26 research outputs found
Approximation of tensor fields on surfaces of arbitrary topology based on local Monge parametrizations
We introduce a new method, the Local Monge Parametrizations (LMP) method, to
approximate tensor fields on general surfaces given by a collection of local
parametrizations, e.g.~as in finite element or NURBS surface representations.
Our goal is to use this method to solve numerically tensor-valued partial
differential equations (PDE) on surfaces. Previous methods use scalar
potentials to numerically describe vector fields on surfaces, at the expense of
requiring higher-order derivatives of the approximated fields and limited to
simply connected surfaces, or represent tangential tensor fields as tensor
fields in 3D subjected to constraints, thus increasing the essential number of
degrees of freedom. In contrast, the LMP method uses an optimal number of
degrees of freedom to represent a tensor, is general with regards to the
topology of the surface, and does not increase the order of the PDEs governing
the tensor fields. The main idea is to construct maps between the element
parametrizations and a local Monge parametrization around each node. We test
the LMP method by approximating in a least-squares sense different vector and
tensor fields on simply connected and genus-1 surfaces. Furthermore, we apply
the LMP method to two physical models on surfaces, involving a tension-driven
flow (vector-valued PDE) and nematic ordering (tensor-valued PDE). The LMP
method thus solves the long-standing problem of the interpolation of tensors on
general surfaces with an optimal number of degrees of freedom.Comment: 16 pages, 6 figure
An adaptive space-time phase field formulation for dynamic fracture of brittle shells based on LR NURBS
We present an adaptive space-time phase field formulation for dynamic fracture of brittle shells. Their deformation is characterized by the Kirchhoff–Love thin shell theory using a curvilinear surface description. All kinematical objects are defined on the shell’s mid-plane. The evolution equation for the phase field is determined by the minimization of an energy functional based on Griffith’s theory of brittle fracture. Membrane and bending contributions to the fracture process are modeled separately and a thickness integration is established for the latter. The coupled system consists of two nonlinear fourth-order PDEs and all quantities are defined on an evolving two-dimensional manifold. Since the weak form requires C1-continuity, isogeometric shape functions are used. The mesh is adaptively refined based on the phase field using Locally Refinable (LR) NURBS. Time is discretized based on a generalized-α method using adaptive time-stepping, and the discretized coupled system is solved with a monolithic Newton–Raphson scheme. The interaction between surface deformation and crack evolution is demonstrated by several numerical examples showing dynamic crack propagation and branching
The multiplicative deformation split for shells with application to growth, chemical swelling, thermoelasticity, viscoelasticity and elastoplasticity
This work presents a general unified theory for coupled nonlinear elastic and
inelastic deformations of curved thin shells. The coupling is based on a
multiplicative decomposition of the surface deformation gradient. The
kinematics of this decomposition is examined in detail. In particular, the
dependency of various kinematical quantities, such as area change and
curvature, on the elastic and inelastic strains is discussed. This is essential
for the development of general constitutive models. In order to fully explore
the coupling between elastic and different inelastic deformations, the surface
balance laws for mass, momentum, energy and entropy are examined in the context
of the multiplicative decomposition. Based on the second law of thermodynamics,
the general constitutive relations are then derived. Two cases are considered:
Independent inelastic strains, and inelastic strains that are functions of
temperature and concentration. The constitutive relations are illustrated by
several nonlinear examples on growth, chemical swelling, thermoelasticity,
viscoelasticity and elastoplasticity of shells. The formulation is fully
expressed in curvilinear coordinates leading to compact and elegant expressions
for the kinematics, balance laws and constitutive relations
A monolithic fluid-structure interaction formulation for solid and liquid membranes including free-surface contact
A unified fluid-structure interaction (FSI) formulation is presented for
solid, liquid and mixed membranes. Nonlinear finite elements (FE) and the
generalized-alpha scheme are used for the spatial and temporal discretization.
