A monolithic approach for interaction of incompressible viscous fluid and an elastic body based on fluid pressure Poisson equation

This paper describes a new monolithic approach based on the fluid pressure Poisson equation (PPE) to solve an interaction problem of incompressible viscous fluid and an elastic body. The PPE is derived so as to be consistent with the coupled equation system for the fluid‐structure interaction (FSI). Based on this approach, we develop two kinds of efficient monolithic methods. In both methods, the fluid pressure is derived implicitly so as to satisfy the incompressibility constraint, and all other unknown variables are derived fully explicitly or partially explicitly. The coefficient matrix of the PPE for the FSI becomes symmetric and positive definite and its condition is insensitive to inhomogeneity of material properties. The arbitrary Lagrangian–Eulerian (ALE) method is employed for the fluid part in order to take into account the deformable fluid‐structure interface. To demonstrate fundamental performances of the proposed approach, the developed two monolithic methods are applied to evaluate the added mass and the added damping of a circular cylinder as well as to simulate the vibration of a rectangular cylinder induced by vortex shedding. Copyright © 2005 John Wiley & Sons, Ltd.     

一种基于压力泊松方程的流体结构紧耦合算法

从流固耦合系统的整体控制方程出发,推导出与流体控制方程一致的耦合等价方程,并得到基于耦合方程的压力泊松方程,通过求解耦合系统压力泊松方程和一致的等价方程就能获得耦合系统的解,而不需要直接求解整体耦合系统的控制方程,有利于降低求解自由度。预估-校正多步迭代格式用于耦合系统的时间推进,克服了传统迭代耦合方法由于时间不同步而产生较大数值误差的不足。应用该方法对附带局部突起的主动脉弓动脉瘤进行流固耦合分析,验证数值方法的可行性。     

The enriched space–time finite element method (EST) for simultaneous solution of fluid–structure interaction

The paper introduces a weighted residual‐based approach for the numerical investigation of the interaction of fluid flow and thin flexible structures. The presented method enables one to treat strongly coupled systems involving large structural motion and deformation of multiple‐flow‐immersed solid objects. The fluid flow is described by the incompressible Navier–Stokes equations. The current configuration of the thin structure of linear elastic material with non‐linear kinematics is mapped to the flow using the zero iso‐contour of an updated level set function. The formulation of fluid, structure and coupling conditions uniformly uses velocities as unknowns. The integration of the weak form is performed on a space–time finite element discretization of the domain. Interfacial constraints of the multi‐field problem are ensured by distributed Lagrange multipliers. The proposed formulation and discretization techniques lead to a monolithic algebraic system, well suited for strongly coupled fluid–structure systems. Embedding a thin structure into a flow results in non‐smooth fields for the fluid. Based on the concept of the extended finite element method, the space–time approximations of fluid pressure and velocity are properly enriched to capture weakly and strongly discontinuous solutions. This leads to the present enriched space–time (EST) method. Numerical examples of fluid–structure interaction show the eligibility of the developed numerical approach in order to describe the behavior of such coupled systems. The test cases demonstrate the application of the proposed technique to problems where mesh moving strategies often fail. Copyright © 2007 John Wiley & Sons, Ltd.     

An efficient parallel computation of unsteady incompressible viscous flow with elastic moving and compliant boundaries on unstructured grids

This paper presents the development and validation of a parallel unstructured‐grid fluid–structure interaction (FSI) solver for the simulation of unsteady incompressible viscous flow with long elastic moving and compliant boundaries. The Navier–Stokes solver on unstructured moving grid using the arbitrary Lagrangian Eulerian formulation is based on the artificial compressibility approach and a high‐order characteristics‐based finite‐volume scheme. Both unsteady flow and FSI are calculated with a matrix‐free implicit dual time‐stepping scheme. A membrane model has been formulated to study fluid flow in a channel with an elastic membrane wall and their interactions. This model can be employed to calculate arbitrary wall movement and variable tension along the membrane, together with a dynamic mesh method for large deformation of the flow field. The parallelization of the fluid–structure solver is achieved using the single program multiple data programming paradigm and message passing interface for communication of data. The parallel solver is used to simulate fluid flow in a two‐dimensional channel with and without moving membrane for validation and performance evaluation purposes. The speedups and parallel efficiencies obtained by this method are excellent, using up to 16 processors on a SGI Origin 2000 parallel computer. A maximum speedup of 23.14 could be achieved on 16 processors taking advantage of an improved handling of the membrane solver. The parallel results obtained are compared with those using serial code and they are found to be identical. Copyright © 2005 John Wiley & Sons, Ltd.     

