A partitioned Newton method for the interaction of a fluid and a 3D shell structure
DOI:
https://doi.org/10.13052/EJCM.19.479-512Keywords:
fluid-structure interaction, 3D shell finite elements, domain decomposition, partitioned schemes, Newton algorithmAbstract
We propose a new fluid-structure algorithm based on a domain decomposition paradigm. The method is based on the principle “linearize first, then decompose” whereas the usual schemes are generally “nonlinear in subdomains”. The proposed approach is more attractive when the complexity of the structure is high, which is the case with the structural model used in this study (nonlinear 3D shell). Another contribution of this paper is to investigate the use of a Neumann-Neumann preconditioner for the linearized problem. In particular, it is shown that when this preconditioner is adequately balanced, it tends to the Dirichlet-Neumann preconditioner because of the heterogeneity of the fluid-structure problem.
Downloads
References
Badia S., Nobile F., Vergara C., “ Fluid-structure partitioned procedures based on Robin transmission
conditions”, J. Comp. Phys., vol. 227, p. 7027-7051, 2008a.
Badia S., Nobile F., Vergara C., “ Robin-Robin preconditioned Krylov methods for fluidstructure
interaction problems”, Comput. Methods Appl. Mech. Engrg., vol. 198, n° 33-36,
p. 2768-2784, 2009.
Badia S., Quaini A., Quarteroni A., “ Modular vs. non-modular preconditioners for fluidstructure
systems with large added-mass effect”, Comput. Methods Appl. Mech. Engrg.,
vol. 197, n° 49-50, p. 4216-4232, 2008b.
Bathe K., Zhang H., “ Finite element developments for general fluid flows with structural interactions”,
Int. J. Num. Meth. Engng., 2004.
Batina J., “ Unsteady Euler airfoil solutions using unstructured dynamic meshes”, AIAA J., vol.
, n° 8, p. 1381-1388, 1990.
Bazilevs Y., Calo V., Hughes T., Zhang Y., Isogeometric Fluid-Structure Interaction: Theory,
Algorithms, and Computations, Technical Report n° 08-16, ICES, 2008.
Bazilevs Y., Calo V., Zhang Y., Hughes T., “ Isogeometric fluid-structure interaction analysis
with applications to arterial blood flow”, Comput. Mech., vol. 38, p. 310-322, 2006.
Burman E., Fernández M., “ Stabilization of explicit coupling in fluid-structure interaction involving
fluid incompressibility”, Comput. Methods Appl. Mech. Engrg., vol. 198, n° 5–8,
p. 766-784, 2009.
Caillerie D., Mourad A., A. R., “ Cell-to-muscle homogenization. Application to a constitutive
law for the myocardium”, Math. Model. Num. Anal., vol. 37, p. 681-698, 2003.
Causin P., Gerbeau J.-F., Nobile F., “ Added-mass effect in the design of partitioned algorithms
for fluid-structure problems”, Comp. Meth. Appl. Mech. Engng., vol. 194, n° 42–44, p. 4506-
, 2005.
Chapelle D., Bathe K., The Finite Element Analysis of Shells - Fundamentals, Springer Verlag,
a.
Chapelle D., Ferent A., “ Modeling of the inclusion of a reinforcing sheet within a 3D medium”,
Math. Models Methods Appl. Sci., vol. 13, n° 4, p. 573-595, 2003b.
Chapelle D., Ferent A., Bathe K., “ 3D-shell elements and their underlying mathematical
model”, Math. Models Methods Appl. Sci., vol. 14, n° 1, p. 105-142, 2004.
Chapelle D., Ferent A., Le Tallec P., “ The treatment of "pinching locking" in 3D-shell elements”,
M2AN Math. Model. Numer. Anal., vol. 37, n° 1, p. 143-158, 2003c.
Deparis S., Numerical Analysis of Axisymmetric Flows and Methods for Fluid-Structure Interaction
Arising in Blood Flow Simulation, PhD thesis, EPFL, Switzerland, 2004.
Deparis S., Discacciati M., Fourestey G., Quarteroni A., “ Fluid-structure algorithms based
on Steklov-Poincaré operators”, Comput. Methods Appl. Mech. Engrg., vol. 195, n° 41-43,
p. 5797-5812, 2006.
Deparis S., Discacciati M., Quarteroni A., “ A domain decomposition framework for fluidstructure
interaction problems”, Proceedings of the Third International Conference on Com-
putational Fluid Dynamics (ICCFD3), 2004.
Dettmer W., Peri´c D., “ A computational framework for fluid-structure interaction: Finite element
formulation and applications”, Comp. Meth. Appl. Mech. Engrg., vol. 195, n° 41-43,
p. 5754-5779, 2006.
Donéa J., Giuliani S., Halleux J. P., “ An arbitrary Lagrangian-Eulerian finite element method
for transient dynamic fluid-structure interactions”, Comp. Meth. Appl. Mech. Engng., vol.
, p. 689-723, 1982.
