Highly Accurate Subcell Integration in the Context of The Finite Cell Method

74 Pages·2013·2.2 MB·English

Faculty of Civil, Geo and Environmental Engineering
Chair for Computation in Engineering
Prof. Dr. rer. nat. Ernst Rank
Highly Accurate Subcell Integration
in the Context of The Finite Cell Method
L´aszl´o Kudela
Master’s thesis
for the Master of Science program Computational Mechanics
Author: La´szlo´ Kudela
Matriculation number: 3627061
Supervisor: Prof. Dr.rer.nat. Ernst Rank
Nils Zander, M.Sc.
Date of issue: 01. April 2013
Date of submission: 20. September 2013

Involved Organisations
Chair for Computation in Engineering
Faculty of Civil, Geo and Environmental Engineering
Technische Universita¨t Mu¨nchen
Arcisstrasse 21
D-80333 Mu¨nchen
Declaration
With this statement I declare, that I have independently completed this Master’s thesis. The
thoughts taken directly or indirectly from external sources are properlymarked as such. This
thesis was not previously submitted to another academic institution and has also not yet
been published.
Mu¨nchen, September 19, 2013
L´aszl´o Kudela
La´szlo´ Kudela
Eggenfeldener Straße 119
D-81929 Mu¨nchen
E-mail: [email protected]

