Abstract

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.