Generalized Robin–Neumann explicit coupling schemes for incompressible fluid‐structure interaction: Stability analysis and numerics

We introduce a new class of explicit coupling schemes for the numerical solution of fluid‐structure interaction problems involving a viscous incompressible fluid and an elastic structure. These methods generalize the arguments reported in [Comput. Methods Appl. Mech. Engrg., 267:566–593, 2013, Numer. Math., 123(1):21–65, 2013] to the case of the coupling with thick‐walled structures. The basic idea lies in the derivation of an intrinsic interface Robin consistency at the space semi‐discrete level, using a lumped‐mass approximation in the structure. The fluid–solid splitting is then performed through appropriate extrapolations of the solid velocity and stress on the interface. Based on these methods, a new, parameter‐free, Robin–Neumann iterative procedure is also proposed for the partitioned solution of implicit coupling. A priori energy estimates, guaranteeing the stability of the schemes and the convergence of the iterative procedure, are established within a representative linear setting. The accuracy and performance of the methods are illustrated in several numerical examples. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

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.     

Immersed stress method for fluid–structure interaction using anisotropic mesh adaptation

This paper presents advancements toward a monolithic solution procedure and anisotropic mesh adaptation for the numerical solution of fluid–structure interaction with complex geometry. First, a new stabilized three‐field stress, velocity, and pressure finite element formulation is presented for modeling the interaction between the fluid (laminar or turbulent) and the rigid body. The presence of the structure will be taken into account by means of an extra stress in the Navier–Stokes equations. The system is solved using a finite element variational multiscale method. We combine this method with anisotropic mesh adaptation to ensure an accurate capturing of the discontinuities at the fluid–solid interface. We assess the behavior and accuracy of the proposed formulation in the simulation of 2D and 3D time‐dependent numerical examples such as the flow past a circular cylinder and turbulent flows behind an immersed helicopter in a forward flight. Copyright © 2013 John Wiley & Sons, Ltd.     

A decoupled augmented IIM strategy for incompressible two‐phase flows with interfaces on irregular domains

A decoupled augmented immersed interface method for solving incompressible two‐phase flows involving both irregular domains and interfaces is presented. In order to impose the prescribed velocity at the boundary of the irregular domain, singular force as one set of augmented variables is introduced. The velocity components at the two‐fluid interface as another set of augmented variables are introduced to satisfy the continuity condition of the velocity across the interface so that the jump conditions for the velocity and pressure are decoupled across the interface. The augmented variables and/or the forces along the interface/boundary are related to the jumps in both pressure and velocity and the jumps in their derivatives across the interface/boundary and applied to the fluid through jump conditions. The resulting augmented equation is a couple system of these two sets of augmented variables, and the direct application of the GMRES is impractical due to larger iterations. In this work, the novel decoupling of two sets of the augmented variables is proposed, and the decoupled augmented equation is then solved by the LU or the GMRES method. The Stokes equations are discretized via the finite difference method with the incorporation of jump contributions on a staggered Cartesian grid and solved by the conjugate gradient Uzawa‐type method. The numerical results show that second‐order accuracy for the velocity is confirmed. The present method has also been applied to solve for incompressible two‐phase Navier–Stokes flow with interfaces on irregular domains. Copyright © 2011 John Wiley & Sons, Ltd.     

A computational approach for predicting the hydroelasticity of flexible structures based on the pressure Poisson equation

This work is concerned with the modeling of the interaction of fluid flow with flexible solid structures. The flow under consideration is governed by the Navier–Stokes equations for incompressible viscous fluids and modeled with low‐order velocity–pressure finite elements. The motion of the fluid domain is accounted for by the arbitrary Lagrangian–Eulerian formulation. The structure is represented by means of an appropriate standard finite element formulation. The spring smooth analogy is used to mesh control. The time integrating algorithm is based on the predictor–multi‐corrector algorithm. An important aspect of the present work is the introduction of a new monolithic approach based on the fluid pressure Poisson equation (PPE) to solve the hydroelasticity problem of an incompressible viscous fluid with an elastic body that is vibrating due to flow excitation. The PPE is derived to be consistent with the coupled system equation for the fluid–structure interaction (FSI). Based on this approach, an efficient monolithic method is adopted to simulate hydroelasticity between the flexible structure and the flow. The fluid pressure is implicitly derived to satisfy the incompressibility constraint, and the other unknown variables are explicitly derived. The coefficient matrix of the PPE for the FSI becomes symmetric and positive definite. To demonstrate the performance of the proposed approach, two working examples, a beam immersed in incompressible fluid and a guide vane of a Francis turbine passage, were used. The results show the validity of the proposed approach. Copyright © 2007 John Wiley & Sons, Ltd.     

