For Fluid–structure Interaction Problems Research Articles

A Robin‐Neumann scheme with quasi‐Newton acceleration for partitioned fluid‐structure interaction

AbstractThe Dirichlet–Neumann scheme is the most common partitioned algorithm for fluid‐structure interaction (FSI) and offers high flexibility concerning the solvers employed for the two subproblems. Nevertheless, it is not without shortcomings: to begin with, the inherent added‐mass effect often destabilizes the numerical solution severely. Moreover, the Dirichlet–Neumann scheme cannot be applied to FSI problems in which an incompressible fluid is fully enclosed by Dirichlet boundaries, as it is incapable of satisfying the volume constraint. In the last decade, interface quasi‐Newton methods have proven to control the added‐mass effect and substantially speed up convergence by adding a Newton‐like update step to the Dirichlet–Neumann coupling. They are, however, without effect on the incompressibility dilemma. As an alternative, the Robin‐Neumann scheme generalizes the fluid's boundary condition to a Robin condition by including the Cauchy stresses. While this modification in fact successfully tackles both drawbacks of the Dirichlet–Neumann approach, the price to be paid is a strong dependency on the Robin weighting parameter, with very limited a priori knowledge about good choices. This work proposes a strategy to merge these two ideas and benefit from their combined strengths. The resulting quasi‐Newton‐accelerated Robin‐Neumann scheme is compared to both Robin‐ and Dirichlet–Neumann variants. The numerical tests demonstrate that it does not only provide faster convergence, but also massively reduces the influence of the Robin parameter, mitigating the main drawback of the Robin‐Neumann algorithm.

A Cartesian Immersed Boundary Method Based on 1D Flow Reconstructions for High-Fidelity Simulations of Incompressible Turbulent Flows Around Moving Objects

The aim of the present numerical study is to show that the recently developed Alternating Direction Reconstruction Immersed Boundary Method (ADR-IBM) (Giannenas and Laizet in Appl Math Model 99:606–627, 2021) can be used for Fluid–Structure Interaction (FSI) problems and can be combined with an Actuator Line Model (ALM) and a Computer-Aided Design (CAD) interface for high-fidelity simulations of fluid flow problems with rotors and geometrically complex immersed objects. The method relies on 1D cubic spline interpolations to reconstruct an artificial flow field inside the immersed object while imposing the appropriate boundary conditions on the boundaries of the object. The new capabilities of the method are demonstrated with the following flow configurations: a turbulent channel flow with the wall modelled as an immersed boundary, Vortex Induced Vibrations (VIVs) of one-degree-of-freedom (2D) and two-degree-of-freedom (3D) cylinders, a helicopter rotor and a multi-rotor unmanned aerial vehicle in hover and forward motion. These simulations are performed with the high-order fluid flow solver Incompact3d which is based on a 2D domain decomposition in order to exploit modern CPU-based supercomputers. It is shown that the ADR-IBM can be used for the study of FSI problems and for high-fidelity simulations of incompressible turbulent flows around moving complex objects with rotors.

Spline-based space-time finite element approach for fluid-structure interaction problems with a focus on fully enclosed domains

Non-Uniform Rational B-Spline (NURBS) surfaces are commonly used within Computer-Aided Design (CAD) tools to represent geometric objects. When using isogeometric analysis (IGA), it is possible to use such NURBS geometries for numerical analysis directly. Analyzing fluid flows, however, requires complex three-dimensional geometries to represent flow domains. Defining a parametrization of such volumetric domains using NURBS can be challenging and is still an ongoing topic in the IGA community.With the recently developed NURBS-enhanced finite element method (NEFEM), the favorable geometric characteristics of NURBS are used within a standard finite element method. This is achieved by enhancing the elements touching the boundary by using the NURBS geometry itself. In the current work, a new variation of NEFEM is introduced, which is suitable for three-dimensional space-time finite element formulations. The proposed method makes use of a new mapping which results in a non-Cartesian formulation suitable for fluid-structure interaction (FSI).This is demonstrated by combining the method with an IGA formulation in a strongly-coupled partitioned framework for solving FSI problems. The framework yields a fully spline-based representation of the fluid-structure interface through a single NURBS.The coupling conditions at the fluid-structure interface are enforced through a Robin-Neumann type coupling scheme. This scheme is particularly useful when considering incompressible fluids in fully Dirichlet-bounded and curved problems, as it satisfies the incompressibility constraint on the fluid for each step within the coupling procedure.The accuracy and performance of the introduced spline-based space-time finite element approach and its use within the proposed coupled FSI framework are demonstrated using a series of two- and three-dimensional benchmark problems.

