Lagrangian FE methods for coupled problems in fluid mechanics

Lagrangian finite element methods emerged in fluid dynamics when the deficiencies of the Eulerian methods in treating free surface flows (or generally domains undergoing large shape deformations) were faced. Their advantage relies upon natural tracking of boundaries and interfaces, a feature particularly important for interaction problems. Another attractive feature is the absence of the convective term in the fluid momentum equations written in the Lagrangian framework resulting in a symmetric discrete system matrix, an important feature in case iterative solvers are utilized. Unfortunately, the lack of the control over the mesh distortions is a major drawback of Lagrangian methods. In order to overcome this, a Lagrangian method must be equipped with an efficient re-meshing tool. This work aims at developing formulations and algorithms where maximum advantage of using Lagrangian finite element fluid formulations can be taken. In particular we concentrate our attention at fluid-structure interaction and thermally coupled applications, most of which originate from practical “real-life” problems. Two fundamental options are investigated - coupling two Lagrangian formulations (e.g. Lagrangian fluid and Lagrangian structure) and coupling the Lagrangian and Eulerian fluid formulations. In the first part of this work the basic concepts of the Lagrangian fluids, the so-called Particle Finite Element Method (PFEM) [1], [2] are presented. These include nodal variable storage, mesh re-construction using Delaunay triangulation/tetrahedralization and alpha shape-based method for identification of the computational domain boundaries. This shall serve as a general basis for all the further developments of this work. Next we show how an incompressible Lagrangian fluid can be used in a partitioned fluid-structure interaction context. We present an improved Dirichlet-Neumann strategy for coupling the incompressible Lagrangian fluid with a rigid body. This is finally applied to an industrial problem dealing with the sea-landing of a satellite capsule. In the following, an extension of the method is proposed to allow dealing with fluid-structure problems involving general flexible structures. The method developed takes advantage of the symmetry of the discrete system matrix and by introducing a slight fluid compressibility allows to treat the fluid-structure interaction problem efficiently in a monolithic way. Thus, maximum benefit from using a similar description for both the fluid (updated Lagrangian) and the solid (total Lagrangian) is taken. We show next that the developed monolithic approach is particularly useful for modeling the interaction with light-weight structures. The validation of the method is done by means of comparison with experimental results and with a number of different methods found in literature. The second part of this work aims at coupling Lagrangian and Eulerian fluid formulations. The application area is the modeling of polymers under fire conditions. This kind of problem consists of modeling the two subsystems (namely the polymer and the surrounding air) and their thermomechanical interaction. A compressible fluid formulation based on the Eulerian description is used for modeling the air, whereas a Lagrangian description is used for the polymer. For the surrounding air we develop a model based upon the compressible Navier-Stokes equations. Such choice is dictated by the presence of high temperature gradients in the problem of interest, which precludes the utilization of the Boussinesq approximation. The formulation is restricted to the sub-sonic flow regime, meeting the requirement of the problem of interest. The mechanical interaction of the subsystems is modeled by means of a one-way coupling, where the polymer velocities are imposed on the interface elements of the Eulerian mesh in a weak way. Thermal interaction is treated by means of the energy equation solved on the Eulerian mesh, containing thermal properties of both the subsystems, namely air and polymer. The developments of the second part of this work do not pretend to be by any means exhaustive; for instance, radiation and chemical reaction phenomena are not considered. Rather we make the first step in the direction of modeling the complicated thermo-mechanical problem and provide a general framework that in the future can be enriched with a more detailed and sophisticated models. However this would affect only the individual modules, preserving the overall architecture of the solution procedure unchanged. Each chapter concludes with the example section that includes both the validation tests and/or applications to the real-life problems. The final chapter highlights the achievements of the work and defines the future lines of research that naturally evolve from the results of this work

Development of New Lagrangian Computational Methods for Ice-Ship Interaction Problems: NICESHIP Project

This document presents the activities carried out to date (04/2019) in the project ‘Development of new Lagrangian computational methods for ice-ship interaction problems’ (NICE-SHIP). The NICE-SHIP project aims at developing a new generation of computational methods, based on the integration of innovative Lagrangian particle-based and finite element procedures for the analysis of the operation of a vessel in an iced sea, taking into account the different possible conditions of the ice. It is expected that the computational analysis techniques to be developed in NICE-SHIP will allow ice-class vessel designers to accurately evaluate the loads acting on the structure of a ship navigating in iced-seas and, in particular, to determine the ice resistance of the ship in different ice conditions

A unified symmetrical formulation for interactions between elastic solids and incompressible fluids

