implementation of a high-order kirchho -love shell: a ... · latter thin-walled structures can be...

Faculty of Civil, Geo and Environmental Engineering

Chair for Computation in Engineering

Prof. Dr. rer. nat. Ernst Rank

Chair of Structural Analysis

Prof. Dr.-Ing. Kai-Uwe Bletzinger

Implementation of a high-order Kirchhoff-Love shell:

a comparison of IGA and p-FEM.

Luca Coradello

Master’s thesis

for the Master of Science program Computational Mechanics

Author: Luca Coradello

Matriculation number: 03646756

Supervisor: Prof. Dr.rer.nat. Ernst Rank

Dr.-Ing. Roland Wuchner

Advisor: Michael Breitenberger, M.Sc.

Prof. Dr.-Ing. Josef Kiendl, TU Braunschweig

Dr.-Ing. Stefan Kollmannsberger

Prof. Dr.-Ing. Alessandro Reali, University of Pavia

Nils Zander, M.Sc.

Date of issue: 01. October 2015

Date of submission: 31. March 2016

Involved Organisations

Chair for Computation in EngineeringFaculty of Civil, Geo and Environmental EngineeringTechnische Universitat MunchenArcisstraße 21D-80333 Munchen

Declaration

With this statement I declare, that I have independently completed this Master’s thesis. Thethoughts taken directly or indirectly from external sources are properly marked as such. Thisthesis was not previously submitted to another academic institution and has also not yetbeen published.

Munchen, March 31, 2016

Luca Coradello

Luca CoradelloArcisstr.21D-80333 Munchene-Mail: [email protected]

III

Acknowledgements

I would like to sincerely thank Nils Zander, who invested a tremendous amount of energy andtime for the completion of this work. His knowledge and cleverness have been a continuoussource of inspiration for me. I am sure one day in the not-so-near future we will both missour Friday afternoon debugging session.

My genuine thanks also go to Dr.-Ing. Stefan Kollmannsberger, for leading and making therealization of this joint project possible. Furthermore, he has always shown sincere interestin my work and my personal development. For this and for the wise guidance I am grateful.

I would like to thank Prof. Ernst Rank and Dr.-Ing. Roland Wuchner for supervising andexamining this work.

Thanks to Prof. Dr.-Ing. Josef Kiendl and Prof. Dr.-Ing. Alessandro Reali for suggestingthe topic of this thesis and for their enthusiasm for the results of this project. Additionally,I would like to thank Michael Breitenberger for his contribution to this work.

I would also express my gratitude to Tino Bog, Laszlo Kudela and the entire research groupof the chair for Computation in Engineering for the irreplaceable support and for alwaystreating me as part of the team.

My gratitude also goes to Massimo Carraturo, Davide D’Angella and Alexandre Mongeaufor the endless brainstorming, the positive interaction and the countless hours spent togetherinside and outside the BGCE-room.

I would also like to thank my fantastic friends for the encouragement and for always be therefor me even though we are often so far away. You know who you are and why you are soimportant to me.

Lastly, my special thanks go to my family, who has always supported me unconditionally inevery decision along this journey. I could not be standing here at this precise moment if itwasn’t for you.

IV

Contents

1 Introduction 31.1 Goal of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Kirchhoff-Love shell formulation 52.1 Continuum mechanics for shells . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 The strong form of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 The weak form of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Bubnov-Galerkin discretization . . . . . . . . . . . . . . . . . . . . . . . . . . 82.5 Rotational boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Isogeometric Analysis (IGA) 113.1 NURBS description of geometries . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.1.1 B-splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.1.2 NURBS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.1.3 Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 IGA discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4 High-order polynomials Finite Element (p-FEM) 214.1 Quasi-Regional mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.2 High-order shape functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.2.1 Lagrange polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.2.2 Legendre polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.2.3 Integrated Legendre polynomials . . . . . . . . . . . . . . . . . . . . . 25

4.3 High-order p-FEM discretization . . . . . . . . . . . . . . . . . . . . . . . . . 27

5 The Finite Cell Method (FCM) 295.1 Introduction to FCM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.2 Adaptive integration schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.2.1 Spacetree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.2.2 Smarttree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.3 FCM for shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.3.1 Trimming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.3.2 Modified weak form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

6 Implementational Aspects 356.1 Introduction to AdHoC++ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356.2 Implementation of IGA into a p-FEM code . . . . . . . . . . . . . . . . . . . 35

1

7 Numerical results 397.1 Elliptic eigenvalue problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

7.1.1 Longitudinal vibration of a one-dimensional elastic rod . . . . . . . . . 407.2 Shell obstacle course . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

7.2.1 Scordelis-Lo roof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467.2.2 Pinched cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

7.3 Trimming of shells and FCM . . . . . . . . . . . . . . . . . . . . . . . . . . . 537.3.1 Modified Scordelis-Lo roof . . . . . . . . . . . . . . . . . . . . . . . . . 537.3.2 Point load on a violin . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

8 Summary and conclusion 598.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2

Chapter 1

Introduction

It is common practice in every field of science and engineering to describe physical phenom-ena by simplified models. Under this category falls the dimensionality reduction technique.Albeit every physical body is a concrete three-dimensional continuum, its description can besimplified, according to its predominant dimensions. Take for instance an object that has adimension significantly smaller than the other two. It can be described by employing reducedtwo-dimensional models, i.e. plates, membranes and shells. It should be noted that all thelatter thin-walled structures can be seen as special cases of shells, where either the shell isflat (plate) or only tangential load can be carried (membrane). Despite their small thickness,shells have a high load-carrying capacity due to their curvature. Furthermore, since they cancarry load efficiently through their membrane state, they are usually considered as lightweightstructure. It comes as no surprise that they are widely used in various engineering areas,spacing from the automotive and aerospace industry to civil engineering.

In the scope of this work, the Kirchhoff-Love mathematical model is employed to describethe mechanical behaviour of shells. This theory was developed in 1888 by August E.H.Love [34] using assumptions firstly proposed by Gustav R. Kirchhoff in 1850 [31]. The mainassumption of the theory resides in neglecting transverse shear deformations, namely alongthe thickness of the shell. For thin shells these effects are often negligible, where an indicatorto determine whether a shell is thin or thick is its slenderness. The slenderness is defined asthe ratio between the radius of curvature of the shell and its thickness. From an heuristicapproach, shells are considered thin if their slenderness is bigger than 20. Another importanttheory widely used in shell analysis is the Reisner-Mindlin model, which takes into accounttransverse shear deformations. The latter approach is often preferred in shell analysis usingthe standard Finite Element Method (FEM).

Nowadays, the Finite Element Method is the most established method for solving complexmechanical problems using computers. Given a problem to solve, the FEM is based on adiscretization into elements of the domain and it results in a numerical approximation of theactual solution. Therefore, arbitrary complex geometries can be analyzed by means of theFEM. Standard elements make use of linear polynomials as shape functions to approximatethe analytical solution, due to their simplicity and versatility. One of the drawback of suchelements is that the continuity between them is bounded to be C0. Subsequently, this hindersa straightforward implementation of the Kirchhoff-Love shell theory into the machinery ofthe FEM, since the former requires at least C1 continuity. This is the main reason why the

1.1. Goal of the thesis 3

Reisner-Mindlin model is predominantly used in FEM shell analysis, since its formulationrequires only C0 inter-element continuity.

In this thesis, two unconventional variants of the FEM are used to derive a Kirchhoff-Loveshell element based only on displacement degrees of freedom (DOFs). These innovative meth-ods are isogeometric analysis (IGA) first proposed by Hughes et al. [22] and the high-orderpolynomial FEM (p-FEM) version established by Babuska et al. [4], respectively. The twomethods are compared through several benchmark examples in the realm of structural vi-brations and static linear elasticity. Furthermore, both formulations are encapsulated intoan embedded domain method, which makes use of high-order shape functions, either non-uniform rational B-Splines (NURBS) or polynomials. This approach is being developed atthe chair for Computation in Engineering, Technical University of Munich, under the nameFinite Cell Method (FCM), firstly introduced by Parvizian et al. [37]. This method allows tohandle easily the mechanical analysis of trimmed geometrical models, which are the standardin CAD software. Indeed, by means of a fictitious domain approach, also the displacementfield is “trimmed” according to the physical geometry.

1.1 Goal of the thesis

At first, the rotation-free Kirchhoff-Love shell element, presented in [29], is implementedinto the high-order Finite Element code used at the chair for Computation in Engineering(AdHoC++ ). The primary goal of this thesis is to perform a comparison between IGA andp-FEM on several benchmark problems. Additionally, given the flexible structure of theframework, a mixed description of the discretization of the geometry and the solution field isinvestigated. Finally, the Kirchhoff-Love shell formulation is extended and brought togetherwith the concepts of FCM, leading to further numerical investigations on trimmed geometries.

1.2 Outline of the thesis

This thesis is structured as follows. Chapter 2 provides a review of continuum mechanicsneeded for the shell kinematics. Furthermore, the strong and weak form for the Kirchhoff-Love formulation are discussed. Subsequently, the weak form is discretized by means of thestandard Bubnov-Galerkin approach.Afterwards, in Chapter 3, the concept of IGA is analyzed, starting from the geometric mean-ing of B-Splines and NURBS and their mathematical formulation.Next, Chapter 4 gives insights into p-FEM, where the idea of Quasi-Regional Mapping for thegeometry description is introduced together with several possible choices of shape functions.The theoretical background of FCM is presented in Chapter 5, where particular focus is puton the challenges related to integration and possible remedies.In Chapter 6 some implementational aspects are given.Finally, in Chapter 7, the numerical investigations are summarized and thoroughly discussed.

4

Chapter 2

Kirchhoff-Love shell formulation

In this chapter, the Kirchhoff-Love shell formulation is introduced, following closely thederivation published in [3, 30, 29]. At first, a review of the fundamentals of continuummechanics is provided. Subsequently, the strong and weak form of the model problem aregiven. Finally, the Bubnov-Galerkin approximation is applied to the problem at hand, leadingto the final system of linear equations. At the very end of the chapter, a possible approachto cope with the imposition of rotational boundary conditions is proposed.

2.1 Continuum mechanics for shells

The main assumption of the Kirchhoff-Love theory states that a vector orthogonal to themiddle surface of the shell in the reference configuration remains normal to the middle surfacein the deformed configuration. Therefore, transverse shear effects are neglected. Furthermore,this allows us to reduce the dimensionality of the problem by describing only the shell middlesurface. Let us define a three-dimensional domain Ω ⊂ R3. Exploiting the previous rationale,the middle surface is described using a convective covariant space Ω0 ⊂ R2 through the linearmap Q : Ω0 → Ω. The following definition is introduced:

u(θ1, θ2) = x(θ1, θ2)−X(θ1, θ2) (2.1)

where u is the unknown displacement field, X is the position vector of the material pointin the reference configuration, x is the position vector of the material point in the currentconfiguration, θ1 and θ2 are the convective curvilinear coordinates defined onto the middlesurface of the shell. In the rest of the chapter, the Einstein summation convention is appliedwhere Greek and Latin letters range from 1, 2 and 1, 2, 3 respectively. Additionally,capital letters refer to quantities in the reference configuration and small letters to quantitiesin the deformed configuration.