Investigation of impact resistance performance of pyramid lattice sandwich structure based on SPH-FEM

In this paper, we have studied the influence of the panel thickness on the underwater impact resistance performance of the pyramid lattice structure based on the SPH-FEM method. Firstly, a benchmark case is simulated and compared with the experimental results to verify the accuracy of SPH-FEM for fluid-structure interaction (FSI) problems. Then, the dynamic response of the pyramid lattice structure subjected to the shock wave is studied by the SPH-FEM coupling method. Also, the sandwich structure's deformation is compared with experimental results to verify the effectiveness of the presented method applied in the FSI problem. Based on this, we study several groups of pyramid lattice structures with different panel thicknesses. In terms of the deflection of the back panel and each part's energy-absorption rate, the influence of the thickness of the panel on the impact resistance of the pyramid lattice sandwich structure is studied in detail. The results of this paper can provide a reference for the design of the ship protection structure.

An immersed boundary-lattice Boltzmann method for fluid-structure interaction problems involving viscoelastic fluids and complex geometries

An immersed boundary-lattice Boltzmann method (IB-LBM) for fluid-structure interaction (FSI) problems involving viscoelastic fluids and complex geometries is presented in this paper. In this method, the fluid dynamics and the constitutive equations of viscoelastic fluids are both solved using the lattice Boltzmann method. In order to enhance numerical stability in solving the constitutive equations, an artificial damping is introduced which does not affect the numerical results if the damping effect is much smaller than the relaxation and the convective effects. The structural dynamics including 2D and 3D capsules, 2D and 3D rigid particles and flags, are solved by the finite difference method (2D capsules, 2D and 3D rigid particles and flags) and the finite element method (3D capsules). The interaction between the solid structure and the fluid is enforced by an immersed boundary method. The overall framework of this method is very simple, enabling modelling FSI problems involving viscoelastic fluids and the inertia of both fluids and structures. It is very efficient for FSI problems involving high Weissenberg numbers flows, large deformations and complicated geometries without any preconditioner. This work uses IB-LBM to solve for the first time, flows involving viscoelastic fluids coupled with non-massless deforming structures. The method is also capable of solving very high Weissenberg number problems, as demonstrated by simulations of flexible particle flows at Wi=100. The present method and models are validated by several cases including a 2D rigid particle migration in a Giesekus Couette flow, a spherical particle rotation in an Oldroyd-B shear flow, a spherical particle settling in a FENE-CR fluid, 2D and 3D capsules deformation in a Newtonian shear flow, and a 3D flag flapping in a Newtonian free stream. In addition, the present method is also applied to simulate the deformation of 2D and 3D capsules in an Oldroyd-B shear flow, a 3D flag flapping in an Oldroyd-B free stream, and elastic capsule movement in a contraction-expansion channel filled with an Oldroyd-B fluid. Deformation of the capsules decreases with the increase of the Weissenberg number and the capsules experience monotonically increasing deformation when the Weissenberg number is above a critical value which is respectively 10 for 2D and 2 for 3D simulations. Viscoelasticity of the Oldroyd-B fluid hinders the flapping motion of the 3D flag. For elastic capsules passing through a periodic contraction-expansion channel, the capsules mix up after a short-term evolution, and then migrate to the bottom of the channel and almost follow two steady trajectories after a long-term evolution. The validations and applications provide extensive data which may be used to expand the currently limited database available for FSI benchmark studies.

