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Linear Systems with Large Uncertainties, with Applications to Truss Structures


Linear systems whose coefficients have large uncertainties arise routinely in finite element calculations for structures with uncertain geometry, material properties, or loads. However, a true worst case analysis of the influence of such uncertainties was previously possible only for very small systems and uncertainties, or in special cases where the coefficients do not exhibit dependence.

This paper presents a method for computing rigorous bounds on the solution of such systems, with a computable overestimation factor that is frequently quite small. The merits of the new approach are demonstrated by computing realistic bounds for some large, uncertain truss structures, some leading to linear systems with over 5000 variables and over 10000 interval parameters, with excellent bounds for up to about 10% input uncertainty.

Also discussed are some counterexamples for the performance of traditional approximate methods for worst case uncertainty analysis.

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  1. ANSYS:

  2. ANSYS Commands Reference: ANSYS 8.0 Help, ANSYS, 2003.

  3. COCONUT, COntinuous CONstraints—Updating the Technology, an IST Project funded by the European Union,

  4. Hansen, E.: Preconditioning Linearized Equations, Computing 58 (1997), pp. 187–196

    Article  MATH  MathSciNet  Google Scholar 

  5. Krawczyk, R. and Neumaier, A.: Interval Slopes for Rational Functions and Associated Centered Forms, SIAM J. Numer. Anal. 22 (1985), pp. 604–616.

    Article  MATH  MathSciNet  Google Scholar 

  6. Muhanna, R. L.: Benchmarks for Interval Finite Element Computations, 2004,

  7. Muhanna, R. L. and Mullen, R. L.: Uncertainty in Mechanics Problems—Interval Based Approach, J. Engrg. Mech. 127 (2001), pp. 557–566.

    Article  Google Scholar 

  8. Muhanna, R. L., Mullen, R. L., and Zhang, H.: Interval Finite Element as a Basis for Generalized Models of Uncertainty in Engineering Mechanics, in: Muhanna, R. L. and Mullen, R. L. (eds), Proc. NSF Workshop Reliable Engineering Computing, Sept. 15–17, 2004, Savannah, pp. 353–370, presentations.html

  9. Neumaier, A.: Interval Methods for Systems of Equations, Cambridge Univ. Press, Cambridge, 1990.

    MATH  Google Scholar 

  10. Neumaier, A.: Introduction to Numerical Analysis, Cambridge Univ. Press, Cambridge, 2001.

    MATH  Google Scholar 

  11. Neumaier, A.: Rigorous Sensitivity Analysis for Parameter-Dependent Systems of Equations, J. Math. Anal. Appl. 144 (1989), pp. 16–25.

    Article  MATH  MathSciNet  Google Scholar 

  12. Neumaier, A.: Second-Order Sufficient Optimality Conditions for Local and Global Nonlinear Programming, J. Global Optim. 9 (1996), pp. 141–151.

    Article  MATH  MathSciNet  Google Scholar 

  13. Neumaier, A.: Taylor Forms—Use and Limits, Reliable Computing 9 (1) (2003), pp. 43–79,

  14. Popova, E.: Improved Solution Enclosures for Over- and Underdetermined Interval Linear Systems, in: Lirkov, I., Margenov, S., and Wasniewski, J. (eds), Proc. 5th Int. Conf. LSSC, Sozopol, 2005, Lecture Notes in Computer Science 3743, 2006, pp. 305–312,

  15. Popova, E.: Parametric Interval Linear Solver, Numerical Algorithms 37 (2004), pp. 345–356.

    Article  MATH  Google Scholar 

  16. Popova, E. D.: Quality of the Solution Sets of Parameter Dependent Interval Linear Systems, ZAMM 82 (2002), pp. 723–727.

    Article  MATH  MathSciNet  Google Scholar 

  17. Pownuk, A.: Efficient Method of Solution of Large Scale Engineering Problems with Interval Parameters Based on Sensitivity Analysis, in: Muhanna, R. L. and Mullen, R. L. (eds), Proc. NSF Workshop Reliable Engineering Computing, Sept. 15–17, 2004, Savannah, pp. 305–316, presentations.html

  18. Pownuk, A. and Neumaier, A.: Worst Case Bounds for Uncertain Finite Element Problems, in preparation.

  19. Rao, S. S. and Berke, L.: Analysis of Uncertain Structural Systems Using Interval Analysis, AIAA Journal 35 (1997), pp. 727–735.

    Article  MATH  Google Scholar 

  20. Rump, S. M.: INTLAB—INTerval LABoratory, in: Csendes, T. (ed.), Developments in Reliable Computing, Kluwer Academic Publishers, Dordrecht, 1999, pp. 77–105, rump/intlab/

    Google Scholar 

  21. Schichl, H. and Neumaier, A.: Exclusion Regions for Systems of Equations, SIAM J. Numer. Anal. 42 (2004), pp. 383–408,

  22. Skalna, I.: A Method for Outer Interval Solution of Parametrized Systems of Linear Equations, in: Muhanna, R. L. and Mullen, R. L. (eds), Proc. NSF Workshop Reliable Engineering Computing, Sept. 15–17, 2004, Savannah, pp. 1–14, recworkshop/presentations/presentations.html

  23. Skalna, I.: Methods for Solving Systems of Linear Equations of Structure Mechanics with Interval Parameters, Computer Assisted Mechanics and Engineering Sciences 10 (2003), pp. 281–293.

    MATH  Google Scholar 

  24. Zienkiewicz, O. C. and Taylor, R. L.: Finite Element Method: Volume 1. The Basis, Butterworth Heinemann, London, 2000.

    MATH  Google Scholar 

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Correspondence to Arnold Neumaier.

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Neumaier, A., Pownuk, A. Linear Systems with Large Uncertainties, with Applications to Truss Structures. Reliable Comput 13, 149–172 (2007).

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  • Large Uncertainty
  • Truss Structure
  • Trial Point
  • Uncertain Linear System
  • Interval Solution