318 Computational Mechanics of Composite Materials The homogenisation method is still the most efficient way of computational modelling of composite systems.Usually it is assumed that there exists some scale relation between composite components and the entire system-two scales are introduced that are related by a scale parameter being some small real value tending most frequently to 0.An essential disadvantage of all these techniques is the impossibility of sensitivity analysis of composite homogenised characteristics with respect to geometrical scales relations. Wavelet analysis became very popular in the area of composite materials modelling because of their multiscale and stochastic nature.The most interesting issue is composite global behaviour,which is more important than the multiphysical phenomena appearing at different levels of their complicated multiscale structure.That is why it is necessary to build an efficient mathematical and numerical multiresolutional algorithm to analyse composite materials and structures. As is known,two essentially different ways are proposed to achieve this goal. First,the composite can be analysed directly using the wavelet decomposition-based FEM approach where the multiresolutional analysis can recover the material properties of any component at practically any geometrical level.The method leads to an exponential increase of the total number of degrees of freedom in the model-each new decomposition level increases this number. Alternatively,a multiscale homogenisation algorithm can be applied to determine effective material parameters of the entire composite and next,to carry out the classical FEM or other related method-based computations.The basic difference between these two approaches is that the wavelet decomposition and construction algorithms are incorporated into the matrix FEM computations in the first method.The second method is based on the determination of the effective material parameters and Finite Element analysis of the equivalent homogeneous system,where the dimensions of the original heterogeneous and homogenised problems are almost the same.An analogous two methodologies had been known before the wavelet analysis was incorporated in engineering computations. However the homogenisation method assumptions dealing with the interrelations between macro-and microscales were essentially less realistic. Considering the above,the aim of this chapter is to demonstrate the use of the wavelet-based homogenisation method in comparison with its preceding classical formulations.Effective material parameters of a periodic composite beam are determined symbolically in MAPLE and next,the temporal and spatial variability of thermal responses of homogenised systems are determined numerically and compared with the real structure behaviour.It is assumed here that material properties are temperature-independent,which should be extended next to the thermal-dependent behaviour.As is verified by the computational experiments,all homogenisation methods (classical and multiresolutional)give a satisfactory approximation of real heat transfer phenomena in the multiscale heterogeneous structure.The approach should be verified next for other types of composites as well as various physical and structural problems in both a deterministic and stochastic context.Separate studies should be carried out for the computer318 Computational Mechanics of Composite Materials The homogenisation method is still the most efficient way of computational modelling of composite systems. Usually it is assumed that there exists some scale relation between composite components and the entire system – two scales are introduced that are related by a scale parameter being some small real value tending most frequently to 0. An essential disadvantage of all these techniques is the impossibility of sensitivity analysis of composite homogenised characteristics with respect to geometrical scales relations. Wavelet analysis became very popular in the area of composite materials modelling because of their multiscale and stochastic nature. The most interesting issue is composite global behaviour, which is more important than the multiphysical phenomena appearing at different levels of their complicated multiscale structure. That is why it is necessary to build an efficient mathematical and numerical multiresolutional algorithm to analyse composite materials and structures. As is known, two essentially different ways are proposed to achieve this goal. First, the composite can be analysed directly using the wavelet decomposition-based FEM approach where the multiresolutional analysis can recover the material properties of any component at practically any geometrical level. The method leads to an exponential increase of the total number of degrees of freedom in the model – each new decomposition level increases this number. Alternatively, a multiscale homogenisation algorithm can be applied to determine effective material parameters of the entire composite and next, to carry out the classical FEM or other related method-based computations. The basic difference between these two approaches is that the wavelet decomposition and construction algorithms are incorporated into the matrix FEM computations in the first method. The second method is based on the determination of the effective material parameters and Finite Element analysis of the equivalent homogeneous system, where the dimensions of the original heterogeneous and homogenised problems are almost the same. An analogous two methodologies had been known before the wavelet analysis was incorporated in engineering computations. However the homogenisation method assumptions dealing with the interrelations between macro- and microscales were essentially less realistic. Considering the above, the aim of this chapter is to demonstrate the use of the wavelet-based homogenisation method in comparison with its preceding classical formulations. Effective material parameters of a periodic composite beam are determined symbolically in MAPLE and next, the temporal and spatial variability of thermal responses of homogenised systems are determined numerically and compared with the real structure behaviour. It is assumed here that material properties are temperature–independent, which should be extended next to the thermal-dependent behaviour. As is verified by the computational experiments, all homogenisation methods (classical and multiresolutional) give a satisfactory approximation of real heat transfer phenomena in the multiscale heterogeneous structure. The approach should be verified next for other types of composites as well as various physical and structural problems in both a deterministic and stochastic context. Separate studies should be carried out for the computer