A particle‐based moving interface method (PMIM) for modeling the large deformation of boundaries in soft matter systems

The mechanics of the interaction between a fluid and a soft interface undergoing large deformations appear in many places, such as in biological systems or industrial processes. We present an Eulerian approach that describes the mechanics of an interface and its interactions with a surrounding fluid via the so‐called Navier boundary condition. The interface is modeled as a curvilinear surface with arbitrary mechanical properties across which discontinuities in pressure and tangential fluid velocity can be accounted for using a modified version of the extended finite element method. The coupling between the interface and the fluid is enforced through the use of Lagrange multipliers. The tracking and evolution of the interface are then handled in a Lagrangian step with the grid‐based particle method. We show that this method is ideal to describe large membrane deformations and Navier boundary conditions on the interface with velocity/pressure discontinuities. The validity of the model is assessed by evaluating the numerical convergence for a axisymmetrical flow past a spherical capsule with various surface properties. We show the effect of slip length on the shear flow past a two‐dimensional capsule and simulate the compression of an elastic membrane lying on a viscous fluid substrate. Copyright © 2015 John Wiley & Sons, Ltd.     

A CBS‐type stabilizing algorithm for the consolidation of saturated porous media

The presented method stems from the works by Zienkiewicz and co‐workers for coupled fluid/thermal problems starting from the early 1990s. They propose algorithms to overcome the difficulties connected to the application of the FEM to the area of fluid mechanics, which include the problems of singular behaviour in incompressibility and the problems connected to convective terms. The major step forward was to introduce the concept of characteristic lines (the particle paths in a simple convection situation): for a class of problems with a single scalar variable, the equations in the characteristic co‐ordinates regain self‐adjointness. The procedure is called characteristic based split algorithm (CBS). We use here a CBS‐type procedure for a saturated deformable elastic porous medium, in which the fluid velocity is governed by Darcy's equation (which comes directly from Navier–Stokes ones). The physical–mathematical model is a fully coupled one and is here used to study an incompressible flow inside a continuum with incompressible solid grains. The power of the adopted algorithm is to treat the basic equations in their strong form and to transform a usual ‘u–p’ problem into a ‘u–v–p’ one, where u generally indicates the displacement of the solid matrix and p and v the pressure and velocity of the fluid, respectively. Attention is focused on the expression of Darcy's velocity which is considered as the starting point of the algorithm. The accuracy of the scheme is checked by comparing the present predictions in a typical consolidation test with available analytical and numerical u–p solutions. A good fitting among different results has been obtained. It is further shown that the procedure eliminates the oscillations at the onset of consolidation, typical for many schemes. The FEM code Ed‐Multifield has been used for implementing and testing the procedure. Copyright © 2005 John Wiley & Sons, Ltd.     

Numerical modeling of Kelvin–Helmholtz instability using smoothed particle hydrodynamics

This paper presents a Smoothed Particle Hydrodynamics (SPH) solution for the Kelvin–Helmholtz Instability (KHI) problem of an incompressible two‐phase immiscible fluid in a stratified inviscid shear flow with interfacial tension. The time‐dependent evolution of the two‐fluid interface over a wide range of Richardson number (Ri) and for three different density ratios is numerically investigated. The simulation results are compared with analytical solutions in the linear regime. Having captured the physics behind KHI, the effects of gravity and surface tension on a two‐dimensional shear layer are examined independently and together. It is shown that the growth rate of the KHI is mainly controlled by the value of the Ri number, not by the nature of the stabilizing forces. It was observed that the SPH method requires a Richardson number lower than unity (i.e. Ri?0.8) for the onset of KHI, and that the artificial viscosity plays a significant role in obtaining physically correct simulation results that are in agreement with analytical solutions. The numerical algorithm presented in this work can easily handle two‐phase fluid flow with various density ratios. Copyright © 2011 John Wiley & Sons, Ltd.     

