Robust Algorithms for Contact Problems with Constitutive Contact Laws

Robuste Algorithmen zur Losung¨ von Kontaktproblemen mit konstitutiven Kontaktgesetzen

Der Technischen Fakultat¨ der Friedrich-Alexander-Universitat¨ Erlangen-Nurnberg¨

zur Erlangung des Doktorgrades

DR.-ING.

vorgelegt von

Saskia Sitzmann aus Munchen¨ Als Dissertation genehmigt von der Technischen Fakultat¨ der Friedrich-Alexander-Universitat¨ Erlangen-Nurnberg¨

Tag der mundlichen¨ Prufung:¨ 27.11.2015

Vorsitzender des Promotionsorgans: Prof. Dr. Peter Greil

Gutachter: Prof. Dr.-Ing. habil. Kai Willner Prof. Dr. rer.nat. habil. Barbara I. Wohlmuth Schriftenreihe Technische Mechanik Band 20 2016 ·

Saskia Sitzmann

Robust Algorithms for Contact Problems with Constitutive Contact Laws

Herausgeber: Prof. Dr.-Ing. habil. Paul Steinmann Prof. Dr.-Ing. habil. Kai Willner

Erlangen 2016 Impressum Prof. Dr.-Ing. habil. Paul Steinmann Prof. Dr.-Ing. habil. Kai Willner Lehrstuhl fur¨ Technische Mechanik Universitat¨ Erlangen-Nurnberg¨ Egerlandstraße 5 91058 Erlangen Tel: +49 (0)9131 85 28502 Fax: +49 (0)9131 85 28503

ISSN 2190-023X

c Saskia Sitzmann

Alle Rechte, insbesondere das der Ubersetzung¨ in fremde Sprachen, vorbehalten. Ohne Genehmigung des Autors ist es nicht gestattet, dieses Heft ganz oder teilweise auf photomechanischem, elektronischem oder sonstigem Wege zu vervielfaltigen.¨ Vorwort

Die vorliegende Arbeit entstand wahrend¨ meiner Tatigkeit¨ als wissenschaftliche Mitar- beiterin am Zentral Institute fur¨ Scientiﬁc Computing und dem Lehrstuhl fur¨ Technis- che Mechanik der Friedrich-Alexander-Universitat¨ Erlangen-Nurnberg¨ in Kooperation mit der MTU Aero Engines AG in Munchen.¨ Mein besonderer Dank gilt meinem Doktorvater, Herrn Prof. Kai Willner, der immer ein offenes Ohr fur¨ mich hatte und mich sowohl fachlich als auch mental wahrend¨ meiner gesamten Promotion bestmoglich¨ betreut hat. Sein Fachwissen auf dem Gebiet der technischen Mechanik hat diese Arbeit erst ermoglicht.¨ Ich bedanke mich besonders fur¨ die vielen intensiven fachlichen Diskussionen und die Geduld, die er mit mir hatte. Desweiteren mochte¨ ich mich bei Frau Prof. Barbara Wohlmuth bedanken, die diese Arbeit von der mathematischen Seite aus betreut hat und auch die Funktion des Zweit- gutachters ubernommen¨ hat. Obwohl sie aufgrund ihrer vielen Aktivitaten¨ sehr einge- spannt ist, hat sie immer die Zeit gefunden, meine fachlichen Fragen schnell zu beant- worten und alle gemeinsamen Veroffentlichungen¨ krititsch bis zum Optimum hin zu korrigieren. Auch bedanke ich mich bei Prof. Ulrich Rude,¨ der diese Arbeit von Seiten der Infor- matik unterstutzt¨ hat und mich am ZISC angestellt hat. Besonders bedanke ich mich bei Dr. Guido Dhondt, der mich innerhalb der MTU Aero Engines AG betreut hat. Er hat mir geduldig geholfen mich in das FEM Pro- gramm CalculiX einzuarbeiten und hat mich mit seinem breiten Fachwissen sowohl in der Mechanik als auch der Numerik immer unterstutzt.¨ An dieser Stelle bedanke ich mich auch bei Yang Li und Samoela Rakotonanahary, die den Grundstock zur Imple- mentierung des Mortarkontakts mit konstitutiven Kontaktgesetzten in CalculiX gelegt haben. Auch mochte¨ ich mich bei meinem Masterstudenten Tobias Langer bedanken, der mich bei der Implementierung der Dynamik unterstutzt¨ hat. Unter den Mitarbeitern des Lehrstuhl fur¨ technische Mechanik bedanke ich mich besonders bei Vera Luchscheider und Dominik Suß¨ , die mir ihre Messdaten zur Verfug-¨ ung gestellt haben und mit denen ich viele spannende Diskussionen uber¨ ihre Arbeit hatte. Ich bedanke mich bei allen Mitarbeitern der Abteilung fur¨ Strukturmechanik der MTU Aero Engines AG, die mir einen spannenden Einblick in den Alltag eines Berechnungsingenieurs gegeben haben und mir bei der Auswahl industriell relevanter Beispiele geholfen haben. Desweiteren bedanke ich mich bei allen Mitarbeitern des ZISC und des Lehrstuhls fur¨ Systemsimulation, die mich in meiner Zeit in Erlangen herzlich aufgenommen haben und mit der einen oder anderen Kaffeepause fur¨ Ab- wechlung gesorgt haben. Hier bedanke ich mich besonders bei Regina Ammer und Daniela Anderl. Abschließend bedanke ich mich von ganzem Herzen bei meinen Freunden, meinen Eltern und besonders bei meinem Freund Alexander, die mich immer unterstutzt¨ haben,

