A monolithic strategy for fluid–structure interaction problems

In this work, we present a new monolithic strategy for solving fluid–structure interaction problems involving incompressible fluids, within the context of the finite element method. This strategy, similar to the continuum dynamics, conserves certain properties, and thus provides a rational basis for the design of the time‐stepping strategy; detailed proofs of the conservation of these properties are provided. The proposed algorithm works with displacement and velocity variables for the structure and fluid, respectively, and introduces no new variables to enforce velocity or traction continuity. Any existing structural dynamics algorithm can be used without change in the proposed method. Use of the exact tangent stiffness matrix ensures that the algorithm converges quadratically within each time step. An analytical solution is presented for one of the benchmark problems used in the literature, namely, the piston problem. A number of benchmark problems including problems involving free surfaces such as sloshing and the breaking dam problem are used to demonstrate the good performance of the proposed method. Copyright © 2010 John Wiley & Sons, Ltd.     

Immersed particle method for fluid–structure interaction

A method for treating fluid–structure interaction of fracturing structures under impulsive loads is described. The coupling method is simple and does not require any modifications when the structure fails and allows fluid to flow through openings between crack surfaces. Both the fluid and the structure are treated by meshfree methods. For the structure, a Kirchhoff–Love shell theory is adopted and the cracks are treated by introducing either discrete (cracking particle method) or continuous (partition of unity‐based method) discontinuities into the approximation. Coupling is realized by a master–slave scheme where the structure is slave to the fluid. The method is aimed at problems with high‐pressure and low‐velocity fluids, and is illustrated by the simulation of three problems involving fracturing cylindrical shells coupled with fluids. Copyright © 2009 John Wiley & Sons, Ltd.     

A Lagrangian finite element approach for the analysis of fluid–structure interaction problems

A Lagrangian finite element method for the analysis of incompressible Newtonian fluid flows, based on a continuous re‐triangulation of the domain in the spirit of the so‐called Particle Finite Element Method, is here revisited and applied to the analysis of the fluid phase in fluid–structure interaction problems. A new approach for the tracking of the interfaces between fluids and structures is proposed. Special attention is devoted to the mass conservation problem. It is shown that, despite its Lagrangian nature, the proposed combined finite element‐particle method is well suited for large deformation fluid–structure interaction problems with evolving free surfaces and breaking waves. The method is validated against the available analytical and numerical benchmarks. Copyright © 2010 John Wiley & Sons, Ltd.     

Topology optimization for stationary fluid–structure interaction problems using a new monolithic formulation

This paper outlines a new procedure for topology optimization in the steady‐state fluid–structure interaction (FSI) problem. A review of current topology optimization methods highlights the difficulties in alternating between the two distinct sets of governing equations for fluid and structure dynamics (hereafter, the fluid and structural equations, respectively) and in imposing coupling boundary conditions between the separated fluid and solid domains. To overcome these difficulties, we propose an alternative monolithic procedure employing a unified domain rather than separated domains, which is not computationally efficient. In the proposed analysis procedure, the spatial differential operator of the fluid and structural equations for a deformed configuration is transformed into that for an undeformed configuration with the help of the deformation gradient tensor. For the coupling boundary conditions, the divergence of the pressure and the Darcy damping force are inserted into the solid and fluid equations, respectively. The proposed method is validated in several benchmark analysis problems. Topology optimization in the FSI problem is then made possible by interpolating Young's modulus, the fluid pressure of the modified solid equation, and the inverse permeability from the damping force with respect to the design variables. Copyright © 2009 John Wiley & Sons, Ltd.     

A new staggered scheme for fluid–structure interaction

Staggered solution procedures represent the most elementary computational strategy for the simulation of fluid–structure interaction problems. They usually consist of a predictor followed by the separate execution of each subdomain solver. Although it is generally possible to maintain the desired order of accuracy of the time integration, it is difficult to guarantee the stability of the overall computation. In the context of large solid over fluid mass ratios, compressible flows and explicit subsolvers, substantial development has been carried out by Felippa, Park, Farhat, Löhner and others. In this work, a new staggered scheme is presented. It is shown that, for a linear model problem, the scheme is second‐order accurate and unconditionally stable. The dependency of the leading truncation error on the solid over fluid mass ratio is investigated. The strategy is applied to two‐dimensional and three‐dimensional fluid–structure interaction problems. It is shown that the conclusions derived from the investigation of the model problem apply. The new strategy extends the applicability of staggered schemes to problems involving relatively small solid over fluid mass ratios and incompressible fluid flow. It is suggested that the proposed scheme has the same range of applicability as the Dirichlet–Neumann or block Gauß–Seidel type strategies. Copyright © 2012 John Wiley & Sons, Ltd.     

