Wavelet-based finite element analysis of composites

Full text: Wavelet analysis became recently very popular in the area of composite materials modeling since their multiscale and stochastic nature. Most of the people including engineers, scientists and even ordinary people involved in designing, manufacturing and the usage of composites are usually interested in their global behavior rather than the multiphysical phenomena appearing at different scales of their complicated multilevel structure. Therefore, the most important topic is to build the efficient mathematical and numerical algorithm to analyze multiscale heterogeneous materials and structures. As it is known, thanks to the homogenization theory we can follow essentially two different paths to achieve this goal. First, the composite can be directly analyzed using the wavelet-based FEM approach. Concurrently, we can use the wavelet-based homogenization algorithm to determine effective material parameters and next, to carry out classical FEM or another related method based computations. The basic difference between those approaches is that the wavelet decomposition and construction algorithms are incorporated into the matrix FEM computations in the first method; therefore, the additional computer code should be modified. The second method is based on rather symbolic computations necessary for determination of the effective material parameters, while the structural analysis is classical. The computational strategy presented by the author is based on the homogenization method where the dynamics of the linear elastic heterogeneous beam is studied for the following general case: ∂/∂x (E(x)∂u/∂x) = n(x) ∂2u/∂t2; where both Young modulus E(x) and the composite mass density p(x) are defined by some wavelets. First, the effective material parameters of the beam are determined; then, the structural behavior of the homogenized system is determined numerically and compared against the real structure vibrations. Analogous analysis is done for the composite with randomly defined material properties - the probabilistic moment of homogenized parameters are determined together with the corresponding characteristics of the beam vibrations. The wavelet-based homogenized material characteristics obtained for both deterministic and probabilistic analyses are compared against those resulting from the classical homogenization theory. The entire discussion is made for different combinations of composite characteristics by computing their sensitivities with respect to various design input parameters; the computational process is performed using the symbolic computations package MAPLE linked with some FEM code. Refs. 3 (author)