# Karhunen-Loève decomposition in solid mechanics

**At LMSSC, Cnam, Paris, January 24th 2007**

Rubens Sampaio

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

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, which 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 modelled 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: