A reduced‐order representation of the Poincaré–Steklov operator: an application to coupled multi‐physics problems

This work investigates a model reduction method applied to coupled multi‐physics systems. The case in which a system of interest interacts with an external system is considered. An approximation of the Poincaré–Steklov operator is computed by simulating, in an offline phase, the external problem when the inputs are the Laplace–Beltrami eigenfunctions defined at the interface. In the online phase, only the reduced representation of the operator is needed to account for the influence of the external problem on the main system. An online basis enrichment is proposed in order to guarantee a precise reduced‐order computation. Several test cases are proposed on different fluid–structure couplings. Copyright © 2016 John Wiley & Sons, Ltd.     

A partitioned approach for the coupling of SPH and FE methods for transient nonlinear FSI problems with incompatible time‐steps

We propose a method to couple smoothed particle hydrodynamics and finite elements methods for nonlinear transient fluid–structure interaction simulations by adopting different time‐steps depending on the fluid or solid sub‐domains. These developments were motivated by the need to simulate highly non‐linear and sudden phenomena requiring the use of explicit time integrators on both sub‐domains (explicit Newmark for the solid and Runge–Kutta 2 for the fluid). However, due to critical time‐step required for the stability of the explicit time integrators in, it becomes important to be able to integrate each sub‐domain with a different time‐step while respecting the features that a previously developed mono time‐step coupling algorithm offered. For this matter, a dual‐Schur decomposition method originally proposed for structural dynamics was considered, allowing to couple time integrators of the Newmark family with different time‐steps with the use of Lagrange multipliers. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

A dual‐primal FETI method for solving a class of fluid–structure interaction problems in the frequency domain

The dual‐primal finite element tearing and interconnecting method (FETI‐DP) is extended to systems of linear equations arising from a finite element discretization for a class of fluid–structure interaction problems in the frequency domain. A preconditioned generalized minimal residual method is used to solve the linear equations for the Lagrange multipliers introduced on the subdomain boundaries to enforce continuity of the solution. The coupling between the fluid and the structure on the fluid–structure interface requires an appropriate choice of coarse level degrees of freedom in the FETI‐DP algorithm to achieve fast convergence. Several choices are proposed and tested by numerical experiments on three‐dimensional fluid–structure interaction problems in the mid‐frequency regime that demonstrate the greatly improved performance of the proposed algorithm over the standard FETI‐DP method. Copyright © 2011 John Wiley & Sons, Ltd.     

The fractal finite element method for added‐mass‐type problems

The fractal finite element method (FFEM), originally developed for calculating stress intensity factors in fracture mechanics problems, has been extended to analyse fluid–structure interaction in the form of added‐mass‐type problems. These include the free vibration of a submerged spherical shell and the interaction between a dam and a reservoir. For the former problem, the numerical solution from the FFEM agrees well with the analytical solution, and the FFEM performed better than conventional finite elements and infinite elements in terms of efficiency. For the latter problem, the FFEM predicted an added mass profile that is different from that based on Westergaard's parabolic solution. Copyright © 2008 John Wiley & Sons, Ltd.     

Improved multi‐level Newton solvers for fully coupled multi‐physics problems

An algorithm is suggested to improve the efficiency of the multi‐level Newton method that is used to solve multi‐physics problems. It accounts for full coupling between the subsystems by using the direct differentiation method rather than error prone finite difference calculations and retains the advantage of greater flexibility over the tightly coupled approaches. Performance of the algorithm is demonstrated by solving a fluid–structure interaction problem. Copyright © 2003 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.     

The particle finite element method: a powerful tool to solve incompressible flows with free‐surfaces and breaking waves

Particle Methods are those in which the problem is represented by a discrete number of particles. Each particle moves accordingly with its own mass and the external/internal forces applied to it. Particle Methods may be used for both, discrete and continuous problems. In this paper, a Particle Method is used to solve the continuous fluid mechanics equations. To evaluate the external applied forces on each particle, the incompressible Navier–Stokes equations using a Lagrangian formulation are solved at each time step. The interpolation functions are those used in the Meshless Finite Element Method and the time integration is introduced by an implicit fractional‐step method. In this manner classical stabilization terms used in the momentum equations are unnecessary due to lack of convective terms in the Lagrangian formulation. Once the forces are evaluated, the particles move independently of the mesh. All the information is transmitted by the particles. Fluid–structure interaction problems including free‐fluid‐surfaces, breaking waves and fluid particle separation may be easily solved with this methodology. Copyright © 2004 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.     

