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We develop a unified continuum modeling framework for viscous fluids andhyperelastic solids using the Gibbs free energy as the thermodynamic potential.This framework naturally leads to a pressure primitive variable formulation forthe continuum body, which is well-behaved in both compressible andincompressible regimes. Our derivation also provides a rational justificationof the isochoric-volumetric additive split of free energies in nonlinearcontinuum mechanics. The variational multiscale analysis is performed for thecontinuum model to construct a foundation for numerical discretization. Wefirst consider the continuum body instantiated as a hyperelastic material anddevelop a variational multiscale formulation for the hyper-elastodynamicproblem. The generalized-alpha method is applied for temporal discretization. Asegregated algorithm for the nonlinear solver is designed and carefullyanalyzed. Second, we apply the new formulation to construct a novel unifiedformulation for fluid-solid coupled problems. The variational multiscaleformulation is utilized for spatial discretization in both fluid and solidsubdomains. The generalized-alpha method is applied for the whole continuumbody, and optimal high-frequency dissipation is achieved in both fluid andsolid subproblems. A new predictor multi-corrector algorithm is developed basedon the segregated algorithm to attain a good balance between robustness andefficiency. The efficacy of the new formulations is examined in severalbenchmark problems. The results indicate that the proposed modeling andnumerical methodologies constitute a promising technology for biomedical andengineering applications, particularly those necessitating incompressiblemodels.

A unified continuum and variational multiscale formulation for fluids, solids, and fluid-structure interaction