A contact formulation based on a volumetric potential: Application to isogeometric simulations of atrioventricular valves

Abstract

This work formulates frictionless contact between solid bodies in terms of a repulsive potential energy term and illustrates how numerical integration of the resulting forces is computationally similar to the "pinball algorithm" proposed and studied by Belytschko and collaborators in the 1990s. We thereby arrive at a numerical approach that has both the theoretical advantages of a potential-based formulation and the algorithmic simplicity, computational efficiency, and geometrical versatility of pinball contact. The singular nature of the contact potential requires a specialized nonlinear solver and an adaptive time stepping scheme to ensure reliable convergence of implicit dynamic calculations. We illustrate the effectiveness of this numerical method by simulating several benchmark problems and the structural mechanics of the right atrioventricular (tricuspid) heart valve. Atrioventricular valve closure involves contact between every combination of shell surfaces, edges of shells, and cables, but our formulation handles all contact scenarios in a unified manner. We take advantage of this versatility to demonstrate the effects of chordal rupture on tricuspid valve coaptation behavior.

Isogeometric hyperelastic shell analysis with out-of-plane deformation mapping

“…Extension to general hyperelastic material can be found in [4]. The formulation has been successfully used in computation of a good number of challenging problems, including wind-turbine fluid-structure interaction (FSI) [3,[5][6][7][8][9], bioinspired flapping-wing aerodynamics [10], bioprosthetic heart valves [11][12][13][14][15], fatigue and damage [16][17][18][19][20][21], and design [22,23].…”

Peridynamic Modeling of Frictional Contact

“…Section 2.1 recapitulates the frictionless contact model of [13], which is essentially equivalent to the "short-range forces" widely used in peridynamic computations [24,Section 3.9], but interpreted at the continuous level, rather than as an algorithmic device to prevent interpenetration of balls surrounding discrete nodes. Section 2.2 then discusses ways to extend the frictionless formulation to include (regularized) Coulomb friction, with Sections 2.2.1 and 2.2.2 reviewing peridynamic frictional contact models from the gray literature and extending them to address various shortcomings.…”

“…), then extend the state of the art with enhancements to the existing models. A secondary purpose is to demonstrate that peridynamic contact is also useful in conjunction with classical local models of solid mechanics, and can be straightforwardly incorporated into finite element and isogeometric analysis, as previously illustrated by [13] in the frictionless setting (albeit without explicitly stating the connection to peridynamics). Related nonlocal frictionless and adhesive contact formulations were studied earlier by Sauer and various collaborators [27][28][29][30] and implemented in the finite element setting.…”

“…That work was based on coarse-graining of inter-atomic potentials; peridynamics may be viewed through a similar lens [26,31]. With a sufficiently liberal interpretation of terminology, one might even consider the "pinball algorithm" [5][6][7] to be a pioneering instance of peridynamic contact in finite element analysis, and its connection to nonlocal modeling is discussed in [13,Section 3.2].…”

Biomechanics of the tricuspid annulus: A review of the annulus' in vivo dynamics with emphasis on ovine data

“…However, given the uncertainty associated with combining data from various sources and species, the quantitative accuracy of their data remains to be confirmed. Kamensky et al, as a numerical example of a novel contact algorithm, performed a similar simulation as Stevanella et al Here the annular geometry was based on micro CT data on isolated porcine hearts. To the best of our knowledge, the annulus was treated as static in their simulations.…”