Nonlinear convergence in contact mechanics

Abstract:

The physics of mechanical deformation in the presence of contact discontinuities is relevant to several geoscientific applications including CO₂ storage, enhanced geothermal, and shale gas production. At the Darcy scale, contacts are referred to as fractures, which can act as preferential conduits for fluid flow. At the pore scale, contacts constitute mineralogical defects in the crystalline structure of rocks, such as micro cracks and grain-to-grain interfaces. Such defects significantly alter the macroscopic mechanical response of a sample to external stimuli (e.g., elastic stiffness). The accurate simulation of contact mechanics requires the resolution of complex boundaries and, potentially intersecting, contact discontinuities (or fractures). The discretization is challenging because sophisticated gridding techniques must be employed to capture geometric details. Moreover, the contact equations are highly nonlinear and lead to frequent divergence of the Newton solver. In this work, we first present an immersed boundary finite-volume formulation of contact mechanics, in which all external and internal boundaries of an object are represented by surfaces embedded within a Cartesian grid. We then focus on the nonlinear convergence of the method and theoretically prove the conditions under which divergence occurs. Finally, we describe how Newton can be modified to guarantee convergence. The conclusions herein are expected to be useful in penalty-based finite element methods. Our algorithm is applicable to geologic and synthetic materials both at the pore scale and the Darcy scale.