A Nitsche-based formulation for fluid-structure interactions with contact

We derive a Nitsche-based formulation for fluid-structure interaction (FSI) problems with contact. The approach is based on the work of Chouly and Hild (SIAM J. Numer. Anal.51(2013) 1295–1307) for contact problems in solid mechanics. We present two numerical approaches, both of them formulating the FSI interface and the contact conditions simultaneously in equation form on a joint interface-contact surface Γ(t). The first approach uses a relaxation of the contact conditions to allow for a small mesh-dependent gap between solid and wall. The second alternative introduces an artificial fluid below the contact surface. The resulting systems of equations can be included in a consistent fashion within a monolithic variational formulation, which prevents the so-called “chattering” phenomenon. To deal with the topology changes in the fluid domain at the time of impact, we use a fully Eulerian approach for the FSI problem. We compare the effect of slip and no-slip interface conditions and study the performance of the method by means of numerical examples.

Flexible slender body fluid interaction: Vector-based discrete element method with Eulerian smoothed particle hydrodynamics

In this paper Eulerian weakly compressible SPH (EWCSPH) is coupled to the Vector-based discrete element model (V-model) to produce a robust mesh-less SPH-V model for fluid-structure interaction (FSI) applications including thin highly flexible bodies. The V-model, originally proposed as a model for granular structures, is extended herein to model non-linear deformation of a continuum representation of elastic structures. Non-linear static validation tests show that the extended V-model provides an accurate representation for problems involving large deformation. Undamped free vibration tests show the V-model is also accurate for transient problems. The SPH-V model with the immersed boundary method and SPH of Eulerian weakly compressible form is applied to established FSI tests for validation. These are oscillation of a flexible plate in quiescent fluids and flow past a cylinder with a rigidly connected flexible flat plate. The complex flow through flexible valve leaflets representative of flow through deep leg veins is demonstrated. Results thus show the SPH-V model is able to accurately capture complex flow features and also non-linear structural mechanics in fully coupled FSI problems.

A sharp-interface immersed smoothed finite element method for interactions between incompressible flows and large deformation solids

A sharp-interface algorithm developed for immersed smoothed finite element method (Sharp-ISFEM) is proposed for fluid–structure interaction (FSI) problems. In Sharp-ISFEM, a line normal approach is used for unstructured meshes to reconstruct the solutions near interface. Advanced smoothed finite element models describe the finite deformation of immersed solid structures. Well-developed characteristic-based split method is employed to simulate the incompressible laminar flow. The coupling of fluid and solid is treated by the modified weak partitioned fictitious domain method with different time steps for two phases. To handle large-scale FSI problems, octree and neighbor-neighbor searching algorithms have been implemented. Meanwhile, parallelization has been implemented for the time-consuming parts of in-house codes. The effectiveness of proposed Sharp-ISFEM is validated by several benchmarking problems, including: fixed rigid body, flow over square cylinder; moving rigid body, flow induced by 2D hovering wing; large deformed solid, a beam with large deformation induced by flow; 3D deformable solid–fluid interaction, and lid-driven cavity flow with a rubber wall. Good agreements are shown in all test cases, comparing with existed experimental data and reference solutions. With the improvement of local velocity reconstruction, advanced searching algorithms and code parallelization, the proposed method has shown great promises for complex engineering flows with higher Reynolds number.A large-scale FSI problem of a hummingbird with flexible wing in hovering is finally simulated using more than one million nodes.

