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 implicit boundary element solution with consistent linearization for free surface flows and non‐linear fluid–structure interaction of floating bodies

In this work, a new comprehensive method has been developed which enables the solution of large, non‐linear motions of rigid bodies in a fluid with a free surface. The application of the modern Eulerian–Lagrangian approach has been translated into an implicit time‐integration formulation, a development which enables the use of larger time steps (where accuracy requirements allow it). Novel features of this project include: (1) an implicit formulation of the rigid‐body motion in a fluid with a free surface valid for both two or three dimensions and several moving bodies; (2) a complete formulation and solution of the initial conditions; (3) a fully consistent (exact) linearization for free surface flows valid for any boundary elements such that optimal convergence properties are obtained when using a Newton–Raphson solver. The proposed framework has been completed with details on implementation issues referring mainly to the computation of the complete initial conditions and the consistent linearization of the formulation for free surface flows. The second part of the paper demonstrates the mathematical and numerical formulation through numerical results simulating large free surface flows and non‐linear fluid structure interaction. The implicit formulation using a fully consistent linearization based on the boundary element method and the generalized trapezoidal rule has been applied to the solution of free surface flows for the evolution of a triangular wave, the generation of tsunamis and the propagation of a wave up to overturning. Fluid–structure interaction examples include the free and forced motion of a circular cylinder and the sway, heave and roll motion of a U‐shaped body in a tank with a flap wave generator. The presented examples demonstrate the applicability and performance of the implicit scheme with consistent linearization. Copyright © 2001 John Wiley & Sons. Ltd.     

Flexible multibody systems–fluid interaction

Modelling the dynamics of a flexible multibody system coupled to a rigid container carrying a fluid with a free surface is addressed. The proposed methodology allows the analyst to implement all sort of non-linearities inherent in the dynamics of the structure. Potential flow with modified Rayleigh damping is used to model the fluid. Non-linear sloshing effects are considered and no simplifications are made on the field equations and boundary conditions. A set of first-order differential equations for the motion of both the structure and the fluid are presented. Emphasis is placed on the point that the motion of the flexible multibody system is not prescribed but is found as part of the solution procedure. Some improvements are presented with respect to a previous introductory work by the authors. Detailed derivations and two numerical examples are presented: a flexible column supporting a rigid water tank (with a comparison using an approximate method) and a double flexible-link pendulum coupled to a rigid container. © 1998 This paper was produced under the auspices of the U.S. Government and it is therefore not subjected to copyright in the U.S.     

Full SPH fluid–shell interaction for leakage simulation in explicit dynamics

This paper is devoted to the modeling of fluid leakage through a shell in case of impact loading. The modeling of the fluid and the shell is based on SPH formulation. The proposed model is devoted to the prediction of failure of a shell filled with fluid. This paper is devoted to the numerical modeling of fluid–structure interaction. The pinballs technique is used for contact analysis. Numerical predictions are compared with analytical as well as with an original experiment. Copyright © 2009 John Wiley & Sons, Ltd.     

DISPLACEMENT/PRESSURE BASED MIXED FINITE ELEMENT FORMULATIONS FOR ACOUSTIC FLUID–STRUCTURE INTERACTION PROBLEMS

We present reliable finite element discretizations based on displacement/pressure interpolations for the analysis of acoustic fluid–structure interaction problems. The finite element interpolations are selected using the inf-sup condition, and emphasis is given to the fact that the boundary conditions must satisfy the mass and momentum conservation. We show that with our analysis procedure no spurious non-zero frequencies are encountered, as heretofore calculated with other displacement-based discretizations. © 1997 by John Wiley & Sons, Ltd.     

