Doctoral Thesis No. 54, 2013

Finite Element Methods forThin Structures with Applicationsin Solid Mechanics

Karl Larsson

Department of Mathematics and Mathematical StatisticsUmea UniversitySE-901 87 Umea, Sweden

Department of Mathematics and Mathematical StatisticsUmea UniversitySE-901 87 Umea, Sweden

In affiliation with the Industrial Doctoral School, Umea University

Copyright c© 2013 Karl LarssonISBN 978-91-7459-653-3ISSN 1102-8300Electronic version:ISBN 978-91-7459-654-0 ( http://umu.diva-portal.org )

Typeset by the author using LATEX 2εPrinted by: Print & Media, Umea University, Umea, 2013

To Elliot

Abstract

Thin and slender structures are widely occurring both in nature and in humancreations. Clever geometries of thin structures can produce strong constructionswhile requiring a minimal amount of material. Computer modeling and analysisof thin and slender structures have their own set of problems, stemming fromassumptions made when deriving the governing equations. This thesis dealswith the derivation of numerical methods suitable for approximating solutions toproblems on thin geometries. It consists of an introduction and four papers.

I. K. Larsson, G. Wallgren, and M.G. Larson, Interactive simulation of acontinuum mechanics based torsional thread, Proceedings of VRIPHYS 10:7th workshop on virtual reality interaction and physical simulation (2010),49–58. 1

II. K. Larsson and M.G. Larson, Continuous piecewise linear finite elementsfor the Kirchhoff-Love plate equation, Numerische Mathematik, Volume121, Number 1 (2012), 65–97. 2

III. K. Larsson and M.G. Larson, A continuous/discontinuous Galerkin methodfor the biharmonic problem on surfaces, Preprint

IV. P. Hansbo, M.G. Larson, and K. Larsson, Intrinsic finite element modelingof curved beams, Preprint

In the first paper we introduce a thread model for use in interactive simulation.Based on a three-dimensional beam model, a corotational approach is used forinteractive simulation speeds in combination with adaptive mesh resolution tomaintain accuracy.

In the second paper we present a family of continuous piecewise linear finiteelements for thin plate problems. Patchwise reconstruction of a discontinuouspiecewise quadratic deflection field allows us to use a discontinuous Galerkinmethod for the plate problem. Assuming a criterion on the reconstructions isfulfilled we prove a priori error estimates in energy norm and L2-norm and providenumerical results to support our findings.

The third paper deals with the biharmonic equation on a surface embed-ded in R3. We extend theory and formalism, developed for the approximationof solutions to the Laplace-Beltrami problem on an implicitly defined surface,to also cover the biharmonic problem. A priori error estimates for a continu-ous/discontinuous Galerkin method is proven in energy norm and L2-norm, andwe support the theoretical results by numerical convergence studies for problemson a sphere and on a torus.

1Reproduced with the kind permission of Eurographics.2Reproduced with the kind permission of Springer.

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In the fourth paper we consider finite element modeling of curved beams inR3. We let the geometry of the beam be implicitly defined by a vector distancefunction. Starting from the three-dimensional equations of linear elasticity, wederive a weak formulation for a linear curved beam expressed in global coordi-nates. Numerical results from a finite element implementation based on theseequations are compared with classical results.

Keywords: a priori error estimation, finite element method, discontinuousGalerkin, corotation, Kirchhoff-Love plate, curved beam, biharmonic equation

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Acknowledgments

Financial support for the work herein is given by Surgical Science AB and theIndustrial Doctoral School at Umea University.

First and foremost I would like to thank my supervisor, Prof. Mats G. Larson,for sharing his knowledge and ideas, for giving me inspiration, and for his gen-erosity. I also would like to send a special thanks to Goran Wallgren at SurgicalScience for sharing his wide knowledge in the field of interactive simulations. Mypast and present colleagues, Fredrik Bengzon, August Johansson, Hakan Jakob-sson, Per Vesterlund, Tor Troeng, Jakob Ohrman, and Martin Bjorklund havemade the time writing this thesis very pleasant, and I am grateful for all theirhelp, encouragement, and friendship.

Finally, I want to thank my parents, Jonas and Ulrika, and my family. I feelvery fortunate for my wonderful son Elliot, and thank you Karolina for your loveand support.

