.CrossRefGoogle Scholar
Bury, K. (1999). Statistical Distributions in Engineering, Cambridge, UK, Cambridge University Press.CrossRefGoogle Scholar
Cacuci, D. G. (2003). Sensitivity and Uncertainty Analysis: Theory, Boca Raton, FL, Chapman & Hall/CRC.CrossRefGoogle Scholar
Cash, J. R. and Karp, A. H. (1990). A variable order Runge-Kutta method for initial-value problems with rapidly varying right-hand sides. ACM Transactions on Mathematical Software. 16(3), 201–222.CrossRefGoogle Scholar
Choi, S.-K., Grandhi, R. V., and Canfield, R. A. (2007). Reliability-based Structural Design, London, Springer-Verlag.Google Scholar
Cullen, A. C. and Frey, H. C. (1999). Probabilistic Techniques in Exposure Assessment: a Handbook for Dealing with Variability and Uncertainty in Models and Inputs, New York, Plenum Press.Google Scholar
Dimov, I. T. (2008). Monte Carlo Methods for Applied Scientists. 2nd edn., Singapore, World Scientific Publishing.Google Scholar
Dorner, D. (1989). The Logic of Failure, Recognizing and Avoiding Error in Complex Situations, Cambridge, MA, Perseus Books.Google Scholar
Fellin, W., Lessmann, H., Oberguggenberger, M., and Vieider, R., eds. (2005). Analyzing Uncertainty in Civil Engineering. New York, Springer.CrossRefGoogle Scholar
Ferson, S. (1996). What Monte Carlo methods cannot do. Human and Ecological Risk Assessment. 2(4), 990–1007.CrossRefGoogle Scholar
Ferson, S. (2002). RAMAS Risk Calc 4.0 Software: Risk Assessment with Uncertain Numbers. Setauket, NY, Applied Biomathematics Corp.
Ferson, S. and Ginzburg, L. R. (1996). Different methods are needed to propagate ignorance and variability. Reliability Engineering and System Safety. 54, 133–144.CrossRefGoogle Scholar
Ferson, S. and Hajagos, J. G. (2004). Arithmetic with uncertain numbers: rigorous and (often) best possible answers. Reliability Engineering and System Safety. 85(1–3), 135–152.CrossRefGoogle Scholar
Ferson, S., Kreinovich, V., Ginzburg, L., Myers, D. S., and Sentz, K. (2003). Constructing Probability Boxes and Dempster–Shafer Structures. SAND2003-4015, Albuquerque, NM, Sandia National Laboratories.CrossRef
Ferson, S., Nelsen, R. B., Hajagos, J., Berleant, D. J., Zhang, J., Tucker, W. T., Ginzburg, L. R., and Oberkampf, W. L. (2004). Dependence in Probabilistic Modeling, Dempster–Shafer Theory, and Probability Bounds Analysis. SAND2004-3072, Albuquerque, NM, Sandia National Laboratories.
Fetz, T., Oberguggenberger, M., and Pittschmann, S. (2000). Applications of possibility and evidence theory in civil engineering. International Journal of Uncertainty. 8(3), 295–309.CrossRefGoogle Scholar
Frank, M. V. (1999). Treatment of uncertainties in space: nuclear risk assessment with examples from Cassini Mission applications. Reliability Engineering and System Safety. 66, 203–221.CrossRefGoogle Scholar
Gehman, H. W., Barry, J. L., Deal, D. W., Hallock, J. N., Hess, K. W., Hubbard, G. S., Logsdon, J. M., Osheroff, D. D., Ride, S. K., Tetrault, R. E., Turcotte, S. A., Wallace, S. B., and Widnall, S. E. (2003). Columbia Accident Investigation Board Report Volume I. Washington, DC, National Aeronautics and Space Administration Government Printing Office.