A hybrid variational Allen-Cahn/ALE scheme for the coupled analysis of two-phase fluid-structure interaction

We present a partitioned iterative formulation for the modeling of fluid-structure interaction (FSI) in two-phase flows. The variational formulation consists of a stable and robust integration of three blocks of differential equations, viz, an incompressible viscous fluid, a rigid or flexible structure, and a two-phase indicator field. The fluid-fluid interface between the two phases, which may have high density and viscosity ratios, is evolved by solving the conservative phase-field Allen-Cahn equation in the arbitrary Lagrangian-Eulerian coordinates. While the Navier-Stokes equations are solved by a stabilized Petrov-Galerkin method, the conservative Allen-Cahn phase-field equation is discretized by the positivity preserving variational scheme. Fully decoupled implicit solvers for the two-phase fluid and the structure are integrated by the nonlinear iterative force correction in a staggered partitioned manner and the generalized-α method is employed for the time marching. We assess the accuracy and stability of the phase-field/ALE variational formulation for two- and three-dimensional problems involving the dynamical interaction of rigid bodies with free surface. We consider the decay test problems of increasing complexity, namely, free translational heave decay of a circular cylinder and free rotation of a rectangular barge. Through numerical experiments, we show that the proposed formulation is stable and robust for high density ratios across fluid-fluid interface and for low structure-to-fluid mass ratio with strong added-mass effects. Overall, the proposed variational formulation produces results with high accuracy and compares well with available measurements and reference numerical data. Using unstructured meshes, we demonstrate the second-order temporal accuracy of the coupled phase-field/ALE method via decay test of a circular cylinder interacting with the free surface. Finally, we demonstrate the three-dimensional phase-field FSI formulation for a practical problem of internal two-phase flow in a flexible circular pipe subjected to vortex-induced vibrations due to external fluid flow.     

3D fluid–structure-contact interaction based on a combined XFEM FSI and dual mortar contact approach

Finite deformation contact of flexible solids embedded in fluid flows occurs in a wide range of engineering scenarios. We propose a novel three-dimensional finite element approach in order to tackle this problem class. The proposed method consists of a dual mortar contact formulation, which is algorithmically integrated into an eXtended finite element method (XFEM) fluid–structure interaction approach. The combined XFEM fluid–structure-contact interaction method (FSCI) allows to compute contact of arbitrarily moving and deforming structures embedded in an arbitrary flow field. In this paper, the fluid is described by instationary incompressible Navier–Stokes equations. An exact fluid–structure interface representation permits to capture flow patterns around contacting structures very accurately as well as to simulate dry contact between structures. No restrictions arise for the structural and the contact formulation. We derive a linearized monolithic system of equations, which contains the fluid formulation, the structural formulation, the contact formulation as well as the coupling conditions at the fluid–structure interface. The linearized system may be solved either by partitioned or by monolithic fluid–structure coupling algorithms. Two numerical examples are presented to illustrate the capability of the proposed fluid–structure-contact interaction approach.     

Unified fractional step method for Lagrangian analysis of quasi‐incompressible fluid and nonlinear structure interaction using the PFEM

The fractional step method (FSM) is an efficient solution technique for the particle finite element method, a Lagrangian‐based approach to simulate fluid–structure interaction (FSI). Despite various refinements, the applicability of the FSM has been limited to low viscosity flow and FSI simulations with a small number of equations along the fluid–structure interface. To overcome these limitations, while incorporating nonlinear response in the structural domain, an FSM that unifies structural and fluid response in the discrete governing equations is developed using the quasi‐incompressible formulation. With this approach, fluid and structural particles do not need to be treated separately, and both domains are unified in the same system of equations. Thus, the equations along the fluid–structure interface do not need to be segregated from the fluid and structural domains. Numerical examples compare the unified FSM with the non‐unified FSM and show that the computational cost of the proposed method overcomes the slow convergence of the non‐unified FSM for high values of viscosity. As opposed to the non‐unified FSM, the number of iterations required for convergence with the unified FSM becomes independent of viscosity and time step, and the simulation run time does not depend on the size of the FSI interface. Copyright © 2016 John Wiley & Sons, Ltd.     

