  Granular materials are heterogenous grains in contact, which are ubiquitous
in many scientific and engineering applications such as chemical engineering,
fluid mechanics, geomechanics, pharmaceutics, and so on. Granular materials
pose a great challenge to predictability, due to the presence of critical
phenomena and large fluctuation of local forces. In this paper, we consider the
quasi-static simulation of the dense granular media, and investigate the
performances of typical minimization algorithms such as conjugate gradient
methods and quasi-Newton methods. Furthermore, we develop preconditioning
techniques to enhance the performance. Those methods are validated with
numerical experiments for typical physically interested scenarios such as the
jamming transition, the scaling law behavior close to the jamming state, and
shear deformation of over jammed states.

Computational Physics;Computational Engineering, Finance, and Science