Recent advances on the use of separated representations

Separated representations based on finite sum decompositions constitute an appealing strategy for reducing the computer resources and the calculation costs by reducing drastically the number of degrees of freedom that the functional approximations involve (the number of degrees of freedom scale linearly with the dimension of the space in which the model is defined instead of the exponential growing characteristic of mesh-based discretization strategies). In our knowledge the use of separated representations is the only possibility for circumventing the terrific curse of dimensionality related to some highly multidimensional models involving hundreds of dimensions, as we proved in some of our former works. Its application is not restricted to multidimensional models, obviously separated representation can also be applied in standard 2D or 3D models, allowing for high resolution computations. Because its early life numerous issues persist, many of them attracting the curiosity of many research groups within the computational mechanics community. In this paper we are focusing in two issues never until now addressed: (i) the imposition of non-homogenous essential boundary conditions and (ii) the consideration of complex geometries. Copyright © 2009 John Wiley & Sons, Ltd.