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This paper proposes a phase field model (PFM) for describing hydraulic fracture propagation in transversely isotopic media. The coupling between the fluid flow and displacement fields is established according to the classical Biot poroelasticity theory, while the phase field model characterizes the fracture behavior. The proposed method applies a transversely isotropic constitutive relationship between stress and strain as well as anisotropy in fracture toughness and permeability. We add an additional pressure-related term and an anisotropic fracture toughness tensor in the energy functional, which is then used to obtain the governing equations of strong form via the variational approach. In addition, the phase field is used to construct indicator functions that transit the fluid property from the intact domain to the fully fractured one. Moreover, the proposed PFM is implemented using the finite element method where a staggered scheme is applied to solve the displacement, fluid pressure, and phase field sequentially. Afterward, two examples are used to initially verify the proposed PFM: a transversely isotropic single-edge-notched square plate subjected to tension and an isotropic porous medium subjected to internal fluid pressure. Finally, numerical examples of 2D and 3D transversely isotropic media with one or two interior notches subjected to internal fluid pressure are presented to further prove the capability of the proposed PFM in 2D and 3D problems. [ABSTRACT FROM AUTHOR]

A new adaptive refinement strategy for phase-field models of brittle fracture is proposed. The approach provides a computationally efficient solution to the high demand in spatial resolution of phase-field models. The strategy is based on considering two types of elements: h-refined elements along cracks, where more accuracy is needed to capture the solution, and standard elements in the rest of the domain. Continuity between adjacent elements of different type is imposed in weak form by means of Nitsche's method. The weakly imposition of continuity leads to a very local refinement in a simple way, for any degree of approximation and both in 2D and 3D. The performance of the strategy is assessed for several scenarios in the quasi-static regime, including coalescence and branching of cracks in 2D and a twisting crack in 3D. [ABSTRACT FROM AUTHOR]

This paper provides a comprehensive overview of a phase-field model of fracture in solid mechanics setting. We start reviewing the potential energy governing the whole process of fracture including crack initiation, branching or merging. Then, a discretization of system of equation is derived, in which the key aspect is that for the correctness of fracture phenomena, a split into tensile and compressive terms of the strain energy is performed, which allows crack to occur in tension, not in compression. For numerical analysis, standard finite element shape functions are used for both primary fields including displacements and phase field. A staggered scheme which solves the two fields of the problem separately is utilized for solution step and illustrated with a segment of Python code. [ABSTRACT FROM AUTHOR]

The staggered numerical scheme is shown to be a robust and simple method for the approximation of the Exner-shallow water equations for bedload sediment modeling. Numerical tests show good convergence properties to an analytical solution and match pretty well data experiments in the case of dam break with erodible bottom. The cases of subcritical steady flow over a bump and transcritical flow over a bump are presented, showing the robustness of the scheme and its interest for engineering applications. [ABSTRACT FROM AUTHOR]

In this contribution we address the issue of efficient finite element treatment for phase-field modeling of brittle fracture. We start by providing an overview of the existing quasi-static and dynamic phase-field fracture formulations from the physics and the mechanics communities. Within the formulations stemming from Griffith's theory, we focus on quasi-static models featuring a tension-compression split, which prevent cracking in compression and interpenetration of the crack faces upon closure, and on the staggered algorithmic implementation due to its proved robustness. In this paper, we establish an appropriate stopping criterion for the staggered scheme. Moreover, we propose and test the so-called hybrid formulation, which leads within a staggered implementation to an incrementally linear problem. This enables a significant reduction of computational cost-about one order of magnitude-with respect to the available (non-linear) models. The conceptual and structural similarities of the hybrid formulation to gradient-enhanced continuum damage mechanics are outlined as well. Several benchmark problems are solved, including one with own experimental verification. [ABSTRACT FROM AUTHOR]

A numerical method (pressure-correction method using a staggered grid) is coupled to a thermodynamic model for compressible liquid hydrazine. The method is applied to the venting of liquid hydrazine into space, during which the fluid undergoes a large pressure drop. Below the saturation pressure vaporisation occurs. This takes place near the outlet and induces variations of temperature, which may cause solidification and pipe clogging. In order to assess the risk of phase changes, numerical simulations of the venting line have been performed using a quasi one-dimensional approach. The numerical method can handle compressible flows of fluids with nonconvex equation of state at the low Mach numbers that occur during hydrazine venting. A numerical study of the liquid behaviour during strong depressurisation is performed. The method is validated using experimental data, and allows prediction of pressure evolution and vaporisation location along the pipe. [ABSTRACT FROM AUTHOR]