A new finite element technique for the analysis of steady viscous flow problems

A new finite element technique for two-dimensional viscous incompressible fluid flow problems is presented in this paper. The vorticity transport equation is integrated in a small control volume, which results in the conservation law of vorticity. The finite element technique is applied to this equation together with the continuity equation, where simple linear triangular elements with three nodes are used for the formulation. Resulting sets of algebraic equations are solved by the use of a kind of relaxation method. Numerical results for viscous flow past a cavity show good agreement with experimental results.     

Investigation of fluid–structure interactions using a velocity‐linked P2/P1 finite element method and the generalized‐ α method

A velocity‐linked algorithm for solving unsteady fluid–structure interaction (FSI) problems in a fully coupled manner is developed using the arbitrary Lagrangian–Eulerian method. The P2/P1 finite element is used to spatially discretize the incompressible Navier–Stokes equations and structural equations, and the generalized‐ α method is adopted for temporal discretization. Common velocity variables are employed at the fluid–structure interface for the strong coupling of both equations. Because of the velocity‐linked formulation, kinematic compatibility is automatically satisfied and forcing terms do not need to be calculated explicitly. Both the numerical stability and the convergence characteristics of an iterative solver for the coupled algorithm are investigated by solving the FSI problem of flexible tube flows. It is noteworthy that the generalized‐ α method with small damping is free from unstable velocity fields. However, the convergence characteristics of the coupled system deteriorate greatly for certain Poisson's ratios so that direct solvers are essential for these cases. Furthermore, the proposed method is shown to clearly display the advantage of considering FSI in the simulation of flexible tube flows, while enabling much larger time‐steps than those adopted in some previous studies. This is possible through the strong coupling of the fluid and structural equations by employing common primitive variables. Copyright © 2012 John Wiley & Sons, Ltd.     

A rheological interface model and its space–time finite element formulation for fluid–structure interaction

This contribution discusses extended physical interface models for fluid–structure interaction problems and investigates their phenomenological effects on the behavior of coupled systems by numerical simulation. Besides the various types of friction at the fluid–structure interface the most interesting phenomena are related to effects due to additional interface stiffness and damping. The paper introduces extended models at the fluid–structure interface on the basis of rheological devices (Hooke, Newton, Kelvin, Maxwell, Zener). The interface is decomposed into a Lagrangian layer for the solid‐like part and an Eulerian layer for the fluid‐like part. The mechanical model for fluid–structure interaction is based on the equations of rigid body dynamics for the structural part and the incompressible Navier–Stokes equations for viscous flow. The resulting weighted residual form uses the interface velocity and interface tractions in both layers in addition to the field variables for fluid and structure. The weak formulation of the whole coupled system is discretized using space–time finite elements with a discontinuous Galerkin method for time‐integration leading to a monolithic algebraic system. The deforming fluid domain is taken into account by deformable space–time finite elements and a pseudo‐structure approach for mesh motion. The sensitivity of coupled systems to modification of the interface model and its parameters is investigated by numerical simulation of flow induced vibrations of a spring supported fluid‐immersed cylinder. It is shown that the presented rheological interface model allows to influence flow‐induced vibrations. Copyright © 2010 John Wiley & Sons, Ltd.     

Fluid–structure interaction problems with strong added‐mass effect

In this paper, the so‐called added‐mass effect is investigated from a different point of view of previous publications. The monolithic fluid–structure problem is partitioned using a static condensation of the velocity terms. Following this procedure the classical stabilized projection method for incompressible fluid flows is introduced. The procedure allows obtaining a new pressure segregated scheme for fluid–structure interaction problems, which has good convergent characteristics even for biomechanical application, where the added‐mass effect is strong. The procedure reveals its power when it is shown that the same projection technique must be implemented in staggered FSI methods. Copyright © 2009 John Wiley & Sons, Ltd.     