Let us define the covariant base vectors as:

Gα = X,α =∂X

∂θαgα = x,α =

∂x

∂θα(2.2)

2.1. Continuum mechanics for shells 5

consequently, the covariant metric coefficients of the surface are given as:

Gαβ = Gα ·Gβ gαβ = gα · gβ (2.3)

from which the contravariant base vectors are obtained as:

Gα = GαβGβ with [Gαβ] = [Gαβ]−1 (2.4)

The curvature tensor coefficients read as follows [7]:

Bαβ =1

2(Gα ·G3,β +Gβ ·G3,α) = Gα,β ·G3 (2.5a)

bαβ =1

2

(gα · g3,β + gβ · g3,α

)= gα,β · g3 (2.5b)

G3 and g3 denote the normalized normal vector of the middle surface and they are definedrespectively as:

G3 =G1 ×G2

‖G1 ×G2‖2g3 =

g1 × g2

‖g1 × g2‖2(2.6)

We additionally define J as the L2 norm of the normal vector in the reference configuration:

J = ‖G3‖2 (2.7)

In this work, Hooke‘s law is used for the description of the constitutive behaviour of the ma-terial. Hence, the second-order tensors related to the second Piola-Kirchhoff stress resultantmeasures n, m are linearly dependent on the membrane and bending strain tensors ε, κ.This dependency reads as follows:

n(u) =E h

1− ν2C : ε(u) = Cm : ε(u) (2.8a)

m(u) =E h3

12(1− ν2)C : κ(u) = Cb : κ(u) (2.8b)

where C is the fourth-order material tensor, E, ν, h are the Young‘s modulus, the Poisson‘sratio and the thickness of the shell, respectively. Note that h is supposed to be constantover the entire domain Ω0, which allows us to perform an analytical pre-integration over thethickness of the shell. Concerning the kinematics of the shell, the membrane and bendingstrain tensors corresponding to the Green-Lagrange strain measure (energetically conjugateto the second Piola-Kirchhoff stress measure) are defined in a non-linear setting as:

ε = εαβGα ⊗Gβ =

1

2

(gα · gβ −Gα ·Gβ

)Gα ⊗Gβ (2.9a)

κ = καβGα ⊗Gβ = (bαβ −Bαβ)Gα ⊗Gβ (2.9b)

2.2. The strong form of the problem 6

The linearized version of the tensor components in equations 2.9a and 2.9b is given by [27]:

εαβ =1

2(Gβ · u,α +Gα · u,β ) (2.10a)

καβ = −G3 · u,αβ +Gα,β ·G31

J((G2 ×G3) · u,1− (G1 ×G3) · u,2 ) (2.10b)

+1

J((Gα,β ×G2) · u,1− (Gα,β ×G1) · u,2 )

2.2 The strong form of the problem

Figure 2.1: Graphical representation of the quantities described in the strong form of the modelproblem [3].

Let us denote the boundary of the problem at hand as Γ = ∂Ω. The latter can be now splitinto two non-overlapping parts ΓD and ΓN , which form the Dirichlet and Neumann portion ofthe boundary, respectively (see Figure 2.1). Finally, the elastic body is also subject to bodyforces denoted by b = bαGα + b3G3. Consequently, the strong form of the Kirchhoff-LoveBoundary Value Problem (BVP) is written as follows:

nαβ|α− qαBβα + bβ = 0 ∀x ∈ Ω0 (2.11a)

nαβBαβ + qα|α + b3 = 0 ∀x ∈ Ω0 (2.11b)

mαβ|α− qβ = 0 ∀x ∈ Ω0 (2.11c)

u = u ∀x ∈ ΓD (2.11d)

ω = ω ∀x ∈ ΓD (2.11e)

nαdα = p ∀x ∈ ΓN (2.11f)

mαdα = r ∀x ∈ ΓN (2.11g)

where d, u, ω, nα = nαβGβ + qαG3, mα = mαβGβ denote the normal to the Neumannboundary ΓN , the prescribed displacement, the prescribed rotation, the traction force and

2.3. The weak form of the problem 7

the traction moment, respectively. Additionally, qα stands for the in-plane contravariantcomponents of the shear force vector [3]. Furthermore, the terms nαβ and mαβ representthe force and moment tensor components of the first Piola-Kirchhoff stress tensor, when ageometrically non-linear formulation is employed. These are related to the aforementionedsecond Piola-Kirchhoff tensor through a pull back operation [9]. In spite of this incongruencein the derivation, due to the geometrically linear assumption, both stress measures concur.Hence, no additional steps are required in the linear case.

2.3 The weak form of the problem

Applying variational calculus to the BVP decribed in Equation 2.11, the weak form of theproblem reads:

Find u ∈ H2(Ω) such that

a(u,v) = F (v) ∀v ∈ H20 (Ω) (2.12)

where H2(Ω) ⊂ L2(Ω) denotes the Sobolev space of functions in L2(Ω) whose first and secondweak derivatives are also a subset of L2(Ω). Note that H2

0 (Ω) has stricter requirements thanH2(Ω). Indeed, H2

0 (Ω) is defined as:

H20 (Ω) =

v ∈ H2(Ω),v = 0 on ΓD and ω(v) = 0 on ΓD

(2.13)

The bilinear form a and the linear functional F can be expanded, respectively, as follows:

a(u,v) =

∫Ωε(v) : n(u) dΩ +

∫Ωκ(v) : m(u) dΩ (2.14a)

F (v) =

∫Ωv · b dΩ +

∫ΓN

v · p+ ω(v) · r dΓ (2.14b)

The space H20 (Ω) is equipped with the energy norm, defined as:

‖v‖E(Ω) =√a(v,v) ∀v ∈ H2

0 (Ω) (2.15)

2.4 Bubnov-Galerkin discretization

To approximate the solution of the model problem 2.11, we use the Bubnov-Galerkin dis-cretization of 2.12.

Let n ∈ N, n > 0, and T (Ω) =

Ω(i)ni=1

be a suitable partition or mesh of Ω ⊂ Rn

into sets Ω(i) called elements. Considering high-order approximations, we associate to eachelement Ω(i) a polynomial degree pi and a diameter hi. Moreover, let S (T , p) denote thecorresponding Finite Element solution space:

S (T , p) = uT ∈ H20 (Ω) ∩ C1 (Ω) : uT |Ω(i) ∈ Pp ∀ Ω(i) ∈ T (Ω) (2.16)

where Pp is the space of rational polynomials of degrees at most pi, namely Pp = spanxα11 ·

. . . · xαdd : α1, . . . , αd ∈ R,max(α1, . . . , αd) ≤ pi. Note that xαii here represents also rational

2.5. Rotational boundary conditions 8

polynomials, i.e. xαii :=xαjj

xαkksuch that αi = max(αj , αk) ≤ pi. Additionally, for any

uT ∈ S (T , p) there exists real coefficients ui such that:

uT =dimS∑i=1

uiφi (2.17)

where (φ(i)j )j , j = 1, . . . , dimS are a set of basis functions at each subdomain Ω(i) and

φ = ∪ni=1 (φ(i)j )j forms a basis of the discrete space S (T , p). It is worth noting that due

to the requirements on the discrete admissible space for the displacement field, the basisfunctions must be at least C1 continuous across element boundaries such that the bendingpart is well-defined. Then, the Finite Element Method for the model problem 2.11 reads

Find uT ∈ S (T , p) such that

a (uT ,vT ) = F (vT ) ∀vT ∈ S (T , p) (2.18)

where the test and trial space of 2.12 have been projected onto the same space S (T , p),following the Bubnov-Galerkin idea. Consequently, this projection leads to a discrete systemof linear equations that reads:

K u = F (2.19)

where K, F and u denote the stiffness matrix with entries Kij = a(φi,φj), the load vector

with entries Fi = F (φi) and the discrete solution vector over the domain Ω, respectively.

2.5 Rotational boundary conditions

The Kirchhoff-Love shell element hereby presented is categorized as a rotation-free formula-tion and therefore only displacement degrees of freedom are necessary for a complete descrip-tion. Nevertheless, cases exist where rotations have to be prescribed at the shell‘s boundary,for instance in the case of clamped edges, symmetry conditions and coupling between ele-ments [29]. Therefore, in all the aforementioned cases, additionally to displacements, alsothe slope of the surface has to be constrained. In the literature, several methods have beenproposed to cope with this issue, e.g. the bending strip method [28] and an approach based onthe Nitsche‘s method [21]. In this work, rotational boundary conditions are imposed weaklyby adding a penalized term to the weak form 2.12. Indeed, the bilinear form a is augmentedas follows:

ap(u,v) = a(u,v) + 〈βω χω(u),χω(v)〉ΓD (2.20)

where 〈•, •〉ΓD denotes the inner product in the L2(ΓD) space. Consequently, the added termreads:

〈βω χω(u),χω(v)〉ΓD =

∫ΓD

βω χω(u) · χω(v) dΓ (2.21)

where χω = ω(v)− ω represents the jump operator for the rotation field along the Dirichletboundary. Additionally, βω is the penalty parameter for the rotation field, which is defined as

2.5. Rotational boundary conditions 9

a strictly positive value. For an analogous problem, it has been shown in [3] that the penaltyform in Equation 2.20 is bilinear, symmetric, bounded and coercive. Hence, by applying theLax-Milgram theorem, there exist one unique solution to the penalized weak form. However,this solution is different from the one of the strong form 2.11 for a finite value of the penaltyparameter. Due to this reason, in the literature the penalty method is often addressed asvariationally inconsistent. Nonetheless, numerical investigations on a benchmark problem(clamped cantilever plate) have shown that values of βω in the range from 106 to 1012 timesbigger than the dominating parameter of the problem (usually the Young‘s modulus) lead toaccurate results without resulting in an extremely ill-conditioned system of equations.

10

Chapter 3

Isogeometric Analysis (IGA)

In this chapter, the basic concepts of Isogeometric Analysis are given, following the derivationin [13]. First, B-splines and NURBS shape functions are reviewed and then they are appliedto the geometrical description of curves and surfaces. In a later section, the concept of IGA isbriefly explained and applied to the discretized form of the Kirchhoff-Love shell formulationpresented in the previous chapter (2).

3.1 NURBS description of geometries

Non-Uniform Rational B-Splines (NURBS) and B-splines are commonly used in computer-aided design (CAD), manufacturing (CAM) and engineering (CAE) to represent geometricalobjects. Due to their accuracy and flexibility, they have been dominating the computer graph-ics scene for decades. However, they have become subject of research in the finite elementcommunity only in recent years. Therefore, we start our description of these mathematicalobjects from a geometrical point of view.

3.1.1 B-splines

Since NURBS are constructed from B-Splines, a brief description of the latter is provided. Fora more detailed description the reader is referred to [13]. In the following we use the definitionof knot spans as elements and patches provided in [13], although in our implementation knotspans are just considered as subdomains for integration. Alternative distinctions have beenproposed, e.g. see [26]. Nevertheless, regardless which definition is used, a finite element codemust perform a loop through the patches during assembly and a loop through the elementsduring numerical quadrature. B-splines are a type of curves based on piecewise polynomials,which are fully defined given a set of control points P i and a knot vector Ξ. Starting fromthe knot vector, these terms are analyzed more in depth in the following sections.

3.1. NURBS description of geometries 11

Knot vector

A one-dimensional knot vector is a sequence of non-decreasing values that corresponds to aset of coordinates in the parameter space, usually denoted as Ξ = ξ1, ξ2, . . . , ξn+p+1, whereξi is the ith knot, p is the polynomial degree and n is the number of shape functions used tobuild up the B-spline curve. The subintervals are referred to as knot spans. If the first andthe last knot values are repeated p+ 1 times the knot vector is said to be open. Open knotvectors are usually the standard in commercial CAD software. Furthermore, the continuityof the shape functions at a knot is defined by the multiplicity of the corresponding knot asCp−k, where p is again the polynomial degree and k the number of repeated knots. It followsthat when an open knot vector is employed, the continuity at the beginning and at the end ofthe parametric space is C−1, and therefore the shape functions are interpolatory at the endsof the knot vector. This is in general not true in the interior part, posing a first fundamentaldifference between knots and nodes as defined in standard Finite Element.

Shape functions

The set of shape functions N is defined recursively by means of the Cox-de Boor formula:

for p = 0:

Ni,0(ξ) =

1 ξi < ξ < ξi+1

0 otherwise(3.1)

For p ≥ 1:

Ni,p(ξ) =ξ − ξiξi+p − ξi

Ni,p−1(ξ) +ξi+p+1 − ξξi+p+1 − ξi+1

Ni+1,p−1(ξ),0

0:= 1 (3.2)

There are some important properties to be pointed out for this set of shape functions. First,the basis forms a partition of unity, which means that the following holds:

n∑i=1

Ni,p(ξ) = 1 ∀ξ ∈ [−1, 1] (3.3)

Additionally, each shape function Ni,p(ξ) is non-zero in the interval [ξi, ξi+p+1] and pointwisenon-negative therein. This is known as the local support property of the B-spline basisfunctions. Finally, it is worth noting that every pth order basis has p−k continuous derivativesacross the knots, related to what has been previously stated about the multiplicity of a knot.An example of shape functions is depicted in Figure 3.1. Note the continuity C−1 at bothends of the parameter space caused by the open knot vector.