Immersed smoothed finite element method for two dimensional fluid–structure interaction problems

A novel method called immersed smoothed FEM using three‐node triangular element is proposed for two‐dimensional fluid–structure interaction (FSI) problems with largely deformable nonlinear solids placed within incompressible viscous fluid. The fluid flows are solved using the semi‐implicit characteristic‐based split method. Smoothed FEMs are employed to calculate the transient responses of solids based on explicit time integration. The fictitious fluid with two assumptions is introduced to achieve the continuous form of the FSI conditions. The discrete formulations to calculate the FSI forces are obtained in terms of the characteristic‐based split scheme, and the algorithm based on a set of fictitious fluid mesh is proposed for evaluating the FSI force exerted on the solid. The accuracy, stability, and convergence properties of immersed smoothed FEM are verified by numerical examples. Investigations on the mesh size ratio indicate that the stability is fairly independent of the wide range of the mesh size ratio. No additional volume correction is required to satisfy the incompressible constraints. Copyright © 2012 John Wiley & Sons, Ltd.     

A comparison of FE–BE coupling schemes for large‐scale problems with fluid–structure interaction

To predict the sound radiation of structures, both a structural problem and an acoustic problem have to be solved. In case of thin structures and dense fluids, a strong coupling scheme between the two problems is essential, since the feedback of the acoustic pressure onto the structure is not negligible. In this paper, the structural part is modeled with the finite element (FE) method. An interface to a commercial FE package is set up to import the structural matrices. The exterior acoustic problem is efficiently modeled with the Galerkin boundary element (BE) method. To overcome the well‐known drawback of fully populated system matrices, the fast multipole method is applied. Different coupling formulations are investigated. They are either based on the Burton–Miller approach or use a mortar coupling scheme. For all cases, iterative solvers with different preconditioners are used. The efficiency with respect to their memory consumption and computation time is compared for a simple model problem. At the end of the paper, a more complex structure is simulated. Copyright © 2008 John Wiley & Sons, Ltd.     

Truly monolithic algebraic multigrid for fluid–structure interaction

The coupling of flexible structures to incompressible fluids draws a lot of attention during the last decade. Many different solution schemes have been proposed. In this contribution, we concentrate on the strong coupling fluid–structure interaction by means of monolithic solution schemes. Therein, a Newton–Krylov method is applied to the monolithic set of nonlinear equations. Such schemes require good preconditioning to be efficient. We propose two preconditioners that apply algebraic multigrid techniques to the entire fluid–structure interaction system of equations. The first is based on a standard block Gauss–Seidel approach, where approximate inverses of the individual field blocks are based on a algebraic multigrid hierarchy tailored for the type of the underlying physical problem. The second is based on a monolithic coarsening scheme for the coupled system that makes use of prolongation and restriction projections constructed for the individual fields. The resulting nonsymmetric monolithic algebraic multigrid method therefore involves coupling of the fields on coarse approximations to the problem yielding significantly enhanced performance. Copyright © 2010 John Wiley & Sons, Ltd.     

Adaptive finite element approximation of multiphysics problems: A fluid–structure interaction model problem

In this paper we consider finite element simulation of the mechanical response of an elastic solid immersed into a viscous incompressible fluid flow. For simplicity, we assume that the mechanics of the solid is governed by linear elasticity and the motion of the fluid by the Stokes equation. For this one‐way coupled multiphysics problem we derive an a posteriori error estimate using duality techniques. Based on the estimate we propose an adaptive algorithm that automatically constructs a suitable mesh for the fluid and solid computational domains given a specific goal quantity for the elastic problem. Copyright © 2010 John Wiley & Sons, Ltd.     

The fixed‐mesh ALE approach applied to solid mechanics and fluid–structure interaction problems

In this paper we propose a method to solve Solid Mechanics and fluid–structure interaction problems using always a fixed background mesh for the spatial discretization. The main feature of the method is that it properly accounts for the advection of information as the domain boundary evolves. To achieve this, we use an Arbitrary Lagrangian–Eulerian (ALE) framework, the distinctive characteristic being that at each time step results are projected onto a fixed, background mesh. For solid mechanics problems subject to large strains, the fixed‐mesh (FM)‐ALE method avoids the element stretching found in fully Lagrangian approaches. For FSI problems, FM‐ALE allows for the use of a single background mesh to solve both the fluid and the structure. Copyright © 2009 John Wiley & Sons, Ltd.     

Analysis of space mapping algorithms for application to partitioned fluid–structure interaction problems

Fluid–structure interactions (FSI) play a crucial role in many engineering fields. However, the computational cost associated with high‐fidelity aeroelastic models currently precludes their direct use in industry, especially for strong interactions. The strongly coupled segregated problem—that results from domain partitioning—can be interpreted as an optimization problem of a fluid–structure interface residual. Multi‐fidelity optimization techniques can therefore directly be applied to this problem in order to obtain the solution efficiently. In previous work, it is already shown that aggressive space mapping (ASM) can be used in this context. In this contribution, we extend the research towards the use of space mapping for FSI simulations. We investigate the performance of two other approaches, generalized space mapping and output space mapping, by application to both compressible and incompressible 2D problems. Moreover, an analysis of the influence of the applied low‐fidelity model on the achievable speedup is presented. The results indicate that output space mapping is a viable alternative to ASM when applied in the context of solver coupling for partitioned FSI, showing similar performance as ASM and resulting in reductions in computational cost up to 50% with respect to the reference quasi‐Newton method. Copyright © 2015 John Wiley & Sons, Ltd.     

