In many natural and industrial situations, fluids in cavities, membranes or pipes of complex shape grow as well as modify the structures through which they flow. This leads to important challenges both fundamental, like biological morphogenesis, and practical, like the motion of nano-electromechanical systems (NEMS). We seek to substantially advance the understanding of the resulting shapes and instabilities. Our approach will focus on numerical methods, validated through theoretical and experimental analysis.Mathematically fluid-structure interactions involve ambitious moving boundary problems, where structure and fluid flow feedback on one another in complex ways. Detailed analysis requires precise modeling of coupling between very strongly de¬forming elasto-plastic solids and fluid flow in intricately curved spaces and solving both iteratively many times. To address this computational challenge, significant innovations will be implemented, including the use of novel erosion laws, the insertion of spatial curvature and metric directly into the equations of motion of the fluid, and special methods to handle the singular behaviour at kinks and constrictions. Our fluid solvers will be new variants of Lattice Boltzmann Models (LBM) coupled to temperature and concentration fields. The accuracy of the methods will be quantitatively validated by experiments.An unconventional hydrodynamic formulation for electronic currents will provide big advantages. We will develop LBM solvers for quantum and relativistic fluids and in particular create a Lattice Wigner model and couple it to the molecular dynamics of the support.Our method will open new horizons for the design of continuously regenerating filters, for shape optimi¬zation of heat exchangers and catalysts and for the engineering of electronic devices. Our approach will also shed light on sand avalanches in oil extraction, on aspects of folding in living matter, and on electromechanical instabilities.