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.     

On time integration of viscoplastic constitutive models suitable for creep

Creep of critical components such as electrical solder connections may occur over long periods of time. Efficient numerical simulations of such problems generally require the use of quasi‐static formulations with conjugate‐gradient techniques for solving the large number of algebraic equations. Implicit in the approach is the need to solve the constitutive equation several times for large time steps and for loading directions that may have no resemblance to the actual solution. Therefore, an unconditionally stable and efficient algorithm for solving the constitutive equation is essential for the overall efficiency of the solution procedure. Unfortunately, constitutive equations suitable for simulating the materials of interest are notoriously difficult to solve numerically and most existing algorithms have a stability limit on the time step which may be several orders of magnitude smaller than the desired time step. Here an algorithm is proposed which is a combination of the use of a trapezoidal rule and an iterative Newton–Raphson method for solving implicitly the non‐linear equations. The key to the success of the proposed approach is to always use an initial guess based on the steady‐state solution to the constitutive equation. A representative viscoplastic constitutive equation is used as a model for illustrating the approach. The algorithm is developed and typical numerical results are provided to substantiate the claim that stability has been achieved. Copyright © 2001 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.     

An extended finite element method approach for structural‐acoustic problems involving immersed structures at arbitrary positions

Noise reduction for passengers' comfort in transport industry is now an important constraint to be taken into account during the design process. This process involves to study several configurations of the structures immersed in a given acoustic cavity in the context of an optimization, uncertainty, or reliability study for instance. The finite element method may be used to model this coupled fluid–structure problem but needs an interface conforming mesh for each studied configuration that may become time consuming. This work aims at avoiding this remeshing step by using noncompatible meshes between the fluid and the structures. The immersed structures are supposed to be thin shells and are localized in the fluid domain by a signed distance level‐set. To take into account the pressure discontinuity from one side of the structures to the other one, the fluid pressure approximation is enriched according to the structures positions by a Heaviside function using a partition of unity strategy (extended finite element method). The same fluid mesh of the empty cavity is then used during the whole parametric study. The method is implemented for a three‐dimensional fluid and tested on academic examples before being applied to an industrial‐like case. Copyright © 2012 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.     

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.     

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.     

Updated Lagrangian/Arbitrary Lagrangian–Eulerian framework for interaction between a compressible neo‐Hookean structure and an incompressible fluid

We propose a numerical method for a fluid–structure interaction problem. The material of the structure is homogeneous, isotropic, and it can be described by the compressible neo‐Hookean constitutive equation, while the fluid is governed by the Navier–Stokes equations. Our study does not use turbulence model. Updated Lagrangian method is used for the structure and fluid equations are written in Arbitrary Lagrangian–Eulerian coordinates. One global moving mesh is employed for the fluid–structure domain, where the fluid–structure interface is an ‘interior boundary’ of the global mesh. At each time step, we solve a monolithic system of unknown velocity and pressure defined on the global mesh. The continuity of velocity at the interface is automatically satisfied, while the continuity of stress does not appear explicitly in the monolithic fluid–structure system. This method is very fast because at each time step, we solve only one linear system. This linear system was obtained by the linearization of the structure around the previous position in the updated Lagrangian formulation and by the employment of a linear convection term for the fluid. Numerical results are presented. Copyright © 2016 John Wiley & Sons, Ltd.     

Robust and efficient monolithic fluid‐structure‐interaction solvers

We construct new robust and efficient preconditioned generalized minimal residual solvers for the monolithic linear systems of algebraic equations arising from the finite element discretization and Newton's linearization of the fully coupled fluid–structure interaction system of partial differential equations in the arbitrary Lagrangian–Eulerian formulation. We admit both linear elastic and nonlinear hyperelastic materials in the solid model and cover a large range of flows, for example, water, blood, and air, with highly varying density. The preconditioner is constructed in form of , where , , and are proper approximations to the matrices L, D, and U in the LDU block factorization of the fully coupled system matrix, respectively. The inverse of the corresponding Schur complement is approximated by applying a few cycles of a special class of algebraic multigrid methods to the perturbed fluid sub‐problem, which is obtained by modifying corresponding entries in the original fluid matrix with an explicitly constructed approximation to the exact perturbation coming from the sparse matrix–matrix multiplications. The numerical studies presented impressively demonstrate the robustness and the efficiency of the preconditioner proposed in the paper. Copyright © 2016 John Wiley & Sons, Ltd.     

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.