Farhat C., van der Zee K., Geuzaine P., “ Provably second-order time-accurate loosely-coupled
solution algorithms for transient nonlinear aeroelasticity”, Comp. Meth. Appl. Mech. En-
gng., vol. 195, n° 17-18, p. 1973-2001, 2006.
Felippa C., Park K., “ Staggered transient analysis procedures for coupled mechanical systems:
formulation”, Computer Methods in Applied Mechanics and Engineering, vol. 24, n° 1,
p. 61-111, 1980.
Felippa C., Park K., Farhat C., “ Partitioned analysis of coupled mechanical systems”, Computer
methods in applied mechanics and engineering, vol. 190, n° 24-25, p. 3247-3270, 2001.
Fernández M., Gerbeau J.-F., Grandmont C., “ A projection semi-implicit scheme for the coupling
of an elastic structure with an incompressible fluid”, Internat. J. Numer. Methods
Engrg., vol. 69, n° 4, p. 794-821, 2007.
Fernández M., Moubachir M., “ A Newton method using exact Jacobians for solving fluidstructure
coupling”, Comp. & Struct., vol. 83, p. 127-142, 2005.
Formaggia L., Gerbeau J.-F., Nobile F., Quarteroni A., “ On the Coupling of 3D and 1D Navier-
Stokes equations for Flow Problems in Compliant Vessels”, Comp. Meth. Appl. Mech. En-
grg., vol. 191, n° 6-7, p. 561-582, 2001.
Förster C., Wall W., Ramm E., “ Artificial Added Mass Instabilities in Sequential Staggered
Coupling of Nonlinear Structures and Incompressible Flows”, Comp. Meth. Appl. Mech.
Engrg., vol. 196, p. 1278-1293, 2006.
Fung Y., Fronek K., Patitucci P., “ Pseudoelasticity of arteries and the choice of its mathematical
expression”, Am. J. Physiol., vol. 237, p. 620-631, 1979.
Gee M., Küttler U., Wall W., “ Truly monolithic algebraic multigrid for fluid-structure interaction”,
Int. J. Numer. Meth. Engng., 2010. DOI: 10.1002/nme.3001.
Gerbeau J.-F., Vidrascu M., “ A quasi-Newton algorithm based on a reduced model for fluidstructure
interactions problems in blood flows”, Math. Model. Num. Anal., vol. 37, n° 4,
p. 631-648, 2003.
Gerbeau J.-F., Vidrascu M., Frey P., “ Fluid-Structure Interaction in Blood Flows on Geometries
based on Medical Imaging”, Comp. & Struct., vol. 83, n° 2-3, p. 155-165, 2005.
Guidoboni G., Glowinski R., Cavallini N., Canic S., “ Stable loosely-coupled-type algorithm
for fluid-structure interaction in blood flow”, Journal of Computational Physics, vol. 228,
n° 18, p. 6916-6937, 2009.
Heil M., “ An efficient solver for the fully coupled solution of large-displacement fluid-structure
interaction problems”, Comput. Methods Appl. Mech. Engrg., vol. 193, n° 1-2, p. 1-23, 2004.
Heil M., Hazel A., Boyle J., “ Solvers for large-displacement fluid-structure interaction problems:
Segregated vs. monolithic approaches”, Comp. Mech., vol. 43, p. 91-101, 2008.
Holzapfel A., Gasser T., R.W. O., “ A new constitutive framework for arterial wall mechanics
and a comparative study of material models”, J. Elasticity, vol. 61, p. 1-48, 2000.
Hron J., Turek S., “ A monolithic FEM/Multigrid solver for an ALE formulation of fluidstructure
interaction with applications in biomechanics”, Fluid-structure interaction - mod-
elling, simulation, optimisation, Computational Science and Engineering, vol. 53, Springer,
p. 146-170, 2006.
Hübner B., Walhorn E., Dinkle D., “ A monolithic approach to fluid-structure interaction using
space-time finite elements”, Comp. Meth. Appl. Mech. Engng., vol. 193, p. 2087-2104,
Küttler U., Gee M., Förster C., Comerford A., Wall W., “ Coupling strategies for biomedical
fluid-structure interaction problems”, Int. J. Numer. Meth. Biomed. Engng., vol. 26, n° 3-4,
p. 305-321, 2009.
Küttler U., Wall W., “ Fixed-point fluid-structure interaction solvers with dynamic relaxation”,
Comp. Mech., vol. 43, n° 1, p. 61-72, 2008.
Le Tallec P., “ Domain decomposition methods in computational mechanics”, Computational
Mechanics Advances, Vol. 1, no.2, North Holland, p. 123-217, 1994.
Le Tallec P., Mouro J., “ Fluid structure interaction with large structural displacements”, Com-
put. Meth. Appl. Mech. Engrg., vol. 190, p. 3039-3067, 2001.
Matthies H., Steindorf J., “ Partitioned but strongly coupled iteration schemes for nonlinear
fluid-structure interaction”, Comp. & Struct., vol. 80, n° 27–30, p. 1991-1999, 2002.
