Simulation of crack propagation using isogeometric analysis applied with NURBS and LR B-splines

Abstract

This report features the isogeometric finite element method applied to the elastodynamic problem in a brittle medium with a potential for cracking. Griffith's theory for fracturing is used. The development of the model is outlined, complete with the Euler-Lagrange equations. The cracking is described with a phase field supplemented with a history field, contrary to the usual way of building the crack directly into the geometry by modification of the basis, facilitating the use of isogeometric analysis even with simplistic basis functions such as Non-Uniform Rational B-Splines (NURBS). The introduction of the crack-phase field results in non-linearity in the coupled problem. The problem is semi-discretized, upon which the spatial sub-problem is treated with isogeometric analysis. The numerical time-stepping solution routine is built around the Newton-Raphson method, but includes both pre-conditioning and correctors and is known as the predictor/multi-corrector time integration scheme. The Jacobian of the semi-discretized system (needed for the Newton-Raphson iteration) is developed analytically. In addition to the numerical tests with NURBS as our basis, we will also test the method with Locally Refined B-splines (LR B-Splines), ensuring better resolution along the crack path. The LR B-spline represents an alternative to the more commonly used T-Spline.