Adaptive FEM with Domain Decomposition Method for Partitioned-Based Fluid–Structure Interaction

A numerical assessment on the solution performance of adaptive finite element methods for fluid–structure interaction (FSI) is presented in this paper. The partitioned-based approach involving separate fluid and structure solvers is considered for adaptive finite element computation with triangular mesh. The performance of the hp-adaptivity, in which error in energy norm is reduced by way of mesh refinement (h) and polynomial order extension (p), is of particular interest. In addition, parallel solution process based on the domain decomposition method is also assessed due to the complexity of two way coupling scheme. Based on its numerical convergence, the hp-adaptive procedure for the FSI problem is shown to yield the fastest convergence as generally expected. Moreover, our domain decomposition parallelization scheme shows reduction in the computation time by up to order 2. Subsequently, the convergence performance is highly dependent on the aspect ratios of the triangular elements. The solution performance strongly suggests the viability of the parallel hp-adaptive method in partitioned-based FSI formulation.

An analysis of a new stable partitioned algorithm for FSI problems. Part I: Incompressible flow and elastic solids

Stable partitioned algorithms for fluid–structure interaction (FSI) problems are developed and analyzed in this two-part paper. Part I describes an algorithm for incompressible flow coupled with compressible elastic solids, while Part II discusses an algorithm for incompressible flow coupled with structural shells. Importantly, these new added-mass partitioned (AMP) schemes are stable and retain full accuracy with no sub-iterations per time step, even in the presence of strong added-mass effects (e.g. for light solids). The numerical approach described here for bulk compressible solids extends the scheme of Banks et al. [1,2] for inviscid compressible flow, and uses Robin (mixed) boundary conditions with the fluid and solid solvers at the interface. The basic AMP Robin conditions, involving a linear combination of velocity and stress, are determined from the outgoing solid characteristic relation normal to the fluid–solid interface combined with the matching conditions on the velocity and traction. Two alternative forms of the AMP conditions are then derived depending on whether the fluid equations are advanced with a fractional-step method or not. The stability and accuracy of the AMP algorithm is evaluated for linearized FSI model problems; the full nonlinear case being left for future consideration. A normal mode analysis is performed to show that the new AMP algorithm is stable for any ratio of the solid and fluid densities, including the case of very light solids when added-mass effects are large. In contrast, it is shown that a traditional partitioned algorithm involving a Dirichlet–Neumann coupling for the same FSI problem is formally unconditionally unstable for any ratio of densities. Exact traveling wave solutions are derived for the FSI model problems, and these solutions are used to verify the stability and accuracy of the corresponding numerical results obtained from the AMP algorithm for the cases of light, medium and heavy solids.

An analysis of a new stable partitioned algorithm for FSI problems. Part II: Incompressible flow and structural shells

Stable partitioned algorithms for fluid–structure interaction (FSI) problems are developed and analyzed in this two-part paper. Part I describes an algorithm for incompressible flow coupled with compressible elastic solids, while Part II discusses an algorithm for incompressible flow coupled with structural shells. The numerical approach described here for structural shells uses Robin (mixed) interface conditions for the pressure and velocity in the fluid which are derived directly from the governing equations. The resulting added-mass partitioned (AMP) algorithm is stable even for very light structures, requires no sub-iterations per time step, and is second-order accurate. The stability and accuracy of the AMP algorithm is evaluated for linearized FSI model problems. A normal mode analysis is performed to show that the new AMP algorithm is stable, even for the case of very light structures when added-mass effects are large. Exact traveling wave solutions are derived for the FSI model problems, and these solutions are used to verify the stability and accuracy of the corresponding numerical results obtained from the AMP algorithm for the cases of light, medium and heavy structures. A summary comparison of the AMP algorithm developed here and the one in Part I is provided.

