A Block-Coupled Finite Volume Method for Incompressible Hyperelastic Solids

This work introduces a block-coupled finite volume method for simulating the large-strain deformation of incompressible hyperelastic solids. Conventional displacement based finite volume solvers for incompressible materials often face stability and convergence challenges, particularly on unstructured meshes and in finite-strain regimes typical of biological tissues. To address these issues, a mixed displacement–pressure formulation is adopted and solved using a block-coupled strategy, enabling simultaneous solution of displacement and pressure increments. This eliminates the need for under-relaxation and improves robustness compared to segregated approaches. The method incorporates several enhancements, including temporally consistent Rhie–Chow interpolation, accurate treatment of traction boundary conditions, and compatibility with a wide range of constitutive models, from linear elasticity to advanced hyperelastic laws such as Holzapfel–Gasser–Ogden and Guccione. Implemented within the solids4Foam toolbox for OpenFOAM, the solver is validated against analytical and finite element benchmarks across diverse test cases, including uniaxial extension, simple shear, pressurized cylinders, arterial wall and idealised ventricle inflation. Results demonstrate second-order spatial and temporal accuracy, excellent agreement with reference solutions, and reliable performance in three-dimensional scenarios. The proposed approach establishes a robust foundation for fluid–structure interaction simulations in vascular and soft tissue biomechanics.