Stabilization approaches for the hyperelastic immersed boundary method for problems of large-deformation incompressible elasticity

Abstract

The immersed boundary method is a model of fluid-structure interaction that describes a structure, or a collection of structures, immersed in fluid. This formulation uses Eulerian coordinates for the momentum, incompressibility, and viscosity of the fluidstructure system and Lagrangian coordinates for the structural deformations and resultant forces. Integral transforms with delta function kernels connect the two frames. In the continuum equations, both the fluid and the structure are typically modeled as incompressible. Upon discretization, however, the structure's incompressibility is only approximately maintained. To obtain a robust method 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 volumetric energy is added by decomposing the strain energy into isochoric and dilatational components, as in standard solid mechanics formulations. Using standard quasi-static solid mechanics benchmarks, we study the performance of the proposed stabilized method with various choices of the finite element basis employed for the structural discretization.

A sharp interface method for an immersed viscoelastic solid

“…In the case of a smooth loading force, extrapolated errors for both the pressure and velocity appear to be converging. This example highlights the method's flexibility in dealing with additional surface forces and volumetric energies with modestly large stabilization parameters [17]. In the final example, we demonstrated the method works in three dimensions with transient flow dynamics and a stress function which depends on a fiber vector field.…”

“…This problem is more challenging because it contains discontinuities in the surface forces that generate solid displacements on the the top and bottom of the block. Further, the material model for the compressed block includes a volumetric energy useful for penalizing compressible deformations [17], but this term prominently contributes to discontinuities at the fluid-structure interface that present additional numerical challenges. The fourth example, inspired by the work of McQueen and Peskin [9], applies the method to an actively contracting thick torus.…”

“…for shear modulus µ e and bulk modulus λ = λ(ν), taken to be a function of a numerical Poisson ratio ν which is detailed in [17]. The dilational energy W dil vanishes at the continuous level because J = 1.…”

“…The dilational energy W dil vanishes at the continuous level because J = 1. In the discretized equations, this term is not necessarily equal to zero, and it can be interpreted as a stabilization that enforces discrete incompressibility [17]. The first Piola-Kirchoff stress takes the form:…”

“…As suggested in Ref. [17], we consider values for the numerical Poisson ratio ν = −1, 0, 0.4. In this test, we include surface force densities on the boundary of the block to weakly impose Dirichlet boundary conditions for the displacement.…”