We present a general Lagrangian formulation for treating elastic solids and quasi/fully incompressible fluids in a unified form. The formulation allows to treat solid and fluid subdomains in a unified manner in fluid-structure interaction (FSI) situations. The use for both fluid and solid of a Lagrangian formulation avoid convective terms allowing to approximate the problem via a Variational Formulation. In our work the FSI problem is solved via the Particle Finite Element Method (PFEM). The PFEM is an effective technique for modeling complex interactions between floating and submerged bodies and free surface flows, accounting for splashing of waves, large motions of the bodies and frictional contact conditions. Applications of the unified Lagrangian formulation to a number of FSI problems are given

A unified monolithic approach for multi-fluid flows and Fluid-Structure Interaction using the Particle Finite Element Method with fixed mesh

This paper describes a strategy to solve multi-fluid and Fluid-Structure Interaction (FSI) problems using Lagrangian particles combined with a fixed Finite Element (FE) mesh . Our approach is an extension of the fluid-only PFEM-2 [1,2] which uses explicit integration over the streamlines to improve accuracy. As a result, the convective term does not appear in the set of equations solved on the fixed mesh. Enrichments in the pressure field are used to improve the description of the interface between phases

Langrangian formulation for incompressible fluids using finite calculus and the finite element method

We present a general formulation for incompressible fluid flow analysis using the finite element method (FEM) and a lagrangian description. The necessary stabilization for dealing with the incompressibility condition is introduced via the so called finite calculus (FIC) method. Both a quasi-implicit algorithm and a fractional step scheme are described. Examples of application of the lagrangian flow description are presented

An adaptive finite point method for aeroelastic analysis

An adaptive Finite Point Method (FPM) for solving aeroelastic compressible flow problems is presented. The numerical methodology is based on a meshless upwind-biased discretization of the Euler equations, written in arbitrary Lagrangian-Eulerian (ALE) form, and integrated in time by means of a dual-time steeping technique. This procedure allows achieving accurate solutions circumventing stability constraints of time marching schemes but profiting from its explicit features. In order to exploit the meshless potential of the method, the domain deformation approach implemented is based on the spring network analogy and ''h''-adaptivity is also employed in the computations. Several numerical examples involving typical aeroelastic problems illustrate the performance of the proposed technique. Moreover, evidence about the computational cost and parallel performance of the method is given

Advances in the simulation of multi-fluid flows with the Particle Finite Element Method. Application to bubble dynamics

In this work we extend the Particle Finite Element Method (PFEM) to multi-fluid flow problems with the aim of exploiting the fact that Lagrangian methods are specially well suited for tracking interfaces. We develop a numerical scheme able to deal with large jumps in the physical properties, included surface tension, and able to accurately represent all types of discontinuities in the flow variables. The scheme is based on decoupling the velocity and pressure variables through a pressure segregation method which takes into account the interface conditions. The interface is defined to be aligned with the moving mesh, so that it remains sharp along time, and pressure degrees of freedom are duplicated at the interface nodes to represent the discontinuity of this variable due to surface tension and variable viscosity. Furthermore, the mesh is refined in the vicinity of the interface to improve the accuracy and the efficiency of the computations. We apply the resulting scheme to the benchmark problem of a two-dimensional bubble rising in a liquid column presented in [[#cite-1|[1]]], and propose two breakup and coalescence problems to assess the ability of a multi-fluid code to model topology changes

Condicion absorbente discreta no local (DNL) en diferencias finitas para modelos elípticos de propagación de ondas en el mar

The finite difference method is used to approximate the solutions of Berkhoff s equation for water radiation and scattering in an unbounded domain. To incorporate the exact far field radiation condition in the numerical scheme an operational method has been developed. The determination of Hemholz discrete operator spectrum over a structured domain allows the design of a non-local perfectly absorbent boundary condition in the discrete medium. Numerical tests validate these conclusions

A general stabilized formulation for incompressible fluid flow using finite calculus and the finite element method

We present a general formulation for incompressible fluid flow analysis using the finite element method (FEM). The necessary stabilization for dealing with convective effects and the incompressibility condition are introduced via the so called finite calculus (FIC) method. The extension of the standard eulerian form of the equations to an arbitrary lagrangian-eulerian (ALE) frame adequate for treating fluid-structure interaction problems is presented. The fully lagrangian form is also discussed. Details of an effective mesh updating procedure are presented together with a method for dealing with free surface effects of importance for ship hydrodynamic analysis and many other fluid flow problems. Examples of application of the eulerian, the ALE and the fully lagrangian flow descriptions are presented