2015/11/26 by Patrick E. Farrell, Farrell, Patrick E., Corrado Maurini +1 · 1 citation
Decision Sciences · Engineering · #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in engineering #Probabilistic and Robust Engineering Design
paper · pdf · doi:10.48550/arxiv.1511.08463
openalex publication_date 2015/11/26 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
The variational approach to fracture is effective for simulating the\nnucleation and propagation of complex crack patterns, but is computationally\ndemanding. The model is a strongly nonlinear non-convex variational inequality\nthat demands the resolution of small length scales. The current standard\nalgorithm for its solution, alternate minimization, is robust but converges\nslowly and demands the solution of large, ill-conditioned linear subproblems.\nIn this paper, we propose several advances in the numerical solution of this\nmodel that improve its computational efficiency. We reformulate alternate\nminimization as a nonlinear Gauss-Seidel iteration and employ over-relaxation\nto accelerate its convergence; we compose this accelerated alternate\nminimization with Newton's method, to further reduce the time to solution; and\nwe formulate efficient preconditioners for the solution of the linear\nsubproblems arising in both alternate minimization and in Newton's method. We\ninvestigate the improvements in efficiency on several examples from the\nliterature; the new solver is 5--6\× faster on a majority of the test\ncases\n