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

Ju Liu1, Alison L Marsden1

  • 1Department of Pediatrics (Cardiology), Bioengineering, and Institute for Computational and Mathematical Engineering, Stanford University, Clark Center E1.3, 318 Campus Drive, Stanford, CA 94305, USA.

Computer Methods in Applied Mechanics and Engineering
|December 4, 2018
PubMed
Summary

We present a unified continuum model using Gibbs free energy for well-behaved simulations in compressible and incompressible regimes. This approach advances fluid-solid coupled problems and nonlinear elasticity for engineering applications.

Keywords:
Fluid-structure interactionGeneralized-α methodGibbs free energyIncompressible solidsNonlinear continuum mechanicsVariational Multiscale Method

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Area of Science:

  • Computational Mechanics
  • Thermodynamics
  • Nonlinear Solid Mechanics

Background:

  • Continuum modeling requires robust thermodynamic potentials.
  • Pressure primitive variable formulations are crucial for handling both compressible and incompressible flows.
  • Additive splits of free energies in nonlinear elasticity lack a unified theoretical basis.

Purpose of the Study:

  • To develop a unified continuum modeling framework based on Gibbs free energy.
  • To provide a theoretical justification for isochoric-volumetric free energy splits.
  • To establish a foundation for numerical discretization of continuum models, including fluid-solid coupled problems.

Main Methods:

  • Utilized Gibbs free energy as the thermodynamic potential for a unified framework.
  • Employed variational multiscale analysis for spatial discretization.
  • Applied the generalized-α method for temporal discretization.
  • Developed segregated and predictor multi-corrector algorithms for nonlinear solvers.

Main Results:

  • A pressure primitive variable formulation that is well-behaved in compressible and incompressible regimes.
  • A rational justification for the isochoric-volumetric additive split of free energies.
  • A novel unified formulation for fluid-solid coupled problems with optimal high-frequency dissipation.
  • Validated the methodology through benchmark problems, showing promise for incompressible models.

Conclusions:

  • The proposed unified continuum modeling and numerical methods offer a promising technology.
  • The framework effectively handles both hyper-elastodynamics and fluid-solid coupled problems.
  • This approach is particularly beneficial for biomedical and engineering applications requiring incompressible models.