A new stabilized finite element method for reaction–diffusion problems: The source‐stabilized Petrov–Galerkin method

This paper proposes a new stabilized finite element method to solve singular diffusion problems described by the modified Helmholtz operator. The Galerkin method is known to produce spurious oscillations for low diffusion and various alternatives were proposed to improve the accuracy of the solution. The mostly used methods are the well‐known Galerkin least squares and Galerkin gradient least squares (GGLS). The GGLS method yields the exact nodal solution in the one‐dimensional case and for a uniform mesh. However, the behavior of the method deteriorates slightly in the multi‐dimensional case and for non‐uniform meshes. In this work we propose a new stabilized finite element method that leads to improved accuracy for multi‐dimensional problems. For the one‐dimensional case, the new method leads to the same results as the GGLS method and hence provides exact nodal solutions to the problem on uniform meshes. The proposed method is a Galerkin discretization used to solve a modified equation that includes a term depending on the gradient of the original partial differential equation. Copyright © 2008 John Wiley & Sons, Ltd.     

FINITE ELEMENT SOLUTION OF INCOMPRESSIBLE FLUID–STRUCTURE VIBRATION PROBLEMS

In this paper we solve an eigenvalue problem arising from the computation of the vibrations of a coupled system, incompressible fluid – elastic structure, in absence of external forces. We use displacement variables for both the solid and the fluid but the fluid displacements are written as curls of a stream function. Classical linear triangular finite elements are used for the solid displacements and for the stream function in the fluid. The kinematic transmission conditions at the fluid–solid interface are taken into account in a weak sense by means of a Lagrange multiplier. The method does not present spurious or circulation modes for non-zero frequencies. Numerical results are given for some test cases. © 1997 by John Wiley & Sons, Ltd.     

Stable and time-dissipative finite element methods for the incompressible Navier–Stokes equations in advection dominated flows

This paper examines a new Galerkin method with scaled bubble functions which replicates the exact artificial diffusion methods in the case of 1-D scalar advection–diffusion and that leads to non-oscillatory solutions as the streamline upwinding algorithms for 2-D scalar advection–diffusion and incompressible Navier–Stokes. This method retains the satisfaction of the Babuska–Brezzi condition and, thus, leads to optimal performance in the incompressible limit. This method, when, combined with the recently proposed linear unconditionally stable algorithms of Simo and Armero (1993), yields a method for solution of the incompressible Navier–Stokes equations ideal for either diffusive or advection-dominated flows. Examples from scalar advection–diffusion and the solution of the incompressible Navier–Stokes equations are presented.     

Stabilizing the retarded potential method for transient fluid–structure interaction problems

The Retarded Potential (RP) method, which is a boundary element technique and non-local in both space and time, is employed to discretize the fluid domain for the analysis of transient fluid–structure interaction problems. The retarded potential analysis program RPFS is coupled to the ABAQUS non-linear finite element code to form ABAQUS/RPFS. The standard RP is inherently unstable for time steps below a critical time step that is equal to the maximum distance in the fluid divided by the wave speed. A technique referred to as the Figueiredo method is used to convert the standard RP differential-delay equations for the fluid to simply delay equations, which are more stable. The Figueiredo approach extends the stability range of the standard RP by a factor of approximately 10–20, but this time step is still not small enough to be useful for analysis. Digital signal processing methods are used to further stablize the response of the fluid by removing the oscillating high-frequency noise in the time histories of the solution without introducing phase shifting or any significant damping. Stability of the coupled system is achieved by not extrapolating the structural accelerations. ABAQUS/RPFS is applied to both a rigid and elastic sphere subjected to a plane wave, and the results using the full time histories required are completely stable and quite accurate. With this procedure, the retarded potential method may yet prove to be a valuable analysis tool for transient fluid–structure interaction problems. © 1997 John Wiley & Sons, Ltd.     

Coupled BEM–FEM solutions for direct time domain soil–structure interaction analysis

A coupled BEM–FEM methodology is presented for 3D wave propagation and soil–structure interaction analysis in the direct time domain. The employed boundary element method (BEM) uses a new generation of the Stokes fundamental solutions that utilize the B-Spline family of polynomials. A standard finite element methodology for dynamic analysis along with direct integration in time is coupled to the BEM through a staggered solution approach. Each method provides initial conditions to the other at the beginning of each time step. Formulation and computational aspects of the proposed coupling scheme are discussed. A number of numerical examples are presented for the validation and demonstration of the general nature of the proposed methodology.     

