Reliable Computing

, Volume 13, Issue 2, pp 149–172 | Cite as

Linear Systems with Large Uncertainties, with Applications to Truss Structures

  • Arnold NeumaierEmail author
  • Andrzej Pownuk


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.


Large Uncertainty Truss Structure Trial Point Uncertain Linear System Interval Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
  2. 2.
    ANSYS Commands Reference: ANSYS 8.0 Help, ANSYS, 2003.Google Scholar
  3. 3.
    COCONUT, COntinuous CONstraints—Updating the Technology, an IST Project funded by the European Union,
  4. 4.
    Hansen, E.: Preconditioning Linearized Equations, Computing 58 (1997), pp. 187–196zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Krawczyk, R. and Neumaier, A.: Interval Slopes for Rational Functions and Associated Centered Forms, SIAM J. Numer. Anal. 22 (1985), pp. 604–616.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Muhanna, R. L.: Benchmarks for Interval Finite Element Computations, 2004,
  7. 7.
    Muhanna, R. L. and Mullen, R. L.: Uncertainty in Mechanics Problems—Interval Based Approach, J. Engrg. Mech. 127 (2001), pp. 557–566.CrossRefGoogle Scholar
  8. 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. 9.
    Neumaier, A.: Interval Methods for Systems of Equations, Cambridge Univ. Press, Cambridge, 1990.zbMATHGoogle Scholar
  10. 10.
    Neumaier, A.: Introduction to Numerical Analysis, Cambridge Univ. Press, Cambridge, 2001.zbMATHGoogle Scholar
  11. 11.
    Neumaier, A.: Rigorous Sensitivity Analysis for Parameter-Dependent Systems of Equations, J. Math. Anal. Appl. 144 (1989), pp. 16–25.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Neumaier, A.: Second-Order Sufficient Optimality Conditions for Local and Global Nonlinear Programming, J. Global Optim. 9 (1996), pp. 141–151.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Neumaier, A.: Taylor Forms—Use and Limits, Reliable Computing 9 (1) (2003), pp. 43–79,
  14. 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. 15.
    Popova, E.: Parametric Interval Linear Solver, Numerical Algorithms 37 (2004), pp. 345–356.zbMATHCrossRefGoogle Scholar
  16. 16.
    Popova, E. D.: Quality of the Solution Sets of Parameter Dependent Interval Linear Systems, ZAMM 82 (2002), pp. 723–727.zbMATHCrossRefMathSciNetGoogle Scholar
  17. 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. 18.
    Pownuk, A. and Neumaier, A.: Worst Case Bounds for Uncertain Finite Element Problems, in preparation.Google Scholar
  19. 19.
    Rao, S. S. and Berke, L.: Analysis of Uncertain Structural Systems Using Interval Analysis, AIAA Journal 35 (1997), pp. 727–735.zbMATHCrossRefGoogle Scholar
  20. 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. 21.
    Schichl, H. and Neumaier, A.: Exclusion Regions for Systems of Equations, SIAM J. Numer. Anal. 42 (2004), pp. 383–408,
  22. 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. 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.zbMATHGoogle Scholar
  24. 24.
    Zienkiewicz, O. C. and Taylor, R. L.: Finite Element Method: Volume 1. The Basis, Butterworth Heinemann, London, 2000.zbMATHGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2007

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität WienWienAustria
  2. 2.Department of Theoretical Mechanics, Faculty of Civil EngineeringSilesian University of TechnologyGliwicePoland

Personalised recommendations