A Jacobian-free Newton-Krylov method for cell-centred finite volume solid mechanics

TL;DR

This work introduces a Jacobian-free Newton-Krylov (JFNK) solver for cell-centered finite-volume solid mechanics, leveraging a compact-stencil preconditioner to enable easy integration into existing segregated frameworks. Using GMRES for the linear solves and a Thornton-style compact Jacobian as preconditioning, the approach achieves substantial speedups over traditional segregated methods in linear and nonlinear elastic cases, with reported speedups reaching up to three orders of magnitude for some meshes. Elastoplastic cases exhibit divergence, highlighting limitations of the elastic-based preconditioner for large plastic deformations and pointing to directions for improvement. The study demonstrates the practical potential of JFNK in finite-volume solid mechanics, provides extensive benchmark results, and releases open-source implementations in the solids4foam/OpenFOAM ecosystem to encourage community adoption and extension.

Abstract

This study investigates the efficacy of Jacobian-free Newton-Krylov methods in 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, encompassing 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 impacts performance: a LU direct solver is fastest in 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.

Figures (22)

• Figure 1: Representative convex polyhedral cell $P$ and neighbouring cell $N_{f_i}$, which share a face $f_i$.
• Figure 2: A cut plane through the cube case geometry showing the magnitude of the manufactured displacement solution (left) and a polyhedral mesh (right). The cut plane passes through the centre of the cube and has the unit normal $\boldsymbol{n} = (1/\sqrt{3} \quad 1/\sqrt{3} \quad 1/\sqrt{3})$.
• Figure 3: Manufactured solution cube case: the accuracy and order of accuracy for displacement magnitude
• Figure 4: Manufactured solution cube case: the predicted $\sigma_{xx}$ stress distribution on a cut-plane for the regular hexahedral mesh with $512\,000$ cells (left).
• Figure 5: Manufactured solution cube case: the accuracy and order of accuracy for stress magnitude