Isogeometric analysis of Cahn-Hilliard phase field-based Binary-Fluid-Structure Interaction based on an ALE variational formulation
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Date
2020
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Abstract
This thesis is concerned with the development of a computational model and simulation technique capable
of capturing the complex physics behind the intriguing phenomena of Elasto-capillarity. Elastocapillarity
refers to the ability of capillary forces or surface tensions to deform elastic solids through
a complex interplay between the energy of the surfaces (interfaces) and the elastic strain energy in the
solid bulk. The described configuration gives rise to a three-phase system featuring a fluid-fluid interface
(for instance the interface of a liquid and an ambient fluid such as air) and two additional interfaces
separating the elastic solid from the first and second fluids, respectively. This setup is encountered in the
wetting of soft substrates which is an emerging young field of research with many potential applications
in micro- and nanotechnology and biomechanics. By virtue of the fact that a lot of physical phenomena
under the umbrella of the wetting of soft substrates (e.g. Stick-slip motion, Durotaxis, Shuttleworth
effect, etc.) are not yet fully understood, numerical analysis and simulation tools may yield invaluable
insights when it comes to understanding the underlying processes. The problem tackled in this work –
dubbed Elasto-Capillary Fluid-Structure Interaction or Binary-Fluid-Structure Interaction (BFSI) – is
of multiphysics nature and poses a tremendous and challenging complexity when it comes to its numerical
treatment. The complexity is given by the individual difficulties of the involved Two-phase Flow
and Fluid-Structure Interaction (FSI) subproblems and the additional complexity emerging from their
complex interplay.
The two-phase flow problems considered in this work are immiscible two-component incompressible
flow problems which we address with a Cahn-Hilliard phase field-based two-phase flow model through
the Navier-Stokes-Cahn-Hilliard (NSCH) equations. The phase field method – also known as the diffuse
interface method – is based on models of fluid free energy and has a solid theoretical foundation in
thermodynamics and statistical mechanics. It may therefore be perceived as a physically motivated
extension of the level-set or volume-of-fluid methods. It differs from other Eulerian interface motion
techniques by virtue of the fact that it does not feature a sharp, but a diffuse interface of finite width
whose dynamics are governed by the joint minimization of a double well chemical energy and a gradientsquared
surface energy – both being constituents of the fluid free energy. Particularly for two-phase flows,
diffuse interface models have gained a lot of attention due to their ability to handle complex interface
dynamics such moving contact lines on wetted surfaces, and droplet coalescence or segregation without
any special procedures.
Our computational model for the FSI subproblem is based on a hyperelastic material model for the solid.
When modeling the coupled dynamics of FSI, one is confronted with the dilemma that the fluid model
is naturally based on an Eulerian perspective while it is very natural to express the solid problem in
Lagrangian formulation. The monolithic approach we take, uses a fully coupled Arbitrary Lagrangian–
Eulerian (ALE) variational formulation of the FSI problem and applies Galerkin-based Isogeometric
Analysis for the discretization of the partial differential equations involved. This approach solves the
difficulty of a common variational description and facilitates a consistent Galerkin discretization of the
FSI problem. Besides, the monolithic approach avoids any instability issues that are associated with
partitioned FSI approaches when the fluid and solid densities approach each other.
The BFSI computational model presented in this work is obtained through the combination of the above
described phase field-based two-phase flow and the monolithic fluid-structure interaction models and
yields a very robust and powerful method for the simulation of elasto-capillary fluid-structure interaction
problems. In addition, we also show that it may also be used for the modeling of FSI with free surfaces,
involving totally or partially submerged solids. Our BFSI model may be classified as “quasi monolithic”
as we employ a two-step solution algorithm, where in the first step we solve the pure Cahn-Hilliard phase
field problem and use its results in a second step in which the binary-fluid-flow, the solid deformation
and the mesh regularization problems are solved monolithically.
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Keywords
Binary-Fluid-Structure Interaction, Elastocapillary Fluid-Structure Interaction