Karl LarssonUmea, April 2013

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Contents

1 Introduction 11.1 Thesis objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Mathematical modeling of elasticity 32.1 Linear elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Geometrically nonlinear elasticity . . . . . . . . . . . . . . . . . . . 42.3 Thin geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.4 Curved geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Finite element methods 63.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Weak problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.3 Conforming finite element spaces . . . . . . . . . . . . . . . . . . . 93.4 Discontinuous Galerkin methods . . . . . . . . . . . . . . . . . . . 11

4 Error estimates 124.1 A priori error estimates . . . . . . . . . . . . . . . . . . . . . . . . 12

5 Summary of papers 15I Interactive simulation of a continuum mechanics based torsional

thread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15II Continuous piecewise linear finite elements for the Kirchhoff–Love

plate equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15III A continuous/discontinuous Galerkin method for the biharmonic

problem on surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 15IV Intrinsic finite element modeling of curved beams . . . . . . . . . . 15

Papers I-IV

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1 Introduction

In solid mechanics we study the behavior of solid materials, especially how theymove and deform under the action of forces. The physical property of a materialto return to its original state when applied forces are removed is called elasticityand the study of elastic materials is of great importance both in industry andacademia. These materials are described by physical laws which are mathemati-cally formulated using partial differential equations (PDE). By finding solutionsto these PDE we are able to predict the response materials will give under vari-ous loadings. However, for almost every real world problem we are interested insolving, it is far too difficult to find closed form solutions. With the developmentof good numerical methods and with readily available computer power, findingsolutions by numerical simulation is both convenient and gives, in most cases,accurate results. Still, the solutions from numerical simulation are only approxi-mate and it is therefore highly interesting to get information about the error inthe form of error estimates.

One of the most versatile numerical methods for approximating solutions toPDE are finite element methods. Built on a solid mathematical foundation finiteelement methods produce approximations by dividing the geometry into smallparts, and on each part let the approximate solution be represented locally by asimple function, typically a polynomial. This approach gives great flexibility insolving problems on complex geometries and the mathematical foundation pro-vides powerful tools for finding various types of error estimates. While simulationand analysis in solid and structural mechanics has been one of the most impor-tant applications driving the early development of finite element methods, thescope of finite elements today has extended far outside this field.

Many constructions in nature and in human creations make use of thin struc-tures to produce strong constructions while requiring a minimal amount of mate-rial. The equations used for modeling such structures put special constraints onthe finite elements used in the numerical simulation of these problems. Methodswhich usually work fine may break down, or ‘lock’, when applied to problems in-volving very thin structures. This calls for especially constructed finite elementmethods which are suitable for simulation of thin structures.

1.1 Thesis objectives

The main objectives of this thesis have been:

• Develop new finite element methods suitable for approximating solutionsto elasticity problems on thin geometries.

• Derive a priori error estimates and provide numerical results for developedmethods.

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• Investigate modeling of problems on embedded geometries with implicitrepresentations of the geometry.

1.2 Main results

• Investigated the use of corotational methods in interactive simulation oftorsional threads. A special beam element with global nodal variables wasapplied in a corotational finite element method. Artifacts due to locallylarge deformation was controlled by an adaptive refinement/coarsening al-gorithm. (Paper I)

• Derived a family of continuous piecewise linear finite element methods forthe fourth order Kirchhoff–Love plate equation. Patchwise reconstructionof a discontinuous piecewise quadratic deflection field enabled the use of adiscontinuous Galerkin method for the plate equation. (Paper II)

• Proved a priori error estimates for the method in Paper II. Due to thepatchwise reconstruction, this analysis involves the use of non-standardtechniques.

• Extended a framework for finite element modeling of second order ellipticPDE on surfaces to a fourth order elliptic PDE. The H2-conformity cri-terion was handled by using a continuous/discontinuous Galerkin method.(Paper III)

• Proved a priori error estimates for the method in Paper III. The estimatestake both the approximation of the geometry and the approximation of thesurface differential operators into account.

• Showcased intrinsic modeling of codimension-two problems by deriving avariational formulation for a three-dimensional curved beam from the equa-tions of linear elasticity. (Paper IV)

1.3 Future work

Future work and extensions of the results herein, include:

• Extend the continuous piecewise linear finite element method in Paper IIto surfaces.

• Extend the results of Paper III to more complex models, for example inelasticity problems.

