Fast multiscale methods for lattice equations

Abstract

The thesis concerns multi-scale analysis of equations defined on very large lattices. A new method for model reduction that allows the resolution scale to vary with spatial position is presented. It leads to fast numerical implementations and comes with strict error bounds. This is obtained by incorporating the averaging of the micro-structure directly into a solver that is based on hierarchical data structures, such as the Fast Multipole Method. For boundary value problems, a lattice analogue of the boundary element method is used. Homogenized equations of arbitrary order are derived and strict error bounds are proved. For mechanical lattice structures, special attention is paid to the question of non-degeneracy and whether rotational degrees of freedom should be incorporated. A computer program that automatically derives the homogenized equations was developed. For any lattice geometry, the lattice Green’s function is determined and its asymptotic expansion in rational poly-harmonic functions is derived. The presented results have applications in many areas of physics and engineering but particular attention is paid to how they can be used to design composite materials with high stiffness-to-weight ratios and prescribed wave-propagation modes. In particular, it is shown how to design materials that have phononic bandgaps, in other words, they block mechanical waves of certain frequencies.