New Finite Difference Methods Based on IIM for Inextensible Interfaces in Incompressible Flows

In this paper, new finite difference methods based on the augmented immersed interface method (IIM) are proposed for simulating an inextensible moving interface in an incompressible two-dimensional flow. The mathematical models arise from studying the deformation of red blood cells in mathematical biology. The governing equations are incompressible Stokes or Navier-Stokes equations with an unknown surface tension, which should be determined in such a way that the surface divergence of the velocity is zero along the interface. Thus, the area enclosed by the interface and the total length of the interface should be conserved during the evolution process. Because of the nonlinear and coupling nature of the problem, direct discretization by applying the immersed boundary or immersed interface method yields complex nonlinear systems to be solved. In our new methods, we treat the unknown surface tension as an augmented variable so that the augmented IIM can be applied. Since finding the unknown surface tension is essentially an inverse problem that is sensitive to perturbations, our regularization strategy is to introduce a controlled tangential force along the interface, which leads to a least squares problem. For Stokes equations, the forward solver at one time level involves solving three Poisson equations with an interface. For Navier-Stokes equations, we propose a modified projection method that can enforce the pressure jump condition corresponding directly to the unknown surface tension. Several numerical experiments show good agreement with other results in the literature and reveal some interesting phenomena.     

Multi‐linearity algorithm for wall slip in two‐dimensional gap flow

Wall slip has been observed in a micro/nanometer gap during the past few years. It is difficult to make a mathematical analysis for the hydrodynamics of the fluid flowing in a gap with wall slip because the fluid velocity at the liquid–solid interface is not known a priori. This difficulty is met especially in a two‐dimensional slip flow due to the non‐linearity of the slip control equation. In the present paper we developed a multi‐linearity method to approach the non‐linear control equation of the two‐dimensional slip gap flow. We used an amended polygon to approximate the circle yield (slip) boundary of surface shear stress. The numerical solution does not need an iterative process and can simultaneously give rise to fluid pressure distribution, wall slip velocity and surface shear stress. We analysed the squeeze film flow between two parallel discs and the hydrodynamics of a finite slider gap with wall slip. Our numerical solutions show that wall slip is first developed in the large pressure gradient zone, where a high surface shear stress is easily generated, and then the slip zone is enlarged with the increase in the shear rate. Wall slip dramatically affects generation of the hydrodynamic pressure. Copyright © 2006 John Wiley & Sons, Ltd.     

Velocity‐based formulations for standard and quasi‐incompressible hypoelastic‐plastic solids

We present three velocity‐based updated Lagrangian formulations for standard and quasi‐incompressible hypoelastic‐plastic solids. Three low‐order finite elements are derived and tested for non‐linear solid mechanics problems. The so‐called V‐element is based on a standard velocity approach, while a mixed velocity–pressure formulation is used for the VP and the VPS elements. The two‐field problem is solved via a two‐step Gauss–Seidel partitioned iterative scheme. First, the momentum equations are solved in terms of velocity increments, as for the V‐element. Then, the constitutive relation for the pressure is solved using the updated velocities obtained at the previous step. For the VPS‐element, the formulation is stabilized using the finite calculus method in order to solve problems involving quasi‐incompressible materials. All the solid elements are validated by solving two‐dimensional and three‐dimensional benchmark problems in statics as in dynamics. Copyright © 2016 John Wiley & Sons, Ltd.     

An augmented incompressible material point method for modeling liquid sloshing problems

The incompressible material point method was proposed for modeling the free surface flow problems based on the operator splitting technique which decouples the solution of the velocity and the pressure in our previous work. To further model the coupling problems between the incompressible fluid and the moving irregular solid bodies, an augmented incompressible material point method is proposed in this paper based on the energy minimization form of operator splitting technique. The interaction between the fluid and the solid is taken into account via the work done by the fluid pressure on the solid bodies. By minimizing the total work done by the fluid pressure, volume-weighted pressure Poisson equations are obtained. The proposed method is validated with liquid sloshing in a rectangular tank subjected to various base-excitations, and is then used to study the optimal height of baffles mounted on the bottom of the tank to mitigate the sloshing wave.     