• Development of locking free methods for the beam formulation in Paper IVby careful construction of the finite element spaces.

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2 Mathematical modeling of elasticity

The mathematical modeling of elastic objects are based on the fundamental con-cepts of stress σ and strain ε. Loosely we can say that the stress describes thedistribution of internal forces in an elastic object while the strain is a measure ofdeformation at a point in the object. Depending on how we define the relation-ship between stress and strain, the so called constitutive relationship, we maymodel various materials, for example viscoelastic materials, plastic materials, orthermoelastic materials. In this thesis we however limit ourselves to the linearelastic isotropic materials described using Hooke’s law, i.e.

σ = 2µε+ λtr(ε)I , (1)

where µ > 0 and λ > 0 are the so called Lame parameters.Let an elastic object in an undeformed reference configuration occupy a do-

main Ω with piecewise smooth boundary ∂Ω. From continuum mechanics [13,32]we have the equation of equilibrium for an elastic object which reads

ρx−∇ · (Fσ) = f in Ω , (2)

where ρ is the density, x = x(x0) is the current position of a material point x0 inΩ, F = ∇x is the deformation gradient, σ is the symmetric 2:nd Piola-Kirchhoffstress tensor, and f is the body load density. In the static case the correspondingequilibrium equation reads

−∇ · (Fσ) = f in Ω . (3)

A strain measure which is work conjugate with the 2:nd Piola-Kirchhoff stress isthe Green strain tensor defined by

ε(u) =1

2

(∇u+ (∇u)T + (∇u)T∇u

), (4)

where u = x− x0 are displacements.While the various boundary conditions in elasticity problems are crucial in

actual applications, we throughout this introduction use simplified boundary con-ditions to keep the presentation clear.

2.1 Linear elasticity

Consider a linear elastic material fixed at the boundary ∂Ω. Assuming strain issmall, i.e. |∇u| 1, we may approximate (3), (1), and (4) by the equations oflinear elasticity which read

−∇ · σ(u) = f in Ω , (5a)

σ = 2µε+ λtr(ε)I in Ω , (5b)

u = 0 on ∂Ω , (5c)

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where the strain is given by the linearized strain tensor

ε(u) =1

2

(∇u+ (∇u)T

). (5d)

In Papers II, III, and IV, we consider equations which more or less explicitlymay be viewed as modeling of problems derived from linear elasticity.

2.2 Geometrically nonlinear elasticity

As deformation is independent of pure translation and rotation of an object it isnatural that also the stress and strain must be invariant under translation androtation in the sense that, if ε = ε(x), we have

ε(Rx+ c) = ε(x) , (6)

where R is a constant rotation matrix and c is a constant vector. The linearizedstrain (5d) is invariant under translation but not under rotation. By the assump-tion that displacements and rotations may be arbitrarily large while deformationmust be locally small, a suitable modification to the linearized strain tensor isthe following, corotational, strain tensor

εCR(x) = ε(RTx− x0) , (7)

whereR is an extractable rotation field which is assumed to be locally constant inspace. This is the foundation of the corotational approach [16, 17, 22]. It allowsfinite elements developed for small strain problems to be applied to problemsinvolving large displacements and rotations as long as the deformations locallyare small.

To interactively simulate soft volumetric materials this approach was success-fully used in [26] and in Paper I we consider the same method for interactivesimulation of a torsional thread. The validity of the assumption that deforma-tions locally are small is controlled by using an adaptive refinement/coarseningprocedure.

2.3 Thin geometries

In the case of very thin structures it is often possible to reduce the dimensionof the problem, many times yielding better results. Usually this reduction isbased on a hypothesis or analysis of how an elastic object deforms in the reduceddimensional direction when a dimension of the elastic object approaches zero.The displacements then no longer depend on derivatives in that direction butcan rather be described using derivatives in the other dimensional directions.For this reason the order of the differential operators typically increase whendimensions are reduced.

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In the following two examples the order of the differential equations hasincreased to four, compared with the second order PDE describing the three-dimensional elastic model (5).

A beam problem. Let Ω be a domain in R. We now consider the one dimen-sional problem for the bending of a thin clamped beam. Using the assumptionthat a plane cross-section normal to the beam midline remains plane and nor-mal to the midline after deformation the Euler–Bernoulli beam equation can bederived. The deflection u of the clamped beam under a distributed load f isdescribed by the following fourth order problem

d4u

dx4= f in Ω , (8a)

u =du

dx= 0 on ∂Ω , (8b)

where f is scaled with the bending stiffness of the beam. A beam model, albeitin three dimensions and for curved geometries, is derived in Paper IV.