Ghanem, R. G. and Spanos, P. D. (2003). Stochastic Finite Elements: a Spectral Approach. Revised, edn., Mineda, NY, Dover Publications.Google Scholar
Haimes, Y. Y. (2009). Risk Modeling, Assessment, and Management. 3rd edn., New York, John Wiley.Google Scholar
Haldar, A. and Mahadevan, S. (2000a). Probability, Reliability, and Statistical Methods in Engineering Design, New York, John Wiley.Google Scholar
Haldar, A. and Mahadevan, S. (2000b). Reliability Assessment Using Stochastic Finite Element Analysis, New York, John Wiley.Google Scholar
Hauptmanns, U. and Werner, W. (1991). Engineering Risks Evaluation and Valuation. 1st edn., Berlin, Springer-Verlag.Google Scholar
Helton, J. C. (1993). Uncertainty and sensitivity analysis techniques for use in performance assessment for radioactive waste disposal. Reliability Engineering and System Safety. 42(2–3), 327–367.CrossRefGoogle Scholar
Helton, J. C. (1994). Treatment of uncertainty in performance assessments for complex systems. Risk Analysis. 14(4), 483–511.CrossRefGoogle Scholar
Helton, J. C. (1997). Uncertainty and sensitivity analysis in the presence of stochastic and subjective uncertainty. Journal of Statistical Computation and Simulation. 57, 3–76.CrossRefGoogle Scholar
Helton, J. C. (1999). Uncertainty and sensitivity analysis in performance assessment for the waste isolation pilot plant. Computer Physics Communications. 117(1–2), 156–180.CrossRefGoogle Scholar
Helton, J. C., Anderson, D. R., Jow, H.-N., Marietta, M. G., and Basabilvazo, G. (1999). Performance assessment in support of the 1996 compliance certification application for the Waste Isolation Pilot Plant. Risk Analysis. 19(5), 959–986.CrossRefGoogle ScholarPubMed
Helton, J. C., Johnson, J. D., and Oberkampf, W. L. (2004). An exploration of alternative approaches to the representation of uncertainty in model predictions. Reliability Engineering and System Safety. 85(1–3), 39–71.CrossRefGoogle Scholar
Helton, J. C., Oberkampf, W. L., and Johnson, J. D. (2005). Competing failure risk analysis using evidence theory. Risk Analysis. 25(4), 973–995.CrossRefGoogle ScholarPubMed
Helton, J. C., Johnson, J. D., Sallaberry, C. J., and Storlie, C. B. (2006). Survey of sampling-based methods for uncertainty and sensitivity analysis. Reliability Engineering and System Safety. 91(10–11), 1175–1209.CrossRefGoogle Scholar
Hoffman, F. O. and Hammonds, J. S. (1994). Propagation of uncertainty in risk assessments: the need to distinguish between uncertainty due to lack of knowledge and uncertainty due to variability. Risk Analysis. 14(5), 707–712.CrossRefGoogle ScholarPubMed
Hora, S. C. and Iman, R. L. (1989). Expert opinion in risk analysis: the NUREG-1150 methodology. Nuclear Science and Engineering. 102, 323–331.CrossRefGoogle Scholar
Jacoby, S. L. S. and Kowalik, J. S. (1980). Mathematical Modeling with Computers, Englewood Cliffs, NJ, Prentice-Hall.Google Scholar
Kaplan, S. and Garrick, B. J. (1981). On the quantitative definition of risk. Risk Analysis. 1(1), 11–27.CrossRefGoogle Scholar
Kleijnen, J. P. C. (1998). Chapter 6: Experimental design for sensitivity analysis, optimization, and validation of simulation models. In Handbook of Simulation: Principles, Methodology, Advances, Application, and Practice. Banks, J., ed. New York, John Wiley: 173–223.CrossRefGoogle Scholar
Klir, G. J. and Wierman, M. J. (1998). Uncertainty-Based Information: Elements of Generalized Information Theory, Heidelberg, Physica-Verlag.Google Scholar
Kloeden, P. E. and Platen, E. (2000). Numerical Solution of Stochastic Differential Equations, New York, Springer.Google Scholar
Kohlas, J. and Monney, P.-A. (1995). A Mathematical Theory of Hints – an Approach to the Dempster–Shafer Theory of Evidence, Berlin, Springer-Verlag.CrossRefGoogle Scholar
Krause, P. and Clark, D. (1993). Representing Uncertain Knowledge: an Artificial Intelligence Approach, Dordrecht, The Netherlands, Kluwer Academic Publishers.CrossRefGoogle Scholar
Kriegler, E. and Held, H. (2005). Utilizing belief functions for the estimation of future climate change. International Journal for Approximate Reasoning. 39, 185–209.CrossRefGoogle Scholar
Krishnamoorthy, K. (2006). Handbook of Statistical Distribution with Applications, Boca Raton, FL, Chapman and Hall.CrossRefGoogle Scholar
Kumamoto, H. (2007). Satisfying Safety Goals by Probabilistic Risk Assessment, Berlin, Springer-Verlag.Google Scholar
Kumamoto, H. and Henley, E. J. (1996). Probabilistic Risk Assessment and Management for Engineers and Scientists. 2nd edn., New York, IEEE Press.Google Scholar