The membrane discretization is based on curvilinear surface elements that can
describe large deformations and rotations, and also provide a straightforward
description for contact. The fluid is described by the incompressible
Navier-Stokes equations, and its discretization is based on stabilized
Petrov-Galerkin FE. The coupling between fluid and structure uses a conforming
sharp interface discretization, and the resulting non-linear FE equations are
solved monolithically within the Newton-Raphson scheme. An arbitrary
Lagrangian-Eulerian formulation is used for the fluid in order to account for
the mesh motion around the structure. The formulation is very general and
admits diverse applications that include contact at free surfaces. This is
demonstrated by two analytical and three numerical examples exhibiting strong
coupling between fluid and structure. The examples include balloon inflation,
droplet rolling and flapping flags. They span a Reynolds-number range from
0.001 to 2000. One of the examples considers the extension to rotation-free
shells using isogeometric FE.Comment: 38 pages, 17 figure
A finite membrane element formulation for surfactants
Surfactants play an important role in various physiological and biomechanical
applications. An example is the respiratory system, where pulmonary surfactants
facilitate the breathing and reduce the possibility of airway blocking by
lowering the surface tension when the lung volume decreases during exhalation.
This function is due to the dynamic surface tension of pulmonary surfactants,
which depends on the concentration of surfactants spread on the liquid layer
lining the interior surface of the airways and alveoli. Here, a finite membrane
element formulation for liquids is introduced that allows for the dynamics of
concentration-dependent surface tension, as is the particular case for
pulmonary surfactants. A straightforward approach is suggested to model the
contact line between liquid drops/menisci and planar solid substrates, which
allows the presented framework to be easily used for drop shape analysis. It is
further shown how line tension can be taken into account. Following an
isogeometric approach, NURBS-based finite elements are used for the
discretization of the membrane surface. The capabilities of the presented
computational model is demonstrated by different numerical examples - such as
the simulation of liquid films, constrained and unconstrained sessile drops,
pendant drops and liquid bridges - and the results are compared with
experimental data.Comment: Some typos are removed. Eqs. 13 and 105 are modified. Eqs. 64 and 73
are added; thus, the rest of equations is renumbered. All the numerical
experiments are repeated. The example of Sec. 6.3 is slightly modifie
Onsager’s variational principle in soft matter : introduction and application to the dynamics of adsorption of proteins onto fluid membranes
This book is the first collection of lipid-membrane research conducted by leading mechanicians and experts in continuum mechanics. It brings the overall intellectual framework afforded by modern continuum mechanics to bear on a host of challenging problems in lipid membrane physics. These include unique and authoritative treatments of differential geometry, shape elasticity, surface flow and diffusion, interleaf membrane friction, phase transitions, electroelasticity and flexoelectricity, and computational modelling.
[Chapter] Lipid bilayers are unique soft materials operating in general in the low Reynolds limit. While their shape is predominantly dominated by curvature elasticity as in a solid shell, their in-plane behavior is that of a largely inextensible viscous fluid. Furthermore, lipid membranes are extremely responsive to chemical stimuli. Because in their biological context they are continuously brought out-of-equilibrium mechanically or chemically, it is important to understand their dynamics. Here, we introduce Onsager’s variational principle as a general and transparent modeling tool for lipid bilayer dynamics. We introduce this principle with elementary examples, and then use it to study the sorption of curved proteins on lipid membranes.Peer ReviewedPostprint (author's final draft
An isogeometric finite element formulation for phase transitions on deforming surfaces
This paper presents a general theory and isogeometric finite element
implementation for studying mass conserving phase transitions on deforming
surfaces. The mathematical problem is governed by two coupled fourth-order
nonlinear partial differential equations (PDEs) that live on an evolving
two-dimensional manifold. For the phase transitions, the PDE is the
Cahn-Hilliard equation for curved surfaces, which can be derived from surface
mass balance in the framework of irreversible thermodynamics. For the surface
deformation, the PDE is the (vector-valued) Kirchhoff-Love thin shell equation.
Both PDEs can be efficiently discretized using -continuous interpolations
without derivative degrees-of-freedom (dofs). Structured NURBS and unstructured
spline spaces with pointwise -continuity are utilized for these
interpolations. The resulting finite element formulation is discretized in time
by the generalized- scheme with adaptive time-stepping, and it is fully
linearized within a monolithic Newton-Raphson approach. A curvilinear surface
parameterization is used throughout the formulation to admit general surface
shapes and deformations. The behavior of the coupled system is illustrated by
several numerical examples exhibiting phase transitions on deforming spheres,
tori and double-tori.Comment: fixed typos, extended literature review, added clarifying notes to
the text, added supplementary movie file