A clamped plate. Let Ω be a domain in R2 with piecewise smooth boundary∂Ω and outward pointing normal n. The Kirchhoff–Love plate equation governingthe deflection of a thin plate is derived from the assumption that a straight linein the material which is normal to the plate midsurface remains straight andnormal to the midsurface after deformation. Using this equation the deflectionfield u of a thin clamped plate under a distributed load f is described by

∆2u = f in Ω , (9a)

u = n · ∇u = 0 on ∂Ω , (9b)

where f is scaled with the flexural rigidity of the plate. This equation is alsodenoted the biharmonic equation and we return to this problem in Papers II andIII.

2.4 Curved geometries

Within solid mechanics a large motivation for considering thin geometries is thesimulation and analysis of shells. The governing equations for shells are typicallydescribed using differential geometry where the geometry of the shell midsurfaceis assumed to be represented by a collection of local parametrizations of thesurface [11].

There are however cases where it is preferred to use an implicit representationof the geometry instead. An example is PDE on evolving surfaces as parametriza-tions may be hard to find and do not naturally deal with changes of topology.

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Implicit representations of the geometry may for example be given through signeddistance functions, level set functions [31], or phase fields [23]. Elasticity mod-els on curved surfaces have been established using signed distance functions torepresent the midsurface [18–20]. The main differential tool in this setting is thetangential gradient ∇Γ = P∇, where P is a projection onto the tangential planeof a surface Γ. In terms of the tangential gradient, the Laplace-Beltrami operatormay be expressed ∆Γ = ∇Γ · ∇Γ. Using this tangential calculus the governingequations of thin elastic structures may be formulated in three dimensions usingglobal coordinates.

Within this thesis we only consider curved geometries which are implicitlydefined. In Paper III we solve the biharmonic equation ∆2

Γu = f on a surface Γembedded in R3, where Γ is defined using a signed distance function. In PaperIV we derive a model for a curved beam in R3, where the beam midline is definedusing a vector distance function. The resulting governing equations are expressedin global coordinates which makes for straightforward implementation. It shouldbe noted that even though implicit definitions of the geometries are used for thederivations and analysis of the methods, we in these papers still use parametricrepresentations in the actual implementations.

While the uses and benefits of implicit surface representations quite clear,the reasons for considering implicit representations for one-dimensional curvesin R3 are less obvious as one-dimensional curves are always easily parametrized.It is however interesting to consider such problems in the same framework asproblems on surfaces and it is easy to envision elastic structures consisting ofshells enforced by beams.

3 Finite element methods

One of the most versatile methods for calculating numerical approximations of so-lutions to PDE are finite element methods (FEM). Based on a variational formu-lation, it approaches the problem by dividing the domain into many smaller partsand on each part approximating the solution with a simple polynomial. Whilethe local approximations are simple, this allows for the global approximations tobe quite intricate and able to capture local effects, also on complex geometries.Further, there is a well developed mathematical framework for the analysis ofFEM. While an introduction to FEM is given below we refer to [8–10,12, 27, 28]for further reading.

3.1 History

In its early stages FEM was developed more or less parallelly, in engineeringand in academia. The early industrial development during the 1950’s was mainlydriven by the aeronautical industry’s need to solve complex problems in elasticity

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and structural analysis. With the emerging access to computer powered calcula-tions, engineers used ideas from structural mechanics to solve complex problemswith the help of computers. A classical publication based on this work is a 1956paper by Turner, Clough, Martin, and Topp [34]. It was also Clough who coinedthe name finite elements in 1960. However, the engineers were not aware of theearlier related developments in mathematics.

In the beginning of the twentieth century variational methods had been usedby Ritz [30] and Galerkin [24] to find approximations to solutions of differentialequations. Many credit the first appearance of FEM in mathematical literatureto Courant in the 1940’s [15]. However, as computers were not yet available forcalculations at that time the true usefulness of the method was not realized untillater. During the late 1960’s and the 1970’s the rigorous mathematical theory ofFEM was developed and formalized in [4,5,12,33]. For a more thorough overviewof the history of finite elements and a more complete set of references, see [6,35].

