A stochastic finite element procedure for moment and reliability analysis
Keywords:
stochastic finite elements, polynomial chaos, finite element reliability, parametric study, foundationAbstract
A new stochastic finite element procedure (SFEP) in the tradition of Ghanem’s work is presented. It allows to deal with any number of input random variables of any type that can model both material properties and loading. The method makes use of Hermite series expansion of the input random variables and polynomial chaos expansion of the response, for which an original implementation is proposed. The link with reliability analysis is also established. Three application examples in geotechnical engineering are given for the sake of illustration. The accuracy and efficiency of SFEP is thoroughly investigated by comparison with well-established approaches.
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Anders M., Hori M., “Stochastic finite element method for elasto-plastic body”, Int. J. Num.
Meth. Eng., vol. 46, n° 11, 1999, p. 1897-1916.
Anders M., Hori M., “Three-dimensional stochastic finite element method for elasto-plastic
body”, Int. J. Numer. Met. Engng., vol. 51, 2001, p. 449-478.
Babuska I., Chatzipantelidis P., “On solving elliptic stochastic partial differential equations”,
Comp. Meth. Appl. Mech. Eng., vol. 191, n° 37-38, 2002, p. 4093-4122.
Berveiller M., Sudret B., Lemaire M., “Comparison of methods for computing the response coefficients
in stochastic finite element analysis”, Proc. AsRANET 2, Barcelona, Spain, 2004a.
BerveillerM., Sudret B., LemaireM., “Presentation of two methods for computing the response
coefficients in stochastic finite element analysis”, Proc. 9th ASCE specialty Conference on
Probabilistic Mechanics and Structural Reliability, Albuquerque, USA, 2004b.
Berveiller M., Sudret B., Lemaire M., “Non linear non intrusive stochastic finite element
method - Application to a fracture mechanics problem”, in , G. Augusti, , G. Schuëller,
, M. Ciampoli (eds), Proc. 9th Int. Conf. Struct. Safety and Reliability (ICOSSAR’2005)
Roma, Italie, Millpress, Rotterdam, 2005.
Berveiller M., Sudret B., Lemaire M., “Stochastic finite element : a non intrusive approach by
regression”, Revue Européenne de Mécanique Numérique, vol. 15, n° 1-3, p. 81-92, 2006.
Choi S., Grandhi R., Canfield R., “Structural reliability under non-Gaussian stochastic behavior”,
Computers and Structures, vol. 82, 2004a, p. 1113-1121.
Choi S., Grandhi R., Canfield R., Pettit C., “Polynomial Chaos Expansion with Latin Hypercube
Sampling for Estimating Response Variability”, AIAA Journal, vol. 45, 2004b, p. 1191-
Deb M., Babuska I., Oden J., “Solution of stochastic partial differential equations using
Galerkin finite element techniques”, Comp. Meth. Appl. Mech. Eng., vol. 190, n° 48, 2001,
p. 6359-6372.
Deodatis G., “The weighted integral method, I : stochastic stiffness matrix”, J. Eng. Mech., vol.
, n° 8, 1991, p. 1851-1864.
Deodatis G., Shinozuka M., “The weighted integral method, II : response variability and reliability”,
J. Eng. Mech., vol. 117, n° 8, 1991, p. 1865-1877.
Der Kiureghian A., Ke J.-B., “The stochastic finite element method in structural reliability”,
Prob. Eng. Mech., vol. 3, n° 2, 1988, p. 83-91.
Der Kiureghian A., Liu P.-L., “Structural reliability under incomplete probability information”,
J. Eng. Mech., ASCE, vol. 112, n° 1, 1986, p. 85-104.
Der Kiureghian A., Taylor R.-L., “Numerical methods in structural reliability”, Proc. 4th Int.
Conf. on Appl. of Statistics and Probability in Soil and Structural Engineering, 1983, p. 769-
Det Norske Veritas, PROBAN user’s manual, V.4.3, 2000.
Ditlevsen O., Madsen H., Structural reliability methods, J. Wiley and Sons, Chichester, 1996.
Field R., Grigoriu M., “On the accuracy of the polynomial approximation”, Prob. Eng. Mech.,
vol. 19, 2004, p. 65-80.
