Fluid–structure interaction using the particle finite element method

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Abstract

In the present work a new approach to solve fluid–structure interaction problems is described. Both, the equations of motion for fluids and for solids have been approximated using a material (Lagrangian) formulation. To approximate the partial differential equations representing the fluid motion, the shape functions introduced by the meshless finite element method (MFEM) have been used. Thus, the continuum is discretized into particles that move under body forces (gravity) and surface forces (due to the interaction with neighboring particles). All the physical properties such as density, viscosity, conductivity, etc., as well as the variables that define the temporal state such as velocity and position and also other variables like temperature are assigned to the particles and are transported with the particle motion. The so called particle finite element method (PFEM) provides a very advantageous and efficient way for solving contact and free-surface problems, highly simplifying the treatment of fluid–structure interactions.

Introduction

Many classifications have been proposed to enclose the numerical formulations that approximate the continuum equations that govern incompressible fluid flows. In particular the one describing the way that convection is treated divides the numerical formulations into two classes, namely, material (or Lagrangian) formulations and spatial (or Eulerian) formulations. The first one describes convection by placing a set of axes over the material particles that move accordingly to the equations of motion. In the Eulerian case the axes are set fixed in space and convection terms are included in the equations describing the transport of the fluid flow. The present work will describe a method that uses a material formulation. The equations of motion for both, the solid and fluid do not present convection terms, implying that the convection effect is directly obtained by moving the discrete domain.

Many authors have taken advantage of Lagrangian formulations to describe different types of problems. The smooth particle hydrodynamics (SPH) method developed by Monaghan [13], [14] should be mentioned as a pioneer method of this kind.

Many other methods have been derived from SPH. One that has shown remarkable results is the moving particle semi-implicit (MPS) method introduced by Koshizuka and Oka [10]. These methods use a kernel function to interpolate the unknowns. SPH uses a weak formulation while MPS uses a strong form of the governing equations.

Ramaswamy [22] proposed a Lagrangian finite element formulation for a 2-D incompressible fluid flow. In that paper the mesh was convected according to the equations of motion but without change of topology, making it rather limiting when the elements got highly distorted. The equations of motion were discretized in space by using the finite element method with linear shape functions.

Another possible classification for numerical formulations may be the one that separates the methods that make use of a standard finite element mesh (like those made of tetrahedra or hexahedra), and the methods that do not need a standard mesh, namely, the meshless methods. The formulation described in this paper can be considered a particular class of meshless method. Again, SPH might be cited as one the first meshless methods.

Indeed, after Monaghans work and in particular in the past 20 years, many have been the attempts to develop a robust meshless method that could approximate PDE’s in 2-D and 3-D with acceptable accuracy, convergence and speed. Among others, the methods based on Moving Least Square interpolations [15], [2], Partition of Unity [5], and the ones based on the natural neighbor interpolation functions [26] may be listed.

In this work the interpolation function used by the meshless finite element method (MFEM) [7] will be implemented. This function uses the Voronoï diagram of the cloud of points to construct the interpolant. The extended Delaunay tessellation (EDT) [9] is applied to connect the neighboring particles. The EDT provides polyhedral elements that are sliver-free in 3-D, avoiding instabilities of the Delaunay tessellation due to distorted tetrahedra. The MFEM shape functions adapt automatically to the polyhedra and in the case that the polyhedron is a simplex, the shape function behaves exactly as the linear finite element shape function.

Fluid–structure interaction (FSI) problems have been of special interest for designers and engineers in the past 20 years. This explains why more robust and stable formulations have been developed to assist the approximation of contact problems. Embedded methods have been developed by Löhner et al. [11] where a single mesh is used to partition the fluid as well as the structure. Also arbitrary Lagrangian–Eulerian (ALE) formulations [25] have given acceptable results when the displacements or the geometry deformations are not excessively large.

The approximation for the FSI problem depends basically on the coupling of the fluid and structure equations. Based on this coupling FSI problems may be divided into problems with weak interaction and problems with strong interaction. The later are found when elastic deformation of the solid takes place. The weak interpolation case happens when large rigid displacements are present. This situation is typical in ship hydrodynamics, when a rigid body moves according to the forces given by the pressure field obtained from the fluid dynamic problem. These forces applied to the rigid body will accelerate it, changing its velocity and therefore, its position.

