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Partitioned Fluid–Solid Coupling for Cardiovascular Blood Flow

Left-Ventricular Fluid Mechanics

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Abstract

We present a 3D code-coupling approach which has been specialized towards cardiovascular blood flow. For the first time, the prescribed geometry movement of the cardiovascular flow model KaHMo (Karlsruhe Heart Model) has been replaced by a myocardial composite model. Deformation is driven by fluid forces and myocardial response, i.e., both its contractile and constitutive behavior. Whereas the arbitrary Lagrangian–Eulerian formulation (ALE) of the Navier–Stokes equations is discretized by finite volumes (FVM), the solid mechanical finite elasticity equations are discretized by a finite element (FEM) approach. Taking advantage of specialized numerical solution strategies for non-matching fluid and solid domain meshes, an iterative data-exchange guarantees the interface equilibrium of the underlying governing equations. The focus of this work is on left-ventricular fluid–structure interaction based on patient-specific magnetic resonance imaging datasets. Multi-physical phenomena are described by temporal visualization and characteristic FSI numbers. The results gained show flow patterns that are in good agreement with previous observations. A deeper understanding of cavity deformation, blood flow, and their vital interaction can help to improve surgical treatment and clinical therapy planning.

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Acknowledgments

The authors want to express their sincere thanks to all the people who have contributed to and worked on the KaHMo project during recent years. Especially to Dipl.-Ing. Stefan Höttges for providing KaHMo MRT reference data as well as Dr.-Ing. Ralf Kröger (ANSYS Germany GmbH) for providing support for the “Implicit Coupling for KaHMo FSI” Fluent engine.

Author information

Authors and Affiliations

  1. Institute for Fluid Mechanics, Karlsruhe Institute of Technology (KIT), Kaiserstrasse 10, 76131, Karlsruhe, Germany

    Sebastian Krittian & Herbert Oertel

  2. Computing Laboratory, University of Oxford, Wolfson Building, Parks Road, Oxford, OX1 3QD, UK

    Sebastian Krittian

  3. Department of Mechanical Engineering, Fluid Mechanics, University of Wuppertal, Gauss-Strasse 20, 42119, Wuppertal, Germany

    Uwe Janoske

  4. Institute of Engineering Mechanics, Karlsruhe Institute of Technology (KIT), Kaiserstrasse 10, 76131, Karlsruhe, Germany

    Thomas Böhlke

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  1. Sebastian Krittian
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  2. Uwe Janoske
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  4. Thomas Böhlke
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Corresponding author

Correspondence to Sebastian Krittian.

Additional information

Associate Editor James B. Bassingthwaighte oversaw the review of this article.

Appendices

Appendix A: Coupling Flow Chart

Figure 14 shows in more detail the flow chart of the coupling steps performed at the back-end of the coupling graphical interface. The major components are:

  • A: Coupling settings

  • B: Time-step settings

  • C: Time-step-looping

  • D: Energy analysis

  • E: Relaxation engine

Figure 14
figure 14

Flow chart of “Implicit Coupling for KaHMo FSI”: explicit algorithm (white) and implicit extension (gray)

Full size image

Whereas white boxes describe the former explicit MpCCI coupling, the grey boxes are implemented for coupling implicitly.

Appendix B: Core Coupling Engine

The coupling algorithm in time-step j and looping k carries out the following steps:

  • A: Coupling settings

    1. 1.

      Initialize fluid domain.

    2. 2.

      Initialize coupling procedure.

  • B: Time-step settings

    1. 3.

      Store fluid domain and mesh moving parameters.

    2. 4.

      Store interface position and stress.

    3. 5.

      Store fluid solution for time-step j and looping k.

  • C: Time-step-looping

    1. 6.

      Perform load manipulation:

      $$ \begin{aligned} &{\it If\,\,load\,\,relaxation\,\,is\,\,chosen}\hbox{:} \\ &{{\mathbf{t}}}^j_{k,solid}=\omega_k\cdot {{\mathbf{t}}}^j_{k,fluid}+(1-\omega_k)\cdot {{\mathbf{t}}}^j_{k-1,solid}\\ &{\it If\,\,position\,\,relaxation\,\,is\,\,chosen\hbox{:}}\\ &{{\mathbf{t}}}^j_{k,solid}={{\mathbf{t}}}^j_{k,fluid}. \end{aligned} $$

      Send load t j k,solid to Abaqus via MpCCI.

    2. 7.

      Calculate interface deformation x j k,solid.

    3. 8.

      Return deformation x j k,solid to Fluent via MpCCI.

    4. 9.

      Perform position manipulation:

      $$ \begin{aligned} &{\it If\,load\,\,relaxation\,\,is\,\,chosen}\hbox{:} \\ &{{\mathbf{x}}}^j_{k,fluid}={{\mathbf{x}}}^j_{k,solid}\\ &{\it If\,\,position\,\,relaxation\,\,is\,\,chosen\hbox{:}}\\ &{{\mathbf{x}}}^j_{k,fluid}=\omega_k\cdot {{\mathbf{x}}}^j_{k,solid}+(1-\omega_k)\cdot {{\mathbf{x}}}^j_{k-1,fluid}. \end{aligned} $$

      Reset spatial and flow field from steps 3. & 5. and perform mesh movement to x j k,fluid.

    5. 10.

      Solve flow field for time-step j → j + 1.

    6. 11.

      If convergence satisfied go to 3, otherwise go to 6.

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Krittian, S., Janoske, U., Oertel, H. et al. Partitioned Fluid–Solid Coupling for Cardiovascular Blood Flow. Ann Biomed Eng 38, 1426–1441 (2010). https://doi.org/10.1007/s10439-009-9895-7

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  • DOI: https://doi.org/10.1007/s10439-009-9895-7

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