0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 3.1: Shape functions (p = 3) for an open knot vector Ξ =

0, 0, 0, 0,

1

4,

2

4,

3

4, 1, 1, 1, 1

.

B-spline curve description

B-spline curves are described as a linear combination of control points and shape functionsover the corresponding parametric space.

The definition of a B-Spline curve reads as:

C(ξ) =n∑i=0

Ni,p(ξ)P i (3.4)

where the control points P i are associated with the corresponding basis Ni,p. An example ofa curve with polynomial order p = 4 and 12 control points is depicted in Figure 3.2

Several properties of a B-Spline curve follow directly from the properties of its shape functions.For instance, a curve of order p has at least p − 1 continuous derivatives if knots or controlpoints are not repeated. Moreover, due to the locality property of the basis, changing theposition of one of the control points influences the geometry of no more than p+1 knot spansmapped onto the curve. There are several other properties, for a broader description thereader is referred to [13].

B-spline surface description

The definition of a B-Spline surface is just a tensor-product extension of Equation 3.4. Let usdefine the knot vectors Ξ = ξ1, ξ2, . . . , ξn+p+1 and Θ = η1, η2, . . . , ηm+q+1. Subsequently,


0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 3.2: Example of a B-Spline curve of order p = 4 with 12 control points (red dots) andcorresponding control polygon. Note the interpolation property caused by the open knot vector atthe first and last control point.

the surface is defined as:

S(ξ, η) =

n∑i=0

m∑j=0

Ni,p(ξ)Mj,q(η)P i,j (3.5)

where P i,j is often referred to as control point net [13, 30], which is built by n ×m controlpoints, and Ni,p(ξ) and Mj,q(η) are the one-dimensional shape functions defined by the knotvectors Ξ and Θ, respectively. In Figure 3.2 an example of B-Spline surface is shown.

Many of the properties of a B-Spline surface are directly derived from those of a B-Splinecurve and the tensor product nature of Equation 3.5.

B-spline solid description

Analogously to surfaces, the description of solids is trivially extended by using a tensorproduct structure with an additional parametric dimension:

S(ξ, η, ζ) =

n∑i=0

m∑j=0

l∑k=0

Ni,p(ξ)Mj,q(η)Lk,r(ζ)P i,j,k (3.6)

Since the focus of this work is on dimensionally reduced models, solids will not be addressedany further here. The reader is refer to [13] for a more in-depth description.


0

100

50

50 100

100

150

0

200

0-50

-100 -100

Figure 3.3: Example of a B-Spline surface of order p = 2 and corresponding control point net.

3.1.2 NURBS

NURBS stands for Non-Uniform Rational B-Splines. The non-uniformity property refersto the knot vector, which can be defined arbitrarily. The term rational addresses the shapefunctions. Indeed, for NURBS, the basis functions are piecewise rational polynomials, definedby means of B-Spline basis. This enrichment in the approximation space allows NURBS todescribe exactly a variety of geometric objects that cannot be represented by rational poly-nomials, e.g. circles and ellipses. From a geometric prospective, a Rn-dimensional NURBS isobtained by a projecting transformation performed on a Rn+1-dimensional B-Spline.

NURBS curve description

Applying what has been mentioned before, a NURBS curve in R3 is the projection of aB-Spline in R4, defined as follows:

C(ξ) =

∑ni=0Ni,p(ξ)wiP i∑ni=0

Ni,p(ξ)wi(3.7)

where wi is the weight associated to control point P i. Additionally, the NURBS shapefunctions can be written as:

Ri,p(ξ) =Ni,p(ξ)wi∑ni=0

Ni,p(ξ)wi(3.8)


Consequently, a NURBS curve can be given analogously to Equation 3.4 as:

C(ξ) =n∑i=0

Ri,p(ξ)P i (3.9)

It is straightforward to prove, that if all the weights wi are equal to one and employing thepartition of unity property, the rational functions in Equation 3.8 reduce to the B-Splinefunctions given in Equation 3.2. Therefore, B-Splines are just a degenerate case of NURBSand thus all the properties that hold for B-Splines are also valid for NURBS.

NURBS surface description

Analogously to B-Splines, a NURBS surface is written as:

S(ξ, η) =

∑ni=0

∑mj=0Ni,p(ξ)Mj,q(η)wi,jP i,j∑n

i=0

∑mj=0

Ni,p(ξ)Mj,q(η)wi,j(3.10)

where the shape functions are defined by:

Rp,qi,j (ξ, η) =Ni,p(ξ)Mj,q(η)wi,j∑n

i=0

∑mj=0

Ni,p(ξ)Mj,q(η)wi,j(3.11)

plugging the latter definition of the basis back into 3.10 leads to the following:

S(ξ, η) =n∑i=0

m∑j=0

Rp,qi,j (ξ, η)P i,j (3.12)

which fully describes a NURBS surface.

0

100

50

50 100

100

150

0

200

0-50

-100 -100

(a) B-Spline surface.

0

100

50

50 100

100

150

0

200

0-50

-100 -100

(b) NURBS surface.

Figure 3.4: Comparison of the same control point net where, on the right, the value of the weightof one control point was changed to 0.5.

It is worth noting that two-dimensional NURBS shape functions are not the tensor productof their one-dimensional counterpart but they can be thought as the ratio of tensor productsof B-Splines basis, weighted by wi. This becomes an issue when computing first and second


derivatives of the shape functions, since they cannot simply be built as the tensor product ofthe derivatives of their corresponding one-dimensional basis.

NURBS solid description

Similarly, a NURBS solid is described by:

S(ξ, η, ζ) =

∑ni=0

∑mj=0

∑lk=0Ni,p(ξ)Mj,q(η)Lk,r(ζ)wi,j,kP i,j,k∑n

i=0

∑mj=0

∑lk=0

Ni,p(ξ)Mj,q(η)Lk,r(ζ)wi,j,k(3.13)

where the shape functions are defined as follows:

Rp,q,ri,j,k (ξ, η, ζ) =Ni,p(ξ)Mj,q(η)Lk,r(ζ)wi,j,k∑n

i=0

∑mj=0

∑lk=0

Ni,p(ξ)Mj,q(η)Lk,r(ζ)wi,j,k(3.14)

3.1.3 Refinement

An interesting feature of NURBS is that the shape functions space can be enhanced withoutmodifying the geometry description and its parametrization. It can be distinguished betweentwo cases, namely knot insertion and order elevation. In both cases, the space is enriched byadding control points to the geometry.

Knot insertion

The first refinement method presented here is knot insertion. This is the IGA counter-part of the classical h-Refinement strategy in standard FEM. Given a knot vector Ξ =ξ1, ξ2, . . . , ξn+p+1, let us define an enriched knot vector:

Ξ =ξ1 = ξ1, ξ2, . . . , ξn+m+p+1 = ξn+p+1

(3.15)

such that Ξ ⊂ Ξ. Subsequently, the new m + n basis are defined recursively as previousintroduced, but this time they are associated to the new knot vector Ξ, see Figure 3.5. Thenew m+n control points are computed as a linear combination of the original control points.The method is applicable directly to B-Splines, whereas for NURBS the same approach canbe used but it has to be applied on the projective Rn+1-dimensional B-Spline entity, fromwhich the NURBS is derived. For a deeper insight and implementation details, the reader isrefer to [13, 38].

Order elevation

The second refinement method is order elevation, which has similarities with the classicalp-Refinement strategy in standard FEM. This method involves increasing the polynomialorder of the shape functions used to represent the geometry. In this approach, the continuityat every knot is preserved and therefore, during order elevation, existing knots multiplicity isincreased by one, without adding any new knot. Analogously to knot insertion, neither the


0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(a) Ξ = 0 0 0 1 1 1

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(b) Ξ = 0 0 0 0.5 1 1 1

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(c) Ξ = 0 0 0 0.5 0.5 1 1 1

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(d) Ξ = 0 0 0 0.5 0.5 0.75 1 1 1

Figure 3.5: Basis functions before and after knot insertion, order p = 2. Note the changes incontinuity at the knots, given as Cp−k.

geometry nor the parametrization are changed after performing order elevation. In Figure 3.6,an example of shape functions after order elevation is depicted.

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(a) Ξ = 0 0 0 0.5 0.5 1 1 1

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(b) Ξ = 0 0 0 0 0 0.5 0.5 0.5 0.5 1 1 1 1 1

Figure 3.6: Basis functions before and after order elevation, from p = 2 to p = 4. Note that thecontinuity at the knots is preserved as well as the number of distinct knot intervals.

3.2. IGA discretization 18

3.2 IGA discretization

Following the idea first introduced in [22], the term Isogeometric Analysis refers to the factthat the analysis model makes use of the same mathematical formulation as for the geometrymodel, e.g. NURBS. This paradigm can be seen as an evolution of the classical isoparametricconcept, where the same functions that are used to exactly represent the geometry are alsoused to approximate the solution field.

Recalling the continuum mechanics review given in 2.1, it can be easily seen that all the state-ments defined on the convective covariant space Ω0 with convective curvilinear coordinates(θ1, θ2) are directly transferable to a NURBS surface defined by its parametric coordinates(ξ, η). Moreover, we can now characterize better the discrete space S (T , p). Indeed, S (T , p)is the space of rational polynomials of order p in the parametric image of partition T (i),

where this parametric image is defined as a rectangular domain (ξ, η) ∈ Ξ(i) ×Θ(i) = Ω(i)0 .

In the latter, Ω(i)0 is the image of subdomain Ω(i) corresponding to partition T (i) under the

inverse mapping of Q(ξ, η), and Ξ(i), Θ(i) denote the knot vectors of the underlying NURBS

patch, see [3] for a detailed description. Furthermore, the set of generic basis functions (φ(i)j )j

defined on the discrete space S (T , p) is here described by the two-dimensional NURBS shapefunctions Ri,j(ξ, η), given previously in 3.11. It is important to bear in mind that, since thevariational index of the problem at hand is two, the required continuity between elements isat least C1 [29, 19]. Therefore, the multiplicity k of the internal knots in Ξ(i) and Θ(i) isbounded to be at most p− 1. In case of meshes partitioned in multiple patches, the couplingbetween the latter becomes obviously an issue, where remedies like those presented in 2.5have to be employed. Finally, the discrete system of linear equations reads as in Equation4.19:

K u = F (3.16)


with entries Fi = F (φi) and the discrete solution vector over the domain Ω, respectively. Itis important to note that in IGA, the discrete solution is computed on the control points ofthe geometry, which are in general non-interpolatory.

19

Chapter 4

High-order polynomials FiniteElement (p-FEM)

In this chapter an overview of p-FEM is proposed, following the derivation published in[48, 15]. First, the concepts behind the Quasi-Regional mapping for describing geometryimplicitly are given. Later in the chapter, several polynomial types for building high-ordershape functions are reviewed. The most important properties of such basis are also dis-cussed. Finally, the Kirchhoff-Love shell formulation, previously introduced in Chapter 2, isdiscretized using the p-FEM paradigm.

4.1 Quasi-Regional mapping

Curved geometries are represented exactly by standard linear isoparametric elements only inthe limit case where h, the element length of the largest element involved in the computation,tends to zero. In the p-version of the Finite Element, it is crucial to be able to representcurved boundaries accurately with very few elements [4], since exponential convergence canbe reached by increasing the polynomial degree. A widely used approach is the linear blend-ing function method, proposed by Gordon et al. [20]. The method is illustrated in thefollowing using a two-dimensional example, but it is directly extendible to three-dimensionalapplications.

Consider a quadrilateral in the xy-plane where edge E2 is curved. Edge E2 can be describedby a parametric equation of the form E2(ξ) where, without any loss of generality, we stateξ ∈ [−1, 1].