FSI problems have been classically solved in a partitioned manner solving iteratively the discretized equations for the flow and the solid domain separately. The solution of both, fluid flow and solid, with the same material formulation, open the door to solve the global coupled problem in a monolithic fashion. Nevertheless, in this paper the rigid solid will still be solved separately from the fluid. A partitioned method [20], [12] or iterative method [23], [24], [27] is chosen to solve the coupling between the fluid and solid. The advantage to use a material formulation for both, solid and fluid parts will be used here only to better reproduce breaking waves or separated drops in the fluid, which are phenomena impossible to reproduce using a spatial formulation.

The layout of the paper is the following: in the next section the basic Lagrangian equations of motion for the fluid and solid domains are given. Next the discretization method chosen to solve the incompressible fluid flow equations and the solid dynamics in time equations are detailed. The algorithm for the recognition of the boundary nodes and the treatment of the free-surface in the fluid is explained. Finally the efficiency of the particle finite element method for solving a variety of fluid–structure interaction problems involving large motion of the free-surface in the fluid is shown.

Section snippets

Fluid dynamic problem: updating the fluid particle positions

The fluid particle positions will be updated via solving the Lagrangian form of the Navier–Stokes equations.

Let Xi the initial position of a particle a time t = t0 and let xi the final position. Been ui(xj, t) = ui the velocity of the particle in the final position the following approximate relation can be written:xi=Xi+f(ui,t,Dui/Dt).Conservation of momentum and mass for incompressible Newtonian fluids in the Lagrangian frame of reference are represented by the Navier–Stokes equations and the

The discrete fluid dynamics problem

The Navier–Stokes equations present three main difficulties:

  • The equations are time dependent and thus a temporal integration needs to be carried out.

  • A spatial dependency is also present and thus the space will be discretized.

  • Finally, Eq. (3) presents a non-linearity, which must be solved iteratively.

Each of the above items will be explained and a solution algorithm will be introduced to obtain a final accurate and robust numerical scheme.

Time integration of the solid dynamics problem

Eqs. (14), (15) that govern the movement of rigid bodies are integrated in time by the explicit Newmark algorithm. It consists in evaluating the velocity by linearizing the acceleration between two time steps:Uin+1=Uin+Δt1-γain+γain+1,Ωin+1=Ωin+Δt1-γαin+γαin+1.The point position for the explicit version of the Newmark algorithm is evaluated byxin+1=xin+ΔtUin+Δt2ain/2.To integrate the angular acceleration in a 3-D system by Newmark algorithm two steps are needed, namely
Predictor stepΩi=Ωin+Δt((1

Free-surface and boundary recognition

The solution of partial differential equations (PDE) requires to prescribe boundary conditions as a necessary step to a well-posed problem. When the PDEs are approximated in space and the domain is partitioned into discrete elements (finite elements, particles, balls, nodes, etc.) the boundary elements should be provided at the initial time step, such that, at run time the algorithm knows where to impose or fix the variables of the analysis (pressure, velocity and their derivatives for

Joining and breaking particles

The idea of h variable mesh is rather different in particle methods than in classical Eulerian formulations. In particle methods, each particle is followed in time and the same particle can cross domains in which the solution need small h in order to represent high gradients or can cross a region with large h where the solution is smooth. The concept of variable h is introduce in particle methods by joining two particles when they are too close to each other or breaking a particle in two when

Validation examples

PFEM was developed as a general-purpose method for solving different kind of problems on which large free surface or interface boundaries changes are involved. The method is well suited to solve a large variety of mechanical problems including mixing fluid and solid materials, wave motion problems, mould filling, coupled thermal–mechanical problems and fluid–solid interaction as well.

In this section some problems are included in order to show the validation of the present approach towards

Ship profile hit by a wave

In the example of Fig. 8.1 the motion of a fictitious rigid ship hit by an incoming wave is analyzed. This is the first example in which the rigid body is moved by the fluid forces in a coupling problem as was explained in the previous chapters.

The dynamic motion of the ship is induced by the resultant of the pressure and the viscous forces acting on the ship boundaries. The horizontal displacement of the mass centre of the ship was fixed to zero. In this way, the ship moves only vertically

Conclusions

The particle finite element method (PFEM) seems ideal to treat problems involving fluids with free-surface and submerged or floating structures within a unified Lagrangian finite element framework. Problems such as the analysis of fluid–structure interactions, large motion of fluid or solid particles, surface waves, water splashing, separation of water drops, etc. can be easily solved with the PFEM.

The success of the method lays in the accurate and efficient solution at each time step of the

Acknowledgement

Thanks are given to Mr. Miguel Angel Calena for computing the numerical results using the PFEM of the wave on a channel presented in 7.2. Thanks also to the Maritime Experimental and Research Channel (CIEM) of the Escuela Técnica de Ingenieros de Canales Caminos y Puertos, University of Catalunya for the experimental results of the same problem.

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