We extend a phase-field/gradient damage formulation for cohesive fracture to the dynamic case. The model is characterized by a regularized fracture energy that is linear in the damage field, as well as non-polynomial degradation functions. Two categories of degradation functions are examined, and a process to derive a given degradation function based on a local stress-strain response in the cohesive zone is presented. The resulting model is characterized by a linear elastic regime prior to the onset of damage, and controlled strain-softening thereafter. The governing equations are derived according to macro-and microforce balance theories, naturally accounting for the irreversible nature of the fracture process by introducing suitable constraints for the kinetics of the underlying microstructural changes. The model is complemented by an efficient staggered solution scheme based on an augmented Lagrangian method. Numerical examples demonstrate that the proposed model is a robust and effective method for simulating cohesive crack propagation, with particular emphasis on dynamic fracture.

A new continuous-discontinuous strategy for the computational modeling of crack propagation within the context of phase-field models of fracture is presented. The method is designed to introduce and update a sharp crack surface within an evolving damage band and to enhance the kinematics of the finite element approximation accordingly. The proposed approach relies on three key elements. First, we propose the use of a crack length functional to provide a trigger for initiating a continuous to discontinuous transition. Next, the crack path identification is addressed by introducing the concept of an auxiliary damage field that varies with an extension of the sharp crack surface. The sharp crack surface is extended through an optimization algorithm, in which the difference between the auxiliary field and the actual damage field stemming from the phase-field framework is minimized. Finally, a strong discontinuity is inserted in the wake of the diffuse crack tip with the extended finite element method, completing the continuous to discontinuous transition. Several benchmark problems in two-dimensional quasi-static fracture mechanics are presented to demonstrate the accuracy and robustness of the method. KEYWORDS crack propagation, finite elements, optimization, phase-field approach, X-FEM Int J Numer Methods Eng. 2018;116:1-20.wileyonlinelibrary.com/journal/nme

An extended/generalized finite element method (XFEM/GFEM) for simulating quasistatic crack growth based on a phase-field method is presented. The method relies on approximations to solutions associated with two different scales: a global scale, that is, structural and discretized with a coarse mesh, and a local scale encapsulating the fractured region, that is, discretized with a fine mesh. A stable XFEM/GFEM is employed to embed the displacement and damage fields at the global scale. The proposed method accommodates approximation spaces that evolve between load steps, while preserving a fixed background mesh for the structural problem. In addition, a prediction-correction algorithm is employed to facilitate the dynamic evolution of the confined crack regions within a load step. Several numerical examples of benchmark problems in two-and three-dimensional quasistatic fracture are provided to demonstrate the approach. K E Y W O R D Scrack growth, extended/generalized FEM, global-local analysis, gradient damage models, multiscale, phase-fields 2534

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This paper proposes a novel approach for learning a data-driven quadratic manifold from highdimensional data, then employing the quadratic manifold to derive efficient physics-based reduced-order models. The key ingredient of the approach is a polynomial mapping between high-dimensional states and a low-dimensional embedding. This mapping comprises two parts: a representation in a linear subspace (computed in this work using the proper orthogonal decomposition) and a quadratic component. The approach can be viewed as a form of data-driven closure modeling, since the quadratic component introduces directions into the approximation that lie in the orthogonal complement of the linear subspace, but without introducing any additional degrees of freedom to the low-dimensional representation. Combining the quadratic manifold approximation with the operator inference method for projectionbased model reduction leads to a scalable non-intrusive approach for learning reduced-order models of dynamical systems. Applying the new approach to transport-dominated systems of partial differential equations illustrates the gains in efficiency that can be achieved over approximation in a linear subspace.