The development of our Particle-in-cell finite element method has been motivated by problems encountered in geology and geomechanics. Rocks in the crust deform viscoelastically over millions of years in response to large-scale lithospheric stresses, and may also fracture if the yield stress is exceeded. Deformation may be slow, but it is also relentless - the accumulated strains are extremely high. In fact, the ultimate source of stresses in the lithosphere is cooling of the Earth by convective flow in the deep mantle where strains are effectively infinite. In order to model a full range of geological processes, a method is required which can deal with very large strain creeping flow - including convection, heat transport, history dependent material properties such as strain softening, viscoelasticity, material interfaces and free surfaces. Classic Lagrangian mesh-based methods are suited to fast, stable solutions of the flow equations, but suffer from the complexity of remeshing at high strain. Meshless methods avoid this difficulty but, with no mesh, accelerated elliptic equation solvers such as multigrid are not very efficient and are horribly complicated. Our approach is a hybrid - lurking somewhere between classic finite element methods and pure particle methods. An Eulerian background mesh is used to solve the flow equations while Lagrangian particles are used for transport of information including material properties, history and interfaces. The two reference frames are coupled by an unusual integration scheme which uses the moving particles to build element matrices. The trick with a hybrid method is to capture as many of the benefits of each of ancestor formulations while leaving behind the disadvantages in each case - all will be revealed and you can judge for yourselves.
 

Convenor: Michael Page