I mir den Rucken¨ freigehalten und mich auch auf andere Gedanken gebracht haben.

Erlangen, im November 2015 Saskia Sitzmann

II Abstract

This thesis is concerned with the solution of contact problems with advanced Coulomb friction in the 3D case using the finite element method. A Lagrange multiplier method modelling the contact traction is employed and the contact conditions are enforced in a weak sense leading to a surface-to-surface discretization. Here more precisely the dual mortar method is used allowing for a static condensation of the additional variables in the system before solving without loosing the optimality of the solution. The discrete contact inequalities are embedded in the algebraic system using a semi-smooth Newton method handling all system non-linearities, geometrical, material and contact within one iteration loop. In contrast to other research in this field, this is the first time, that the dual Lagrange method is combined with constitutive contact laws considering the surface roughness on the micro scale. In normal direction this is achieved via a perturbed Lagrange method using a regularization of the contact conditions being able to model a transition from soft to hard contact. In tangential direction the transition from elastic stick over micro slip to macro slip needed for frictional damping is considered. The elastic stick is modelled again with a perturbed Lagrange method while the micro slip satisfying the Masing rules for arbitrary load path is modelled with the serial-parallel Iwan model. Thus a perturbed Lagrange dual mortar method for contact problems is derived and studied in detail. The method is first derived for the quasi-static case using linear finite elements. Then it is extended to quadratic finite elements using a transformation of the basis functions. Also extensions suitable for complex applications including cyclic symmetry and directional blocking or hanging nodes are presented. In the last part of this work the fully dynamical contact problem with friction is considered. The generalized α-method is used for discretization in time and the algebraic system is derived with respect to accelerations. Again, the contact conditions are embedded using a semi-smooth Newton method and the dual Lagrange multiplier can be condensed from the system enabling the calculation of frictional damping as needed in structural dynamics. Numerical examples including various contact zones, non-linear material laws and comparisons with measurements show the wide applicability of the derived algorithms.

Keywords: contact mechanics, FEM, dual mortar method, perturbed Lagrange, constitutive contact laws, surface roughness.