Frangopol D., Imaia K., “Geometrically nonlinear finite element reliability analysis of structural
systems. II: applications”, Comput. Struc., vol. 77, n° 6, 2000, p. 693-709.
Frangopol D.M., Maute K. (Editors), “Advances in Probabilistic Mechanics and Structural Reliability”,
Comput. Struct., 2004.
Ghanem R., “Probabilistic characterization of transport in heterogeneous media”, Comp. Meth.
Appl. Mech. Eng., vol. 158, 1998, p. 199-220.
Ghanem R., “Ingredients for a general purpose stochastic finite elements implementation”,
Comp. Meth. Appl. Mech. Eng., vol. 168, 1999a, p. 19-34.
Ghanem R., “Stochastic finite elements with multiple random non-Gaussian properties”, J. Eng.
Mech., ASCE, vol. 125, n° 1, 1999b, p. 26-40.
Ghanem R., “Iterative solution of systems of linear equations arising in the context of stochastic
finite elements”, Advances in Engineering Software, vol. 31, 2000, p. 607-616.
Ghanem R., Kruger R., “Numerical solution of spectral stochastic finite element systems”,
Comp. Meth. Appl. Mech. Eng., vol. 129, n° 3, 1996, p. 289-303.
Ghanem R., Spanos P., Stochastic finite elements - A spectral approach, Springer Verlag, 1991.
Ghanem R., Spanos P.-D., “Polynomial chaos in stochastic finite elements”, J. App. Mech.,
ASME, vol. 57, 1990, p. 197-202.
Ghiocel D., Ghanem R., “Stochastic finite element analysis of seismic soil-structure interaction”,
J. Eng. Mech., (ASCE), vol. 128, 2002, p. 66-77.
Hisada T., Nakagiri S., “Stochastic finite element method developed for structural safety and
reliability”, Proc. 3rd Int. Conf. Struct. Safety and Reliability, ICOSSAR’81, 1981, p. 395-
Hisada T., Nakagiri S., “Role of stochastic finite element method in structural safety and reliability”,
Proc. 4th Int. Conf. Struct. Safety and Reliability, ICOSSAR’85, 1985, p. 385-395.
Imaia K., Frangopol D., “Geometrically nonlinear finite element reliability analysis of structural
systems. I: theory”, Comput. Struc., vol. 77, n° 6, 2000, p. 677-691.
Isukapalli S. S., Uncertainty Analysis of Transport-Transformation Models, PhD thesis, The
State University of New Jersey, 1999.
Keese A., Matthies H., “Efficient solvers for non linear stochastic problems”, H. Mang, ,
F. Rammerstorfer, , J. Eberhardsteiner (eds), Proc. 5th World Cong. Comp. Mech., 2002.
Kleiber, M. (Editor), “Computational stochastic mechanics”, Comp. Meth. Appl. Mech. Eng.,
Kleiber M., Hien T.-D., The stochastic finite element methods - Basic perturbation technique
and computational implementation, J. Wiley and sons, Chichester, 1992.
Lemaire M., “Finite element and reliability : combined methods by surface response”,
G.N. Frantziskonis (ed.), Probamat-21st Century, Probabilities and Materials : Tests, Models
and Applications for the 21st century, Kluwer Academic Publishers, 1998, p. 317-331.
Lemaire M., Mohamed A., “Finite Element and Reliability: A Happy Marriage?”, Proc. 9th
IFIP WG 7.5 Working Conference on Reliability and Optimization of Structural Systems
(Keynote lecture), Nowak A. and Szerszen M., 2000, p. 3-14.
Li C., Der Kiureghian A., “Optimal discretization of random fields”, J. Eng. Mech., vol. 119,
n° 6, 1993, p. 1136-1154.
Liu W., Belytschko T., Mani A., “Probabilistic finite elements for non linear structural dynamics”,
Comp. Meth. App. Mech. Eng., vol. 56, 1986a, p. 61-86.
Liu W., Belytschko T., Mani A., “Random field finite elements”, Int. J. Num. Meth. Eng., vol.
, n° 10, 1986b, p. 1831-1845.
Malliavin P., Stochastic Analysis, Springer, 1997.