ACKNOWLEDGEMENTS
I would like to offer my special thanks to Nils Zander, my main supervisor, for his continous
support in the last six months. The completion of this thesis would have been not possible
without his help.
I would like to thank Prof. Ernst Rank and Dr. Stefan Kollmannsberger for offering me the
opportunity of writing this thesis.
I thank Tino Bog for his valuable and constructive inputregarding implementation questions
and for his help in the revision of this work.
Advice and assistance given by Felix Frischmann in the field of geometric modeling is greatly
appreciated.
Ithankmycolleague, MohamedElhaddadforthefruitfuldiscussionswehadaboutthetheory
and implementation of FEM.
Thanks to all people from the Chair for Computation in Engineering for the supportand the
great atmosphere. It is a pleasure to work with them.
IamparticularlygratefulforthefinancialsupportoftheGermanAcademicExchangeService
(DAAD) in the last two years.
Last but not least, I would like to thank my family for supporting me during all the years of
my studies. Especially, I want to thank my girlfriend Emese for her loving understanding in
the past times.
ABSTRACT
The Finite Cell Method is an immersed boundary method using higher order finite elements.
In the FCM, the material parameters of the fictitious domain are scaled down, thus a dis-
continuity is introduced in the cut finite cells. This introduces an error in the discretized
weak form of the governing physcial equations. Typically, the integrals are evaluated using
composed Gaussian quadrature, based on a space tree decomposition. The drawback of the
spacetree decomposition is that it introduces a high number of quadrature points and yields
only a 0th order aproximation of the geometry. This thesis proposes a different apporach
to deal with integration: the cut cells are decomposed into quadrilaterals and triangles with
curved sides and the quadrature points are distributed in their parameter spaces. To this
end, an algorithm is proposed that performs this decomposition in two dimensions. The
algorithm is able to work on complex geometries. The results show that the blended subcell
integration is able to produce results with higher precision using less integration points than
the spacetree approach. Thus, the overall computational effort of FCM analyses is consider-
ably reduced in comparison to the spacetree integration. This makes the proposed approach
a good fit into the design-through-analysis pipeline.
VII
Contents
1 Introduction and Motivation 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Theoretical background 3
2.1 The p-version of the Finite Element Method . . . . . . . . . . . . . . . . . . . 3
2.2 The blending function method . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.1 Parametric curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.2 Trimmed curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.3 NURBS curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.4 Blended mappings for quadrilaterals . . . . . . . . . . . . . . . . . . . 12
2.2.5 Blended mappings for triangles . . . . . . . . . . . . . . . . . . . . . . 14
2.2.6 Remark on the consistency of blended elements in p-FEM . . . . . . . 14
2.3 Numerical considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3.1 Numerical integration of parametric curves . . . . . . . . . . . . . . . 19
2.3.2 Numerical integration of blended elements . . . . . . . . . . . . . . . . 21
2.4 The Formulation of the Finite Cell Method . . . . . . . . . . . . . . . . . . . 22
2.4.1 Variational formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4.2 Adaptive integration of cut cells . . . . . . . . . . . . . . . . . . . . . 24
3 Algorithmic partitioning of cut cells 27
3.1 Marching squares approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 Triangulated cut cell approach . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2.1 Numbering and naming conventions . . . . . . . . . . . . . . . . . . . 29
3.2.2 Determining whether the cell is cut . . . . . . . . . . . . . . . . . . . . 31
3.2.3 Determining which edges are cut . . . . . . . . . . . . . . . . . . . . . 32
3.2.4 Creating the auxiliary point and line, intersecting the edges . . . . . . 32
3.2.5 Create blended triangles and quadrilaterals . . . . . . . . . . . . . . . 33
3.2.6 Mapping to the parameter space of the finite cell . . . . . . . . . . . . 35
3.2.7 Subdivision for too complex geometries . . . . . . . . . . . . . . . . . 36
3.3 Implementation of the concept . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4 Patch test of the integration cells 37
4.1 Curved domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.2 Rectangular domain with moving circle . . . . . . . . . . . . . . . . . . . . . 39
4.3 Violin with f-holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5 Finite Cell Method examples 45
5.1 Perforated plate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.2 Poisson problem on a curved domain . . . . . . . . . . . . . . . . . . . . . . . 48
5.3 Compression wave on a violin . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6 Summary, conclusion and outlook 55
6.1 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
1
Chapter 1
Introduction and Motivation
The Finite Element Method (FEM) has been used by scientists and engineers for decades as
a standard approach for solving partial differential equations (PDE) numerically. As a result
of the growth of the computational power in the second half of the 20th century, the method
allowed its users to solve problems of increasing complexity and size. However, the core idea
of FEM remains the same. The domain is subdivided by a suitable spatial discretization
method into a set of smaller subdomains, often called mesh, and the solution is represented
by the linear combination of shape functions whose supportis local to the these subdomains.
One major issue with the standard FEM approach is the generation of a proper mesh. The
method requires the elements of the mesh to conform with geometric boundaries. While this
requirement is easy to fulfill in case of simple geometries, it constitutes a severe bottleneck
for domains with higher geometric complexity. Translating the geometric data that is usu-
ally available in the form of CAD files into a conforming FEM discretization is often time
consuming and expensive. This translation is hard to fully-automate, and the resulting mesh
might need manual adjustments in order to be able to perform numerical computations on
it. Studies show that the time required for creating an analysis suitable geometry and its
computational mesh accounts for about 80% of the overall analysis time [1].
ThedesiretoovercomemeshingproblemsandtobringCADandFEAclosertogetherresulted
in the method of Isogeometric Analysis (IGA), in which the spline basis functions describing
the geometry are directly used as the shape functions of the solution field [1].
Another approach to avoid expensivemesh generation of complex geometries is the immersed
boundary and fictitious domain methods [2]. This group of methods extend the domain of
interest with complex boundaries to a larger embedding domain whose geometry is simpler
and thus can be meshed easily with a structured grid. The influence of the fictitious domain
extension is eliminated by scaling down its material parameters by a penalty factor such that
the consistency with the original problem is preserved.
The Finite Cell Method (FCM) is an immersed boundary method combined with higher
orderfiniteelements (p-FEM)[3]. Themethodshowsexponentialconvergence characteristics
in the energy norm known from p-FEM, and allows accurate numerical calculations to be
performed without having to mesh complex geometries. The method is also suitable for
analyses onimplicitly definedorvoxel basedgeometries thataretypicaline.g. biomechanical
applications [4]. Its biggest advantage lies in fast convergence at almost no meshing costs.
2 1. Introduction and Motivation
1.1 Motivation
The concept of FCM implies that the effort of a structural analysis is shifted from mesh
generation to the level of integration. The penalty factor introduces a discontinuity at the
boundary of the fictitious and physical domain that has to be taken into account at the
construction time of the stiffness matrix. As the numerical quadrature is usually chosen
to be the Gaussian quadrature that does not generally perform well along discontinuities,
the quadrature points have to be distributed in a better way. In the context of FCM this
distribution is usually done by an adaptive spacetree approach, which recursively refines the
finite cell mesh in the cut cells until a pre-defined depth, thus the discontinuity is captured
by a higher density of integration points along the boundary.
Although the spacetree approach converges as the depth of the tree is increased, it has
some drawbacks. On the one hand, the number of integration points increases exponentially
with every new level of spacetree decomposition, thus making the analysis computationally
expensive. On the other hand, using a spacetree results in a 0th order approximation of the
boundary which does not balance well with the higher order shape functions of p-FEM [5].
These drawbacks indicate the need of another quadrature point distribution approach that
should take the geometric description of the boundary into account. Investigations have
already been made in this direction by approximating the cut cells with Lagrangian elements
[5]. A similar approach hasalso been applied to XFEMapplications, wheretheelements that
arecutby theinterface arepartitioned suchthattheedges of theresultingsubcellsalign with
the interface [6]. However, the possibility of directly using the geometrical description of the
boundaryfor the cutcell integration in the context of FCM is still an openquestion and shall
be addressed in this work. To this end, the blending function method as a possible tool for
the quadrature point distribution is going to be investigated.
1.2 Outline
Chapter 2 presents the theoretical background of the Finite Cell Method and the geometric
representation of curves, as well as the blending function method. Chapter 3 discusses the
algorithmic approach to subdivide cut cells into blended subcells. An integration test of
the resulting integration mesh is performed in Chapter 4. The results of the new approach
are presented and compared with the conventional spacetree method in Chapter 5. The
discussion is closed by some concluding remarks and an outlook in Chapter 6 .