3.2 Weak problems

It is very useful to reformulate differential equations such as equations (5), (8),and (9) in weak form. Solutions to the weak form of a problem are not requiredto fulfill the differential equation in an absolute sense but rather when multipliedby certain test functions and integrated over the domain.

To present weak formulations of the previously mentioned problems we firstneed to introduce the following Sobolev spaces: Let Hk(Ω), with k ≥ 0, be theset of square integrable functions whose weak derivatives up to order k also aresquare integrable. We equip this space with the norm

‖u‖Hk(Ω) =

∑|α|≤k

‖Dαu‖L2(Ω)

1/2

, (10)

the seminorm

|u|Hk(Ω) =

∑|α|=k

‖Dαu‖L2(Ω)

1/2

, (11)

and note that Hk(Ω) is a Hilbert space.We first consider Poisson’s problem where Ω ∈ R2: Given f , find u such that

−∆u = f in Ω and u = 0 on ∂Ω . (12)

Multiplying both sides of the left equation by a test function v ∈ V , where inthis case V = v ∈ H1(Ω) : v = 0 on ∂Ω, and integrating over Ω we get the

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weak formulation of the differential equation as

−∫

Ω

v∆u dx =

∫Ω

vf dx for all v ∈ V . (13)

By using a Green’s formula (integration by parts) we may rewrite the left side as

−∫

Ω

v∆u dx =

∫Ω

∇v · ∇u dx−∫∂Ω

vn · ∇u dx , (14)

where n is the outward pointing normal on ∂Ω and the last term vanishes asv = 0 on ∂Ω. Defining the symmetric bilinear form a(u, v) =

∫Ω∇u · ∇v dx

and linear functional l(v) =∫

Ωfv dx, we can express the problem in an abstract

setting: Given f ∈ L2(Ω), find u ∈ V such that

a(u, v) = l(v) for all v ∈ V . (15)

This symmetric bilinear form is continuous and coercive, i.e. we have

|a(u, v)| ≤ c1‖u‖Hm(Ω)‖v‖Hm(Ω) for all u, v ∈ V , (16)

a(v, v) ≥ c2‖v‖2Hm(Ω) for all v ∈ V , (17)

where Hm(Ω) is the underlying Sobolev space of V , in this case m = 1.We continue by presenting weak forms of the example differential equations

given in Section 2. They are formulated in the abstract setting (15), and hencewe provide the linear functional, bilinear form, and suitable Sobolev space foreach problem.

Linear elasticity. The weak form for linear elasticity (5) is given by the linearfunctional l(v) =

∫Ωf · v dx, where f ∈ [L2(Ω)]3, and the bilinear form

a(u,v) =

∫Ω

σ(u) : ε(v) dx , (18)

where the contraction operator is defined by A : B =∑

AijBij . The suitableSobolev space is given by V = v ∈ [H1(Ω)]3 : v = 0 on ∂Ω.

A clamped beam. For the one-dimensional clamped beam (8) the appropriateSobolev space is V = v ∈ H2(Ω) : v = dv/dx = 0 on ∂Ω, the linear functionalis l(v) =

∫Ωfv dx, where f ∈ L2(Ω), and the bilinear form is given by

a(u, v) =

∫Ω

d2u

dx2

d2v

dx2dx . (19)

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A clamped plate. The weak formulation of the two-dimensional clamped plateproblem (9) is given by the linear functional l(v) =

∫Ωfv dx, where f ∈ L2(Ω),

and the bilinear form

a(u, v) =

∫Ω

ν∆u∆v + (1− ν)H(u) : H(v) dx , (20)

where ν is the Poisson’s ratio and the Hessian is given by Hij(v) = ∂2v∂xi∂xj

. The

appropriate space for this weak formulation is

V = v ∈ H2(Ω) : v = n · ∇v = 0 on ∂Ω . (21)

3.3 Conforming finite element spaces

As the function spaces V defined above are infinite dimensional we typicallycan only find solutions in special cases or in especially constructed problems.By rather seeking the solution within a finite dimensional subspace Vh ⊂ V wemay numerically calculate the best approximation uh ∈ Vh to the solution. Theproblem of finding the approximate solution reads: Find uh ∈ Vh such that

a(uh, v) = l(v) for all v ∈ Vh . (22)

