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Stability and Conservation Properties of Collocated Constraints in Immersogeometric Fluid-Thin Structure Interaction Analysis

Part of:Coupling of solid mechanics with other effectsPartial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  15 October 2015

David Kamensky,
John A. Evans and
Ming-Chen Hsu
David Kamensky
Affiliation:
Center for Cardiovascular Simulation, Institute for Computational Engineering and Sciences, The University of Texas at Austin, 201 East 24th St, Stop C0200, Austin, TX 78712, USA
John A. Evans
Affiliation:
Center for Cardiovascular Simulation, Institute for Computational Engineering and Sciences, The University of Texas at Austin, 201 East 24th St, Stop C0200, Austin, TX 78712, USA
Ming-Chen Hsu*
Affiliation:
Department of Mechanical Engineering, Iowa State University, 2025 Black Engineering, Ames, IA 50011, USA
*
*Corresponding author. Email addresses: kamensky@ices.utexas.edu (D. Kamensky), john.a.evans@colorado.edu (J. A. Evans), jmchsu@iastate.edu (M.-C. Hsu)

Abstract

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The purpose of this study is to enhance the stability properties of our recently-developed numerical method [D. Kamensky, M.-C. Hsu, D. Schillinger, J.A. Evans, A. Aggarwal, Y. Bazilevs, M.S. Sacks, T.J.R. Hughes, “An immersogeometric variational framework for fluid-structure interaction: Application to bioprosthetic heart valves”, Comput. Methods Appl. Mech. Engrg., 284 (2015) 1005–1053] for immersing spline-based representations of shell structures into unsteady viscous incompressible flows. In the cited work, we formulated the fluid-structure interaction (FSI) problem using an augmented Lagrangian to enforce kinematic constraints. We discretized this Lagrangian as a set of collocated constraints, at quadrature points of the surface integration rule for the immersed interface. Because the density of quadrature points is not controlled relative to the fluid discretization, the resulting semi-discrete problem may be over-constrained. Semi-implicit time integration circumvents this difficulty in the fully-discrete scheme. If this time-stepping algorithm is applied to fluid-structure systems that approach steady solutions, though, we find that spatially-oscillating modes of the Lagrange multiplier field can grow over time. In the present work, we stabilize the semi-implicit integration scheme to prevent potential divergence of the multiplier field as time goes to infinity. This stabilized time integration may also be applied in pseudo-time within each time step, giving rise to a fully implicit solution method. We discuss the theoretical implications of this stabilization scheme for several simplified model problems, then demonstrate its practical efficacy through numerical examples.

Keywords

Immersogeometric analysisfluid-structure interactionisogeometric analysiscollocationstabilized methodsaugmented Lagrangian

MSC classification

Secondary:65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods65M85: Fictitious domain methods74F10: Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)

Information

Type
Research Article
Information
Communications in Computational Physics , Volume 18 , Issue 4 , October 2015, pp. 1147 - 1180
Copyright
Copyright © Global-Science Press 2015 

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