line search algorithm for multiphysics problems with fracture deformation

Models for multiphysics problems often involve significant nonlinearities. When fracture contact mechanics are incorporated, discontinuous derivatives arise at the interfaces between open and closed fractures, or between sliding and sticking fractures. The resulting system of equations is highly challenging to solve. The naïve choice of Newton’s method frequently fails to converge, calling for more refined solution techniques such as line search methods. When dealing with strong nonlinearities and discontinuous derivatives, a global line search based on the magnitude of the residual of all equations is at best costly to evaluate and at worst fails to converge. We therefore suggest a cheap and reliable approach tailored to the discontinuities. Utilizing adaptive variable scaling, the algorithm uses a line search to identify the transition between contact states for each nonlinear iteration. Then, a solution update weight is chosen to ensure that fracture cells which change state do not move far beyond the transition point. We demonstrate the algorithm on a series of test cases for poromechanics and thermoporomechanics in fractured porous media. We consider both single- and multifracture cases, and study the importance of proper scaling of variables and equations.