Immersed-boundary/soft-sphere method for particle–particle-fluid interaction in a viscous flow: An OpenFOAM solver

We developed a stable OpenFOAM solver for Immersed Boundary Method based on direct forcing and regularized delta function. The soft-sphere model and a lubrication model were implemented to consider particle–particle collision in a viscous flow. We proposed a fluid–structure interaction (FSI) coupling method to accurately calculate the fluid forcing term and particle velocity. Our solver was validated for fixed and moving bodies, including rotation. The accuracy of various FSI schemes was evaluated in predicting the solid and fluid flow behavior in a viscous flow. It was demonstrated that neglecting or simplifying the fluid momentum change affects the accuracy of the solid velocity and fluid flow dynamic; for higher solid-to-fluid density ratios, a larger deviation was predicted. Furthermore, the FSI schemes highly influenced the behavior of the formed vortices.The solver was validated to predict the effective restitution coefficient of particles in a viscous flow as a function of the Stokes number. We also thoroughly analyzed the dynamic flow behavior of colliding particles through the pressure and velocity field and fluid force. This analysis helped us accurately determine the rebound velocity of particles in case of high Stokes numbers when the effect of viscous force is significant.     

Pressure and flow in fluid films constrained between translating and vibrating flexible surfaces

In this paper, the dynamic pressure and flow developed in a two-dimensional, viscous fluid film constrained between flexible surfaces are analyzed. The problem formulation assumes that the response of the flexible surface is governed by linear equations of motion, and the fluid motion is governed by linearized momentum equations including the unsteady inertia. Three states of the model are developed to describe the coupled fluid-structural response problem. The fluid dynamic pressure is derived in the frequency domain as a function of the fluid impedances and the surface transverse vibrations. The perturbed, coupled problem is described by an integral equation (in state vector form) that governs the coupled responses of the flexible surfaces. The integral equation is solved by a discretization method. The analysis is applied to a rigid slider bearing with a flexible, translating plate surface under the excitation of a harmonic point load. The accuracy of the discretization method is evaluated, and numerical results for the dynamic pressure and the plate response are presented.     

Monolithic cut finite element–based approaches for fluid-structure interaction

Cut finite element method–based approaches toward challenging fluid-structure interaction (FSI) are proposed. The different considered methods combine the advantages of competing novel Eulerian (fixed grid) and established arbitrary Lagrangian-Eulerian (moving mesh) finite element formulations for the fluid. The objective is to highlight the benefit of using cut finite element techniques for moving-domain problems and to demonstrate their high potential with regard to simplified mesh generation, treatment of large structural motions in surrounding flows, capturing boundary layers, their ability to deal with topological changes in the fluid phase, and their general straightforward extensibility to other coupled multiphysics problems. In addition to a pure fixed-grid FSI method, advanced fluid-domain decomposition techniques are also considered, leading to highly flexible discretization methods for the FSI problem. All stabilized formulations include Nitsche-based weak coupling of the phases supported by the ghost penalty technique for the flow field. For the resulting systems, monolithic solution strategies are presented. Various two- and three-dimensional FSI cases of different complexity levels validate the methods and demonstrate their capabilities and limitations in different situations.     

A three dimensional immersed smoothed finite element method (3D IS-FEM) for fluid–structure interaction problems

A three-dimensional immersed smoothed finite element method (3D IS-FEM) using four-node tetrahedral element is proposed to solve 3D fluid–structure interaction (FSI) problems. The 3D IS-FEM is able to determine accurately the physical deformation of the nonlinear solids placed within the incompressible viscous fluid governed by Navier-Stokes equations. The method employs the semi-implicit characteristic-based split scheme to solve the fluid flows and smoothed finite element methods to calculate the transient dynamics responses of the nonlinear solids based on explicit time integration. To impose the FSI conditions, a novel, effective and sufficiently general technique via simple linear interpolation is presented based on Lagrangian fictitious fluid meshes coinciding with the moving and deforming solid meshes. In the comparisons to the referenced works including experiments, it is clear that the proposed 3D IS-FEM ensures stability of the scheme with the second order spatial convergence property; and the IS-FEM is fairly independent of a wide range of mesh size ratio.     