A modular, operator‐splitting scheme for fluid–structure interaction problems with thick structures

SUMMARYWe present an operator‐splitting scheme for fluid–structure interaction (FSI) problems in hemodynamics, where the thickness of the structural wall is comparable to the radius of the cylindrical fluid domain. The equations of linear elasticity are used to model the structure, while the Navier–Stokes equations for an incompressible viscous fluid are used to model the fluid. The operator‐splitting scheme, based on the Lie splitting, separates the elastodynamics structure problem from a fluid problem in which structure inertia is included to achieve unconditional stability. We prove energy estimates associated with unconditional stability of this modular scheme for the full nonlinear FSI problem defined on a moving domain, without requiring any sub‐iterations within time steps. Two numerical examples are presented, showing excellent agreement with the results of monolithic schemes. First‐order convergence in time is shown numerically. Modularity, unconditional stability without temporal sub‐iterations, and simple implementation are the features that make this operator‐splitting scheme particularly appealing for multi‐physics problems involving FSI. Copyright © 2013 John Wiley & Sons, Ltd.

A heterogeneous flow model based on DD method for free surface fluid–structure interaction problems

SUMMARYAn efficient heterogeneous flow model that combines incompressible viscous flow and potential flow(PF)(both) with free‐surface boundaries for fluid–structure interaction(FSI) problems was developed and solved numerically. In the model, the near field(viscous) fluid subdomain of the FSI simulation is described by the Navier–Stokes equations(NSE), whereas the far field(inviscid) subdomain is described by PF. This method is particularly suitable for modeling a large fluid domain while avoid solving the NSE at the far field, where the inviscid flow assumption is appropriate and adopting a simplified(e.g., PF) flow description reduces computing cost. Numerically, the NSE are solved using a FEM, and the PF is solved using a BEM. Free surface tracking is achieved through a hybrid level set method for the NSE and the mixed Eulerian–Lagrangian method for the PF. The coupling is based on a non‐overlapping DD method similar to the Dirichlet–Neumann method. At the interface between the two flow model subdomains, velocity and pressure are matched to ensure continuity of the corresponding variables at the common boundary. In this framework of the heterogeneous DD method, an explicit scheme is developed and implemented. The relation between this scheme and the classical Dirichlet–Neumann method for a homogeneous problem is revealed. Numerical examples demonstrating the capability of the method to model nonlinear wave phenomena and FSI problems with flexible structures are presented. Copyright © 2013 John Wiley & Sons, Ltd.

Generalized fictitious methods for fluid–structure interactions: Analysis and simulations

We present a new fictitious pressure method for fluid–structure interaction (FSI) problems in incompressible flow by generalizing the fictitious mass and damping methods we published previously in [1]. The fictitious pressure method involves modification of the fluid solver whereas the fictitious mass and damping methods modify the structure solver. We analyze all fictitious methods for simplified problems and obtain explicit expressions for the optimal reduction factor (convergence rate index) at the FSI interface [2]. This analysis also demonstrates an apparent similarity of fictitious methods to the FSI approach based on Robin boundary conditions, which have been found to be very effective in FSI problems. We implement all methods, including the semi-implicit Robin based coupling method, in the context of spectral element discretization, which is more sensitive to temporal instabilities than low-order methods. However, the methods we present here are simple and general, and hence applicable to FSI based on any other spatial discretization. In numerical tests, we verify the selection of optimal values for the fictitious parameters for simplified problems and for vortex-induced vibrations (VIV) even at zero mass ratio (“for-ever-resonance”). We also develop an empirical a posteriori analysis for complex geometries and apply it to 3D patient-specific flexible brain arteries with aneurysms for very large deformations. We demonstrate that the fictitious pressure method enhances stability and convergence, and is comparable or better in most cases to the Robin approach or the other fictitious methods.

