An energy‐stable mixed formulation for isogeometric analysis of incompressible hyperelastodynamics

Abstract

Summary We develop a mixed formulation for incompressible hyperelastodynamics based on a continuum modeling framework recently developed in the work of Liu and Marsden and smooth generalizations of the Taylor‐Hood element based on nonuniform rational B‐splines (NURBSs). This continuum formulation draws a link between computational fluid dynamics and computational solid dynamics. This link inspires an energy stability estimate for the spatial discretization, which favorably distinguishes the formulation from the conventional mixed formulations for finite elasticity. The inf‐sup condition is utilized to provide a bound for the pressure field. The generalized‐α method is applied for temporal discretization, and a nested block preconditioner is invoked for the solution procedure. The inf‐sup stability for different pairs of NURBS elements is elucidated through numerical assessment. The convergence rate of the proposed formulation with various combinations of mixed elements is examined by the manufactured solution method. The numerical scheme is also examined under compressive and tensile loads for isotropic and anisotropic hyperelastic materials. Finally, a suite of dynamic problems is numerically studied to corroborate the stability and conservation properties.

Numerical investigation of abdominal aortic aneurysm hemodynamics using the reduced unified continuum formulation for vascular fluid-structure interaction

“…Under the above assumption, we do not distinguish between the reference and current frames, allowing the total and partial time derivatives in (2.1)-(2.3) to be used interchangeably. Furthermore, β θ (p s ) = 1/κ s , where κ s is the solid bulk modulus, and (2.2) can be integrated to yield p s = −κ s ∇ • u s [19].…”

“…For improved representation of the Schur complement, we apply the nested block preconditioner [25] that algorithmically defines the action of the Schur complement on a vector in a matrix-free fashion. We have demonstrated enhanced robustness and scalability of our block preconditioner as compared to alternative preconditioners in applications spanning hyperelasticity, viscous fluids, and FSI [19].…”

A continuum and computational framework for viscoelastodynamics: finite deformation linear models

“…For fully incompressible materials, it can be shown that the volumetric part of the free energy adopts the form G ∞ vol = P/ρ 0 [55], which leads to ρ = ρ 0 . In the formulation, we adopt a modified constitutive relation for the density, that is ρ(J) = ρ 0 /J [60]. Apparently, at the continuum level, this relation is equivalent to ρ(J) = ρ 0 , as the divergence-free condition for the velocity guarantees J = 1.…”

“…The aforementioned Gibbs free energy can be viewed as the thermodynamic explanation of the effectiveness of the pressure primitive variables. A theory for finite elasticity based on the Gibbs free energy has been established [55]; the saddle-point nature has been exploited to design preconditioners [58,59]; it can also be shown to enjoy a provably nonlinearly stable semi-discrete formulation using inf-sup stable elements [60]; it can be conveniently utilized to construct a strongly-coupled fluid-structure interaction formulation [55,61]. In this work, we follow our prior approach and use the Gibbs free energy to construct a mixed formulation for viscoelastodynamics.…”

“…the energy stability property. The resulting mixed formulation necessitates inf-sup stable element pairs, and we adopt a suite of smooth generalization of the Taylor-Hood element by NURBS that has been numerically demonstrated to be stable [60,62,63,64]. In particular, the same set of basis functions is utilized to represent the geometry and to construct the spaces for the displacement and velocity fields, which makes it fitted into the paradigm of NURBS-based isogeometric analysis [65,66].…”