Mathematics of Planet Earth

Impact cratering on planetary surfaces is one of the most important geological processes in the solar system. The cratered landscapes such as on the Moon, Mars or Mercury testify to the importance of collision events during the evolution of planets. Although remnants of meteorite impacts are rare on Earth it is generally accepted that impact events played an important role in the evolution of the biosphere. On the other hand impacts pose a threat to life on earth. 65 Mio years ago the dinosaurs were wiped out by the impact of a 10 km diameter asteroid striking the Earth at approx. 20 km/s. A quantitative understanding of impact processes can be obtained by the analysis of remnants of impacts from the geological record, by analogue experiments, and numerical modelling. The latter is the main topic of this presentation. Numerical simulations of meteorite impacts require a special type of computer codes, so-called “hydroxides”. A hydrocode may be loosely defined as a code designed to solve large deformation, finite strain transient problems that occur on a short time scale. While material strength is neglected in Eulerian codes used for gas dynamics, it is a key component of hydrocodes. In contrast to structural analysis codes, the energy equation is integrated in time, and the deviatoric and pressure terms in the stress tensor are usually modelled separately. The solution is advanced in time using an explicit integration scheme because stress waves and shocks are an important part of the solution, and they must be resolved accurately in both space and time. We will describe modelling approaches to investigate hypervelocity impact processes and shock wave propagation on different scale ranging from millimetres to thousands of kilometres. Besides an efficient parallelized numerical solution of partial differential equations by finite volume technique on Eulerian grids the parameterisation of material properties poses the biggest challenge in the simulation of impact processes. Material modelling in terms thermodynamic behaviour and mechanical response of large, rapid deformation is key for realistic description of the processes.