Variational implementation of immersed finite element methods

Dirac-δ distributions are often crucial components of the solid–fluid coupling operators in immersed solution methods for fluid–structure interaction (FSI) problems. This is certainly so for methods like the immersed boundary method (IBM) or the immersed finite element method (IFEM), where Dirac-δ distributions are approximated via smooth functions. By contrast, a truly variational formulation of immersed methods does not require the use of Dirac-δ distributions, either formally or practically. This has been shown in the finite element immersed boundary method (FEIBM), where the variational structure of the problem is exploited to avoid Dirac-δ distributions at both the continuous and the discrete level.In this paper, we generalize the FEIBM to the case where an incompressible Newtonian fluid interacts with a general hyperelastic solid. Specifically, we allow (i) the mass density to be different in the solid and the fluid, (ii) the solid to be either viscoelastic of differential type or purely elastic, and (iii) the solid to be either compressible or incompressible. At the continuous level, our variational formulation combines the natural stability estimates of the fluid and elasticity problems. In immersed methods, such stability estimates do not transfer to the discrete level automatically due to the non-matching nature of the finite dimensional spaces involved in the discretization. After presenting our general mathematical framework for the solution of FSI problems, we focus in detail on the construction of natural interpolation operators between the fluid and the solid discrete spaces, which guarantee semi-discrete stability estimates and strong consistency of our spatial discretization.

Partitioned iterative solution methods for fluid–structure interaction

AbstractThe basic subiteration method for fluid–structure interaction (FSI) problems is based on a partitioning of the fluid–structure system into a fluidic part and a structural part. The effect of the fluid on the structure can be represented by an added mass to the structural operator. This added mass can be identified as an upper bound on the norm or spectral radius of the Poincar'e–Steklov operator of the fluid. The convergence behavior of the subiteration method depends sensitively on the ratio of the added mass to the actual structural mass. For FSI problems with large added‐mass effects, the subiteration method is either unstable or its convergence behavior is prohibitively inefficient. In recent years, several more advanced partitioned iterative solution methods have been proposed for this class of problems, which use subiteration as a component. The rudimentary characterization of the Poincaré–Steklov operator provided by the added mass is, however, inadequate to analyze these methods. Moreover, this characterization is inappropriate for compressible flows. In this paper, we investigate the fine properties of the Poincaré–Steklov operators and of the corresponding subiteration operators for incompressible‐ and compressible flow models and for two distinct structural operators. Based on the characteristic properties of the subiteration operators, we subsequently examine the convergence behavior of several partitioned iterative solution methods for FSI, viz. subiteration, subiteration in conjunction with underrelaxation, the modified‐mass method, Aitken's method, and interface‐GMRES and interface‐Newton–Krylov methods. Copyright © 2010 John Wiley & Sons, Ltd.

Fluid–structure interaction of prismatic line-like structures, using LES and block-iterative coupling

The design of long-span bridges to resist wind loading often requires wind tunnel testing of sectional or full aeroelastic models. Recently, efforts have been made to realise a reliable computational alternative to these physical tests. In the current work, a novel computational scheme for fluid–structure interaction (FSI) is presented. Large eddy simulation (LES) for 3D viscous turbulent incompressible flow has been coupled to the response of prismatic line-like structures. LES is chosen because of the inherent unsteadiness in FSI problems and the capability of LES to maintain the turbulence structure in the flow, in contrast to the over-dissipativeness of the traditional Reynolds averaged Navier–Stokes equations (RANS)-based turbulence models. A Gauss–Seidel-type block-iterative algorithm is adopted to address both the field coupling and the non-linearity simultaneously. At global convergence, this gives solutions identical to those obtained using direct coupling, but with field modularity preserved, storage requirements well under control and computational effort significantly reduced. Numerical examples are presented for elastically supported rigid circular and rectangular cylinders (sectional model tests); the method is readily extendable to flexible structures (full aeroelastic models) via modal analysis.

ALE finite element method for FSI problems with free surface using mesh re-generation method based on background mesh

This paper presents an arbitrary Lagrangian–Eulerian (ALE) finite element method for fluid-structure interaction (FSI) problems with a free-surface using the mesh re-generation method. The mesh re-generation method using the background mesh is introduced to improve the applicability to the complicated FSI problem with large deformed interface. The incompressible Navier–Stokes equation based on ALE description is used as the governing equation of fluid. The stabilization method based on the SUPG/PSPG method is employed. The presented method is applied to several numerical examples to show the validity and efficiency of the method.