We let the approximate solution uh be written as a linear combination of Nlinearly independent basis functions ϕi(x) spanning Vh, i.e.

uh =

N∑i=1

ϕi(x)ξi , (23)

where the coefficients ξi are the degrees of freedom describing uh. The problemmay now be formulated as solving the linear system of equations

N∑i=1

a(ϕi, ϕj)ξi = l(ϕj) for j = 1..N , (24)

for the N unknown coefficients ξi.Next we turn to a construction of finite dimensional spaces Vh which consti-

tutes the finite element method. By partitioning the domain into a finite numberof ‘elements’, i.e. a mesh, and choosing basis functions on each element we definethe finite element space Vh.

Mesh

The finite dimensional function spaces Vh are closely related to a partition of thedomain Ω into a set T of disjoint subdomains K such that Ω =

⋃K∈T K. This

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is what we call a mesh. In one dimension, i.e. Ω ∈ R, the mesh is just a partitionof the interval defining Ω. For Ω ∈ R2 and Ω ∈ R3 we typically partition thedomain using triangles and tetrahedrons respectively, but any polyhedron shapeis possible. From here on we for convenience assume a triangle mesh unlessotherwise stated.

We require the triangle mesh to be shape regular, i.e. there must exist aconstant γ such that

diameter of K

diameter of the largest ball in K≤ γ , (25)

for all K ∈ T . This requirement ensures the quality of the mesh and is neededfor various theoretical results to hold.

Let the mesh size h be given by the largest diameter of all triangles K in themesh. By choosing a smaller mesh size h, we get a more detailed approximationspace Vh and typically a better approximation uh.

Basis functions

The finite element space Vh is spanned by the finite element basis functionsϕi defined on each triangle K. These describe polynomial functions on K,usually by using nodal values as degrees of freedom. The triangle K, the spacespanned by the basis functions ϕi on K, and the set of nodal variables NKtogether constitute what we call a finite element. A common family of finiteelements is Lagrange elements and we illustrate the nodal variables defining theapproximation on each triangle in Figure 1.

Lagrange elements are easily constructed on both triangles in R2 and on tetra-hedrons in R3. The resulting finite element spaces are by construction continuouswhich is convenient as C0(Ω) ⊂ H1(Ω) and thus they can define appropriate ap-proximation spaces for many problems, for example in linear elasticity.

In the case of the clamped beam or the clamped plate the requirement isthat Vh ⊂ V ⊂ H2(Ω) which in practice means that the deflections must beC1-continuous as C1(Ω) ⊂ H2(Ω). For one-dimensional problems such as theclamped beam it is straightforward to create finite elements of any regularity.We may for example construct our finite element space using cubic Hermite

Figure 1: Linear, quadratic, and cubic triangular elements.

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splines giving Vh ⊂ C1(Ω). Creating C1(Ω) finite element spaces on triangularmeshes is however nontrivial. Classical C1(Ω) finite elements include the fifthorder Argyris triangle [1] and the Hsieh-Clough-Tocher macro element [14].

3.4 Discontinuous Galerkin methods

An alternate approach to constructing conforming finite element spaces, whereVh ⊂ V , is using a method which allows the use of nonconforming approximationspaces, i.e. Vh /∈ V . Such a method is the discontinuous Galerkin (dG) methodwhich comprises features from both finite element methods and finite volumemethods. There are various flavors of dG methods but in this thesis we refer tothe classical method of Nitsche [29], also known as interior penalty methods [2].For a summary of discontinuous Galerkin methods for elliptic problems see [3]and the references therein.

To formulate a dG method we first need to introduce some notation. Let Ebe the set of interior element edges in T and on each edge in E define a normaldirection n. The jump between elements is denoted J·K and the average is denoted·. We may now formulate a dG method for Poisson’s problem (12) using a finiteelement space Vh which is discontinuous between elements, i.e. Vh /∈ C0(Ω). Themethod is defined by the following bilinear form

a(u, v) =∑K∈T

∫K

∇u · ∇v dx (26a)

−∑E∈E

∫E

JuKn · ∇v+ n · ∇uJvK dx (26b)

+ β∑E∈E

∫E

JuKJvK dx , (26c)

where β is a penalty parameter which penalizes jumps in the solution. Note thatif β →∞ and if it is allowed by the basis functions, the space becomes continuousand the method coincides with the conforming finite element method.

