 |
|
|
Stochastic finite elements
Ghanem R., Spanos P., Springer-Verlag New York, Inc., New York, NY, 1991. Type: Book (9780387974569) |
|
|
|
|
|
|
|
|
|
The two major mathematical techniques the authors have used in the study of stochastic structural mechanics are Karhuen-Loeve expansion and the Fourier-Volterra-Wiener series of polynomial chaos. Karhuen-Loeve expansion is used to develop random coefficients of differential equations as an infinite series in an effort to solve dynamic equations of structural mechanics using Neumann expansion. Second-order random processes and, in general, random fields { &xgr; ( x ) , x ∈ D } , D ⊆ R n, can be completely described in terms of their means E { &xgr; ( x ) } ≡ &xgrmacr; ( x ) and covariance kernels C ( x , y ) ≡ E { ( &xgr; ( x ) - &xgrmacr; ( x ) ) ( &xgr; ( y ) - &xgrmacr; ( y ) ) } Here E { . } stands for the mathematical expectation. Given the kernel C, consider the homogeneous Fredholm integral equation of the second kind ∫D C ( x , y ) ϕ ( y ) d y = &lgr; ϕ ( x ) written in the operator notation as C ϕ = &lgr; ϕ. Since the underlying process is second-order, the kernel C ( . , . ) ∈ L 2 ( D × D ) and one can consider this equation in the Hilbert space L 2 ( D ). Note that the trace of C, given by TraceC ≡ ∫ C ( x , x ) d x, is positive and finite. The operator C is a symmetric, positive trace class (compact) operator in the Hilbert space L 2 ( D ) and hence it has a countable set of eigenvalues and eigenfunctions { &lgr; i , ϕ i } ∈ ( 0 , ∞ ) × L 2 ( D ) solving C ϕ i = &lgr; i ϕ i. The family { ϕ i } is orthogonal and can be normalized to obtain an orthonormal set. The process { &xgr; } can then be developed in terms of generalized Fourier series, giving which converges in the mean square sense where the { &ggr; i } are mutually independent second-order real random variables with zero mean and unit variance. Note that the TraceC = ∫ C ( x , x ) d x = ∑ &lgr; i. This is Karhuen-Loeve expansion. If &xgr; is Gaussian then { &ggr; i } is also Gaussian. Further, the kernel C can be expanded into a uniformly convergent series in terms of its eigenfunctions C ( x , y ) = ∑ &lgr; i &ggr; i ϕ i ( y ). In general, the family { ϕ i } is not complete in L 2, but if the operator C is positive definite then it is complete in the class L 2 ( D ). Though Karhuen-Loeve expansion is a powerful tool for the representation of second-order random processes, its application to dynamical systems is limited. The authors also discuss Wiener’s polynomial chaos. In the 1950s, Wiener developed a complete set of orthogonal functionals using the Volterra series and the Wiener measure. Let W ≡ C ( D ) denote the space of continuous functions on D with the usual sup norm, and let B denote the Borel field of subsets of the set W and μ W the Wiener measure on B. The space ( W , B , μ W ) (abbreviated ( W , μ W )) is called the Wiener measure space. Let L 2( W , μ W ) denote a Wiener measure space with square integrable random variables. If B is completed, then L 2( W , μ W ) is a Hilbert space. One can then use the functional polynomials { f n ( &xgr; ) } given by f n ( &xgr; ) ≡ ∫D n K n ( x 1 , x 2 ,..., x n ) &xgr; ( d x 1 ) &xgr; ( d x 2 ) ... &xgr; ( d x n ) called the homogeneous chaos, and the measure μ W to construct orthogonal functionals using the classical Gram-Schmidt procedure. This approach is similar to the construction of Hermite functions using the weighting function exp - ( ½ ) | x | 2. The kernels K n ∈ L 2 ( D n ), where L 2 ( D n ) is the Hilbert space of square integrable functions on D n and &xgr; is the Wiener process or field. Use of symmetric kernels makes the set { f n } orthogonal. The Wiener-Ito multiple integrals simplify Wiener’s version of orthogonal functions [1]. Thus for K n ∈ L s2 ( D n ) equivalent to the space of symmetric L 2 kernels, we have The set { f n } is complete in the class L 2 ( W , μ W ), so for any f ∈ L 2 ( W , μ W ), one can find a sequence of kernels R n ∈ L s2 ( D n ) such that where f n is the multiple Wiener-Ito integral corresponding to the kernel R n. Substantial generalizations of this work in the last decade have led to generalized functionals on the Wiener measure space [2]. After giving some physical motivation in chapter 1, the authors introduce Karhuen-Loeve expansion and the homogeneous chaos in chapter 2. The presentation is intuitive, avoiding mathematical rigor. Chapter 3 discusses finite element techniques, touching on the Galerkin approach and Neumann expansion. Chapter 4 uses polynomial chaos expansion to represent statistical moments of a response process, which the authors claim are useful in reliability theory. Chapter 5 uses these techniques for numerical analysis of static beam and plate equations and a dynamic one-dimensional beam equation with random coefficients. The authors compare the results from a truncated Karhuen-Loeve expansion with the results of chaos expansions. In the absence of experimental results, it is difficult to assess the practical engineering value of this method. The book is certainly not for theoreticians, but is probably a useful reference for specialists and graduate students in structural mechanics.
|
|
|
Reviewer:
N. U. Ahmed |
Review #: CR115316 |
|
|
| 1) |
Hida, T. Generalized multiple Wiener integrals. Proc. Japan. Acad. series A, 54 (1978), 55–58. |
|
| 2) |
Ahmed, N. U. A new class of generalized nonlinear functionals of white noise with applications to random integral equations. Stochastic Anal. Appl. 1, 2 (1983), 139–158. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| Other reviews under "Finite Element Methods": |
Date |
|
Mixed and hybrid finite element methods
Brezzi F., Fortin M., Springer-Verlag New York, Inc., New York, NY, 1991. Type: Book (9780387975825) |
Jun 1 1992 |
|
A modular program for Poisson’s equation with linear boundary and domain elements
Rudolphi T., Petersen V. Advances in Engineering Software 9(1): 7-28, 1987. Type: Article |
Oct 1 1987 |
|
Finite element analysis on microcomputers
Baran N., McGraw-Hill, Inc., New York, NY, 1988. Type: Book (9789780070336940) |
Mar 1 1989 |
|
| more... |
|
|
|
|
|
|
|
 E-Mail
This
 Printer-Friendly
|
|
|
|
|
|
|