A parallel hp‐FEM infrastructure for three‐dimensional structural acoustics

This paper describes a parallel three‐dimensional numerical infrastructure for the solution of a wide range of time‐harmonic problems in structural acoustics and vibration. High accuracy and rate of error‐convergence, in the mid‐frequency regime,is achieved by the use of hp‐finite and infinite element approximations. The infrastructure supports parallel computation in both single and multi‐frequency settings. Multi‐frequency solves utilize concurrent factoring of the frequency‐dependent linear algebraic systems and are naturally scalable. Scalability of large‐scale single‐frequency problems is realized by using FETI‐DP—an iterative domain‐decomposition scheme. Numerical examples are presented to cover applications in vibratory response of fluid‐filled elastic structures as well as radiation and scattering from elastic structures submerged in an infinite acoustic medium. We demonstrate both the numerical accuracy as well as parallel scalability of the infrastructure in terms of problem parameters that include wavenumber and number of frequencies, polynomial degree of finite/infinite element approximations as well as the number of processors. Scalability and accuracy is evaluated for both single and multiple frequency sweeps on four high‐performance parallel computing platforms: SGI Altix, SGI Origin, IBM p690 SP and Linux‐cluster. Results show good performance on shared as well as distributed‐memory architecture. Copyright © 2006 John Wiley & Sons, Ltd.     

A unified formulation for continuum mechanics applied to fluid–structure interaction in flexible tubes

This paper outlines the development of a new procedure for analysing continuum mechanics problems with a particular focus on fluid–structure interaction in flexible tubes. A review of current methods of fluid–structure coupling highlights common limitations of high computational cost and solution instability. It is proposed that these limitations can be overcome by an alternative approach in which both fluid and solid components are solved within a single discretized continuum domain. A single system of momentum and continuity equations is therefore derived that governs both fluids and solids and which are solved with a single mesh using finite volume discretization schemes. The method is validated first by simulating dynamic oscillation of a clamped elastic beam. It is then applied to study the case of interest—wave propagation in highly flexible tubes—in which a predicted wave speed of 8.58 m/s falls within 2% of an approximate analytical solution. The method shows further good agreement with analytical solutions for tubes of increasing rigidity, covering a range of wave speeds from those found in arteries to that in the undisturbed fluid. Copyright © 2005 John Wiley & Sons, Ltd.     

Partitioned vibration analysis of internal fluid‐structure interaction problems

A partitioned, continuum‐based, internal fluid–structure interaction (FSI) formulation is developed for modeling combined sloshing, acoustic waves, and the presence of an initial pressurized state. The present formulation and its computer implementation use the method of localized Lagrange multipliers to treat both matching and non‐matching interfaces. It is shown that, with the context of continuum Lagrangian kinematics, the fluid sloshing and acoustic stiffness terms originate from an initial pressure term akin to that responsible for geometric stiffness effects in solid mechanics. The present formulation is applicable to both linearized vibration analysis and nonlinear FSI transient analysis provided that a convected kinematics is adopted for updating the mesh geometry in a finite element discretization. Numerical examples illustrate the capability of the present procedure for solving coupled vibration and nonlinear sloshing problems. Copyright © 2012 John Wiley & Sons, Ltd.     

FETI‐DP: a dual–primal unified FETI method—part I: A faster alternative to the two‐level FETI method

The FETI method and its two‐level extension (FETI‐2) are two numerically scalable domain decomposition methods with Lagrange multipliers for the iterative solution of second‐order solid mechanics and fourth‐order beam, plate and shell structural problems, respectively.The FETI‐2 method distinguishes itself from the basic or one‐level FETI method by a second set of Lagrange multipliers that are introduced at the subdomain cross‐points to enforce at each iteration the exact continuity of a subset of the displacement field at these specific locations. In this paper, we present a dual–primal formulation of the FETI‐2 concept that eliminates the need for that second set of Lagrange multipliers, and unifies all previously developed one‐level and two‐level FETI algorithms into a single dual–primal FETI‐DP method. We show that this new FETI‐DP method is numerically scalable for both second‐order and fourth‐order problems. We also show that it is more robust and more computationally efficient than existing FETI solvers, particularly when the number of subdomains and/or processors is very large. Copyright © 2001 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.     

Analysis of transient flow in pipelines with fluid–structure interaction using method of lines

A mathematical model is presented for transient flow in a pipeline with fluid–structure interaction. Water hammer theory and equations of axial motion for the pipeline are employed and the Poisson, junction and transient shear stress couplings are taken into account, which give rise to four coupled non‐linear, first‐order hyperbolic partial differential equations governing the fluid flow and pipe motion. These equations are discretized in space using the Keller box scheme and the method of lines is employed to reduce the partial differential equations to a system of ordinary differential equations. The resulting system is solved using a backward differentiation formulation method. The effect of transient shear stress on transient flow is investigated and the mechanisms underlying this effect are explored. The results revealed that the influence of transient shear stress can be significant and varies considerably, depending on the boundary conditions, viz, valve closure time. Copyright © 2005 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.