The linear blending mapping from the local parameter space (ξ, η) to the global space (x, y)reads as the standard bilinear mapping plus an additional term f2(ξ):

Q(ξ, η) =1

4

((1− ξ)(1− η)X1 + (1 + ξ)(1− η)X2 +

(1 + ξ)(1 + η)X3 + (1− ξ)(1 + η)X4

)+ f2(η)

(4.1)

4.2. High-order shape functions 20

This extra term stems from the difference between the contribution of E2 and the line con-necting the two corresponding vertices of the quadrilateral, see Figure 4.1, blended out by alinear interpolation that reaches the value one where the curved edge is located and zero onthe opposite edge, in this example E4.

Figure 4.1: Blended quadrilateral with one curved edge [32].

The blending term ensures that the contribution of the curved edge does not influence theopposite edge. Hence, f2(η) from this example reads as follows:

f2(η) =1

2(1 + ξ)

(E2(ξ)− 1

2((1− ξ)X2 + (1 + ξ)X3)

)(4.2)

Substituting the latter into Equation 4.1 leads, after some simplifications, to:

Q(ξ, η) =1

4

((1− ξ)(1− η)X1 + (1− ξ)(1 + η)X4

)+

1

2E2(η)(1 + ξ) (4.3)

This procedure can be easily generalized to the case where all edges are curved. After sometedious algebra, the complete blended mapping for a quadrilateral is given by:

Q(ξ, η) =1

2

((1− η)E1(ξ) + (1 + ξ)E2(η) +

(1 + η)E3(ξ) + (1− ξ)E4(η)

)−

4∑i=1

Ni(ξ, η)Xi

(4.4)

The inverse of the mapping, Q−1(ξ, η), cannot be directly computed in a general case, sinceit is a non-linear function. Therefore, an iterative scheme has to be employed, e.g. a Newton-Raphson method. For a more detailed discussion on the inverse mapping and on the consis-tency of the blending method the reader is referred to [47, 32, 45].

4.2 High-order shape functions

In order to introduce the properties of the different sets of shape functions, let us considera simple one-dimensional wave equation model problem of linear elastodynamics, for a more


in-depth formulation the reader is referred to [17, 18, 42]. The discretization in space of theweak form leads to the following semi-discrete system of linear equations:

Mu(t) +Ku(t) = F (t) (4.5)

where u = u(x, t), u =∂2u

∂t2,M ,K and F are the displacement vector, the acceleration

vector, the mass matrix, the stiffness matrix and the force vector, respectively. Additionally,the entries of the element stiffness matrix K(i) and the element mass matrix M (i) are given:

K(i)jk =

2

h

∫ 1

−1

dNj

dξ

dNk

dξdξ (4.6a)

M(i)jk =

hρ

2

∫ 1

−1NjNkdξ (4.6b)

where h represents the length of the element and ρ is the mass density, respectively. Note thatany damping effect is being neglected. Moreover, in the following, we focus the discussiononly on the spatial discretization of the problem at hand.

4.2.1 Lagrange polynomials

Following the definition given in [48] and similarly to what previously defined in Equation2.16, let Sp ∈ H1 be the finite element subspace of continuous piecewise polynomials of degreep denoted by Pp:

Sp =φ | φ ∈ C0(Ω), φ ∈ Pp(Ωi)

(4.7)

In p-type element based on Lagrange interpolation, also known in the literature as spectralelement method, the solution variable φh is extended within each element in terms of high-order Lagrangian interpolants, Ni(ξ) ∈ Pp:

Ni(ξ) ∈ Pp, Ni(ξj) = δij ∀i, j ∈ 1, . . . , p+ 1 (4.8)

If a Gauss-Lobatto quadrature scheme is used, the solution variables are all nodal [48, 4],i.e. φi = φh(ξi) and the formed basis is not hierarchical. After performing a numericalintegration, the element stiffness and mass matrix are written, respectively, as:

K(i)jk =

2

h

p+1∑q=1

N ′j(ξq)N′k(ξq)wq (4.9a)

M(i)jk =

hρ

2

p+1∑q=1

Nj(ξq)Nk(ξq)wq (4.9b)

By employing a Gauss-Lobatto scheme, the element mass matrix is diagonal and underin-tegrated. The element stiffness matrix is exactly integrated in the one-dimensional case,whereas for two- and three-dimensional elements it is also underintegrated. Gauss-Lobatto


points ξq and their corresponding weights wq can be found in tables, e.g. refer to [25]. InFigure 4.2 the one-dimensional Lagrange shapes up to p = 4 are shown, note that the basisis not hierarchical.

-1 -0.5 0 0.5 1

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Figure 4.2: One-dimensional shape functions based on Lagrange polynomials for p=4.

4.2.2 Legendre polynomials

The Legendre polynomials satisfy the Legendre differential equation [15]:

((1− x2)y′) + n(n+ 1)y = 0, x ∈ (−1, 1), n = 0, 1, 2, . . . (4.10)

They can be computed by using either the Bonnet recursive formula:

Pn(x) =1

n[(2n− 1)xLn−1(x)− (n− 1)Ln−2(x)] , x ∈ (−1, 1), n = 2, 3, 4, . . .

(4.11)

or the Rodriguez formula:

Pn(x) =1

2n n!

dn

dxn(x2 − 1)n, x ∈ (−1, 1), n = 0, 1, 2, . . . (4.12)

Additionally, another important property of these polynomials is that they are orthogonal onthe interval I = (−1, 1) with respect to the constant weighting function w(x) = 1. Therefore,


the following holds:

∫ 1

−1Pn(x)Pm(x)dx =

2

2n+ 1if n = m

0 otherwise(4.13)

In Figure 4.3 the one-dimensional Legendre shapes up to p = 4 are depicted.

-1 -0.5 0 0.5 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Figure 4.3: One-dimensional Legendre polynomials up to p=4.

4.2.3 Integrated Legendre polynomials

As a first step, we define the linear shape functions which are identical to the p = 1 Lagrangecase:

Ni(ξ) =1

2(1 + ξiξ) i = 1, 2 (4.14)

Additionally, we introduce internal hierarchical shape functions defined as integrals of Leg-endre polynomials as follows:

Ni(ξ) :=1

‖Pi−2‖

∫ ξ

−1Pi−2(ξ′)dξ′ i = 3, . . . , p+ 1 (4.15)

where the norm of the Legendre polynomial is given by:

‖Pi−2‖2 =2

2i− 3(4.16)


Furthermore, by construction, the following holds: Ni(±1) = 0, i > 2. Therefore, only φ1

and φ2 define discrete nodal values, while the φi, i > 2 form a set of internal variables. If weconsider the derivatives of these hierarchical functions, the following property is true:∫ 1

−1

dNi

dξ

dNj

dξdξ = δij i, j = 3, 4, . . . , p+ 1 (4.17)

From this orthogonality condition, it can be proven that the element stiffness matrix isdiagonal after the block contribution given by the linear modes (i = 1, 2). This does nothold anymore if two- and three-dimensional elements are considered, although generally thematrices are nearly diagonal, therefore leading to a better conditioning number of the systemof linear equations, compared to standard finite elements [47, 52, 15, 48]. The hierarchicalnature has to be understood as follows, the matrix corresponding to the subspace Sp iscontained into the one related to subspace Sp+1.

In Figure 4.4 the one-dimensional integrated Legendre shapes up to p = 4 are illustrated.Note that the internal modes are either even or odd functions, with this pattern repeatingitself according to the polynomial order.

-1 -0.5 0 0.5 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Figure 4.4: One-dimensional integrated Legendre shape functions up to p=4.

Since all the numerical results of this thesis related to p-FEM have been obtained withintegrated Legendre polynomials, the extension of the basis to two dimension is here provided.In the following, we consider the tensor product space Spξ,pη formed by all polynomials on(ξ, η) ∈ [−1, 1]× [−1, 1] spanned by the following set of monomials:

ξiηj where i = 0, 1, . . . , pξ j = 0, 1, . . . , pη (4.18)

By construction, the two-dimensional shape functions are categorized into three distinct

4.3. High-order p-FEM discretization 25

groups, which are nodal modes, edge modes and face modes, respectively. In Figure 4.5,anexample for each group is depicted. It should be noted that face modes are entirely local toan element, an therefore they can be statically condensed out of the system [15]. This featurehas been thoroughly investigated in the literature, leading to preconditioners for iterativesolvers based on static condensation [35, 36, 2, 46].

(a) Nodal mode.

(b) Edge mode. (c) Face mode.

Figure 4.5: Two-dimensional shape functions based on integrated Legendre polynomials up to orderpξ = pη = 2. Note the locality of the face mode.

4.3 High-order p-FEM discretization

Analogously to what has been stated in 3.2, all the definitions given in convective curvi-linear coordinates (θ1, θ2) are directly transferable to a blended quadrilateral defined by itsparametric coordinates (ξ, η). Similarly, we can characterize the discrete space S (T , p) asthe space of polynomials of order p in the parametric image of partition T (i), where this

parametric image is defined as a rectangular domain (ξ, η) ∈ [−1, 1] × [−1, 1] = Ω(i)0 . In the

latter, Ω(i)0 is the image of subdomain Ω(i) corresponding to partition T (i) under the inverse

mapping of Q(ξ, η), and [−1, 1] × [−1, 1] denotes the classical Finite Element parametric

space. Furthermore, the set of generic basis functions (φ(i)j )j defined on the discrete space

S (T , p) can be described by any set of shape functions Ni(ξ, η) presented in this chapter. In

4.3. High-order p-FEM discretization 26

this work, the set based on integrated Legendre polynomials (4.2.3) is chosen. The discretesystem of linear equations reads as in Equation 4.19:

K u = F (4.19)


with entries Fi = F (φi) and the discrete solution vector over the domain Ω, respectively.Here, the discrete solution is defined with respect to the modes, e.g. external and internal[15].

27

Chapter 5

The Finite Cell Method (FCM)

In this chapter the basic concepts of the Finite Cell Method are introduced, following thederivation published in [37, 16]. At first, the basic ideas behind FCM are summarized, wherea particular focus in put on the challenges of integration. In the later sections, trimming isbriefly explained and motivated in the context of shells for FCM. Finally, the modified shellweak form is presented.

5.1 Introduction to FCM

The FCM can be classified as an embedded domain method, which makes use of high-orderpolynomial shape functions. In the following, an introduction to the topic is provided bymeans of a standard linear elasticity problem. For a deeper insight the reader is referred to[16, 37, 44].

One of the main motivation behind FCM is to tackle and reduce the time-consuming opera-tion of generating a proper mesh, specially for complex geometries. Quantitatively, the lattertask takes about 20% of the overall modelling and computation time [13]. FCM copes withthis challenging operation by performing the computation on unfitted structured meshes.The physical geometry is therefore embedded into a simpler domain, which is afterwardssubdivided into a regular grid of cells. Furthermore, these cells are discretized by means ofhigh-order shape functions. Additionally to drastically mitigate the mesh generation phase,the method shows high accuracy and computational efficiency and it has been applied toseveral practical problems beside linear elasticity, e.g. thermoelasticity [51], transient elas-todynamics [18], biomechanical applications [41], elastoplasticity [1], contact mechanics [8].Moreover, this method can be used in combination with adaptive schemes [43, 50].

Let us introduce the formulation of linear elasticity on a two-dimensional physical domainΩphy . We assume the boundary Γ of such domain to consist of two disjoint parts ΓD and ΓN ,representing Dirichlet and Neumann boundary, respectively. The strong form of the problemreads:

5.1. Introduction to FCM 28

∇ · σ + b = 0 ∀x ∈ Ωphy (5.1)

σ = C : ε ∀x ∈ Ωphy

ε =1

2

[∇u+ (∇u)T

]∀x ∈ Ωphy

σ · n = t ∀x ∈ ΓN

u = u ∀x ∈ ΓD

(5.2)

where σ, b, C, ε, t, u, denote the stress tensor, the body force, the material tensor, thestrain tensor, the prescribed traction and the prescribed displacement, respectively.

Introducing a test function v (often in the literature also denoted as δu, which represents avirtual displacement in engineering terms), the weak form of the equilibrium can be writtenas follows:

Find u ∈ H1(Ωphy) such that

a(u,v) = F (v) ∀v ∈ H10 (Ωphy) (5.3)

where

a(u,v) =

∫Ωphy

σ : (∇symv) dΩphy (5.4)

F (v) =

∫Ωphy

b · v dΩphy +

∫ΓN

t · v dΓ

H10 (Ωphy) =

v ∈ H1(Ωphy),v = 0 on ΓD

where ∇sym stands for the symmetric part of the gradient. Here, H1(Ωphy) ⊂ L2(Ωphy)denotes the Sobolev space of functions in L2(Ωphy) whose weak derivatives up to order oneare also a subset of L2(Ωphy).

