Implicit coupling of partitioned fluid–structure interaction problems with reduced order models

doi:10.1016/j.compstruc.2006.11.006

Computers & Structures

Volume 85, Issues 11–14, June–July 2007, Pages 970-976

https://doi.org/10.1016/j.compstruc.2006.11.006 Get rights and content

Abstract

In this paper a newly developed technique for strongly coupled fluid–structure interaction problems is presented. In order to achieve strong coupling the Jacobian of the fluid and/or structural problem is needed or has to be approximated. A technique is presented which uses the Jacobian from reduced order models that are built up during the coupling iterations. As validation, pressure wave propagation in a blood vessel is computed and as a second example growth and detachment of a gas bubble from a vertical needle submerged in a liquid is simulated. Both examples illustrate the algorithmic performance and show accurate results.

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 ρ_f = 1050 kg/m³ and dynamic viscosity μ_f = 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).

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Implicit coupling of partitioned fluid–structure interaction problems with reduced order models

Abstract

Introduction

Section snippets

Method

Pressure wave in an elastic blood vessel

Conclusion

Acknowledgements

Comput Methods Appl Mech Eng

Comput Methods Appl Mech Eng

Comput Methods Appl Mech Eng

J Am Soc Echocardiogr

J Biomech

Comput Struct

Comput Struct

Comput Methods Appl Mech Eng

Comput Methods Appl Mech Eng

Comput Methods Appl Mech Eng

J Comput Phys

J Comput Phys

Comput Struct

Comput Methods Appl Mech Eng

Comput Struct

Comput Methods Appl Mech Eng