Nodal velocity derivatives of finite element solutions: The FiND method for incompressible Navier–Stokes equations

This paper presents the development and application of the finite node displacement (FiND) method to the incompressible Navier–Stokes equations. The method computes high‐accuracy nodal derivatives of the finite element solutions. The approach imposes a small displacement to individual mesh nodes and solves a very small problem on the patch of elements surrounding the node. The only unknown is the value of the solution ( u , p) at the displaced node. A finite difference between the original and the perturbed values provides the directional derivative. Verification by grid refinement studies is shown for two‐dimensional problems possessing a closed‐form solution: a Poiseuille flow and a flow mimicking a boundary layer. For internal nodes, the method yields accuracy slightly superior to that of the superconvergent patch recovery (SPR) technique of Zienkiewicz and Zhu (ZZ). We also present a variant of the method to treat boundary nodes. The local discretization is enriched by inserting an additional mesh point very close to the boundary node of interest. Computations show that the resulting nodal derivatives are much more accurate than those obtained by the ZZ SPR technique. Copyright © 2008 John Wiley & Sons, Ltd.     

A local RBF collocation method for band structure computations of 2D solid/fluid and fluid/solid phononic crystals

In this paper, an efficient local radial basis function collocation method (LRBFCM) is presented for computing the band structures of the two‐dimensional (2D) solid/fluid and fluid/solid phononic crystals. Both systems of solid scatterers embedded in a fluid matrix (solid/fluid phononic crystals) and fluid scatterers embedded in a solid matrix (fluid/solid phononic crystals) are investigated. The solid–fluid interactions are taken into account by properly formulating and treating the continuity/equilibrium conditions on the solid–fluid interfaces, which require an accurate computation of the normal derivatives of the displacements and the pressure on the fluid–solid interfaces and the unit‐cell boundaries. The developed LRBFCM for the mixed wave propagation problems in 2D solid/fluid and fluid/solid phononic crystals is validated by the corresponding results obtained by the finite element method (FEM). To the best knowledge of the authors, the present LRBFCM has yet not been applied to the band structure computations of 2D solid/fluid and fluid/solid phononic crystals. For different lattice forms, scatterers' shapes, acoustic impedance ratios, and material combinations (solid scatterers in fluid matrix or fluid scatterers in solid matrix), numerical results are presented and discussed to reveal the efficiency and the accuracy of the developed LRBFCM for calculating the band structures of 2D solid/fluid and fluid/solid phononic crystals. A comparison of the present numerical results with that of the FEM shows that the present LRBFCM is much more efficient than the FEM for the band structure computations of the considered 2D solid/fluid and fluid/solid phononic crystals. Copyright © 2016 John Wiley & Sons, Ltd.     

Seismic response and fragility evaluation for an Eastern US NPP including soil–structure interaction effects

The paper discusses methodological aspects involved in a probabilistic seismic soil–structure interaction (SSI) analysis for a Seismic Probabilistic Risk Assessment (SPRA) review. An example of an Eastern US nuclear power plant (NPP) is presented. The approach presented herein follows the current practice of the Individual Plant Examination for External Events (IPEEE) program in the US. The NPP is founded on a relatively soft soil deposit, and thus the SSI effects on seismic responses are significant. Probabilistic models used for the idealization of the seismic excitation and the surrounding soil deposit are described. Using a lognormal format, computed random variability effects were combined with those proposed in the SPRA methodology guidelines. Probabilistic floor response spectra and structural fragilities for different NPP buildings were computed. Structural capacities were determined following the current practice which assumes independent median safety factors for strength and inelastic absorption. Limitations of the IPEEE practice for performing SPRA are discussed and alternate procedures, more rigorous and simple to implement, are suggested.