Employing the FCM fundamental idea of an embedded domain approach, we can extend theclassical definition of Ωphy by a fictitious domain Ωfict . Note that the entire computationaldomain is now defined as Ω = Ωphy ∪Ωfict , such that Ωphy ∩Ωfict = ∅. Furthermore, note thatnow the mesh is generated on the embedding domain Ω instead of the physical domain Ωphy .Since the geometry of Ω can be chosen arbitrarily, simple shapes are advantageous becausethe domain can be easily discretized by a Cartesian grid.

With these definitions at hand, the weak form in Equation 5.3 can be reformulated as:

aα(u,v) =

∫Ωphy

ασ : (∇symv) dΩ (5.5)

Fα(v) =

∫Ωα b · v dΩ +

∫ΓN

t · v dΓ

α(x) =

1 if x ∈ Ωphy

0 if x ∈ Ωfict

5.2. Adaptive integration schemes 29

Due to numerical reasons, the value of α on Ωfict is chosen to be as small as possible, withoutleading to extremely ill-conditioned element stiffness matrices for the cells that are partlylying in Ωfict , also known as cut cells. A proper value for α is typically taken in the rangefrom 10−4 to 10−15 [37].

5.2 Adaptive integration schemes

It can be noticed from the derivation presented in the previous section (5.1) that the domainstep function α introduces a discontinuity in the integrals of the weak form (Equation 5.5)for those cells that are cut. Hence, the challenges arising from the generation of conformingmeshes are shifted towards the accurate and efficient computation of these integrals. For com-pleteness, the standard Gaussian quadrature applied to a generic one-dimensional functionf(x) is briefly recalled. Note that the numerical integration is performed on the normalizeddomain [−1, 1], therefore we have to refer to quantities in the parameter space ξ and mapthem back to the physical space x. Assume it exists a linear map Q(ξ) such that x = Q(ξ).Hence, the scheme reads as follows:∫

Ωf(x) dx =

∫ 1

−1f(Q(ξ))det(J) dξ '

nGP∑i=1

f(x(ξi))wi det(J)|ξi (5.6)

where det(J) is the determinant of the Jacobian of the mapping Q(ξ) and ξi and wi arethe coordinates and weights of the Gauss points, respectively. The extension to higher di-mensionality is straightforward, for details see [25]. Since the standard Gaussian quadraturerequires smoothness of the integrated functions, this scheme cannot be directly applied toFCM. Nevertheless, the use of adaptive integration schemes mitigates the problem by re-cursively partitioning the integration domain into smaller cells refined towards the physicalboundaries. By doing that, the whole integral is computed by hierarchically summing up theintegrals over each cell. Therefore, the area of influence of the discontinuity can be reducedarbitrarily.

5.2.1 Spacetree

A first scheme for evaluating integrals over cut cells makes use of spacetree-like partitioning.Let us consider an example in two dimensions, where the domain Ω is decomposed using aquadtree structure, e.g. see Figure 5.1. At first, every cell of the initial mesh (partitioningdepth k = 0) is checked whether or not it is cut by the physical boundary. If it is cut, everycell is split into four sub-cells whose partitioning depth is updated to k += 1. If the cell is notcut, a standard Gaussian quadrature scheme is performed on its domain. The partitioning isrecursively repeated until a user-defined depth k = m is reached. Once all the Gauss pointsare gathered, they have to be mapped into the parameter space of the initial cell, since thenumerical integration is defined on this space. Therefore, the term det(J) in the standardGaussian quadrature (Equation 5.6) is here given by the product of the determinant of theJacobian of the mappings Q(ξ, η) and Q(ξi, ηi). The former is the well-known mapping fromthe parameter space of a cell to the physical space, whereas the latter recursively maps theintegration cell of depth i to the level i− 1 up to zero, which corresponds to the local spaceof the finite cell.


Figure 5.1: Recursive partitioning of a cut cell. Each cell is subdivided until a predefined depth isreached [44].

It should be pointed out that one main disadvantage of the method is the potential exponen-tial growth in the number of Gauss points as the cells are partitioned, which could impactdramatically the computational cost. Moreover, since the scheme is a 1th order approximationof the geometry, it worsens the superior convergence behaviour achievable with high-orderpolynomials [32]. Nevertheless, the method has been proven to be an applicable approachfor performing simulation. Furthermore, the spacetree decomposition algorithm is easy toimplement and can be performed on arbitrary geometries, being therefore an attractive choiceas an integration scheme. An example of the Gauss points distribution for a tree depth k = 4is shown in Figure 5.2

Figure 5.2: Gauss points distribution and integration cells on one quarter of the modified Scordelis-Lo roof example, partitioning depth k = 4, polynomial order p = 3. The red boundary represents thetrimmed circular hole.


5.2.2 Smarttree

Another possibility for evaluating integrals is to decompose the cut finite cells into blendedquadrilaterals and triangles, as proposed in [32] and similar to what published in [10]. Theintegration points are therefore distributed in the parametric space of the blended elementsand subsequently mapped into the parametric space of the cell. Furthermore, this methodexploits directly the parametric description of the boundary, contrary to a spacetree approach.Hence, an accurate approximation of the geometry is reachable with fewer integration points.An idea of how the algorithm works is given in Figure 5.3, for a detailed description thereader is referred to [32].

Figure 5.3: Partitioning of a cut cell into blended elements [32].

The integration points distribution for the Scordelis-Lo roof example is illustrated in Figure5.4. It can be noticed that fewer points are used, leading nonetheless to a higher accuracy inthe integration compared to a spacetree approach.

Figure 5.4: Gauss points distribution on one quarter of the modified Scordelis-Lo roof example usingthe blended partitioner, polynomial order p = 3, 4×4 knot spans. Note the efficient integration pointsdistribution around the trimmed circular hole.

5.3. FCM for shells 32

5.3 FCM for shells

In this work, an extension of the classical FCM concepts is used. Following the idea proposedin [40], the description of the structural geometry is separated from the fictitious domainapproach of FCM. By doing this, the method is well suited for thin-walled structures likeshells and allows trimmed patches to be easily described. This motivates the choice of startingthis section with a brief description of trimming.

5.3.1 Trimming

Trimming refers to the operation of creating trimmed surfaces, for instance by performing aboolean operation [10, 12]. A trimmed surface is formed by two parts: an underlying surfacethat defines the geometric shape and a set of properly ordered trimming curves that markssections of the underlying surfaces that are clipped. Trimming is a standard operation inCAD software that allows a simplified representation of many complex objects.

5.3.2 Modified weak form

Recalling the definition of the bilinear form a and the linear functional F for the Kirchhoff-Love model problem presented in 2.14 and applying the FCM concepts summarized in thischapter, the weak form can be reformulated as follows:

Find u ∈ H2(Ω) such that

aα(u,v) = Fα(v) ∀v ∈ H20 (Ω) (5.7)

where:

aα(u,v) =

∫Ωα ε(v) : n(u) dΩ +

∫Ωακ(v) : m(u) dΩ (5.8a)

Fα(v) =

∫Ωα v · b dΩ +

∫ΓN

v · p+ ω(v) · r dΓ (5.8b)

α(x) =

1 if x ∈ Ωphy

0 if x ∈ Ωfict

Using an embedded domain approach such as FCM on the Kirchhoff-Love shell mitigates thedifficulties of simulating trimmed NURBS patches. Indeed, we can easily assign a fictitiousdomain index to the trimmed parts by performing an inside-outside test, therefore leadingto the penalization of their contribution. Hence, geometric trimming is directly translatedinto a trimming of the FEM solution by means of the domain indicator α, which shows oneof the main advantages of using FCM for shell analysis.

33

Chapter 6

Implementational Aspects

In this chapter, some of the most important implementational aspects are discussed. A briefintroduction to the in-house code used during this work is provided. Moreover, particularfocus is put on how IGA has been embedded into a high-order Finite Element frameworkwithout fundamentally changing the structure of the latter.

6.1 Introduction to AdHoC++

AdHoC++ is a multi-functional, object-oriented, high-order Finite Element code, which hasbeen developed at the Chair for Computation in Engineering in recent years. All the resultspresented in this work have been obtained by exploiting and expanding this framework, ifnot stated otherwise. One of the most interesting features of AdHoC++, which is particularlyrelevant for this thesis, is its flexibility. Indeed, the data structure and the code architecturemake possible the usage of different discretization techniques for the geometry and the solutionspace, respectively. Furthermore, the pipeline for performing FCM-based simulations is alsoimplemented into this code.

6.2 Implementation of IGA into a p-FEM code

The following section is meant to describe a possible strategy for implementing IsogeometricAnalysis into a high-order FEM code. Bear in mind that, in our implementation, we considerIGA patches as the counterpart of p-FEM elements. A first, pivotal difference resides inthe handling of the Degrees Of Freedom (DOFs). In low-order standard FEM codes, DOFsare commonly associated only with nodes. Analogously, in IGA the DOFs of the problemare related to the control points of the underlying geometry. This concept is expandedin the p-Version of the FEM, indeed all topological components are capable of “bearing”DOFs. Therefore, we not only have nodal DOFs, but also DOFs associated to edges, facesand volumes, depending on the polynomial order and on the dimensionality of the problem.Obviously, this difference poses an issue when constructing the local-to-global DOFs LocationMap (LM) for IGA. As an example, let us write a sub-routine that assemble the LM for atwo-dimensional element, as shown in pseudo-code in Algorithm 1.

6.2. Implementation of IGA into a p-FEM code 34

Algorithm 1 Scatter element matrix into the global system

1: for each element Ωi ∈ T (Ω) do2: procedure Get DOF Location Map3: Initialize DOF Location Map of element Ωi.4: Get topological components t of element Ωi.5: for each topological component tj of t in Ωi do6: Get DOFs of current topology tj .7: for each DOF k of tj do8: Obtain global ID of k.9: Store global ID into DOF Location Map.

10: end for11: end for12: end procedure13: Obtain element matrix K(i).14: procedure Scatter into global system(K(i), DOF Location Map)15: end procedure16: end for

In the p-FEM case, the topological components t are stored in the following sequence: firstall the nodes, then edges, faces and finally volume, in a row-wise fashion. Using this strategyand the consequent numbering leads to a better bandwidth of the system matrices. Moreover,this approach follows the “natural” way shape functions are defined, as mentioned in Section4.2.3. However, recalling Section 3.1.1, a NURBS surface is described by its control pointnet P i,j . A convenient way to store this net is in a Cartesian fashion, e.g by choosing thefirst parametric direction as predominant. Unfortunately, this causes a mismatch betweenthe DOFs naturally associated to the control point net and the numbering dictated by thetopological components sequence. Therefore, we have to find a “mapping”, that transformsa generic control point into a nodal control point, edge control point or face control point,respectively. Algorithm 1 can be modified to account for this, as proposed in version 2.Note that additional care is needed when dealing with edge and face DOFs in IGA. In fact,their number depends not only on the polynomial order (as in the p-FEM case) of the patch,but also on the number of the inserted knots. The same procedure could be extended tohigher dimensions, where also the volume part is accounted for, namely by properly changingthe rearrangement of the topological components.

6.2. Implementation of IGA into a p-FEM code 35

Algorithm 2 Scatter IGA-patch matrix into the global system

1: for each patch Ωi ∈ T (Ω) do2: procedure Get DOF Location Map3: Initialize DOF Location Map of patch Ωi.4: procedure Build topological sequence of patch Ωi(Ωi)5: Get topological components t of patch Ωi.6: Rearrange topological components (sequence: node lower left, bottom edge,

node lower right, left edge, face, right edge, node upper left, top edge, node upper right).7: end procedure8: for each topological component tj of t in Ωi do9: Get DOFs of current topology tj .

