[Paper Review] Stabilization approaches for the hyperelastic immersed boundary method for problems of large-deformation incompressible elasticity

Ben Vadala-Roth, Shashank Acharya+3 more|arXiv (Cornell University)|Nov 15, 2018

Lattice Boltzmann Simulation Studies45 references4 citations

TL;DR

This paper proposes a volumetric stabilization approach for the hyperelastic immersed boundary method to improve volume conservation in large-deformation incompressible elasticity. By decomposing the strain energy into isochoric and dilatational components and augmenting the discrete Lagrange multiplier for incompressibility, the method achieves accuracy comparable to stabilized finite element methods with minimal computational cost, significantly reducing volume errors in quasi-static and dynamic fluid-structure interaction benchmarks.

ABSTRACT

The immersed boundary method is a mathematical framework for modeling fluid-structure interaction. This formulation describes the momentum, viscosity, and incompressibility of the fluid-structure system in Eulerian form, and it uses Lagrangian coordinates to describe the structural deformations, stresses, and resultant forces. Integral transforms with Dirac delta function kernels connect the Eulerian and Lagrangian frames. The fluid and the structure are both typically treated as incompressible materials. Upon discretization, however, the incompressibility of the structure is only maintained approximately. To obtain an immersed method for incompressible hyperelastic structures that is robust under large structural deformations, we introduce a volumetric energy in the solid region that stabilizes the formulation and improves the accuracy of the numerical scheme. This formulation augments the discrete Lagrange multiplier for the incompressibility constraint, thereby improving the original method's accuracy. This volumetric energy is incorporated by decomposing the strain energy into isochoric and dilatational components, as in standard solid mechanics formulations of nearly incompressible elasticity. We study the performance of the stabilized method using several quasi-static solid mechanics benchmarks, a dynamic fluid-structure interaction benchmark, and a detailed three-dimensional model of esophageal transport. The accuracy achieved by the stabilized immersed formulation is comparable to that of a stabilized finite element method for incompressible elasticity using similar numbers of structural degrees of freedom.

Motivation & Objective

• To address volume conservation errors in the immersed boundary method when simulating large-deformation incompressible elasticity.
• To develop a stabilization technique that maintains incompressibility in the solid region despite discretization-induced errors.
• To improve numerical accuracy and convergence without compromising stability or increasing computational cost.
• To evaluate the method’s performance across quasi-static and dynamic fluid-structure interaction benchmarks, including a 3D esophageal transport model.
• To establish a robust default formulation for hyperelastic IB methods using volumetric penalization and modified invariants.

Proposed method

• Introduce a volumetric energy term in the solid region to stabilize the incompressibility constraint, decomposing the strain energy into isochoric and dilatational components.
• Augment the discrete Lagrange multiplier for incompressibility using the volumetric energy, improving enforcement of the constraint in the Eulerian frame.
• Use modified invariants (e.g., $\bar{I}_4$, $\bar{I}_5$) in the elastic energy functional to better capture material response under large deformations.
• Apply the stabilization only in the solid region, preserving the original IB framework’s structure and coupling operators.
• Implement the method via a deviatoric projection formulation, ensuring consistency with nearly incompressible elasticity theory.
• Use regularized Dirac delta functions for coupling between Eulerian and Lagrangian domains, maintaining smooth velocity transfer.

Experimental results

Research questions

• RQ1 How does volumetric stabilization improve volume conservation in hyperelastic immersed boundary simulations under large deformations?
• RQ2 What is the impact of using modified invariants versus unmodified invariants on numerical accuracy and deformation quality?
• RQ3 Can the stabilized method achieve accuracy comparable to stabilized finite element methods with similar degrees of freedom?
• RQ4 How does the method perform in dynamic fluid-structure interaction problems, such as the elastic band benchmark and esophageal transport?
• RQ5 Does the stabilization technique introduce numerical instability or significant computational overhead?

Key findings

• The stabilized method reduces volume errors by up to 59% in the torsion test compared to the unstabilized case with unmodified invariants.
• Volume conservation is significantly improved across all benchmarks, including the anisotropic Cook’s membrane and 3D esophageal transport model.
• The method achieves accuracy comparable to stabilized finite element methods using the same number of structural degrees of freedom.
• Pointwise volume conservation is maintained with only a small reduction, and element inversion is prevented in the esophageal transport simulation.
• The method shows no adverse effect on stability and incurs negligible computational cost, making it practical for large-scale simulations.
• Using modified invariants in combination with volumetric penalization yields superior results compared to unmodified invariants with zero penalization.