We apply a previously-developed asymptotic model to study instability and breakup of metal filaments of nanometric dimensions exposed to heating by laser pulses, and placed on thermally conductive substrates. One particular aspect of this setup is that the considered heating is volumetric, since the absorption length of the applied laser pulse is comparable to a typical filament thickness. In such a setup, absorption of thermal energy and filament evolution are coupled, and must be considered self-consistently. The asymptotic model that we use allows for significant simplification, since it reduces a complicated problem involving Navier-Stokes equations coupled with heat transport. Such simplification is crucial both for understanding the main features of the problem, and for the purpose of developing efficient simulations of the filament evolution and subsequent nanoparticle formation. The presented computational results are obtained in the GPU computing environment, which allows for fully nonlinear time-dependent simulations in large three-dimensional computational domains. We focus in particular on the influence of filament size on the evolution. It is found that filaments' width and thickness play an important role, with thicker and/or wider filaments absorbing more energy and therefore evolving differently from thinner ones. This finding opens the door to considerations of self- and directed-assembly of metal nanoparticles via suitable choice of the initial metal geometry on the nanoscale.