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

Abstract

We develop a unified continuum modeling framework for viscous fluids and hyperelastic solids using the Gibbs free energy as the thermodynamic potential. This framework naturally leads to a pressure primitive variable formulation for the continuum body, which is well-behaved in both compressible and incompressible regimes. Our derivation also provides a rational justification of the isochoric-volumetric additive split of free energies in nonlinear continuum mechanics. The variational multiscale analysis is performed for the continuum model to construct a foundation for numerical discretization. We first consider the continuum body instantiated as a hyperelastic material and develop a variational multiscale formulation for the hyper-elastodynamic problem. The generalized-alpha method is applied for temporal discretization. A segregated algorithm for the nonlinear solver is designed and carefully analyzed. Second, we apply the new formulation to construct a novel unified formulation for fluid-solid coupled problems. The variational multiscale formulation is utilized for spatial discretization in both fluid and solid subdomains. The generalized-alpha method is applied for the whole continuum body, and optimal high-frequency dissipation is achieved in both fluid and solid subproblems. A new predictor multi-corrector algorithm is developed based on the segregated algorithm to attain a good balance between robustness and efficiency. The efficacy of the new formulations is examined in several benchmark problems. The results indicate that the proposed modeling and numerical methodologies constitute a promising technology for biomedical and engineering applications, particularly those necessitating incompressible models.

References in corpus (1)

• Immersed boundary model of aortic heart valve dynamics with physiological driving and loading conditions

Cited by in corpus (11)

• Fluid-structure interaction modeling of blood flow in the pulmonary arteries using the unified continuum and variational multiscale formulation
• A continuum and computational framework for viscoelastodynamics: finite deformation linear models
• The nested block preconditioning technique for the incompressible Navier-Stokes equations with emphasis on hemodynamic simulations
• A note on the accuracy of the generalized-$\alpha$ scheme for the incompressible Navier-Stokes equations
• An energy-stable mixed formulation for isogeometric analysis of incompressible hyper-elastodynamics
• A robust and efficient iterative method for hyper-elastodynamics with nested block preconditioning
• A reduced unified continuum formulation for vascular fluid-structure interaction
• A structure-preserving integrator for incompressible finite elastodynamics based on a grad-div stabilized mixed formulation with particular emphasis on stretch-based material models
• A continuum and computational framework for viscoelastodynamics: III. A nonlinear theory
• Neural inference of fluid-structure interactions from sparse off-body measurements
• Modeling finite viscoelasticity based on the Green-Naghdi kinematic assumption and generalized strains