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Immersogeometric Analysis of Bioprosthetic Heart Valves, Using the Dynamic Augmented Lagrangian Method

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Frontiers in Computational Fluid-Structure Interaction and Flow Simulation

Abstract

In the mid-2010s, we began applying a combination of isogeometric analysis and immersed boundary methods to the problem of bioprosthetic heart valve (BHV) fluid–structure interaction (FSI). This chapter reviews how our research on BHV FSI (1) crystallized the emerging concept of immersogeometric analysis, (2) introduced a new semi-implicit numerical method for weakly enforcing constraints in time dependent problems, which we refer to as the dynamic augmented Lagrangian approach, and (3) clarified the important role of mass conservation in immersed FSI analysis. We illustrate these contributions with selected numerical results and discuss future improvements to, and applications of, the computational FSI techniques we have developed.

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Notes

  1. 1.

    In practice, immersogeometric methods must frequently approximate integrals over the domain geometry, which may be considered a type of geometrical approximation [76, Sections 4.3 and 4.4], but this is conceptually distinct from the direct alteration of domain geometry that occurs in traditional mesh generation.

  2. 2.

    The word “immersogeometric” was originally coined in 2014 by T. J. R. Hughes, while traveling in Italy; it is derived from the Italian word immerso, meaning “immersed.”

  3. 3.

    We use of the term “VMS” in this chapter to refer to the specific VMS formulation explained in Sect. 3.1.1, applied to equal-order pressure–velocity discretizations. Our choice of terminology should not be taken to mean that the concept of VMS analysis is incompatible with div-conforming B-splines, which is demonstrably [82] not true.

  4. 4.

    For readers unfamiliar with the construction and basic properties of B-splines, a comprehensive explanation can be found in [88].

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Acknowledgements

The work summarized in this chapter was supported by the National Heart, Lung, and Blood Institute of the National Institutes of Health (NIH/NHLBI) under award number R01HL129077. We thank the Texas Advanced Computing Center (TACC) at the University of Texas at Austin for providing HPC resources that have contributed to the research results reported in this chapter.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Iowa State University, Ames, IA, USA

    Ming-Chen Hsu

  2. Department of Structural Engineering, University of California, San Diego, La Jolla, CA, USA

    David Kamensky

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Correspondence to Ming-Chen Hsu .

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  1. Mechanical Engineering, Rice University, Houston, TX, USA

    Tayfun E. Tezduyar

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Hsu, MC., Kamensky, D. (2018). Immersogeometric Analysis of Bioprosthetic Heart Valves, Using the Dynamic Augmented Lagrangian Method. In: Tezduyar, T. (eds) Frontiers in Computational Fluid-Structure Interaction and Flow Simulation. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-96469-0_5

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