Mathworks T., Manuel de réference du logiciel Matlab, 2001.
Matthies G., Brenner C., Bucher C., Guedes Soares C., “Uncertainties in probabilistic numerical
analysis of structures and solids - Stochastic finite elements”, Struct. Safety, vol. 19, n° 3,
, p. 283-336.
Mohamed A., Pendola M., Heinfling G., Defaux G., “Etude de l’intégrité des aéro-réfrigérants
par couplage éléments finis et fiabilité”, Revue Européenne des Eléments Finis, vol. 1, 2002,
p. 101-126.
PendolaM.,Mohamed A., LemaireM., Hornet P., “Combination of finite element and reliability
methods in nonlinear fracture mechanics”, Reliability Engineering & System Safety, vol. 70,
n° 1, 2000, p. 15-27.
Puig B., Poirion F., Soize C., “Non-Gaussian simulation using Hermite polynomial expansion:
convergences”, Prob. Eng. Mech., vol. 17, 2002, p. 253-264.
Schuëller, G. (Editor), “A state-of-the-art report on computational stochastic mechanics”, Prob.
Eng. Mech., 1997. IASSAR report.
Sudret B., “Des éléments finis stochastiques spectraux aux surfaces de réponse stochastiques :
une approche unifiée”, Proc. 17e Congrès Français de Mécanique - Troyes, 2005.
Sudret B., Berveiller M., Lemaire M., “Application of a Stochastic Finite Element Procedure to
Reliability Analysis”, Proc 11th IFIP WG7.5 Conference on Reliability and Optimization
of Structural Systems, Banff, Canada, 2003, p. 319-327.
Sudret B., Berveiller M., Lemaire M., “Eléments finis stochastiques en élasticité linéaire”, C.
R. Mécanique, vol. 332, 2004, p. 531-537.
Sudret B., Defaux G., Pendola M., “Time-variant finite element reliability analysis - application
to the durability of cooling towers”, Struct. Safety, vol. 27, 2005, p. 93-112.
Sudret B., Der Kiureghian A., Stochastic finite elements and reliability : A state-of-the-art
report, Technical Report n° no UCB/SEMM-2000/08, University of California, Berkeley,
173 pages.
Sudret B., Der Kiureghian A., “Comparison of finite element reliability methods”, Prob. Eng.
Mech., vol. 17, 2002, p. 337-348.
Takada T., “Weighted integral method in multidimensional stochastic finite element analysis”,
Prob. Eng. Mech., vol. 5, n° 4, 1990a, p. 158-166.
Takada T., “Weighted integral method in stochastic finite element analysis”, Prob. Eng. Mech.,
vol. 5, n° 3, 1990b, p. 146-156.
Van Den Nieuwenhof B., Coyette J.-P., “Modal approaches for the stochastic finite element
analysis of structures with material and geometric uncertainties”, Comp. Meth. Appl. Mech.
Eng., vol. 192, n° 33-34, 2003, p. 3705-3729.
Webster M., Tatang M., McRae G., Application of the Probabilistic Collocation Method for an
Uncertainty Analysis of a Simple Ocean Model, Technical report, MIT Joint Program on
the Science and Policy of Global Change Reports Series No. 4, Massachusetts Institute of
Technology, 1996.
Xiu D., Karniadakis G. E., “The Wiener-Askey polynomial chaos for stochastic differentiel
equations”, J. Sci. Comput., vol. 24, n° 2, 2002, p. 619-644.
Xiu D., Karniadakis G. E., “A new stochastic approach to transient heat conduction modeling
with uncertainty”, Int. J. Heat Mass Transfer, vol. 46, 2003, p. 4681-4693.
Yu Y., Coupling a Stochastic Finite element Solver with ANSYS and Visualization of the Results,
Technical Report n°Matr.-Nr.: 2663324, Institute of scientific Computing. Technische
Universität Carolo-Wilhelmina zu Braunschweig, 2003.
Zhang J., Ellingwood B., “Orthogonal series expansion of random fields in reliability analysis”,
J. Eng. Mech., ASCE, vol. 120, n° 12, 1994, p. 2660-2677.
Zienkiewicz O.-C., Taylor R.-L., The finite element method, Butterworth Heinemann, 5th edition,
