Parallel inexact Newton–Krylov and quasi-Newton solvers for nonlinear elasticity

In this work, we address the implementation and performance of inexact Newton–Krylov and quasi-Newton algorithms, more specifically the BFGS method, for the solution of the nonlinear elasticity equations, and compare them to a standard Newton–Krylov method. This is done through a systematic analysis of the performance of the solvers with respect to the problem size, the magnitude of the data and the number of processors in both almost incompressible and incompressible mechanics. We consider three test cases: Cook's membrane (static, almost incompressible), a twist test (static, incompressible) and a cardiac model (complex material, time dependent, almost incompressible). Our results suggest that quasi-Newton methods should be preferred for compressible mechanics, whereas inexact Newton–Krylov methods should be preferred for incompressible problems. We show that these claims are also backed up by the convergence analysis of the methods. In any case, all methods present adequate performance, and provide a significant speed-up over the standard Newton–Krylov method, with a CPU time reduction exceeding 50% in the best cases.