Goal-Oriented Mesh Adaptivity for Fluid-Structure Interaction with Application to Heart-Valve Settings

Open access

Goal-Oriented Mesh Adaptivity for Fluid-Structure Interaction with Application to Heart-Valve Settings

We apply a fluid-structure interaction method to simulate prototypical dynamics of the aortic heart-valve. Our method of choice is based on a monolithic coupling scheme for fluid-structure interactions in which the fluid equations are rewritten in the ‘arbitrary Lagrangian Eulerian’ (ALE) framework. To prevent the backflow of structure waves because of their hyperbolic nature, a damped structure equation is solved on an artificial layer that is used to prolongate the computational domain. The increased computational cost in the presence of the artificial layer is resolved by using local mesh adaption. In particular, heuristic mesh refinement techniques are compared to rigorous goal-oriented mesh adaption with the dual weighted residual (DWR) method. A version of this method is developed for stationary settings. For the nonstationary test cases the indicators are obtained by a heuristic error estimator, which has a good performance for the measurement of wall stresses. The results for prototypical problems demonstrate that heart-valve dynamics can be treated with our proposed concepts and that the DWR method performs best with respect to a certain target functional.

Quarteroni A.: What mathematics can do for the simulation of blood circulation. MOX Report, 2006.

Figueroa C. A., Vignon-Clementel I. E., Jansen K. E., Hughes T. J. R., Taylor C. A.: A coupled momentum method for modeling blood ow in three-dimensional deformable arteries. Comput. Methods Appl. Mech. Engrg., 2006, Vol. 195, pp. 5685-5706.

Nobile F., Vergara C.: An Effective Fluid-Structure Interaction Formulation for Vascular Dynamics by Generalized Robin Conditions. SIAM J. Sci. Comput., 2008, Vol. 30, No. 2, pp. 731-763.

Janela J., Moura A., Sequeira A.: Absorbing boundary conditions for a 3D non-Newtonian fluid-structure interaction model for blood flow in arteries. Int. J. Engrg. Sci., 2010.

Formaggia L., Quarteroni A., Veneziani A.: Cardiovascular Mathematics: Modeling and simulation of the circulatory system, Springer-Verlag, Italia, Milano, 2009.

Formaggia L., Veneziani A., Vergara Ch.: Flow rate boundary problems for an incompressible fluid in deformable domains: formulations and solution methods. Comput. Meth. Appl. Mech. Engrg., 2010, Vol. 199, pp. 677-688.

Wick T.: An energy absorbing layer for the structure outflow boundary for fluid-structure interactions applied to valve dynamics, in review, 2011.

Wick T.: Adaptive finite element simulation of fluid-structure interaction with application to heart valve dynamics, PhD thesis, 2011.

Jianhai Z., Dapeng C., Shengquan Z.: ALE finite element analysis of the opening and closing process of the artificial mechanical valve. Applied Math. Mech., 2006, Vol. 17, No. 5, pp. 403-412.

Le Tallec P., Mouro J.: Fluid structure interaction with large structural displacements. Comput. Meth. Appl. Mech. Engrg., 2001, Vol. 190, pp. 3039-3067.

Peskin C.: The immersed boundary method. Acta Numerica, Cambridge University Press, 2002, pp. 1-39.

Diniz Dos Santos N., Gerbeau J.-F., Bourgat J. F.: A partitioned fluid-structure algorithm for elastic thin valves with contact. Comp. Meth. Appl. Mech. Engng., 2008, Vol. 197, No. 19-20, pp. 1750-1761.

Vierendeels J., Dumont K., Verdonck P. R.: A partitioned strongly coupled fluid-structure interaction method to model heart valve dynamics. J. Comp. Appl. Math., 2008.

Baaijens F. P. T.: A fictitious domain/mortar element method for fluid-structure interaction, Int. J. Num. Methods Fluids, 2001, Vol. 35, pp. 743-761.

Causin P., Gerbeau J.-F., Nobile F.: Added-mass effect in the design of partitioned algorithms for fluid-structure problems. Comput. Methods Appl. Mech. Engrg., 2005, Vol. 194, pp. 4506-4527.

An optimal control approach to error control and mesh adaptation in finite element methods. Acta Numerica 2001, (A. Iserles, ed.), Cambridge University Press, 2001.

Dunne T.: An Eulerian approach to fluid-structure interaction and goal-oriented mesh adaption, Int. J. Numer. Methods in Fluids, 2006, Vol. 51, pp. 1017-1039.

Numerical Simulation of Fluid-Structure Interaction Based on Monolithic Variational Formulations. Numerical Fluid Structure Interaction, G. P. Galdi, R. Rannacher et. al, Springer, 2010.

Richter T.: Goal oriented error estimation for fluid-structure interaction problems, Computer Methods in Applied Mechanics and Engineering 223-224, pp. 38-42, 2012.

van der Zee K. G., van Brummelen E. H., de Borst R.: Goal-oriented error estimation for Stokes flow interacting with a flexible channel. Int. J. Numer. Meth. Fluids, 2008, Vol. 56, pp. 1551-1557.

