Table Of Content121
Stéphane P. A. Bordas · Erik Burman
Mats G. Larson · Maxim A. Olshanskii
Editors
Geometrically Unfi tted
Finite Element Methods
and Applications
Editorial Board
T. J. Barth
M. Griebel
D. E. Keyes
R. M. Nieminen
D. Roose
T. Schlick
Lecture Notes
in Computational Science 121
and Engineering
Editors:
Timothy J. Barth
Michael Griebel
David E. Keyes
Risto M. Nieminen
Dirk Roose
Tamar Schlick
More information about this series at http://www.springer.com/series/3527
Stéphane P.A. Bordas • Erik Burman •
Mats G. Larson • Maxim A. Olshanskii
Editors
Geometrically Unfitted Finite
Element Methods
and Applications
Proceedings of the UCL Workshop 2016
123
Editors
Stéphane P.A. Bordas Erik Burman
Faculté des Sciences, de la Technologie Department of Mathematics
et de la Communication University College London
Université du Luxembourg UK
Luxembourg
School of Engineering
Cardiff University
Cardiff Wales
Mats G. Larson Maxim A. Olshanskii
Department of Mathematics Department of Mathematics
and Mathematical Statistics University of Houston
Umeå University USA
Sweden
ISSN 1439-7358 ISSN 2197-7100 (electronic)
Lecture Notes in Computational Science and Engineering
ISBN 978-3-319-71430-1 ISBN 978-3-319-71431-8 (eBook)
https://doi.org/10.1007/978-3-319-71431-8
Library of Congress Control Number: 2018932361
Mathematics Subject Classification (2010): 65N12, 65N15, 65N85, 65N30, 74G15, 74S05, 74M15
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Contents
Deriving Robust Unfitted Finite Element Methods from Augmented
Lagrangian Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Erik Burman and Peter Hansbo
Cut Finite Element Methods for Linear Elasticity Problems . . . . . . . . . . . . . . . 25
Peter Hansbo, Mats G. Larson, and Karl Larsson
A Higher Order Isoparametric Fictitious Domain Method for Level
Set Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Christoph Lehrenfeld
An Overview of Recent Results on Nitsche’s Method for Contact
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Franz Chouly, Mathieu Fabre, Patrick Hild, Rabii Mlika, Jérôme Pousin,
and Yves Renard
Stability and Optimal Convergence of Unfitted Extended Finite
Element Methods with Lagrange Multipliers for the Stokes Equations . . . 143
Michel Fournié and Alexei Lozinski
Penalty-Free Nitsche Method for Interface Problems. . . . . . . . . . . . . . . . . . . . . . . . 183
Thomas Boiveau, Erik Burman, and Susanne Claus
Trace Finite Element Methods for PDEs on Surfaces . . . . . . . . . . . . . . . . . . . . . . . . 211
Maxim A. Olshanskii and Arnold Reusken
A Cut Discontinuous Galerkin Method for Coupled Bulk-Surface
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
André Massing
A Space-Time Cut Finite Element Method with Quadrature in Time . . . . . 281
Sara Zahedi
v
vi Contents
Well Conditioned Extended Finite Elements and Vector Level Sets
for Three-Dimensional Crack Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
Konstantinos Agathos, Giulio Ventura, Eleni Chatzi,
and Stéphane P.A. Bordas
Unfitted FEM for Modelling the Interaction of Multiple Fractures
in a Poroelastic Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
Bianca Giovanardi, Luca Formaggia, Anna Scotti, and Paolo Zunino
Introduction
The question of how to design methods where the integration of geometric data
into large-scale computations is facilitated has been receiving increasing attention
from the scientific community lately. Indeed computational methods are typically
defined on a mesh, a decomposition of the physical domain in smaller geometric
entities, so-called elements, often consisting of tetrahedra or cubes. The construction
of such a mesh is a challenging problem in itself that can fail if the geometry of
the computational domain is sufficiently complex. Different approaches have been
proposed to allow for a simplification of the problems related to meshing and several
have developed into their own subfields. We will mention some below.
The difficulty of the meshing problem increases with the number of constraints
that are put on the elements. In particular close to the boundary the restriction to
a too small class of shapes may lead to very irregular meshes that can result in
approximation spaces with poor properties. One idea to solve this problem is to
modify the computational method so that it can use more general polyhedral or
polygonal meshes. This has led to methods such as the Mimetic Finite Element
Method [13], the Hybridised Discontinuous Galerkin Method [28], the Virtual
Element Method [12] or the High Order Hybridised Method [30].
On the other hand in some engineering applications the computational geometry
comes from computer-aided design and is then generally described using splines.
To make the transition from geometric description to computation as seamless as
possible in this context it has been proposed to use the same type of functions that
are used in the geometric parametrisation for the computation. This led to the field of
Isogeometric Analysis [46], where many results known for finite element methods
now find new applications in the context of these new approximation spaces.
In the essays presented in this volume, the focus is on the so-called geometrically
unfitted finite element methods, that is, methods where the solution method of
the PDE typically remains a standard finite element method, but the variational
formulation is modified so that constraints on interfaces and boundaries can be
integrated in the computation even if the mesh is not fitted to the geometry.
