Methods for solving systems of linear equations of structure mechanics with interval parameters

• Autorzy: Skalna I.
• Języki publikacji: EN

Abstrakty

EN

Interval analysis permits to calculate guaranteed a posteriori bounds for the solutions of problems with uncertain (interval) input data. Most of the methods of interval analysis assume that all input data vary independently within the given lower and upper bounds. In many practical applications it need not be a case, and the assumption of independence may lead to large overestimation of the set of solutions. The subject of this work is the problem of solving systems of linear interval equations with coefficients linearly dependent on a set of interval parameters called coefficient dependence problem. The purpose of this work is to present methods producing sharp bounds for the set of solutions of systems with dependent input data. The paper starts with an introduction to systems of linear interval equations and the problem of data dependencies in such systems. A parametric formulation of the coefficient dependence problem follows next. Finally, three algorithms to calculate tighter bounds for problems with linearly dependent coefficients, namely the Rump's method, its improved version developed by the author, and the IPM method based on the results from Neumaier [8] are presented and discussed. The algorithms are evaluated and compared using some examples of truss structure analysis.

Słowa kluczowe

EN

linear equations

PL

równanie liniowe

• Wydawca: Instytut Podstawowych Problemów Techniki PAN
• Czasopismo: Computer Assisted Mechanics and Engineering Sciences
• Rocznik: 2003
• Tom: Vol. 10, No. 3
• Strony: 281--293
• Opis fizyczny: Bibliogr. 17 poz., tab.,rys., wykr.

Twórcy

autor

Skalna I.

• Department of Management, Academy of Mining and Metallurgy, [AGH], ul.Gramatyka 10, 30-067 Cracow, Poland

Bibliografia

• [1] G. Alefeld, J. Herzberger. Introduction to interval computations. Academic Press Inc., New York, USA, 1983. Transl. by J. Rokne from original German 'Einführung In Die Intervallrechnung'.
• [2] D.J. Hartfiel. Concerning the solution set of Ax = B where P ≤ A ≤ Q and p ≤ b ≤ q. Numerische Mathematik, 35: 355-359, 1980.
• [3] C. Jansson. Interval linear systems with symmetric matrices, skew-symmetric matrices and dependencies in the right hand side. Computing, 46(3): 265-274, 1991.
• [4] Z. Kulpa, A. Pownuk, I. Skalna. Analysis of linear mechanical structures with uncertainties by means of interval methods. GAMES, 5(4): 443-477, 1998.
• [5] Z. Kulpa, A. Radomski, O. Gajl, M. Kleiber, I. Skalna. Hybrid expert system for qualitative/ quantitative analysis of mechanical structures (in Polish). Knowledge Engineering and Expert Systems, Z Bubniacki, A. Grzech eds. 2: 135-142, 1997.
• [6] R.L. Mullen, R.L. Muhanna. Bounds of structural response for all possible loading combinations. Journal of Structural Engineering, 125(1): 98-106, 1999.
• [7] A. Neumaier. Overestimation in linear interval equations. SIAM Journal of Numerical Analisys, 24(1): 207-214, 1987.
• [8] A. Neumaier. Interval methods for systems of equations. Cambridge University Press, Cambridge, UK, 1990.
• [9] A.M. Ostrowski. Über die Determinanten mit iiberwiegender Hauptdiagonale. Comment. Math. Helv., 10: 69-96, 1937.
• [10] S.S. Rao, L. Berke. Analysis of uncertain structural systems using interval analysis. AIAA Journal, 35(4): 727-735, 1997.
• [11] J. Rohn. Linear interval equations: computing sufficiently accurate enclosures is NP-hard?, (621): 7, 1995.
• [12] J. Rohn. NP-hardness results for some linear and qudratic problems. (619): 11, 1995.
• [13] J. Rohn. Linear interval equations: Computing enclosures with bounded relative overestimation is NP-Hard, in R.B. Kearfott, V. Kreinovich (eds.), Applications of interval computations, Papers presented at an international workshop in El Paso. Texas, February 23-25, 1995, vol. 3 of Applied optimization, pp. 81-90, Norwell, MA, USA, and Dordrecht, The Netherlands, 1996. Kluwer Academic Publishers Group.
• [14] J. Rohn, V. Kreinovich. Computing exact componentwise bounds on solutions of linear systems with interval data is NP-Hard. SIAM Journal on Matrix Analysis and Applications, 16(2): 415-420, 1995.
• [15] S.M. Rump. Kleine Fehlerschranken Bei Matrixproblemen. Dissertation. Universität Karlsruhe, 1980.
• [16] S.M. Rump. Verification methods for dense and sparse systems of equations. Elsevier, 5: 63-135, The Netherlands, Amsterdam 1994.
• [17] D.I. Schwartz, S.S. Chen. Towards a unified framework for interval based qualitative computational matrix structural analysis. Computing Systems in Engineering, 5(2): 147-158, 1994.