Implicit coupling of partitioned fluid–structure interaction problems with reduced order models
Introduction
The computation of fluid–structure interaction (FSI) problems has gained a lot of interest in the past decade. Two of the main interest areas are aeronautical [1], [2], [3], [4], [5], [6], [7] and biomechanical applications [8], [9], [10], [11], [12], [13], [14], [15], [16], but there are also other examples like parachute dynamics [17], [18] amongst other [19], [20].
In aeronautical applications FSI computations are mainly used for flutter analysis. In this application there exists a loose coupling between the fluid and the structural problem. Therefore explicit coupling or implicit coupling with block Jacobi or block Gauss Seidel coupling iterations can be used when calculating a time step. For this application typically the arbitrary Lagrangian–Eulerian (ALE) method is used where the flow equations are discretized on a moving grid that follows the motion of the structure.
Another technique for FSI calculations applied in this field is based on the use of reduced order models [3], [4], [5]. Therefore, first, calculations are run with the fluid solver with prescribed movement of the boundaries in order to gather response data of the fluid problem. From this data, a reduced order model for the fluid behaviour is then constructed and the FSI computations are run with the structural solver and the reduced order model for the fluid. Proper orthogonal decomposition techniques are mainly used to compose the reduced order model. This technique is very interesting in aeroelasticity studies because a lot of runs have to be performed for nearly identical geometries as seen from the fluid side.
In the biomedical application field typical compliant structures like vessels or the heart wall interact with the blood flow, which result in strongly coupled problems. Different coupling methods for this FSI problem have been used. Both in the immersed boundary method [21], [22], [8] and in the fictitious domain method [10], [13] a fixed grid is used for the flow calculation. The influence of the structure is introduced through sources in the momentum equations of the fluid. The fluid velocity field is used to integrate the displacement of the ‘immersed’ boundaries. The disadvantage of these methods is that the details of the flow near the boundary (e.g. shear stresses) cannot be computed easily. An ALE method is more appropriate to use if the details of the flow near the boundary is of interest.
If strong interaction is present between the fluid and the solid, a monolithic scheme can be used [23], [24], [25], [26]. However partitioned schemes can also be used for these applications. In order to study FSI problems in blood vessels and the left ventricle, Vierendeels et al. [9], [11] used a partitioned procedure and reached stabilization of the coupling iterations by introducing artificial compressibility in the fluid solver during the coupling iterations. Recently strongly coupled partitioned methods were developed [12], [27], [20], [15], [28] using approximate or exact Jacobians of the fluid and structural solver.
In this paper a coupling method for strongly coupled FSI problems with partitioned solvers is presented which makes use of reduced order models for the fluid and the structural problem. Different from the use of reduced order models mentioned before, where the models are used to replace one of both subproblems, here the reduced order models are only used to drive a coupling algorithm, which computes the fully coupled solution of a black box fluid solver and a black box structural solver each time step. Due to the use of the reduced order models, which are constructed at negligible computational cost during the coupling iterations, a coupled solution can be obtained for strongly coupled problems with partitioned solvers with a small number of coupling iterations. As fluid solver the commercial CFD software package Fluent 6.2 (Fluent Inc.) and as structural dynamics package Abaqus 6.5 (Abaqus Inc.) is used to illustrate the practical applicability of the method.
Section snippets
Method
A method is presented to solve coupled problems with partitioned solvers. However not limited to fluid–structure interaction applications, the method is explained for the coupling of a fluid and a structural solver. At time level n, the position of the interface between the fluid and the structure is denoted by Xn and the stress distribution at this interface by Pn. Given the state at time level n, the appropriate boundary conditions at time level n + 1 and the new position of the interface at
Pressure wave in an elastic blood vessel
The geometry described is a two-dimensional axisymmetric straight tube shown in Fig. 1. The tube has length L = 50 mm, radius R = 14.5 mm and wall thickness h = 0.85 mm, a geometry that matches the human aorta.
The fluid domain is meshed using 3468 nodes and 6674 triangular, linear cells. The structural domain was meshed using 1107 nodes and 300 quadratic, hybrid, reduced, rectangular cells.
Blood is modelled as a Newtonian fluid with density ρf = 1050 kg/m3 and dynamic viscosity μf = 2.678 × 10−3 kg/ms. The wall
Conclusion
We presented a coupling method for strongly coupled fluid–structure interaction problems with partitioned solvers. The method uses reduced order models for the fluid and the structural solver that are built up during the coupling iterations. If the structural solver is accessible, only a reduced order model for the fluid solver is needed which can be used as a dynamic boundary condition for the structural solver. Two examples are given. First, the pressure wave propagation in an elastic blood
Acknowledgements
Lieve Lanoye is financed by BOF-grant 011D09403 from the Ghent University. Joris Degroote is financed by a Ph.D. fellowship of the Research Foundation – Flanders (FWO).
References (33)
- et al.
Partitioned procedures for the transient solution of coupled aeroelastic problems. I. model problem, theory and two-dimensional application
Comput Methods Appl Mech Eng
(1995) - et al.
Load and motion transfer algorithms for fluid/structure interaction problems with non-matching discrete interfaces: momentum and energy conservation, optimal discretization and application to aeroelasticity
Comput Methods Appl Mech Eng
(1998) - et al.
Provably second-order time-accurate loosely-coupled solution algorithms for transient nonlinear computational aeroelasticity
Comput Methods Appl Mech Eng
(2006) - et al.
Hydrodynamics of color M-mode Doppler flow wave propagation velocity V(p): a computer study
J Am Soc Echocardiogr
(2002) - et al.
A three-dimensional computational analysis of fluid–structure interaction in the aortic valve
J Biomech
(2003) - et al.
A Newton method using exact Jacobians for solving fluid–structure coupling
Comput Struct
(2005) - et al.
Fluid–structure interaction in blood flows on geometries based on medical imaging
Comput Struct
(2005) - et al.
Parachute fluid–structure interactions: 3-D computation
Comput Methods Appl Mech Eng
(2000) - et al.
Fluid structure interaction with large structural displacements
Comput Methods Appl Mech Eng
(2001) An efficient solver for the fully coupled solution of large-displacement fluid–structure interaction problems
Comput Methods Appl Mech Eng
(2004)
Flow patterns around heart valves: a numerical method
J Comput Phys
Modeling prosthetic heart valves for numerical analysis of blood flow in the heart
J Comput Phys
Finite element analysis of fluid flows fully coupled with structural interactions
Comput Struct
A monolithic approach to fluid–structure interaction using space-time finite elements
Comput Methods Appl Mech Eng
Partitioned strong coupling algorithms for fluid–structure interaction
Comput Struct
Fluid–structure algorithms based on Steklov–Poincaré operators
Comput Methods Appl Mech Eng
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