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The thesis presents the development, the mathematical analysis, as well as applications of a computational framework for the simulation of curvilinear crack propagation. At the core of the computational framework lies a novel finite element method, named the Mapped Finite Element Method (MFEM), for the optimal convergence of singular solutions. The main challenges in solving numerically the mathematical description of a propagating fracture can be identified in: the continuously evolving (cracked) domain, the singular nature of the elasticity fields, and the computation of the stress intensity factors for the prediction of crack growth. Current state-of-the-art methods are plagued by low order of accuracy, high computational cost, and complex data structures. The work herein addresses the aforementioned challenges by developing a computationally efficient, rapidly convergent, and non-intrusive algorithm consisting of three key ingredients: Universal Meshes, Mapped Finite Element Methods, and Interaction Integrals. First, Universal Meshes are introduced as a computationally efficient and robust meshing algorithm for the generation of conforming subdivisions of the evolving domain. Second, Mapped Finite Element Methods (MFEM) will be developed for the solution of the singular elasticity fields. The methods are shown to converge with optimal order for the same computational cost, preserving well conditioning and sparsity properties, and with no alteration to the data structure of standard Lagrange finite element methods (known to converge sub-optimally for this class of problems). The optimality of convergence is supported by mathematical analysis and applications of MFEM are showcased beyond brittle fracture (e.g. the resolution of boundary layers in flows around moving obstacle, real-space Kohn-Sham density functional theory calculations, etc). Third, the thesis presents the construct of a family of linear and affine functionals, named Interaction Integral functionals, for the rapidly convergent computation of the stress intensity factors (SIFs) for curvilinear fractures. The distinct feature of the Interaction Integral functionals is their ability to double the rate of convergence of the energy norm of the solution in the evaluation of the SIFs. Sketches of the mathematical analysis are provided to support the observed rapid rates of convergence. The propagation algorithm that combines the developed tools (Universal Meshes, Mapped Finite Element Methods, and Interaction Integrals) is presented and shown to be consistent (in the sense of being able to replicate observed experimental results) and predictive (in the sense of yielding crack paths that converge to a unique solution with refinement of the discretization). Lastly, the capabilities of the developed algorithm are exploited to study the formation of wavy crack patterns when brittle heat conductors are rapidly cooled.

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