An arbitrary Lagrangian‐Eulerian framework with exact mass conservation for the numerical simulation of 2D rising bubble problem

An arbitrary Lagrangian‐Eulerian framework, which combines the advantages of both Lagrangian and Eulerian methods, is presented to solve incompressible multiphase flow problems. The incompressible Navier‐Stokes equations are discretized using the side‐centered unstructured finite volume method, where the velocity vector components are defined at the midpoint of each cell face, while the pressure term is defined at element centroids. The pressure field is treated to be discontinuous across the interface with the discontinuous treatment of density and viscosity. The surface tension term at the interface is treated as a force tangent to the interface and computed with several different approaches including the use of Legendre polynomials. In addition, the several different discretizations of interface kinematic boundary conditions are investigated. For the application of the interface kinematic boundary condition, a special attention is given to satisfy both local and global discrete geometric conservation law to conserve the total mass of both species at machine precision. The mesh vertices are deformed by solving the linear elasticity equations due to the normal displacement of interface. The resulting algebraic equations are solved in a fully coupled manner, and a one‐level restricted additive Schwarz preconditioner with a block‐incomplete factorization within each partitioned subdomain is used for the resulting fully coupled system. The method is validated by simulating the classical benchmark problem of a single rising bubble in a viscous fluid due to buoyancy. The results of numerical simulations are found out to be in an excellent agreement with the earlier results in the literature. The mass of the bubble is conserved, and discontinuous pressure field is obtained to avoid errors due to the incompressibility condition in the vicinity of the interface, where the density and viscosity jumps occur.     

Extended space–time finite elements for landslide dynamics

The paper introduces a methodology for numerical simulation of landslides experiencing considerable deformations and topological changes. Within an interface capturing approach, all interfaces are implicitly described by specifically defined level‐set functions allowing arbitrarily evolving complex topologies. The transient interface evolution is obtained by solving the level‐set equation driven by the physical velocity field for all three level‐set functions in a block Jacobi approach. The three boundary‐coupled fluid‐like continua involved are modeled as incompressible, governed by a generalized non‐Newtonian material law taking into account capillary pressure at moving fluid–fluid interfaces. The weighted residual formulation of the level‐set equations and the fluid equations is discretized with finite elements in space and time using velocity and pressure as unknown variables. Non‐smooth solution characteristics are represented by enriched approximations to fluid velocity (weak discontinuity) and fluid pressure (strong discontinuity). Special attention is given to the construction of enriched approximations for elements containing evolving triple junctions. Numerical examples of three‐fluid tank sloshing and air–water‐liquefied soil landslides demonstrate the potential and applicability of the method in geotechnical investigations. 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.     

NURBS based least‐squares finite element methods for fluid and solid mechanics

This contribution investigates the performance of a least‐squares finite element method based on non‐uniform rational B‐splines (NURBS) basis functions. The least‐squares functional is formulated directly in terms of the strong form of the governing equations and boundary conditions. Thus, the introduction of auxiliary variables is avoided, but the order of the basis functions must be higher or equal to the order of the highest spatial derivatives. The methodology is applied to the incompressible Navier–Stokes equations and to linear as well as nonlinear elastic solid mechanics. The numerical examples presented feature convective effects and incompressible or nearly incompressible material. The numerical results, which are obtained with equal‐order interpolation and without any stabilisation techniques, are smooth and accurate. It is shown that for p and h refinement, the theoretical rates of convergence are achieved. Copyright © 2014 John Wiley & Sons, Ltd.     

On singularities in the solution of three‐dimensional Stokes flow and incompressible elasticity problems with corners

In this paper, a numerical procedure is presented for the computation of corner singularities in the solution of three‐dimensional Stokes flow and incompressible elasticity problems near corners of various shape. For obtaining the order and mode of singularity, a neighbourhood of the singular point is considered with only local boundary conditions. The weak formulation of this problem is approximated using a mixed u , p Galerkin–Petrov finite element method. Additionally, a separation of variables is used to reduce the dimension of the original problem. As a result, the quadratic eigenvalue problem ( P +λ Q +λ² R ) d = 0 is obtained, where the saddle‐point‐type matrices P , Q , R are defined explicitly. For a numerical solution of the algebraic eigenvalue problem an iterative technique based on the Arnoldi method in combination with an Uzawa‐like scheme is used. This technique needs only one direct matrix factorization as well as few matrix–vector products for finding all eigenvalues in the interval ??(λ) ∈ (?0.5, 1.0), as well as the corresponding eigenvectors. Some benchmark tests show that this technique is robust and very accurate. Problems from practical importance are also analysed, for instance the surface‐breaking crack in an incompressible elastic material and the three‐dimensional viscous flow of a Newtonian fluid past a trihedral corner. Copyright © 2004 John Wiley & Sons, Ltd.