Law, A. M. (2006). Simulation Modeling and Analysis. 4th edn., New York, McGraw-Hill.Google Scholar
Lawson, D. (2005). Engineering Disasters – Lessons to be Learned, New York, ASME Press.Google Scholar
LeGore, T. (1990). Predictive software validation methodology for use with experiments having limited replicability. In Benchmark Test Cases for Computational Fluid Dynamics. Celik, I. and Freitas, C. J., eds. New York, American Society of Mechanical Engineers. FED-Vol. 93: 21–27.Google Scholar
Leijnse, A. and Hassanizadeh, S. M. (1994). Model definition and model validation. Advances in Water Resources. 17, 197–200.CrossRefGoogle Scholar
Mason, R. L., Gunst, R. F., and Hess, J. L. (2003). Statistical Design and Analysis of Experiments, with Applications to Engineering and Science. 2nd edn., Hoboken, NJ, Wiley Interscience.Google Scholar
Melchers, R. E. (1999). Structural Reliability Analysis and Prediction. 2nd edn., New York, John Wiley.Google Scholar
Modarres, M., Kaminskiy, M., and Krivtsov, V. (1999). Reliability Engineering and Risk Analysis; a Practical Guide, Boca Raton, FL, CRC Press.Google Scholar
Moller, B. and Beer, M. (2004). Fuzz Randomness: Uncertainty in Civil Engineering and Computational Mechanics, Berlin, Springer-Verlag.CrossRefGoogle Scholar
Morgan, M. G. and Henrion, M. (1990). Uncertainty: a Guide to Dealing with Uncertainty in Quantitative Risk and Policy Analysis. 1st edn., Cambridge, UK, Cambridge University Press.CrossRefGoogle Scholar
Mosey, D. (2006). Reactor Accidents: Institutional Failure in the Nuclear Industry. 2nd edn., Sidcup, Kent, UK, Nuclear Engineering International.Google Scholar
NASA (2002). Probabilistic Risk Assessment Procedures Guide for NASA Managers and Practitioners. Washington, DC, NASA.
Neelamkavil, F. (1987). Computer Simulation and Modelling. 1st edn., New York, John Wiley.Google Scholar
Nikolaidis, E., Ghiocel, D. M., and Singhal, S., eds. (2005). Engineering Design Reliability Handbook. Boca Raton, FL, CRC Press.Google Scholar
NRC (1990). Severe Accident Risks: an Assessment for Five U.S. Nuclear Power Plants. NUREG-1150, Washington, DC, US Nuclear Regulatory Commission, Office of Nuclear Regulatory Research, Division of Systems Research.
NRC (2009). Guidance on the Treatment of Uncertainties Assoicated with PRAs in Risk-Informed Decision Making. Washington, DC, Nuclear Regulator Commission.
Oberkampf, W. L. and Helton, J. C. (2005). Evidence theory for engineering applications. In Engineering Design Reliability Handbook. Nikolaidis, E., Ghiocel, D. M. and Singhal, S., eds. New York, NY, CRC Press: 29.Google Scholar
Oberkampf, W. L., DeLand, S. M., Rutherford, B. M., Diegert, K. V., and Alvin, K. F. (2000). Estimation of Total Uncertainty in Computational Simulation. SAND2000-0824, Albuquerque, NM, Sandia National Laboratories.
Oberkampf, W. L., DeLand, S. M., Rutherford, B. M., Diegert, K. V., and Alvin, K. F. (2002). Error and uncertainty in modeling and simulation. Reliability Engineering and System Safety. 75(3), 333–357.CrossRefGoogle Scholar
Oksendal, B. (2003). Stochastic Differential Equations: an Introduction with Applications. 6th edn., Berlin, Springer-Verlag.CrossRefGoogle Scholar
Paté-Cornell, M. E. (1990). Organizational aspects of engineering system failures: the case of offshore platforms. Science. 250, 1210–1217.CrossRefGoogle ScholarPubMed
Pegden, C. D., Shannon, R. E., and Sadowski, R. P. (1990). Introduction to Simulation Using SIMAN. 1st edn., New York, McGraw-Hill.Google Scholar
Petroski, H. (1994). Design Paradigms: Case Histories of Error and Judgment in Engineering, Cambridge, UK, Cambridge University Press.CrossRefGoogle Scholar
Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P. (2007). Numerical Recipes in FORTRAN. 3rd edn., New York, Cambridge University Press.Google Scholar
Raczynski, S. (2006). Modeling and Simulation: the Computer Science of Illusion, New York, Wiley.CrossRefGoogle Scholar
Reason, J. (1997). Managing the Risks of Organizational Accidents, Burlington, VT, Ashgate Publishing Limited.Google Scholar
Roache, P. J. (1998). Verification and Validation in Computational Science and Engineering, Albuquerque, NM, Hermosa Publishers.Google Scholar
Ross, S. M. (2006). Simulation. 4th edn., Burlington, MA, Academic Press.Google Scholar
Ross, T. J. (2004). Fuzzy Logic with Engineering Applications. 2nd edn., New York, Wiley.Google Scholar
Rubinstein, R. Y. and Kroese, D. P. (2008). Simulation and the Monte Carlo Method. 2nd edn., Hoboken, NJ, John Wiley.Google Scholar
Salguero, D. E. (1999). Trajectory Analysis and Optimization Software (TAOS). SAND99-0811, Albuquerque, NM, Sandia National Laboratories.
