Isogeometric methods for the study of fracture mechanics via phase-field modeling

The design of structural elements and industrial components must consider fracturephenomena, which can render a device or structure inoperative. Preventing fracture induced failure is, therefore, a critical constraint in engineering design. Numerical simulation of fracture provides a valuable decision-making tool in this context, reducing the need for costly and time-consuming experimental testing. Among the various numerical models proposed for fracture analysis, the phase-field method has gained significant popularity due to its ability to elegantly simulate complex crack behaviours, including initiation, propagation, merging, and branching. Notably, this method eliminates the need for explicit or implicit crack location definitions, as the crack path is automatically tracked without requiring mesh updates or ad-hoc tracking techniques. Despite its advantages, the phase-field method presents several modeling and computational challenges. A primary limitation is its high computational cost, especially for large-scale engineering applications, due to the requirement for extremely fine meshes around crack paths to resolve the internal length parameter. Specifically, when using classical finite elements, at least four elements are typically required within the length parameter as a general guideline. High-order functional counterparts address this issue by incorporating a high-order term alongside the local and gradient-based nonlocal terms, resulting in higher regularity in the phase field solution. This enhanced regularity improves the convergence rate of numerical solutions and enables higher-order accuracy for fracture problems. Therefore; in this thesis, we first follow Borden et al. in "A higher-order phase-field model for brittle fracture: Formulation and analysis within the isogeometric analysis framework" (Comp. Meth. Appl. Mech. Eng., 273, 100-118, 2014) and examine two distinct types of phase-field functionals: (i) a second-order AT2 model and (ii) a fourth-order AT2 model. The latter approach involves the bi-Laplacian of the phase field, which necessitates continuously differentiable basis functions in the resulting Galerkin formulation, a condition we satisfy through Isogeometric Analysis (IGA), highlighting the capability of IGA to propose novel and straightforward simulation frameworks. We extensively compare the considered formulations by conducting several tests that progressively increase the complexity of the crack patterns. We propose an image-based algorithm featuring an automatic skeletonization technique capable of tracking complex fracture patterns to measure the fracture length required for our accuracy evaluations. Our findings indicate that, for the same mesh, the fourth-order model is always more accurate than the second-order one. At the same time, we observe a comparable accuracy with meshes typically twice coarser for the fourth-order model. This turns into a remarkable reduction in the total computational time required by the overall analysis. Furthermore, we investigate a different strategy to impose the initial pre-field that gives higher accuracy results concerning a state-of-the-art method. After studying the advantages of the AT2 high-order model; we decided to extend our investigation to the AT1 regularization, aiming to combine the favourable physical properties of the AT1 model with the high regularity of the high-order model. Generally, AT1 functionals, compared to AT2, exhibit a clear elastic limit before the onset of crack growth and have narrower transition regions. High-order functionals demonstrate comparable accuracy to low-order ones while using larger mesh sizes, reducing computational effort. To leverage both advantages, we propose a novel fourth-order AT1 model, again utilizing an isogeometric framework, which allows for a straightforward discretization of the high-order term in the crack surface density functional. In the introduced function, the coefficient weighing [...]