Matthies H., Steindorf J., “ Partitioned strong coupling algorithms for fluid-structure interaction”,
Comp. & Struct., vol. 81, p. 805-812, 2003.
Michler C., van Brummelen E., de Borst R., “ An interface Newton-Krylov solver for fluidstructure
interaction”, Int. J. Num. Meth. Fluids, vol. 47, n° 10-11, p. 1189-1195, 2005.
Michler C., van Brummelen E., de Borst R., “ Error-amplification analysis of subiterationpreconditioned
GMRES for fluid-structure interaction”, Comp. Meth. Appl. Mech. Engng.,
vol. 195, p. 2124-2148, 2006.
Mok D. P., Wall W. A., “ Partitioned analysis schemes for the transient interaction of incompressible
flows and nonlinear flexible structures”, in K. S.W.A.Wall, K.U. Bletzinger (ed.),
Trends in computational structural mechanics, CIMNE, Barcelona, 2001a.
Mok D. P., Wall W. A., Ramm E., “ Partitioned analysis approach for the transient, coupled
response of viscous fluids and flexible structures”, in W. Wunderlich (ed.), Proceedings of
the European Conference on Computational Mechanics, ECCM’99, TU Munich, 1999.
Mok D. P.,WallW. A., Ramm E., “ Accelerated iterative substructuring schemes for instationary
fluid-structure interaction”, in K. Bathe (ed.), Computational Fluid and Solid Mechanics,
Elsevier, p. 1325-1328, 2001b.
Nobile F., Numerical approximation of fluid-structure interaction problems with application to
haemodynamics, PhD thesis, EPFL, Switzerland, 2001.
Oijen C. H. v., Mechanics and design of fiber-reinforced vascular prostheses, PhD thesis, Technische
Universiteit Eindhoven, 2003.
Park K., Felippa C., DeRuntz J., “ Stabilization of staggered solution procedures for fluidstructure
interaction analysis”, Computational methods for fluid-structure interaction prob-
lems, p. 95-124, 1977.
Piperno S., Farhat C., Larrouturou B., “ Partitioned procedures for the transient solution of coupled
aeroelastic problems. Part I:Model problem, theory and two-dimensional application”,
Comp. Meth. Appl. Mech. Engrg., vol. 124, p. 79-112, 1995.
Quaini A., Quarteroni A., “ A semi-implicit approach for fluid-structure interaction based on an
algebraic fractional step method”, Math. Models Methods Appl. Sci., vol. 17, n° 6, p. 957-
, 2007.
Rugonyi S., Bathe K., “ On finite element analysis of fluid flows coupled with structural interaction”,
CMES - Comp. Modeling Eng. Sci., vol. 2, n° 2, p. 195-212, 2001.
Salsac A.-V., Fernández M., Chomaz J., Le Tallec P., “ Effects of the flexibility of the arterial
wall on the wall shear stresses and wall tension in abdominal aortic aneurysms”, Bulletin of
the American Physical Society, 2005.
Salsac A.-V., Sparks S., Chomaz J., Lasheras J., “ Evolution of the wall shear stresses during the
progressive enlargement of symmetric abdominal aortic aneurysms”, J. Fluid Mech., vol.
, p. 19-51, 2006.
Tezduyar T., “ Finite Element Methods for Fluid Dynamics with Moving Boundaries and Interfaces”,
Arch. Comput. Methods Engrg., vol. 8, p. 83-130, 2001.
Tezduyar T., Sathe S., Keedy R., Stein K., “ Space-time finite element techniques for computation
of fluid-structure interactions”, Comput. Meth. Appl. Mech. Engng., vol. 195, p. 2002-
, 2006.
Thomas P., Lombard C., “ Geometric conservation law and its application to flow computations
on moving grids”, AIAA J., vol. 17, n° 10, p. 1030-1037, 1979.
van Brummelen E., “ Added mass effects of compressible and incompressible flows in fluidstructure
interaction”, Journal of Applied Mechanics, vol. 76, n° 2, p. 021206-07, 2009.
van Brummelen E., van der Zee K., de Borst R., “ Space/time multigrid for a fluid-structure
interaction problem”, Applied Numerical Mathematics, vol. 58, n° 12, p. 1951-1971, 2008.
van der Zee K., van Brummelen E., Akkerman I., de Borst R., “ Goal-oriented error estimation
and adaptivity for fluid-structure interaction using exact linearized adjoint”, 2010, to appear
in Comp. Meth. Appl. Mech. Engng.
Vierendeels J., “ Implicit coupling of partioned fluid-structure interaction solvers using reduced
order models”, in M. S. H.J. Bungartz (ed.), Fluid-Structure interaction, Modelling, Simu-
lation, Optimization, Springer, p. 1-18, 2006.
Zhang H., Zhang X., Ji S., Guo Y. Ledezma G., Elabbasi N., deCougny H., “ Recent development
of fluid-structure interaction capabilities in the ADINA system”, Computers & Struc-
tures, vol. 81, n° 8-11, p. 1071-1085, 2003.
