Three important decompositions in mechanics

Rubens Sampaio
Professor, Department of Mechanical Engineering, Pontifícia Universidade Católica, Rio de Janeiro, Brazil

Karhunen-Loève : The Karhunen-Loève, KL, decomposition establishes that a second order random process can be expanded as a series involving a sequence of deterministic orthogonal functions with orthogonal random coefficients. The KL decomposition, that originally appeared in the mathematic literature, has been used in Mechanics as a tool to obtain reduced models and is a strong candidate to extend modal analysis to non-linear systems. The KL method can be viewed as a statistical procedure. One initially supposes that the observed system dynamics can be modeled as a second-order ergodic stochastic process. The method consists then in constructing a spatial autocorrelation tensor from data obtained through numerical or physical experiments and performing its spectral decomposition. The autocorrelation tensor is by definition Hermitian and positive semi-definite. Therefore, its decomposition provides a set of orthogonal eigenfunctions (called proper orthogonal modes, POMs). These POMs can then be used as a basis for the dynamics projection and in the construction of a reduced-order model by truncation. Usually the KL basis is presented for Rn vectors and in this framework it is equivalent to the Principal Component Analysis and to the Singular Value Decomposition. Lately the number of papers dealing with KL increased very much and the use of the decomposition is not exactly as I have described above. The propose of the presentation will be:

Smooth Decomposition : The Smooth Decomposition (SD) was defined for systems with finite degree of freedom (DOF) and discrete time. The definition was generalized for continuous time under the name of Smooth Karhunen-Loève Decomposition to analyze random processes. The SD is obtained solving a generalized eigenproblem defined from the covariance matrix of the random process and the covariance matrix of the associated timederivative random process. We will present a new non-trivial generalization for continuous processes in space and time, classically named random fields. This generalization is a major step since one now deals with operators in infinite-dimensional spaces and not matrices. The techniques used so far to show the relation between the smooth modes and the normal modes rely on the possibility to write both problems, the covariance definition of SD and the eigenvalue problem defining the normal modes, in matrix form and to reduce one form to the other under convenient hypothesis. The technique can be applied only because the number of degrees of freedom is finite. This technique cannot be used in the random-field case, that is, to write in a global form the covariance operator and the eigenvalue problem for continuous systems. The main properties of the SD for the random-field case are described and compared to the classical Karhunen-Loève decomposition.