10: for each DOF k of tj do11: Obtain global ID of k.12: Store global ID into DOF Location Map.13: end for14: end for15: end procedure16: Obtain patch matrix K(i).17: procedure Scatter into global system(K(i), DOF Location Map)18: end procedure19: end for

36

Chapter 7

Numerical results

In this chapter, the main numerical results obtained throughout this thesis are presented anddiscussed. At first, we analyze a one-dimensional eigenvalue problem related to structuralvibrations of an elastic rod. Afterwards, the proposed shell formulation is validated againstthe well-known pinched cylinder and Scordelis-Lo roof benchmarks. Finally, several moreexamples of shell trimming in the context of FCM are given. Bear in mind that all theexamples aim at assessing and comparing the performance of IGA and p-FEM.

7.1 Elliptic eigenvalue problem

Following the formulation presented in [24, 48, 23], the elliptic eigenvalue problem fundamen-tals are briefly summarized. Examples of BVP that are categorized as elliptic include, forinstance, free vibration and linearized buckling in the context of elasticity and the Helmholtzequation in the context of wave propagation. Recalling Equation 4.5 for linear elastodynam-ics, let us analogously define the undamped, unforced, semi-discrete equations of motion fora free-vibrating structural system:

Mu(t) +Ku(t) = 0 (7.1)

where u = u(x, t), u =∂2u

∂t2,M and K are the displacement vector, the acceleration vector,

the mass matrix and the stiffness matrix, respectively. Applying the technique of separationof variables, let us assume that the free vibrations of the system u(x, t) can be written asfollows:

u(x, t) =∑n

φn(x)qn(t) (7.2)

where φn is the nth natural mode vector and qn is a harmonic function, related to the nth

natural frequency ωn, which can be expressed as:

qn(t) = An cos(ωn t) +Bn sin(ωn t) (7.3)

7.1. Elliptic eigenvalue problem 37

Plugging Equation 7.3 into 7.2 yields:

u(x, t) =∑n

φn(x)(An cos(ωn t) +Bn sin(ωn t)) (7.4)

Consequently, after differentiation, it follows:

u = −ω2nu (7.5)

Substituting the last result into Equation 7.1 leads to the following system of linear equations:

(K − ω2nM)φnqn = 0 (7.6)

The non-trivial solutions are found by solving the following generalized eigenvalue problem:

det(K − ω2nM) = 0 (7.7)

where the solutions ωn, n = 1, . . . , N (N being the number of unconstrained DOFs of thesystem), represent the natural frequencies at which the system vibrates. Once those valuesare known, it is possible to calculate the corresponding natural modes φn by solving:

(K − ω2nM)φn = 0 (7.8)

The resulting φn are defined up to a multiplicative constant, whose normalization is actuallyarbitrary [24].

7.1.1 Longitudinal vibration of a one-dimensional elastic rod

The results presented in this section have been obtained with the MatLab toolbox FCMLab,for details see [49].

The problem at hand tackles the structural free-vibrations of a one-dimensional elastic rodof unit length, constrained at both ends with homogeneous Dirichlet boundary conditions.Assuming unit material parameters, the governing equation for its natural frequencies andmodes, arising from the one-dimensional Laplace equation, writes:

∆u+ ω2u = 0 ∀x ∈ ]0, 1[ (7.9a)

u(0) = u(1) = 0 (7.9b)

where ∆ denotes the Laplace operator and u is a scalar value representing the longitudinaldisplacement along the rod. There exists an exact solution in terms of natural frequencies(i.e. square root of the eigenvalues), given as follows:

ω2n = nπ where n = 1, 2, . . . ,∞ (7.10)

Subsequently, the standard machinery of the FEM is applied to the problem at hand, namelyby discretizing its weak form using the Bubnov-Galerkin method. This leads to a generalizedeigenvalue problem of the form 7.6. Note that the derivation up to this point writes the sameindependently of the choice of shape functions (NURBS or polynomials).


Results

The numerical solution of the generalized eigenproblem 7.6 are discussed in this section. Weshow the data by presenting the numerical natural frequencies, ωhn, normalized by its an-alytical counterpart ωn, denoted in the following as normalized eigenfrequency. Figure 7.1illustrates the discrete spectra for different polynomial orders for a system with 49 DOFs com-posed by p-FEM elements. On the abscissa, we plot the frequency number normalized by thetotal number of DOFs. We can observe the appearance of the branching phenomenon, thatdivides the spectrum in an acoustical branch and an optical branch. This effect substantiallyworsens the p-FEM approximation in the high-frequency part of the spectrum, for details see[48]. In particular, we can distinguish p branches, where the approximation shows a suddenjump. Indeed, at these points the solution is not purely propagating but it is damped. In theextreme case of a single p = 48 element, we observe branching for every frequency throughoutthe entire spectrum. Nevertheless, before diverging, the approximation of more than half ofthe spectrum is highly accurate.

0.0 0.2 0.4 0.6 0.8 1.01.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

ωi / number of DOFs

Nor

mal

ized

eige

nfr

equ

ency

p = 1p = 2p = 3p = 4p = 48

(a) p-FEM, p-refinement.

0.0 0.2 0.4 0.6 0.8 1.01.0

1.1

1.2

1.3

1.4

1.5

1.6


Nor

mal

ized

eige

nfr

equ

ency

p = 1p = 2p = 3p = 4p = 48

(b) p-FEM, p-refinement, zoom-in.

Figure 7.1: Longitudinal vibration of an elastic rod. Normalized discrete spectra for differentpolynomial degrees, p-FEM C0-continuous elements, number of DOFs = 49.

In Figure 7.2, the same problem setup is shown for IGA. Namely, in 7.2a order elevation isperformed on a single-patch mesh, whereas in 7.2b the behaviour of multiple-patch meshes oforder p = 2 is depicted. The results on a single patch are similar to what published in [14],where we can recognize the superior approximation power of IGA for the entire spectrum forthe problem at hand, except for some frequencies at the very end of the spectrum, known asoutliers. The outliers constitute a discrete optical branch, which effects worsen by increasingp. A way to get rid of such spurious frequencies consists in reparametrizing the isogeometricmapping, as shown in [14].


0.0 0.2 0.4 0.6 0.8 1.01.00

1.05

1.10

1.15

1.20

1.25

1.30


Norm

aliz

edei

gen

freq

uen

cy

p = 1, C0 continuity




(a) IGA single-patch, p-refinement.

0.0 0.2 0.4 0.6 0.8 1.01.0

1.1

1.2

1.3

1.4

1.5


Norm

aliz

edei

gen

freq

uen

cy

24 patches16 patches12 patches2 patches

(b) IGA multi-patch, p = 2.

Figure 7.2: Longitudinal vibration of an elastic rod. Normalized discrete spectra for single-patchand multi-patch IGA setups, number of DOFs = 49.

Interestingly, a continuous optical branch seems to reappear when multiple patches are em-ployed, up to the limit case where p-FEM and IGA coincide (same number of elements /single-knot patches). These effects are related to the type of stencil for the problem at hand.Indeed, using p-FEM elements of order p = 2 yields to a stencil which can be split into equa-tions related to nodal modes and internal modes. The latter can be subsequently staticallycondensed out, leading to a unique stencil, but their influence to the appearance of branch-ing still contributes. This phenomenon worsens with p-Refinement, since every increase inpolynomial order introduces a new type of equations into the stencil. This is not the case forh-Refinement in IGA, since all the modes produce the same type of stencil, except for a fewmodes at the ends of the knot span that are responsible for the outliers.

In the second part of this section on vibrations, we shift our focus to the convergence prop-erties of the first three eigenmodes. This is motivated by what stated in [24] for ellipticproblems. Indeed, due to the best approximation property in the energy norm of FEM,the contribution of the inaccurate higher modes is suppressed. Note that this might not bethe case for parabolic and hyperbolic initial-value problems, which are not considered in thescope of this work. In Figure 7.3 the convergence behaviour for h- and p-Refinement on ap-FEM mesh is illustrated. As expected, we observe an exponential convergence rate forp-Refinement, whereas we get an optimal algebraic rate of slope 2p [48, 5] for h-Refinement.Note that all these plots are shown in a double logarithmic scale. Hence, p-Refinement yieldsto an accuracy that is several orders of magnitude better than h-Refinement, for a chosenfrequency with the same number of DOFs.

A comparison of the h-Refinement convergence between p-FEM and IGA is depicted in Figure


101

10−15

10−10

10−5

10−2

Number of DOFs

Nor

mali

zed

eige

nfr

equ

ency

-1

1st mode, p-Refinement

2nd mode, p-Refinement

3rd mode, p-Refinement

1st mode, p = 1, h-Refinement

2nd mode, p = 1, h-Refinement

3rd mode, p = 1, h-Refinement

1st mode, p = 2, h-Refinement

2nd mode, p = 2, h-Refinement

3rd mode, p = 2, h-Refinement

Figure 7.3: Longitudinal vibration of an elastic rod. Order of convergence of the first three eigen-modes performing p-Refinement and h-Refinement on standard p-FEM, C0-continuous elements.

7.4. We note the same algebraic rate of convergence in the asymptotic behaviour, namely 4 forp = 2. However, in the pre-asymptotic regime IGA shows superior approximation properties,given by the higher continuity between elements (Cp−k, k = 1) compared to standard C0

continuous p-FEM elements, which yields to a better stencil.

Lastly, we present the results of p-Refinement for different multi-patch IGA meshes (Figure7.5). In all cases we perform an adequate h-Refinement beforehand, such that all curves startfrom the same number of DOFs. We observe that the best rate of convergence is reached bythe maximum achievable continuity, i.e. with a single patch.


1

4

101 10210−15

10−10

10−7

10−4

Number of DOFs

Nor

mali

zed

eige

nfr

equ

ency

-1

1st mode, IGA

2nd mode, IGA

3rd mode, IGA

1st mode, p-FEM

2nd mode, p-FEM

3rd mode, p-FEM

Figure 7.4: Longitudinal vibration of an elastic rod. Order of convergence of the first three eigen-modes performing h-Refinement, p = 2, p-FEM vs IGA.

10110−12

10−9

10−5

10−1

Number of DOFs

Nor

mal

ized

eige

nfr

equ

ency

-1

1 patch, p-Refinement5 patches, p-Refinement10 patches, p-Refinement

Figure 7.5: Longitudinal vibration of an elastic rod. Order of convergence of the third eigenmodeperforming p-Refinement for multi-patch meshes, p-Refinement, IGA.

Finally, the results of our numerical experiments on the vibration of an elastic rod are sum-marized as follows:


• performing h-Refinement in IGA on a single patch gives the best results in terms ofapproximation of the entire discrete spectrum.

• performing p-Refinement leads to a diverging behaviour of the high-frequency part ofthe spectrum for p-FEM, whereas it yields to a converging, accurate approximation incase of IGA except for a few number of frequencies at the very end of the spectrum(outliers).

• for the first three frequencies, p-Refinement gives an exponential rate of convergence,with the best rate given by the maximum reachable continuity.

• for the first frequencies, knot insertion keeping the maximum continuity and h-Refinementgive the same algebraic rate of convergence O(h2p), with IGA approximating better theanalytical solution in the pre-asymptotic regime due to the higher continuity (Cp−k

against the C0 of classical p-FEM).

• in case of multiple IGA patches, we observe optical branching analogous to the p-FEMcase

7.2. Shell obstacle course 43

7.2 Shell obstacle course

The shell obstacle course is a well-known set of geometrically linear benchmark problems forthe assessment of shell analysis in complex strain states. The problem description was firstproposed in [6]. Two of these problems are here presented, i.e. the Scordelis-Lo roof and thepinched cylinder examples. The convergence is determined by comparing the displacementof certain points in the shell with reference solutions presented in [6, 13, 29]. Moreover, theconvergence of the error in the energy norm is also assessed for the the Scordelis-Lo roof.