Possibly the oldest approach to the integration of geometry in computation,
aiming to reduce the complexity of the meshing problem, is the fictitious domain
vii
viii Introduction
method or volume penalty method [3, 15, 35, 36, 49, 52, 56]. In such approaches
the computational mesh used is independent of the physical domain. Instead
the physical problem is extended to the domain defined by the computational
mesh. Boundary or interface conditions are imposed through a modification of the
variational form, using Lagrange multipliers, or penalty terms either on the volume
or on the boundary. Although very successful in applications the classical variants of
this approach can have some shortcomings. The solution may present a kink over the
interfaces that typically cannot be represented by the finite element approximation
space leading to loss of accuracy or the choice of the penalty parameter may
result in a trade-off between a severely ill-conditioned linear system and loss of
accuracy. As a remedy to these disadvantages, the so-called unfitted finite element
method was introduced in a series of papers by Barrett and Elliott [7–9]. Typically a
weight function was introduced in the variational formulation in such a way that the
equations were approximated in the physical domain only. Another approach to the
solution of this problem was the introduction of the extended finite element method
by Belytschko et al. [32]. In this case the finite element space was enriched locally
so that it could represent the unfitted singular behaviour of the solution over the
interface better. This approach has been very successful in practice and resulted
in a vast literature [14, 31, 38, 47, 57], but enforcing conformity often leads to
complicated approximation spaces due to nonlocal enrichment functions, through
so-called blending. It has also proved difficult to prove optimal approximation
results and the stability results necessary in some applications (typically inf-sup
stability for incompressible elasticity). In the seminal works [40, 41], Hansbo and
Hansbo showed that the use of Nitsche’s method [58] for the interface coupling
and using an enrichment inspired by discontinuous Galerkin methods led to an
XFEM-type method that could be given a solid foundation in theory. The approach
based on Nitsche’s method was then developed to include overlapping meshes [42],
time-dependent problems [67], incompressible elasticity/flow [11, 43], transport
models [51], fictitious domain methods [20], high contrast problems [4, 5, 21, 26],
discontinuous Galerkin methods (in the bulk) [48, 55] and contact problems [34].
The efficient solution of the resulting linear system has also been considered for
instance in [50].
In parallel an approach using Lagrange multipliers was developed. Also in this
case the elements were cut and the equations integrated only over the physical
domain, but here instead of using Nitsche’s method the matching conditions
were imposed using Lagrange multipliers. To obtain stable discretisations the
approximation spaces must be carefully chosen, or a stabilisation term must be
added. Examples of such methods are proposed in [2, 6, 10, 17, 19, 45, 68].
Another important problem is how to compute solutions to partial differential
equations set on a manifold, for example a two-dimensional surface embedded
in three space dimensions. The original approach is to construct a surface mesh
on which approximation spaces can be constructed (see Dzuik and Elliott [33]).
Here also the design of a mesh is non-trivial, in particular if the surface evolves
in time and interacts with a system defined on a bulk mesh. An unfitted finite
element method for the discretisation of PDEs on surfaces, called “TraceFEM,” was
introduced by Olshanskii et al. [61]. In this case the surface PDE is approximated on
Introduction ix
a surface that cuts through the elements of a bulk mesh, using the degrees of freedom
associated with the volume mesh. These results were then extended in a series of
papers to adaptive methods [27, 29], time-dependent problems [44, 60, 64], high
order approximation [37, 65], transport problems [24, 63] and more realistic models
[45]. Combining the different unfitted methods in the bulk and on the surface leads
to a versatile framework for multiphysics problems including bulk-surface coupling
[25, 39] and fluid-structure interaction [1, 18, 54]. An important contribution to the
application of these techniques to real world applied problems has been made by
Wall and co-workers; see for instance, the recent PhD thesis of Schott [66], the
contents of which were presented at the UCL workshop.
When the mesh is not fitted to the interface it is known that unfortunate cuts,
resulting in elements with very small intersections with the physical domain, can
lead to ill-conditioned linear systems and sometimes affect accuracy. Different
solutions to this problem have been proposed in the literature, such as precondi-
tioning [50, 59], weakly consistent penalty, so-called ghost penalty [16, 20, 22, 53],
dimensional extension of surface PDEs into the bulk [62] or agglomeration of
elements [45, 48].
For an overview of the ideas combining discretisation of PDEs and geometry
using unfitted finite element methods, in the bulk and on immersed surfaces, we
refer to [23].
This volume collects eleven essays on topics related to the type of geometrically
unfitted finite element methods discussed above. They have been inspired by a
workshop held at UCL, 6–8th of January 2016, and funded by EPSRC grant
EP/J002313/2. Three articles have a review character and are dedicated to cutFEM
(Hansbo et al.), Trace FEM (Olshanskii and Reusken) and contact problems (Chouly
et al.). Both Nitsche’s method (Burman and Hansbo; Boiveau et al.; Lehrenfeld)
and Lagrange multiplier methods (Burman and Hansbo; Fournie and Lozinski)
are discussed in original papers, and recent results on trace FEMs are reported
(Massing; Sahedi). Finally there are two papers that delve deeper into the application
of geometrically unfitted methods to more realistic physical models, such as cracks
and poroelastic media (Agathos et al.; Giovanardi et al.).
The intention has been to create a snapshot of the state of the art in this rapidly
evolving field that can serve both as an introduction to some central topics and as a
discussion probing the research frontier.
London, UK Erik Burman
2017
References
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an incompressible fluid with immersed thin-walled structures. Comput. Methods Appl. Mech.
Eng. 301, 300–335 (2016). http://dx.doi.org/10.1016/j.cma.2015.12.015
