Stability and Conservation Properties of Collocated Constraints in Immersogeometric Fluid-Thin Structure Interaction Analysis (Communications in Computational Physics, 2020)
The purpose of this study is to enhance the stability properties of our recently-developednumerical method [D. Kamensky, M.-C. Hsu, D. Schillinger, J. A. Evans, A.Aggarwal, Y. Bazilevs, M. S. Sacks, T. J. R. Hughes, "An immersogeometric variationalframework for fluid-structure interaction: Application to bioprosthetic heart valves",Comput. Methods Appl. Mech. Engrg., 284 (2015) 1005–1053] for immersing spline-basedrepresentations of shell structures into unsteady viscous incompressible flows.In the cited work, we formulated the fluid-structure interaction (FSI) problem usingan augmented Lagrangian to enforce kinematic constraints. We discretized this Lagrangianas a set of collocated constraints, at quadrature points of the surface integrationrule for the immersed interface. Because the density of quadrature points is notcontrolled relative to the fluid discretization, the resulting semi-discrete problem maybe over-constrained. Semi-implicit time integration circumvents this difficulty in thefully-discrete scheme. If this time-stepping algorithm is applied to fluid-structure systemsthat approach steady solutions, though, we find that spatially-oscillating modesof the Lagrange multiplier field can grow over time. In the present work, we stabilizethe semi-implicit integration scheme to prevent potential divergence of the multiplierfield as time goes to infinity. This stabilized time integration may also be applied inpseudo-time within each time step, giving rise to a fully implicit solution method. Wediscuss the theoretical implications of this stabilization scheme for several simplifiedmodel problems, then demonstrate its practical efficacy through numerical examples.