Bathe K.-J., Grätsch T.: Goal-oriented error estimation in the analysis of fluid flows with structural interactions. Comp. Methods Appl. Mech. Engrg., 2006, Vol. 195, pp. 5673-5684.

Fung Y. C.: Biodynamics: Circulation, first ed., Springer-Verlag, 1984.

Holzapfel G. A.: Nonlinear Solid Mechanics: A continuum approach for engineering, John Wiley and Sons, LTD, 2000.

Holzapfel G. A., Ogden R. W.: Mechanics of Biological Tissue, Springer, Heidelberg, 2006.

Humphrey J. D.: Cardiovascular Solid Mechanics: Cells, Tissues, and Organs, Springer, New York, 2002.

Wloka J.: Partielle Differentialgleichungen, B. G. Teubner, Stuttgart, 1987.

Berenger J.-P.: A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys., 1994, Vol. 114, No. 185.

Formaggia L., Moura A., Nobile F.: On the stability of the coupling of 3D and 1D fluid-structure interaction models for blood flow simulations. Technical Report at MOX, 2006, Vol. 94.

Vignon-Clementel I. E., Figueroa C. A., Jansen K. E., Taylor Ch.A.: Outflow boundary conditions for three-dimensional finite element modeling of blood flow and pressure in arteries. Comput. Meth. Appl. Mech. Engrg., 2006, Vol. 195, pp. 3776-3796.

Stein K., Tezduyar T., Benney R.: Mesh moving techniques for fluid-structure interactions with large displacements, J. Appl. Math., 2003, Vol. 70, pp. 58-63.

Tezduyar T. E., Behr M., Mittal S., Johnson A. A.: Computation of Unsteady Incompressible Flows With the Finite Element MethodsSpace- Time Formulations, Iterative Strategies and Massively Parallel Implementations. ASME: New Methods in Transient Analysis, 1992, Vol. 143, pp. 7-24.

Wick T.: Fluid-Structure Interactions using Different Mesh Motion Techniques, Comput. Struct., 2011, Vol. 89, pp. 1456-1467.

Brooks A. N., Hughes T. J. R.: Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 1982, Vol. 32, No. 1-3, pp. 199-259.

Wall Wolfgang A.: Fluid-Struktur-Interaktion mit stabilisierten Finiten Elementen, PhD Thesis, University of Stuttgart, 1999.

Braack M., Burman E., John V., Lube G.: Stabilized finite element methods for the generalized Oseen equations. Comput. Methods Appl. Mech. Engrg., 2007, Vol. 196, No. 4-6, pp. 853-866.

Besier M.: Adaptive Finite Element methods for computing nonstationary incompressible Flows, University of Heidelberg, 2009.

Besier M., Wollner W.: On the dependence of the pressure on the time step in incompressible flow simulations on varying spatial meshes, Int. J. Num. Methods in Fluids, 2011.

Becker R., Rannacher R.: A feed-back approach to error control in finite element methods: basic analysis and examples. East-West J. Numer. Math., 1996, Vol. 4, pp. 237-264.

Wick T.: Adaptive Finite Elements for Fluid-Structure Interactions on a Prolongated Domain: Applied to Valve Simulations, Proc. Comput. Methods Mech., Warsaw in Poland, May 9-12, 2011.

Zienkiewicz O. C., Zhu J. Z.: The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique, Int. J. of Numer. Methods Engrg., 1992, Vol. 33, pp. 1331-1364.

Zienkiewicz O. C., Zhu J. Z.: The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity, Int. J. of Numer. Methods Engrg., 1992, Vol. 33, pp. 1365-1382.

Bangerth W., Rannacher R.: Adaptive Finite Element Methods for Differential Equations. Lectures in Mathematics, ETH Zuerich, Birkhäuser Verlag, 2003.

Bangerth W., Hartmann R., Kanschat G.: Differential Equations Analysis Library. Technical Reference, 2010. http://www.dealii.org

Wick T.: Solving Monolithic Fluid-Structure Interaction Problems in Arbitrary Lagrangian Eulerian Coordinates with the deal. II Library. IWR-Preprint, 2011, in review for publication in Archive of Numerical Software.

Heywood J., Rannacher R., Turek S.: Artificial boundaries and flux and pressure conditions for the incompressible Navier-Stokes equations. Int. J. Num. Meth. Fluids., 1996, Vol. 22, pp. 325-353.

Archive of Mechanical Engineering

The Journal of Committee on Machine Building of Polish Academy of Sciences

Journal Information


CiteScore 2016: 0.44

SCImago Journal Rank (SJR) 2016: 0.162
Source Normalized Impact per Paper (SNIP) 2016: 0.459

Metrics

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 214 156 8
PDF Downloads 88 73 4