In this thesis we use dG methods to handle the H2(Ω) conformity requirementinduced by fourth order problems. The underlying method in Paper II is a dGmethod for the plate equation [25] using a finite element space which is piecewisequadratic and continuous only in the mesh’s vertices. In Paper III the finiteelement space used is continuous but not C1(Ω) as needed for conformity. Herethe method is based on a continuous/discontinuous Galerkin method for fourthorder elliptical PDE [21]. The penalty terms in the bilinear forms of the dGmethods in both these papers read

β∑E∈E

∫E

Jn · ∇uKJn · ∇vK dx , (27)

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and thus β here penalizes jumps in the normal gradient.

4 Error estimates

An important area of mathematical research within the field of numerical meth-ods for approximating solutions to PDE:s is the derivation of error estimates.Such estimates can be divided into two categories; a priori estimates and a pos-teriori estimates.

A priori error estimates give upper bounds for the error based on the exactsolution and space dependent quantities such as the mesh size h or the polynomialorder of approximation p. Within this thesis we derive a priori error estimates inPapers II and III and in the next section we give some general notes on a priorierror estimates.

A posteriori error estimates instead bound the error by a numerical solutionand space dependent quantities. This is useful in the design of adaptive algo-rithms to iteratively create an optimal finite element space with regards to theerror in some desired goal quantity. Such an algorithm can for example auto-matically refine the mesh or increase the polynomial order of approximation inareas of the domain where it is most meaningful. While a highly interesting fieldof research, it is outside the scope of this thesis and we refer to [7] for moreinformation.

4.1 A priori error estimates

By a priori error estimates we show that a numerical method converges to theexact solution. This means that the error e = u − uh in some norm goes tozero as the approximation space Vh is refined. We also get information aboutthe asymptotic behavior of the method, i.e. how fast the approximations willapproach the exact solution. In this thesis we refine Vh by decreasing the meshsizeh, but it is also possible to refine Vh by increasing the order p of the polynomialapproximation, or by a combination of these techniques. Below we will mentionsome standard results and techniques related to a priori estimates which we makeuse of in Papers II and III.

We first consider the case of conforming finite element spaces. Let the energynorm be given by ‖v‖V =

√a(v, v), and let Hm(Ω) be the underlying Sobolev

space of V . Combining (15) and (22) we have that

a(u− uh, v) = 0 for all v ∈ Vh , (28)

a property known as the Galerkin orthogonality. A consequence of the Galerkinorthogonality is the abstract a priori error estimate

‖u− uh‖V = infvh∈Vh

‖u− vh‖V , (29)

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which means that in energy norm the finite element approximation uh is the bestpossible approximation within Vh. By using continuity (16) and coercivity (17)this estimate instead gives an inequality in Hm(Ω)-norms

‖u− uh‖Hm(Ω) ≤c1c2

infvh∈Vh

‖u− vh‖Hm(Ω) , (30)

known as Cea’s lemma.Such abstract error estimates may be transformed into explicit error estimates

depending on the exact solution u and space dependent variables by the useof interpolation theory. Consider the case where Vh is the space of continuouspiecewise polynomial functions of order p defined on a triangle mesh. If πh : V →Vh is the Lagrange interpolant we from interpolation theory have the estimate

|u− πhu|Hk(Ω) ≤ Chp+1−k|u|Hp+1(Ω) , (31)

and as infwh∈Vh‖u−wh‖Hm(Ω) ≤ ‖u−πhu‖Hm(Ω) we in combination with Cea’s

lemma (30) have the error estimate

‖u− uh‖Hm(Ω) ≤ Chp+1−m|u|Hp+1(Ω) . (32)

By using a duality argument known as Nitsche’s trick we can also prove errorestimates in L2-norm: Let φ ∈ V be the solution to the weak problem

a(v, φ) =

∫Ω

(u− uh)v dx for all v ∈ V . (33)

Choosing v = u − uh, we by (33), Galerkin orthogonality (28), and continuity(16) have

‖u− uh‖2L2(Ω) = a(u− uh, φ) (34)

= a(u− uh, φ− wh) (35)

≤ c1‖u− uh‖Hm(Ω)‖φ− wh‖Hm(Ω) (36)

for all wh ∈ Vh. This gives the abstract estimate

‖u− uh‖L2(Ω) ≤ c1(

infwh∈Vh

‖φ− wh‖Hm(Ω)

‖u− uh‖L2(Ω)

)‖u− uh‖Hm(Ω) , (37)

where we can control the first term by using an interpolation estimate (31) forwh = πhφ and showing that the stability estimate

|φ|Hp+1(Ω) ≤ ‖u− uh‖L2(Ω) (38)

holds.

