Numerical methods for fluid-structure interaction problems with valves
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This thesis is motivated by the modelling and the simulation of fluid-structure interaction phenomena in the vicinity of heart valves. On the one hand, the interaction of the vessel wall is dealt with an Arbitrary Lagrangian Eulerian (ALE) formulation. On the other hand the interaction of the valves is treated with the help of Lagrange multipliers in a Fictitious Domains-like (FD) formulation. After a synthetic presentation of the several methods available for the fluid-structure interaction in blood flows, we describe a method that permits capture the dynamics of a valve immersed in an incompressible fluid. The coupling algorithm is partitioned which allows the fluid and structure solvers to remain independent. In order to follow the vessel walls, the fluid mesh is mobile, but it remains none the less independent of the valve mesh. In this way we allow large displacements without the need to perform remeshing. We propose a strategy to manage contact between several immersed structures. The algorithm is completely independent of the structure solver and is well adapted to the partitioned fluid-structure coupling. Lastly we propose a semi-implicit coupling scheme allowing to mix, effectively, the ALE and FD formulations. The methods considered are followed with several numerical tests in 2D and 3D.Keywords:
Fluid–structure interaction
Solver
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An immersed boundary method (IBM) in a moving frame is proposed in this paper to study fluid–structure interaction problems. This solver includes the predictor and corrector steps. In the predictor step, the intermediate flow field is predicted on a moving Cartesian grid by Arbitrary Lagrangian–Eulerian(ALE) methods. In the corrector step, velocity correction is made by the implicit immersed boundary method to accurately satisfy the no-slip boundary condition. The motion of rigid body is obtained by solving the governing ordinary differential equations using the forth-order Runge–Kutta method. By enforcing the speed of the Cartesian grid the same as the translational velocity of the rigid body, the present solver is able to study a freely large movement object in a large flow domain. It not only extends the applicability of fixed grid-based solver but also considerably reduces the number of grid points and computational efforts. In addition, the re-meshing process, which is commonly used in the conventional arbitrary-Lagrangian-Eulerian (ALE) for Body-fitted or moving mesh approaches, is avoided. Several benchmarks, including an actively moving cylinder, freely falling quadrilateral with finite aspect ratios and a semi-active movement of a flapping foil, are studied to examine the reliability of the proposed solver. The obtained results compare well with theoretical and/or experimental data, which successfully demonstrate the capability of the proposed solver for fluid–structure interaction problems.
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Fluid–structure interaction
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A computational method is developed to solve the coupled fluid-structure interaction problem, where the viscous incompressible fluid and a rigid body-spring system interact with each other. In order to incorporate the effect of the moving surface of the rigid body, the arbitrary Lagrangian-Eulerian formulation is employed as the basis of the finite element spatial discretization. The predictor-corrector method is then used for the time integration of the equations of the motion.
Fluid–structure interaction
Basis (linear algebra)
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A semi-implicit coupling strategy under the arbitrary Lagrangian–Eulerian description is presented for the incompressible fluid flow past a geometrically nonlinear solid in this paper. The incompressible fluid is solved by means of the characteristic-based split (CBS) finite element method while the cell-based smoothed finite element method is employed to settle the governing equation of the geometrically nonlinear solid. Because of the CBS fluid solver, the present coupling strategy is performed in a semi-implicit fashion. In particular, the first step of the CBS scheme is explicitly treated whereas the others are implicitly coupled with the structural motion. The computational cost is hence reduced because no subiterations are included in the explicit coupling step and the fluid mesh is frozen in the implicit coupling step. A classic cantilever problem is dealt with to validate the structural solver, and then flow-induced vibrations of a restrictor flap in a uniform channel flow is analyzed in detail. The obtained results agree well with the existing data.
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