Abstract
There is currently an increasing interest in developing efficient solvers for variational phase-field models of brittle fracture. The governing equations for this problem originate from a constrained minimization of a non-convex energy functional, and the most commonly used solver is a staggered solution scheme. This is known to be robust compared to the monolithic Newton method, however, the staggered scheme often requires many iterations to converge when cracks are evolving. The focus of our work is to accelerate the solver through a scheme that sequentially applies Anderson acceleration and over-relaxation, switching back and forth depending on the residual evolution, and thereby ensuring a decreasing tendency. The resulting scheme takes advantage of the complementary strengths of Anderson acceleration and over-relaxation to make a robust and accelerating method for this problem. The new method is applied as a post-processing technique to the increments of the solver. Hence, the implementation merely requires minor modifications to already available software. Moreover, the cost of the acceleration scheme is negligible. The robustness and efficiency of the method are demonstrated through numerical examples.
Highlights
Mathematical modeling of brittle fracture propagation is an important and challenging topic in engineering sciences
The acceleration method alternates between Anderson acceleration and over-relaxation according to a switch that depends on the norms of the previous residuals of the scheme
Over-relaxation, on the other hand, works well within regimes of brutal crack propagation, but might struggle when the iterates get close to the solution within a single loading step
Summary
Mathematical modeling of brittle fracture propagation is an important and challenging topic in engineering sciences. The main difficulty arises in the transition between the distinct material properties in the fracture and the bulk domain. We consider a variational phase-field model, as introduced by Bourdin, Francfort, and Marigo [1,2]. A smooth indicator that marks the broken and unbroken parts of the material regularizes the sharp crack topology. This enables modeling of fractures without conforming meshes or path-tracking algorithms (as in XFEM [3]). Fine meshes are needed to resolve the regularized region between the fracture and the bulk domain
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