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We now turn to a priori error estimates when using nonconforming approxi-mation spaces. For the dG methods in Paper II and III we extend the bilinearform so that we can formulate the weak problem by: Find u ∈ V + Vh such that

a(u, v) = l(v) for all v ∈ V + Vh . (39)

The dG method reads: Find uh ∈ Vh such that

a(uh, v) = l(v) for all v ∈ Vh . (40)

By proving that fundamental properties such as Galerkin orthogonality, conti-nuity, and coercivity also holds for the extended bilinear form in some norm onV + Vh, we may prove a priori error estimates for the dG methods similarly toestimates for conforming finite element spaces.

However, in Paper III we formulate our method using approximate bilinearforms and linear functionals which depend on the approximation space Vh. Thebilinear form a(·, ·) in this case contains differential operators which depend on asurface Γ embedded in R3 and integration is also done over Γ. Approximating Γby a discrete surface Γh gives both an approximation ah(·, ·) to the bilinear form,and an approximation lh(·) to the linear functional. The resulting approximatemethod is formulated: Find uh ∈ Vh such that

ah(uh, v) = lh(v) for all v ∈ Vh . (41)

While Cea’s lemma only regard the error introduced by the approximation space,we here must also consider the errors introduced by the approximations of thebilinear form and the linear functional. By proving that ah(·, ·) is coercive andcontinuous with respect to some norm ‖ · ‖h we can derive the abstract a priorierror estimate

‖u− uh‖h ≤ C

(inf

vh∈Vh

(‖u− vh‖h + sup

wh∈Vh

|a(vh, wh)− ah(vh, wh)|‖wh‖h

)

+ supwh∈Vh

|l(wh)− lh(wh)|‖wh‖h

). (42)

This estimate is known as Strang’s first lemma and it is the foundation for theproof of the a priori error estimate in Paper III.

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5 Summary of papers

I Interactive simulation of a continuum mechanics basedtorsional thread

In this paper we explore the use of corotational FEM for interactive simulation ofa torsional thread. By using a slender three-dimensional element featuring globaldegrees of freedom we can apply the same corotational algorithm as successfullyused on tetrahedral elements. We use a hierarchical mesh structure with adaptiverefinement/coarsening to avoid breaking the corotational assumption of locallysmall strains.

II Continuous piecewise linear finite elements for theKirchhoff–Love plate equation

In this paper we present a family of continuous piecewise linear finite elementsfor the Kirchhoff–Love plate equation. We use patchwise reconstruction of adiscontinuous piecewise quadratic deflection field and apply a dG method for theplate equation. Given a criterion on the reconstructions we prove a priori errorestimates in a discrete energy norm and in L2-norm. We provide three examplereconstructions and show that the simplest reconstruction yields the Basic PlateElement. This particular reconstruction does however not fulfill the criterion forthe a priori estimate. Numerical results on unstructured meshes indicate thatonly the example reconstructions fulfilling the criterion converge.

III A continuous/discontinuous Galerkin method for thebiharmonic problem on surfaces

In this paper we extend formalism used for approximating solutions to the Laplace–Beltrami problem on implicitly defined surfaces, to the biharmonic equation onsurfaces. To deal with the H2-conformity requirement imposed by the fourthorder PDE we use a continuous/discontinuous Galerkin method on a facet ap-proximation of the surface. We prove a priori error estimates in a discrete energynorm and in L2-norm and give numerical results for problems on a sphere andon a torus to confirm the sharpness of our estimates.

IV Intrinsic finite element modeling of curved beams

Models for curved beams are often established using local equilibrium equationswhere the local coordinates are defined through a parametrized curve and theSerret–Frenet formulas. In this paper we instead let the geometry be implicitlydefined though a vector distance function. From the three-dimensional equations

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of linear elasticity we derive a weak formulation for the governing equations ofa linear curved beam expressed in three dimensions. We discuss compatibilityrequirements for very thin beams and compare numerical and classical results.

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