Stochastic Finite Element Expansion for Random Media

A new method for the solution of problems involving material variability is proposed. The material property is modeled as a stochastic process. The method makes use of the Karhunen‐Loeve expansion to represent the random material property. The expansion is a representation of the process in terms of a finite set of uncorrelated random variables. The resulting formulation is compatible with the finite element method. A Neumann expansion scheme is subsequently employed to obtain a convergent expansion of the response process. The response is thus obtained as a homogeneous multivariate polynomial in the uncorrelated random variables. From this representation various statistical quantities may be derived. The usefulness of the proposed method, in terms of accuracy and efficiency, is exemplified by considering a cantilever beam with random rigidity. The derived results pertaining to the second‐order statistics of the response are found in good agreement with those obtained by a Monte Carlo simulation solution ...

[1]  Armen Der Kiureghian,et al.  The stochastic finite element method in structural reliability , 1988 .

[2]  Ted Belytschko,et al.  Transient probabilistic systems , 1988 .

[3]  M. A. Lawrence,et al.  Basis random variables in finite element analysis , 1987 .

[4]  Christos C. Chamis,et al.  Probabilistic structural analysis methods for space propulsion system components , 1987 .

[5]  Haym Benaroya,et al.  PARAMETRIC RANDOM EXCITATION. I: EXPONENTIALLY CORRELATED PARAMETERS. , 1987 .

[6]  Sami F. Masri,et al.  Orthogonal decomposition and transmission of nonstationary random processes , 1986 .

[7]  Wing Kam Liu,et al.  Probabilistic finite elements for nonlinear structural dynamics , 1986 .

[8]  Mircea Grigoriu,et al.  STOCHASTIC FINITE ELEMENT ANALYSIS OF SIMPLE BEAMS , 1983 .

[9]  Sami F. Masri,et al.  Compact Probabilistic Representation of Random Processes , 1982 .

[10]  J. Akin.  Application and Implementation of Finite Element Methods , 1982 .

[11]  George Adomian,et al.  Inversion of stochastic partial differential operators—The linear case , 1980 .

[12]  E. Vanmarcke.  Probabilistic Modeling of Soil Profiles , 1977 .

[13]  E. Vanmarcke.  Properties of Spectral Moments with Applications to Random Vibration , 1972 .

[14]  Masanobu Shinozuka,et al.  Random Eigenvalue Problems in Structural Analysis , 1971 .

[15]  Uriel Frisch,et al.  WAVE PROPAGATION IN RANDOM MEDIA. , 1970 .

[16]  J. D. Collins,et al.  The eigenvalue problem for structural systems with statistical properties. , 1969 .

[17]  J. Mercer.  Functions of Positive and Negative Type, and their Connection with the Theory of Integral Equations , 1909 .

[18]  Josef Kittler,et al.  Pattern recognition : a statistical approach , 1982 .

[19]  J. Z. Zhu,et al.  The finite element method , 1977 .

[20]  M. Shinozuka,et al.  A probabilistic model for spatial distribution of material properties , 1976 .

[21]  A. T. Bharucha-Reid,et al.  Probabilistic methods in applied mathematics , 1968 .