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

Technical Summary: A Jacobian-free Newton-Krylov Method for Cell-Centred Finite Volume Solid Mechanics

Problem Statement
Traditional finite volume (FV) methods for solid mechanics predominantly rely on segregated implicit approaches (Picard iterations) or fully explicit schemes. While segregated methods offer implementation simplicity and memory efficiency, they often suffer from slow convergence, particularly in problems involving geometric nonlinearity, large rotations, or complex coupling. Conversely, Newton-Raphson-type methods, common in the finite element community, offer quadratic convergence but require the explicit assembly, storage, and inversion of the full Jacobian matrix. This requirement is computationally expensive and memory-intensive, and adapting existing segregated FV frameworks to block-coupled Newton methods is non-trivial due to the complexity of deriving and storing the true Jacobian. Consequently, there is a need for a robust, efficient block-coupled solver that can be integrated into existing segregated FV frameworks without major code refactoring.

Methodology
The authors propose a Jacobian-free Newton-Krylov (JFNK) approach tailored for cell-centred finite volume solid mechanics. The core methodology avoids the explicit formation of the Jacobian matrix ($J$) by approximating its action on a vector ($v$) using finite differences:
$$Jv \approx \frac{R(u + \epsilon v) - R(u)}{\epsilon}$$
where $R$ is the residual function, $u$ is the solution vector, and $\epsilon$ is a perturbation scalar. This approximation is combined with the Generalised Minimal Residual (GMRES) Krylov subspace solver to solve the linear systems arising within the Newton iterations.

To ensure efficiency and robustness, the method employs a preconditioning strategy. Instead of using the exact Jacobian (which is unavailable), the authors utilize a compact-stencil approximate Jacobian derived from the inertia term and a diffusion-like term (based on the Rhie-Chow stabilisation used in segregated solvers). This preconditioner ($P$) is conceptually similar to the matrix used in traditional segregated approaches, allowing the new method to leverage existing matrix assembly and storage routines within frameworks like OpenFOAM's solids4foam toolbox.

The implementation uses the PETSc library for the nonlinear solver (SNES) and linear algebra, interfaced with the solids4foam FV solver. Key algorithmic features include:

• Preconditioners: Comparison of LU direct solvers (MUMPS), Incomplete LU (ILU), and Algebraic Multigrid (HYPRE BoomerAMG).
• Globalisation: Use of line search and transient continuation (predictor-corrector) to steer the solution toward the quadratic convergence domain.
• Stabilisation: A Rhie-Chow-type term is added to the residual to suppress zero-energy modes (checkerboarding), with a global scaling factor ($\alpha$) tuned for convergence speed.

Key Contributions

1. First Application to FV Solid Mechanics: This work represents the first proposal of a JFNK method specifically for finite volume solid mechanics, bridging a gap between fluid dynamics applications and solid mechanics.
2. Seamless Integration: By using a compact-stencil approximate Jacobian as a preconditioner, the method enables the adoption of Newton-type convergence within existing segregated FV frameworks with minimal code modification.
3. Comprehensive Benchmarking: The method is rigorously tested against a standard segregated approach across a suite of cases varying in dimensionality (2-D/3-D), material behavior (linear elasticity, hyperelasticity, elastoplasticity), and physics (static, dynamic, fluid-structure interaction).

Results
The study benchmarks the JFNK method against the segregated approach using metrics of computational time, memory usage, and robustness:

• Efficiency: The JFNK method outperforms the segregated approach in all linear and nonlinear elastic cases. Speedups of one to two orders of magnitude were observed in many instances, with a maximum speedup of 341x in a 2-D linear elastic Cook's membrane case.
• Memory: For problems with fewer than $10^4$ degrees of freedom, memory usage is comparable to the segregated method. For larger problems, JFNK memory usage increases, reaching a maximum of 3.6 times that of the segregated method (in a spherical cavity case), primarily due to the storage requirements of the GMRES solver and multigrid preconditioner.
• Robustness and Limitations:
  • The JFNK method demonstrated superior robustness in hyperelastic cases (large strains/rotations) where the segregated method failed to converge.
  • However, the JFNK method diverged in elastoplastic cases (Cook's membrane with plasticity). The authors attribute this to the compact-stencil preconditioner, which is based on small-strain elasticity, being a poor approximation of the Jacobian for large plastic strains.
• Preconditioner Choice:
  • LU Direct Solvers were fastest for small-to-moderate sized problems and 2-D cases.
  • Multigrid (AMG) was more effective for larger 3-D problems, offering better scalability and lower memory growth than LU.
  • ILU(k) was generally the slowest and less robust.
• Stabilisation: Increasing the Rhie-Chow stabilisation factor ($\alpha$) generally improved convergence speed and robustness, though at the cost of pre-asymptotic accuracy.

Significance and Claims
The paper claims that Jacobian-free Newton-Krylov methods are a promising advancement for finite volume solid mechanics, particularly for existing segregated frameworks where minimal modifications enable their adoption. The study demonstrates that JFNK can achieve significant speedups and improved robustness for elastic problems without the prohibitive memory cost of full Jacobian assembly.

The authors remain modest regarding the method's current limitations, explicitly noting that the approach is not yet robust for elastoplasticity with the current preconditioning strategy. They identify the development of enhanced preconditioners for plasticity and the inclusion of non-orthogonal corrections as necessary future work. The implementation is made publicly available in the solids4foam toolbox to encourage community exploration and extension.