The study of complex multi-physics phenomena requires advanced modeling and simulation. These problems are insoluble by traditional theoretical and experimental approaches, hazardous to study in the laboratory, or time-consuming and expensive to solve by classical means. Our objective is to simplify the numerical modeling of problems involving complex unsteady geometries and multi-scale physical phenomena. Rather than using extremely optimized but non-scalable schemes, we adopt robust alternatives that bypass the difficulties linked to unsteady grid generation, a prohibitive task when the boundaries are moving and the topology is complex and unsteady. Hierarchical Cartesian schemes allow the multi-scale solution of PDEs on non body-fitted meshes with a drastic reduction of the computational setup overhead. Thanks to exemples relative to fluid-structure interaction, high-speed impacts, rarefied flows and material science, we plan to show how appropriate mathematical modeling, hierarchical Cartesian schemes and HPC can contribute to the simulation of new challenging complex phenomena in physics. Work in collaboration with M. Bergmann, F. Bernard, A. de Brauer, M. Cisternino, T. Milcent, A. Raeli, F. Tesser, L. Weynans