Penalty-Based Solution for the Interval Finite-Element Methods
Publication: Journal of Engineering Mechanics
Volume 131, Issue 10
Abstract
A new approach for achieving guaranteed reliable results within the context of finite-element approximation of mechanical systems is developed. A reliable analysis requires that all the sources of uncertainty and errors be accommodated. The appropriateness of a partial differential equation to a given physical problem is beyond the scope of this work. Parameter uncertainty is treated as intervals in this work and guaranteed bounds on the “unknown” true solutions are obtained. In this paper an element-by-element penalty-based interval finite-element analysis of linear elastic structural mechanics and solid mechanics problem is introduced. Material and load uncertainties are handled simultaneously. Presented numerical examples illustrate the ability of the method to maintain very sharp solution enclosures even when the number of the interval parameters or the size of the problems is increased.
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ACKNOWLEDGMENT
The writers would like to acknowledge support from the Center for Reliable Engineering Computing (REC).
References
Akpan, U. O., Koko, T. S., Orisamolu, I. R., and Gallant, B. K. (2001). “Practical fuzzy finite element analysis of structures.” Finite Elem. Anal. Design, 38, 93–111.
Alefeld, G., and Herzberger, J. (1983). Introduction to interval computations, Academic, New York.
Apostolatos, N., and Kulisch, U. (1968). “Grundzüge einer Intervallrechtung für Matrizen und einige Anwwendungen.” Elektron. Rechenanlagen, 10, 73–83 (in German).
Babuska, I., Strouboulis, T., and Gangaraj, S. K. (1999). “Guaranteed computable bounds for the exact error in the finite element solution.” Comput. Methods Appl. Mech. Eng., 176, 51–79.
Bathe, K. (1996). Finite element procedures, Prentice-Hall, Upper Saddle River, N.J.
Chen, S. H., Lian, H. D., and Yang, X. W. (2002). “Interval static displacement analysis for structures with interval parameters.” Int. J. Numer. Methods Eng., 53, 393–407.
Cook, R. D., Malkus, D. S., and Plesha, M. E. (1989). Concepts and applications of finite element analysis, Wiley, New York.
Dempster, A. P. (1967). “Upper and lower probabilities induced by a multi-valued mapping.” Ann. Math. Stat., 38, 325–339.
Dessombz, O., Thouverez, F., Laîné, J.-P., and Jézéquel, L. (2001). “Analysis of mechanical systems using interval computations applied to finite element methods.” J. Sound Vib., 239(5), 949–968.
Gallagher, R. H. (1975). Finite element analysis fundamentals, Printice-Hall, Englewood Cliffs, N.J.
Ganzerli, S., and Pantelides, C. P. (1999). “Load and resistance convex models for optimum design.” Struct. Optim., 17, 259–268.
Gay, D. M. (1982). “Solving interval linear equations.” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal., 19(4), 858–870.
Hansen, E. (1965). “Interval arithmetic in matrix computation.” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal., I(2), 308–320.
Hansen, E. (1992). Global optimization using interval analysis, Marcel Dekker, New York.
Hansen, E. (2004). “Methods for interval linear equations.” Proc., NSF workshop on reliable engineering computing, Savannah, Ga.
Jansson, C. (1991). “Interval linear system with symmetric matrices, skew-symmetric matrices, and dependencies in the right hand side.” Computing, 46, 265–274.
Koyluoglu, U., Cakmak, S., Ahmet, N., and Soren, R. K. (1995). “Interval algebra to deal with pattern loading and structural uncertainties.” J. Eng. Mech., 121(11), 1149–1157.
Mayer, O. (1970). “Algebraische und metrische Strukturen in der Intervallrechung und eingine Anwendungen.” Computing, 5, 144–162 (in German).
Möller, B., Graf, W., and Beer, M. (2000). “Fuzzy structural analysis using -level-optimization.” Comput. Mech., 26(6), 547–565.
Moore, R. E. (1966). Interval Analysis, Prentice-Hall, Englewood Cliffs, N.J.
Moore, R. E. (1979). Methods and applications of interval analysis, SIAM, Philadelphia.
Muhanna, R. L., and Mullen, R. L. (1995). “Development of interval based methods for fuzziness in continuum mechanics.” Proc., ISUMA-NAFIPS’95, 145–150.
Muhanna, R. L., and Mullen, R. L. (1999). “Formulation of fuzzy finite element methods for mechanics problems.” Comput. Aided Civ. Infrastruct. Eng., 14, 107–117.
Muhanna, R. L., and Mullen, R. L. (2001). “Uncertainty in mechanics problems—interval-based approach.” J. Eng. Mech., 127(6), 557–566.
Mullen, R. L., and Muhanna, R. L. (1996). “Structural analysis with fuzzy-based load uncertainty.” Proc., 7th ASCE EMD/STD Joint Spec. Conf. on Probabilistic Mech. and Struct. Reliability, Mass., 310–313.
Mullen, R. L., and Muhanna, R. L. (1999). “Bounds of structural response for all possible loadings.” J. Struct. Eng., 125(1), 98–106.
Neumaier, A. (1989). “Rigorous sensitivity analysis for parameter-dependent systems of equations.” J. Math. Anal. Appl., 144, 16–25.
Neumaier, A. (1990). Interval methods for systems of equations, Cambridge University Press.
Oden, J. T., and Feng, Y. (1996). “Local and pollution error estimates for finite element approximations of elliptical boundary value problems.“ J. Comput. Appl. Math., 74, 245–293.
Pantelides, C. P., and Ganzerli, S. (2001). “Comparison of fuzzy set and convex model theories in structural design.” Mech. Syst. Signal Process., 15(3), 499–511.
Rao, S. S., and Berke, L. (1997). “Analysis of uncertain structural systems using interval analysis.” AIAA J., 35(4), 727–735.
Rao, S. S., and Chen, L. (1998). “Numerical solution of fuzzy linear equations in engineering analysis.” Int. J. Numer. Methods Eng., 43, 391–408.
Rohn, J., and Rex, G. (1998). “Enclosing solutions of linear equations.” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal., 35(2), 525–539.
Rump, S. M. (1983). “Solving algebraic problems with high accuracy.” A new approach to scientific computation, U. Kulisch and W. Miranker, eds., Academic, New York.
Rump, S. M. (1992). “On the solution of interval linear systems.” Computing, 47, 337–353.
Rump, S. M. (2001). “Self-validating methods.” Linear Algebr. Appl., 324, 3–13.
Shafer, G. (1976). A mathematical theory of evidence, Princeton University Press, Princeton, N.J.
Sun Microsystems. (2002). Interval arithmetic in high performance technical computing. (A White Paper).
Walster, G. W., and Hansen, E. (2002). “Computing interval parameters bounds from fallible measurements using overdetermined (tall) systems of nonlinear equations.” Proc. COCOS 2002, Valbonne-Sophia Antipolis, France, 171–177.
Zadeh, L. A. (1978). “Fuzzy sets as a basis for a theory of possibility.” Fuzzy Sets Syst., 1, 3–28.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 131 • Issue 10 • October 2005
Pages: 1102 - 1111
Copyright
© 2005 ASCE.
History
Received: Mar 10, 2004
Accepted: Oct 26, 2004
Published online: Oct 1, 2005
Published in print: Oct 2005
Notes
Note. Associate Editor: Gerhart I. Schueller
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