7.2.1 Scordelis-Lo roof

The Scordelis-Lo roof is a structure undergoing its self-weight load. Both ends are supportedby rigid diaphragms (constrain on the in-plane vertical and horizontal displacements) andthe sides are free edges, see Figure 7.6. The geometry is described either via a single NURBSpatch or via a single blended p-FEM element. At first, convergence is checked against thevertical displacement of the middle point of one free edge. This is depicted in Figure 7.7 fordifferent p- and h-Refinement in an IGA setting, where the displacement is normalized bythe reference value uref = 0.3024 and on the abscissa the corresponding number of DOFs isplotted. It can be observed that the best rate of convergence is achieved with the maximumcontinuity, i.e. by performing p-Refinement on a single-knot-span patch. In all the the othercases where h-Refinement is performed, the convergence rate is slower, particularly for p = 2,where locking phenomena occur.

Figure 7.6: Scordelis-Lo roof. Problem description and parameters [29].

In Figure 7.8, the p-Refinement convergence of the tensor-product combinations of geometryand solution space descriptions is illustrated. No significant difference in the behaviour canbe observed, yielding to the claim that the high-order nature of the employed shape functionsis the leading factor for the problem at hand, but not the chosen basis itself.


0 50 100 150 200 250 300 350 4000.0

0.2

0.4

0.6

0.8

1.0

Number of DOFs

Nor

mali

zed

dis

pla

cem

ent

IGA p-Refinement, one knot span

IGA p = 2, h-Refinement, C1 continuity


IGA p = 4, h-Refinement, C3 continuityReference

Figure 7.7: Scordelis-Lo roof. Convergence comparison of h-Refinement (knot insertion) againstp-Refinement (order elevation).

0 50 100 150 2000.0

0.2

0.4

0.6

0.8

1.0

Number of DOFs

Nor

mal

ized

dis

pla

cem

ent

QRM geometry, polynomial approximationNURBS geometry, NURBS approximationQRM geometry, NURBS approximationNURBS geometry, polynomial approximationReference

Figure 7.8: Scordelis-Lo roof. Convergence of standard IGA using NURBS against Quasi-RegionalMapping (QRM) with polynomial field approximation (p-FEM) and their mixed geometry-solutionfield descriptions, p-Refinement.


In the classic literature for the shell obstacle course, only displacement values are given asreference. However, in [29], the distribution of membrane and bending stress resultants n11

and m11 are given and compared against an Abaqus solution, where θ1 represents the ringsurface direction. Our results are presented in Figure 7.9, showing good agreement with whatis published in [29] in both stress distribution and limit values.

(a) Membrane stress n11. (b) Bending stress m11.

Figure 7.9: Stress resultants distribution for membrane and bending state in the transversal direc-tion.

Another aspect of interest is the comparison between the condition number of the system oflinear equations in case of IGA and p-FEM. Let us define the condition number κ of a matrixA as follows:

κ(A) =|λmax(A)||λmin(A)|

(7.11)

where λmax and λmin are the maximum and minimum eigenvalues of A, respectively. Notethat for the problem at hand the condition number is always numerical infinity, since thetranslation along the cylinder axis is not fixed, leading to a zero eigenvalue in the stiffnessmatrix. Therefore, the number of iterations needed by an iterative solver to reach a certainprecision threshold is used as an indicator. In Figure 7.10 the results are illustrated for p-Refinement. Due to the orthogonality condition of the integrated Legendre shapes (4.2.3),the solution of the system requires less iterations for p-FEM.

Lastly, the convergence of the error in the energy norm is presented. The latter is defined as:

e(%) =

√Πref −Πnum

Πref· 100 (7.12)

Where Π = ‖u‖E(Ω). The reference values are obtained by an extrapolated overkill solutionfor a three-dimensional shell [39] and by an IGA Kirchhoff-Love shell overkill solution in thetwo-dimensional case (p = 14, 40 h-Refinement steps). It can be observed (Figure 7.11a)that within the method we reach the optimal rate of convergence in case of h-Refinement,i.e. p − 1, and an exponential rate in case of p-Refinement. We can still detect the same


0 50 100 150 2000

50

100

150

200

250

300

350

Number of DOFs

Nu

mb

erof

iter

ati

ons

p-FEM, p-RefinementIGA, p-Refinement

Figure 7.10: Scordelis-Lo roof. Number of iterations needed by an iterative solver (conjugategradient with diagonal preconditioning) to solve the system of equations within a tolerance of 10−16.

behaviour in Figure 7.11b for the first few points. However, all the curves are leveling off ataround 4% error. Indeed, in the Kirchhoff-Love assumption transversal shear deformationsare neglected, which yields to a modelling error. This error is particularly pronounced for theproblem at hand, where a complex strain state through the thickness is expected [33, 11].

11

1

2

101 102 10310−3

10−2

10−1

100

101

102

Number of DOFs

Err

orin

ener

gyn

orm

[%]

p-FEM, p-RefinementIGA, p-RefinementIGA p = 2, h-RefinementIGA p = 3, h-Refinement

(a) Convergence against a 2D shell reference so-lution.

111

2

101 102100

101

102

Number of DOFs

Err

orin

ener

gyn

orm

[%]

p-FEM, p-RefinementIGA p = 2, h-RefinementIGA p = 3, h-Refinement

(b) Convergence against a 3D shell reference so-lution.

Figure 7.11: Scordelis-Lo roof. Convergence of the error in the energy norm for p-Refinement(order elevation) and h-Refinement (knot insertion). On the left, the reference value is given by atwo-dimensional Kirchhoff-Love shell overkill. On the right, the reference value is taken from [39],computed from an overkill volumetric shell solution.


7.2.2 Pinched cylinder

The second benchmark here presented is the pinched cylinder. The structure is constrainedat both ends by rigid diaphragms and undergoes two opposite point loads in the middle,see Figure 7.12. The reference radial displacement under the point loads is given as uref =1.8248 · 10−5. Analogously to the Scordelis-Lo example, in Figure 7.13 the convergence fordifferent p- and h-Refinement in an IGA setting is illustrated, where on the ordinate thenormalized displacement and on the abscissa the number of DOFs is plotted. The overallbehaviour is similar to the previous benchmark, indeed the best rate of convergence is achievedwith the maximum continuity. These results are surprising to a certain degree, given thatthe structure is subjected to a point load, therefore yielding to a singularity.

Figure 7.12: Pinched cylinder. Problem description and parameters [29].

In Figure 7.14, the convergence of p-Refinement for all possible combinations of geometry-solution field discretization is depicted. Analogously to the Scordelis-Lo roof, no substantialdifference is observed in the convergence rate.

Finally, the distribution of membrane and bending stress resultants n11 and m11 along theloaded edge is presented, see Figure 7.15. Note that 300 corresponds to the point where theforce is applied. Furthermore, for all cases the number of DOFs is the same, namely 768.To the best of the authors knowledge, there is no reference value in the literature regardingthe stress distribution in the pinched cylinder. Nonetheless, we claim that the presenteddistribution seems reasonable, showing again that the maximum reachable continuity yieldsto smoother and more accurate results even in the secondary field. Additionally, it is worthnoting the influence of the singularity on the bending resultant m11 (7.15b), whereas thiseffect does not appear for the membrane part n11 (7.15a).


0 200 400 600 800 1000 1200 14000.0

0.2

0.4

0.6

0.8

1.0

Number of DOFs

Nor

mali

zed

dis

pla

cem

ent

IGA p-Refinement, one knot span




IGA p = 5, h-Refinement, C4 continuityReference

Figure 7.13: Pinched cylinder. Convergence comparison of h-Refinement (knot insertion) againstp-Refinement (order elevation).

0 100 200 300 400 5000.0

0.2

0.4

0.6

0.8

1.0

Number of DOFs

Nor

mal

ized

dis

pla

cem

ent

QRM geometry, polynomial approximationNURBS geometry, NURBS approximationQRM geometry, NURBS approximationNURBS geometry, polynomial approximationReference

Figure 7.14: Pinched cylinder. Convergence of standard isogeometric analysis (IGA) against Quasi-Regional Mapping (QRM) with polynomial field approximation (p-FEM) and their mixed geometry-solution field descriptions, p-elevation.


100 150 200 250 300−0.08

−0.06

−0.04

−0.02

0.0

0.02·10−2

Length along line

Str

ess

resu

ltantn

11

p = 15p = 2, 13 knotsp = 3, 12 knots

(a) Membrane stress n11.

200 225 250 275 300−0.3

−0.2

−0.1

0.0

0.1

Length along line

Str

ess

resu

ltantm

11

p = 15p = 2, 13 knotsp = 3, 12 knots

(b) Bending stress m11.

Figure 7.15: Pinched cylinder. Comparison of stress resultants distribution for membrane andbending state along the loaded edge for different p- and h-Refinement, keeping the number of DOFsconstant.

As a conclusion of this section, the numerical results on the shell obstacle course (Scordelis-Loroof and pinched cylinder) are outlined as follows:

• for IGA the best rate of convergence of displacements, in terms of DOFs, is achievedby p-Refinement on a single-knot-span patch.

• any reduction in continuity inside the patch worsens the convergence rate.

• analogous results are achieved with p-Refinement on a single p-FEM element.

• all the possible combinations geometry-field description with NURBS and polynomialslead to similar results, suggesting that the benefits of IGA are primarily related to thehigh-order nature of the basis functions.

• the optimal rate of convergence for the error in the energy norm is achieved for p-Refinement and h-Refinement within the method, but when compared to a three-dimensional shell solution the Kirchhoff-Love assumption yields to a non-negligiblemodelling error.

7.3. Trimming of shells and FCM 50

7.3 Trimming of shells and FCM

In this section, the Finite Cell Method is combined with the Kirchhoff-Love shell formulationto easily perform simulations of trimmed surfaces. The first example here presented is amodified version of the classical Scordelis-Lo roof, where four circular holes are drilled inthe structure. The convergence of the displacement of the middle point of the free edge isstudied. Additionally, the convergence of the error in the energy norm is also presented.Next, a comparison of accuracy and total number of integration points for the spacetree andsmarttree integration is discussed. Finally, the robustness and applicability of the method isshown by computing the solution of the cover plate of a violin subjected to a point load.

7.3.1 Modified Scordelis-Lo roof

The first example of this section is a modified version of the Scordelis-Lo roof, previouslypresented in section 7.2.1, where a circular hole is drilled in each quarter of the roof. InFigure 7.16 the problem description and its parameters are depicted. Only one quarter ofthe entire structure is meshed due to symmetry.

Figure 7.16: Modified Scordelis-Lo roof. Problem description and parameters [39].

Following the same methodology used for the shell obstacle course, convergence in terms ofthe vertical displacement of the mid point of the free edge (or tip displacement since weare only modelling one quarter of the structure) is first studied. There is no reference valuefor this example, nevertheless both p- and h-Refinement show asymptotic convergence to thesame value. The results shown in Figure 7.17 are again similar to what is presented in Section7.2. Indeed, p-Refinement on a single p-FEM element (similarly for a single-knot-span patch)achieves the best rate of convergence compared to h-Refinement.

Next, the convergence of the error in the energy norm is depicted in Figure 7.18. Thereference value is again obtained from an overkill p-FEM solution with three-dimensionalvolumetric shell elements, for details see [39]. It can be observed that h-Refinement yieldsto the optimal rate of convergence p− 1 for the problem at hand, up to the point where themodelling error causes a level-off. Performing p-Refinement leads to an exponential rate ofconvergence, similar to the data for the volumetric shell, up to the plateaux at around 8%


1020.0

0.2

0.4

0.6

0.8

1.0·10−2

Number of DOFs

Tip

dis

pla

cem

ent

p-Refinement, p-FEMh-Refinement, p = 2, IGAh-Refinement, p = 3, IGA

Figure 7.17: Modified Scordelis-Lo roof. Convergence of the tip displacement for p-FEM p-Refinement against IGA h-Refinement, semi-logarithmic scale.

of error. Note that before the flattening of the convergence, the curves associated to theKirchhoff-Love shell are shifted to the left part of the graph, since practically only half of theDOFs per element are needed compared to a three-dimensional formulation.

Finally, the integration schemes presented in Section 5.2 are compared. The convergencein the energy norm is depicted in Figure 7.19, on the left with respect to the DOFs of theproblem and on the right with regards to the number of integration points distributed by thealgorithm. It can be seen that by using a smarttree partitioner the accuracy is improved andat the same time the number of evaluated Gauss points is reduced.


