A Hermite cubic collocation scheme for plane strain hydraulic fractures
We describe a novel cubic Hermite collocation scheme for the solution of the coupled integro-partial differential equations governing the propagation of a hydraulic fracture in a state of plane strain. Special blended cubic Hermite-powerlaw basis functions, with arbitrary index 0<?<1, are developed to treat the singular behavior of the solution that typically occurs at the tips of a hydraulic fracture. The implementation of blended infinite elements to model semi-infinite crack problems is also described. Explicit formulae for the integrated kernels associated with the cubic Hermite and blended basis functions are provided. The cubic Hermite collocation algorithm is used to solve a number of different test problems with two distinct propagation regimes and the results are shown to converge to published similarity and asymptotic solutions. The convergence rate of the cubic Hermite scheme is determined by the order of accuracy of the tip asymptotic expansion as well as the O(h4) error due to the Hermite cubic interpolation. The errors due to these two approximations need to be matched in order to achieve optimal convergence. Backward Euler time-stepping yields a robust algorithm that, along with geometric increments in the time-step, can be used to explore the transition between propagation regimes over many orders of magnitude in time.
hydraulic fractures integro-partial differential equations hermite cubic interpolation collocation methods