Saltelli, A., Tarantola, S., Campolongo, F., and Ratto, M. (2004). Sensitivity Analysis in Practice: a Guide to Assessing Scientific Models, Chichester, England, John Wiley.Google Scholar
Saltelli, A., Ratto, M., Andres, T., Campolongo, F., Cariboni, J., Gatelli, D., Saisana, M., and Tarantola, S. (2008). Global Sensitivity Analysis: the Primer, Hoboken, NJ, Wiley.Google Scholar
Schrage, M. (1999). Serious Play: How the World's Best Companies Simulate to Innovate, Boston, MA, Harvard Business School Press.Google Scholar
Serrano, S. E. (2001). Engineering Uncertainty and Risk Analysis: a Balanced Approach to Probability, Statistics, Stochastic Modeling, and Stochastic Differential Equations, Lexington, KY, HydroScience Inc.Google Scholar
Severance, F. L. (2001). System Modeling and Simulation: an Introduction, New York, Wiley.Google Scholar
Singh, V. P., Jain, S. K., and Tyagi, A. (2007). Risk and Reliability Analysis: a Handbook for Civil and Environmental Engineers, New York, American Society of Civil Engineers.CrossRefGoogle Scholar
Singpurwalla, N. D. (2006). Reliability and Risk: a Bayesian Perspective, New York, NY, Wiley.CrossRefGoogle Scholar
Stockman, C. T., Garner, J. W., Helton, J. C., Johnson, J. D., Shinta, A., and Smith, L. N. (2000). Radionuclide transport in the vicinity of the repository and associated complementary cumulative distribution functions in the 1996 performance assessment for the Waste Isolation Pilot Plant. Reliability Engineering and System Safety. 69(1–3), 369–396.CrossRefGoogle Scholar
Storlie, C. B. and Helton, J. C. (2008). Multiple predictor smoothing methods for sensitivity analysis: description of techniques. Reliability Engineering and System Safety. 93(1), 28–54.CrossRefGoogle Scholar
Suter, G. W. (2007). Ecological Risk Assessment. 2nd edn., Boca Raton, FL, CRC Press.Google Scholar
Taylor, H. M. and Karlin, S. (1998). An Introduction to Stochastic Modeling. 3rd edn., Boston, Academic Press.Google Scholar
Tenner, E. (1996). Why Things Bite Back, New York, Alfred A. Knopf.Google Scholar
Tung, Y.-K. and Yen, B.-C. (2005). Hydrosystems Engineering Uncertainty Analysis, New York, McGraw-Hill.Google Scholar
Vaughan, D. (1996). The Challenger Launch Decision: Risky Technology, Culture, and Deviance at NASA, Chicago, IL, The University of Chicago Press.Google Scholar
Vinnem, J. E. (2007). Offshore Risk Assessment: Principles, Modelling and Applications of QRA Studies, Berlin, Springer-Verlag.Google Scholar
Vose, D. (2008). Risk Analysis: a Quantitative Guide. 3rd edn., New York, Wiley.Google Scholar
Yee, H. C. and Sweby, P. K. (1995). Dynamical approach study of spurious steady-state numerical solutions of nonlinear differential equations II. Global asymptotic behavior of time discretizations. Computational Fluid Dynamics. 4, 219–283.CrossRefGoogle Scholar
Yee, H. C. and Sweby, P. K. (1996). Nonlinear Dynamics & Numerical Uncertainties in CFD. Rept. No. 110398, Moffett Field, CA, NASA/Ames Research Center.
Yee, H. C. and Sweby, P. K. (1998). Aspects of numerical uncertainties in time marching to steady-state numerical solutions. AIAA Journal. 36(5), 712–724.CrossRefGoogle Scholar
Yee, H. C., Torczynski, J. R., Morton, S. A., Visbal, M. R., and Sweby, P. K. (1997). On spurious behavior of CFD simulations. 13th AIAA Computational Fluid Dynamics Conference, AIAA Paper 97-1869, Snowmass, CO, American Institute of Aeronautics and Astronautics.CrossRef
Zeigler, B. P., Praehofer, H., and Kim, T. G. (2000). Theory of Modeling and Simulation: Integrating Discrete Event and Continuous Complex Dynamic Systems. 2nd edn., San Diego, CA, Academic Press.Google Scholar

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