1

1

101 102 103 104100

101

102

Number of DOFs

Err

orin

ener

gyn

orm

[%]

2D shell, 1 p-FEM element, p-Refinement2D shell, 1 IGA patch, p = 2, h-Refinement3D shell, 4 FCM cells, p-Refinement3D shell, 24 FCM cells, p-Refinement

Figure 7.18: Modified Scordelis-Lo roof. Convergence of the error in the energy norm against thevalues published in [39], h- and p-Refinement. Spacetree integration is performed in all cases.

1

1

102 103

101

102

Number of DOFs

Err

orin

ener

gyn

orm

[%]

smarttreespacetree, depth k = 1spacetree, depth k = 2spacetree, depth k = 6

(a) Comparison with respect to the number ofDOFs.

102 103 104100

101

102

Number of integration points

Err

or

inen

ergy

nor

m[%

]

smarttreespacetree, k = 6

(b) Comparison with respect to the number ofintegration points.

Figure 7.19: Modified Scordelis-Lo roof. Comparisons of convergence of the error in the energynorm between spacetree and smarttree partitioner for integration, h-Refinement.


7.3.2 Point load on a violin

In this section, the robustness and range of applicability of FCM for shells are tested. Tothis end, the untrimmed cover plate of a violin is modelled by a single patch of order p = 3,formed by 40×40 knot spans. The final model is obtained by trimming the latter patch witha set of curves. In Figure 7.20, the trimmed and untrimmed surfaces are visualized. All thegeometric models were created and exported from the commercial CAD software Rhinoceros.As boundary conditions, the displacements in all directions of the outer physical boundaryof the violin are set to zero. Additionally, a point load in the negative z-direction is appliedin the middle of the surface.

Figure 7.20: Trimmed and untrimmed model of the cover plate of a violin [39].

In Figure 7.21 the integration cells given by a spacetree partitioner (partitioning depth k = 2)are illustrated. The resulting displacements are shown in Figure 7.22, where we can observetheir smooth distribution. Additionally, it is worth noting the influence of the f-holes, whichis particularly relevant for the x-component. Indeed, it can be observed that the displacementin the x-direction given by the point load are not transmitted towards the external boundary,as physically expected. These results demonstrate the versatility of our approach, which issuccessfully applicable to the analysis of complex, trimmed geometries, even in the presenceof kinks.


Figure 7.21: Quadtree partitioning of depth k = 2. The physical boundaries of the violin arehighlighted in red.

(a) Displacement distribution in the x-direction.


(b) Displacement distribution in the y-direction.

(c) Displacement distribution in the z-direction.

Figure 7.22: Point load on a violin. Distribution of the displacement in the three directions. Notethat the point load is applied in the negative z-direction.

56

Chapter 8

Summary and conclusion

8.1 Summary

The main goal of this thesis was to compare the performance of a rotation-free shell elementbased on the Kirchhoff-Love theory with respect to two different high-order discretizationfor the geometry and the solution field. Therefore, several different benchmark problems ofstatic linear elasticity were studied by using either NURBS or integrated Legendre polyno-mials as shape functions. Additionally, the methods were also assessed on a problem relatedto structural vibrations.

To this end, the shell element was first implemented following a test-driven development intoAdHoC++, a high-order finite element code used for research-related engineering tasks at thechair for Computation in Engineering. Next, the machinery of IGA for dimensionally re-duced problems was also implemented into the code, as briefly described in Chapter 6. This,together with the p-Version of FEM already at hand in the code, allowed for a first numericalinvestigation and comparison of the methods on the pinched cylinder and Scordelis-Lo roofproblems, well-known benchmarks taken from the shell obstacle course. For the latter, weobserved that the best rate of convergence in terms of DOFs, both for displacement anderror in the energy norm, is achieved by keeping the maximum reachable continuity, i.e. byperforming p-Refinement on a single element / single-knot-span patch. Surprisingly, whencompared to three-dimensional volumetric shell solution, we obtained a significant discrep-ancy given by the modeling error in the transversal direction.

In the following step, we shifted our attention to the analysis of structural vibrations, namelyby analyzing the eigenvalues of a free-vibrating, one-dimensional elastic rod. This study didnot necessarily relate to the shell element assessment, however it let us gain a deeper knowl-edge into the behavior of the basis functions for a different class of Boundary Value Problem.For this example, h-Refinement in the IGA sense gave the best approximation for the entireeigenfrequency discrete spectrum. This is related to the stencil that corresponds to an IGAdiscretization, which only yields to a discrete optical branching. In the p-FEM case, theassociated stencil leads to a continuous optical branching, which deteriorates the quality ofthe results for the high-frequency part of the spectrum. Nonetheless, a study performed on

8.2. Conclusion 57

the first three frequencies of the spectrum showed that p-Refinement achieves the best rate ofconvergence in terms of DOFs. Additionally, in the asymptotic regime both IGA and p-FEMachieve the theoretical rate of convergence expected for h-Refinement.

Finally, the concepts of FCM were applied to the element formulation at hand. This allowedus to easily tackle the issue of translating the geometrical concept of trimming (standardin CAD) into FEM. Indeed, by employing an immersed boundary method such as FCM,the trimming operation was elegantly included in the weak form of the problem, as shownin Chapter 5. As a consequence, complex untrimmed geometries could be exported fromcommercial CAD softwares, e.g. Rhinoceros, and directly used for computation with noadditional effort required. The accuracy and range of applicability of the method were testedby performing computation on several different complex models.

8.2 Conclusion

As a conclusion to this work, the primary aspects which were achieved are summarized asfollows:

• implemented and fully tested a rotation-free Kirchhoff-Love shell element in AdHoC++,which provided flexibility and accurate results for linear shell analysis

• implemented and fully tested IGA for dimensionally reduced problems in AdHoC++

• performed numerical investigations on the convergence behavior of the aforementionedhigh-order shell element using either classical p-FEM, IGA or a mixed discretization ofthe geometry and solution field

• numerically investigated the performance of NURBS and integrated Legendre shapefunctions on a one-dimensional elliptic eigenvalue problem

• extended the formulation of the Kirchhoff-Love shell element within the context of FCM,therefore providing a strategy for Finite Element computation on trimmed geometries

LIST OF FIGURES 58

List of Figures

2.1 Graphical representation of the quantities described in the strong form of themodel problem [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.1 Shape functions (p = 3) for an open knot vector Ξ =

0, 0, 0, 0,

1

4,2

4,3

4, 1, 1, 1, 1

. 13

3.2 Example of a B-Spline curve of order p = 4 with 12 control points (red dots)and corresponding control polygon. Note the interpolation property caused bythe open knot vector at the first and last control point. . . . . . . . . . . . . 14

3.3 Example of a B-Spline surface of order p = 2 and corresponding control pointnet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.4 Comparison of the same control point net where, on the right, the value of theweight of one control point was changed to 0.5. . . . . . . . . . . . . . . . . . 16

3.5 Basis functions before and after knot insertion, order p = 2. Note the changesin continuity at the knots, given as Cp−k. . . . . . . . . . . . . . . . . . . . . 18

3.6 Basis functions before and after order elevation, from p = 2 to p = 4. Notethat the continuity at the knots is preserved as well as the number of distinctknot intervals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.1 Blended quadrilateral with one curved edge [32]. . . . . . . . . . . . . . . . . 22

4.2 One-dimensional shape functions based on Lagrange polynomials for p=4. . . 24

4.3 One-dimensional Legendre polynomials up to p=4. . . . . . . . . . . . . . . . 25

4.4 One-dimensional integrated Legendre shape functions up to p=4. . . . . . . . 26

4.5 Two-dimensional shape functions based on integrated Legendre polynomialsup to order pξ = pη = 2. Note the locality of the face mode. . . . . . . . . . . 27

5.1 Recursive partitioning of a cut cell. Each cell is subdivided until a predefineddepth is reached [44]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.2 Gauss points distribution and integration cells on one quarter of the modifiedScordelis-Lo roof example, partitioning depth k = 4, polynomial order p = 3.The red boundary represents the trimmed circular hole. . . . . . . . . . . . . 32

LIST OF FIGURES 59

5.3 Partitioning of a cut cell into blended elements [32]. . . . . . . . . . . . . . . 33

5.4 Gauss points distribution on one quarter of the modified Scordelis-Lo roofexample using the blended partitioner, polynomial order p = 3, 4 × 4 knotspans. Note the efficient integration points distribution around the trimmedcircular hole. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

7.1 Longitudinal vibration of an elastic rod. Normalized discrete spectra for dif-ferent polynomial degrees, p-FEM C0-continuous elements, number of DOFs= 49. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

7.2 Longitudinal vibration of an elastic rod. Normalized discrete spectra for single-patch and multi-patch IGA setups, number of DOFs = 49. . . . . . . . . . . . 42

7.3 Longitudinal vibration of an elastic rod. Order of convergence of the first threeeigenmodes performing p-Refinement and h-Refinement on standard p-FEM,C0-continuous elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

7.4 Longitudinal vibration of an elastic rod. Order of convergence of the first threeeigenmodes performing h-Refinement, p = 2, p-FEM vs IGA. . . . . . . . . . 44

7.5 Longitudinal vibration of an elastic rod. Order of convergence of the thirdeigenmode performing p-Refinement for multi-patch meshes, p-Refinement, IGA. 44

7.6 Scordelis-Lo roof. Problem description and parameters [29]. . . . . . . . . . . 46

7.7 Scordelis-Lo roof. Convergence comparison of h-Refinement (knot insertion)against p-Refinement (order elevation). . . . . . . . . . . . . . . . . . . . . . . 47

7.8 Scordelis-Lo roof. Convergence of standard IGA using NURBS against Quasi-Regional Mapping (QRM) with polynomial field approximation (p-FEM) andtheir mixed geometry-solution field descriptions, p-Refinement. . . . . . . . . 47

7.9 Stress resultants distribution for membrane and bending state in the transver-sal direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

7.10 Scordelis-Lo roof. Number of iterations needed by an iterative solver (conju-gate gradient with diagonal preconditioning) to solve the system of equationswithin a tolerance of 10−16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

7.11 Scordelis-Lo roof. Convergence of the error in the energy norm for p-Refinement(order elevation) and h-Refinement (knot insertion). On the left, the referencevalue is given by a two-dimensional Kirchhoff-Love shell overkill. On the right,the reference value is taken from [39], computed from an overkill volumetricshell solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

7.12 Pinched cylinder. Problem description and parameters [29]. . . . . . . . . . . 50

7.13 Pinched cylinder. Convergence comparison of h-Refinement (knot insertion)against p-Refinement (order elevation). . . . . . . . . . . . . . . . . . . . . . . 51

LIST OF FIGURES 60

7.14 Pinched cylinder. Convergence of standard isogeometric analysis (IGA) againstQuasi-Regional Mapping (QRM) with polynomial field approximation (p-FEM)and their mixed geometry-solution field descriptions, p-elevation. . . . . . . . 51

7.15 Pinched cylinder. Comparison of stress resultants distribution for membraneand bending state along the loaded edge for different p- and h-Refinement,keeping the number of DOFs constant. . . . . . . . . . . . . . . . . . . . . . . 52

7.16 Modified Scordelis-Lo roof. Problem description and parameters [39]. . . . . . 53

7.17 Modified Scordelis-Lo roof. Convergence of the tip displacement for p-FEMp-Refinement against IGA h-Refinement, semi-logarithmic scale. . . . . . . . 54

7.18 Modified Scordelis-Lo roof. Convergence of the error in the energy normagainst the values published in [39], h- and p-Refinement. Spacetree inte-gration is performed in all cases. . . . . . . . . . . . . . . . . . . . . . . . . . 55

7.19 Modified Scordelis-Lo roof. Comparisons of convergence of the error in theenergy norm between spacetree and smarttree partitioner for integration, h-Refinement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

7.20 Trimmed and untrimmed model of the cover plate of a violin [39]. . . . . . . 56

7.21 Quadtree partitioning of depth k = 2. The physical boundaries of the violinare highlighted in red. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

7.22 Point load on a violin. Distribution of the displacement in the three directions.Note that the point load is applied in the negative z-direction. . . . . . . . . 58

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implementation of a high-order kirchho -love shell: a ... · latter thin-walled structures can be...

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