A Jacobian‐Free Newton‐Krylov Method for Cell‐Centred Finite Volume Solid Mechanics

ABSTRACT This study proposes a Jacobian‐free Newton‐Krylov approach for finite‐volume solid mechanics. Traditional Newton‐based approaches require explicit Jacobian matrix formation and storage, which can be computationally expensive and memory‐intensive. In contrast, Jacobian‐free Newton‐Krylov methods approximate the Jacobian's action using finite differences, combined with Krylov subspace solvers such as the generalised minimal residual method (GMRES), enabling seamless integration into existing segregated finite‐volume frameworks without major code refactoring. This work proposes and benchmarks the performance of a compact‐stencil Jacobian‐free Newton‐Krylov method against a conventional segregated approach on a suite of test cases that span varying geometric dimensions, nonlinearities, dynamic responses and material behaviours. Key metrics, including computational cost, memory efficiency and robustness, are evaluated, along with the influence of preconditioning strategies and stabilisation scaling. Results show that the proposed Jacobian‐free Newton‐Krylov method outperforms the segregated approach in all linear and nonlinear elastic cases, achieving order‐of‐magnitude speedups in many instances; however, divergence is observed in elastoplastic cases, highlighting areas for further development. It is found that preconditioning choice affects performance: a LU direct solver is fastest for small to moderately sized cases, while a multigrid method is more effective for larger problems. The findings demonstrate that Jacobian‐free Newton‐Krylov methods are promising for advancing finite‐volume solid mechanics simulations, particularly for existing segregated frameworks where minimal modifications enable their adoption. The described implementations are available in the solids4foam toolbox for OpenFOAM, inviting the community to explore, extend and compare these procedures.