A robust monolithic solver for phase-field fracture integrated with fracture energy based arc-length method and under-relaxation

A micromorphic phase-field model for brittle and quasi-brittle fracture

“…Finally, the novel micromorphic phase-field fracture model opens a plethora of future research extensions, particularly, in multi-physics applications, composite laminates [61]. Other studies may include the implementation of a dissipation-based arc-length method [55,56] or quasi-Newton methods [62,63] for addressing the non-convexity of energy functional.…”

“…As such, conventional incremental solution techniques, like the Newton-Raphson method fails to achieve convergence in the softening regime, possibly due to an indefinite stiffness matrix. Efforts to circumvent this issue include the novel line search technique proposed in [34], the use of arc-length solvers [54][55][56], trust-region methods [57] and convexification via extrapolation of the phase-field for the momentum balance equation [36]. These aforementioned techniques are within the framework of monolithic solution techniques.…”

On the Geometry Transferability of the Hybrid Iterative Numerical Solver for Differential Equations

“…x (x, y) = ∂ux(x,y) ∂x y (x, y) = ∂ux(x,y) ∂y γ xy (x, y) = ∂ux(x,y) ∂y + ∂uy(x,y) ∂x (7) with subscripts i, j ∈ {1, 2} refer to the two in-plane directions, u i is the displacement in the i direction, ε ij is the strain component measured in j direction due to displacement in i direction, σ ij is the stress component, and f i is the body force in the i direction. The Lamé parameters, µ = E 2(1+ν) and λ = Employing HINTS to solve the Elasticity problem on source domain…”

“…Numerical simulations play a crucial role in scientific and engineering applications such as mechanics of materials and structures [1,2,3,4,5,6,7], bio-mechanics [8,9], fluid dynamics [10,11,12], etc. The simulation approach is based on solving linear/nonlinear partial differential equations (PDEs).…”

Fast staggered schemes for the phase-field model of brittle fracture based on the fixed-stress concept

“…On numerical implementation, using the monolithic scheme where the displacements and the phase variable are solved simultaneously, Gerasimov and De Lorenzis [12] proposed a line search assisted approach to overcome the iterative convergence issues of non-convex minimization and improve the computation efficiency. Subsequently, several solvers have been applied or developed to solve the monolithic problem more efficiently, including modified Newton methods using the Jacobian matrix with dynamic modification [13] or with inertia correction [14], the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm as a quasi-Newton method [15][16][17][18], and a fracture energy-based arc-length method with adaptive under-relaxation [19].…”