Robin‐Neumann transmission conditions for fluid‐structure coupling: Embedded boundary implementation and parameter analysis

Partitioned procedures are appealing for solving complex fluid‐structure interaction (FSI) problems, as they allow existing computational fluid dynamics (CFD) and computational structural dynamics algorithms and solvers to be combined and reused. However, for problems involving incompressible flow and strong added‐mass effect (eg, heavy fluid and slender structure), partitioned procedures suffer from numerical instability, which typically requires additional subiterations between the fluid and structural solvers, hence significantly increasing the computational cost. This paper investigates the use of Robin‐Neumann transmission conditions to mitigate the above instability issue. Firstly, an embedded Robin boundary method is presented in the context of projection‐based incompressible CFD and finite element–based computational structural dynamics. The method utilizes operator splitting and a modified ghost fluid method to enforce the Robin transmission condition on fluid‐structure interfaces embedded in structured non–body‐conforming CFD grids. The method is demonstrated and verified using the Turek and Hron benchmark problem, which involves a slender beam undergoing large transient deformation in an unsteady vortex‐dominated channel flow. Secondly, this paper investigates the effect of the combination parameter in the Robin transmission condition, ie, α_f, on numerical stability and solution accuracy. This paper presents a numerical study using the Turek and Hron benchmark problem and an analytical study using a simplified FSI model featuring an Euler‐Bernoulli beam interacting with a two‐dimensional incompressible inviscid flow. Both studies reveal a trade‐off between stability and accuracy: smaller values of α_f tend to improve numerical stability, yet deteriorate the accuracy of the partitioned solution. Using the simplified FSI model, the critical value of α_f that optimizes this trade‐off is derived and discussed.     

A coupled BEM and arbitrary Lagrangian–Eulerian FEM model for the solution of two‐dimensional laminar flows in external flow fields

This paper describes a new computational model developed to solve two‐dimensional incompressible viscous flow problems in external flow fields. The model based on the Navier–Stokes equations in primitive variables is able to solve the infinite boundary value problems by extracting the boundary effects on a specified finite computational domain, using the pressure projection method. The external flow field is simulated using the boundary element method by solving a pressure Poisson equation that assumes the pressure as zero at the infinite boundary. The momentum equation of the flow motion is solved using the three‐step finite element method. The arbitrary Lagrangian–Eulerian method is incorporated into the model, to solve the moving boundary problems. The present model is applied to simulate various external flow problems like flow across circular cylinder, acceleration and deceleration of the circular cylinder moving in a still fluid and vibration of the circular cylinder induced by the vortex shedding. The simulation results are found to be very reasonable and satisfactory. Copyright © 2001 John Wiley & Sons, Ltd.     

An algorithm for modelling the interaction of a flexible rod with a two‐dimensional high‐speed flow

We present an algorithm for modelling coupled dynamic interactions of a very thin flexible structure immersed in a high‐speed flow. The modelling approach is based on combining an Eulerian finite volume formulation for the fluid flow and a Lagrangian large‐deformation formulation for the dynamic response of the structure. The coupling between the fluid and the solid response is achieved via an approach based on extrapolation and velocity reconstruction inspired in the Ghost Fluid Method. The algorithm presented does not assume the existence of a region exterior to the fluid domain as it was previously proposed and, thus, enables the consideration of very thin open boundaries and structures where the flow may be relevant on both sides of the interface. We demonstrate the accuracy of the method and its ability to describe disparate flow conditions across a fixed thin rigid interface without pollution of the flow field across the solid interface by comparing with analytical solutions of compressible flows. We also demonstrate the versatility and robustness of the method in a complex fluid–structure interaction problem corresponding to the transient supersonic flow past a highly flexible structure. Copyright © 2005 John Wiley & Sons, Ltd.