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

TL;DR

This work extends Jacobian-free Newton–Krylov methods to high-order (third- and fourth-order) cell-centered finite-volume formulations for solid mechanics, achieving accurate stress and deformation fields via local least-squares gradient reconstructions and Gaussian quadrature. An alpha-stabilisation scheme is introduced to suppress high-frequency modes on irregular meshes, while a compact-stencil second-order approximate Jacobian acts as an efficient preconditioner within the JFNK framework, enabling robust convergence without assembling a full Jacobian. The method is implemented in the solids4foam toolbox for OpenFOAM and validated across 2D/3D linear and nonlinear benchmarks, demonstrating clear accuracy gains over second-order schemes with competitive efficiency. The study confirms the feasibility and potential of combining high-order FV discretisations with JFNK solvers for solid mechanics, while outlining avenues for curved-boundary treatment and further efficiency improvements. Open-source releases of the solver and benchmarks support community adoption and extension.

Abstract

This work extends the application of Jacobian-free Newton-Krylov (JFNK) methods to higher-order cell-centred finite-volume formulations for solid mechanics. While conventional schemes are typically limited to second-order accuracy, we present third- and fourth-order formulations employing local least-squares reconstructions for gradient evaluation and Gaussian quadrature at cell faces. These schemes enable accurate resolution of complex stress and deformation fields in linear and nonlinear solids while retaining the flexibility of finite-volume methods. A key contribution is a JFNK solution strategy for these higher-order schemes, eliminating the need to assemble complex Jacobian matrices. A compact-stencil approximate Jacobian is used as a preconditioner, providing efficiency gains similar to second-order frameworks. To enhance robustness on irregular meshes, an alpha-stabilisation scheme is incorporated, damping high-frequency error modes without compromising formal accuracy. The proposed methodology is benchmarked across a suite of two- and three-dimensional test problems involving elastic and nonlinear materials, where key performance metrics, including accuracy, computational cost, memory usage, and robustness, are systematically evaluated. Results confirm that the higher-order formulations deliver substantial accuracy improvements over second-order schemes, while the JFNK approach achieves strong performance with only minimal modifications to existing segregated frameworks. These findings underscore the potential of combining higher-order finite-volume methods with JFNK solvers to advance the state of the art in computational solid mechanics. The implementations are openly released in the solids4foam toolbox for OpenFOAM, supporting further exploration and adoption by the community.

Figures (28)

• Figure 1: Representative convex polyhedral cell $P$ and neighbouring cell $N$. Quantities calculated at internal and boundary faces are denoted by the subscripts $f$ and $b$, respectively.
• Figure 1: Convergence of the $L_2$ and $L_\infty$ displacement error norm for case \ref{['case:mms']} with over-integration as stabilisation.
• Figure 2: Volume integration, cell decomposition into tetrahedral elements.
• Figure 3: Interpolation stencils for internal $f$ and boundary $b$ face quadrature points, $\boldsymbol{x}_{f,q}$ and $\boldsymbol{x}_{b,q}$, respectively.
• Figure 4: Manufactured solution square (2D case): error convergence for displacement magnitude.