III Kurzfassung

Diese Arbeit befasst sich mit der Losung¨ des 3D Mehrkorperkontaktproblems¨ mit kom- plexen Coulomb Reibgesetzten, das mit Hilfe der Finite Elemente Methode gelost¨ wird. Die Kontaktbedingungen werden mittels eines Lagrange Multiplikators, der den Kontak- tspannungsvektor modelliert, im schwachen Sinne erzwungen, was zu einer flachen-¨ basierten Kopplung der Kontaktflachen¨ fuhrt.¨ Es wird die duale Mortar Methode ange- wandt mit dem Vorteil, dass die zusatzlichen¨ Variablen des linearen Systems vor dem Losen¨ kondensiert werden konnen¨ ohne auf der anderen Seite die Optimalitat¨ der Losung¨ zu verlieren. Die diskreten Kontaktbedingungen werden in das algebraische System mit Hilfe einer semi-smooth Newton Methode eingebunden, die alle Nichtlin- earitaten¨ des Systems, geometrische, Material und Kontakt in einer Iterationsschleife auflost.¨ Im Gegensatz zu anderen Arbeiten auf diesem Feld ist dies die erste Arbeit, die die duale Lagrange Multiplikator Methode mit konstitutiven Kontaktgesetzen kombiniert und somit die Oberflachenrauhigkeiten¨ auf der Mikroebene miteinbeziehen. In Normalen- richtung wird dies mittels einer Perturbed Lagrange Methode erreicht, die eine regular- isierte Form der Kontaktbedingungen verwendet und so den Ubergang¨ von weichen zu harten Kontakt modellieren kann. In Tangentialrichtung wird der Ubergang¨ von elastis- chen Haften uber¨ Mikroschlupf zu Makroschlupf betrachtet, der fur¨ eine physikalisch ko- rrekte Beschreibung von Reibdampfung¨ benotigt¨ wird. Das elastische Haften wird aber- mals mit einer Perturbed Lagrange Methode modelliert, wahrend¨ der Mikroschlupf mit einem seriell-parallelen Iwan Model erzwungen wird. Dieses garantiert das Einhalten der Masing Regeln fur¨ beliebige Lastpfade. In dieser Arbeit wird somit eine perturbed Lagrange duale Mortar Methode zur Kontaktbehandlung hergeleitet und studiert. Die Methode wird dabei zuerst fur¨ den quasi-statischen Fall mit linearen Finiten El- ementen hergeleitet. Dann wird sie auf quadratische Finite Elemente mit Hilfe einer Transformation der Basisfunktionen erweitert. Des weiteren werden Erweiterungen fur¨ komplexe Anwendungen vorgestellt, die zum Beispiel zyklische Symmetrie, Blockieren von Richtungen sowie hangende¨ Knoten beinhalten. Im letzten Teil der Arbeit wird auf das voll dynamische Kontaktproblem mit Reibung eingegangen. Die generalisierte α-Methode wird zur Diskretisierung der Zeit verwendet und das algebraische System wird in Bezug auf die Beschleunigungen hergeleitet. Wieder werden die Kontaktbedingungen mittels eines semi-smooth Newton eingebunden und der duale Lagrange Multiplier kann aus dem System kondensiert werden, was die Berechnung von Reibdampfung¨ ermoglicht,¨ die vor allem in der Strukturdynamik gebraucht wird. Verschiedene numerische Beispiele, die zum Teil mehrere Kontaktzonen, nichtlineare Materialgesetze und Vergleiche mit tatsachlichen¨ Messungen beinhalten, unter- streichen die vielfaltigen¨ Anwendungsmoglichkeiten¨ der hergeleiteten Algorithmen. Schlagworter:¨ Kontaktmechanik, FEM, duale Mortar Methode, perturbed Lagrange,

IV konstitutive Kontaktgesetze, Oberﬂachenrauigkeiten.¨

Contents

1. Introduction 1 1.1. Motivation and state of the art ...... 1 1.2. Aim of this work ...... 3 1.3. Outline ...... 4

2. Continuum solid mechanics 7 2.1. Kinematics and balance laws ...... 7 2.2. Material laws ...... 10 2.3. Contact kinematics ...... 11 2.3.1. Normal contact with microscopic rough surfaces ...... 12 2.3.2. Tangential contact with microscopic rough surfaces ...... 14 2.4. Problem deﬁnition ...... 16 2.5. Methods for contact constraint enforcement ...... 19 2.5.1. Lagrange method ...... 20 2.5.2. Penalty method ...... 21 2.5.3. Perturbed Lagrange method ...... 21 2.5.4. Augmented Lagrange method ...... 22 2.6. Discretization methods ...... 23 2.6.1. Node-to-surface (NTS) ...... 23 2.6.2. Surface-to-surface (STS) ...... 24 2.6.3. Gauss Point-to-surface (GPTS) ...... 24 2.6.4. Mortar method ...... 24

3. Quasi-static dual mortar with CCL 27 3.1. Introduction of the dual Lagrange multiplier ...... 28 3.2. Regularization of the contact conditions ...... 29 3.2.1. Normal contact conditions ...... 29 3.2.2. Tangential contact conditions ...... 32 Linear regularization ...... 33 Iwan regularization ...... 33 3.2.3. Weak formulation of the contact conditions and summary of the weak problem ...... 34

VII Contents

3.3. Discretization ...... 35 3.3.1. Balance equation ...... 36 3.3.2. Normal contact conditions ...... 38 3.3.3. Tangential contact conditions ...... 40 Linear regularization ...... 40 Iwan regularization ...... 40 3.4. Semi-smooth Newton for contact equations ...... 41 3.4.1. Normal direction ...... 41 3.4.2. Tangential direction ...... 43 Linear regularization ...... 43 Iwan model ...... 46 3.5. Full algebraic linear system and condensation of the Lagrange multiplier . 48 3.6. Numerical examples ...... 50 3.6.1. Lamination stack of an electric motor ...... 51 3.6.2. Simpliﬁed disk blade foot contact ...... 51 3.6.3. Menq example ...... 56 3.6.4. Resonator ...... 61

4. Dual mortar methods facing industrial applications 65 4.1. Quadratic dual mortar ...... 66 4.2. Petrov–Galerkin quadratic dual mortar ...... 75 4.3. Modiﬁcation for SPCs/MPCs and overlap ...... 78 4.4. Numerical examples ...... 81 4.4.1. Cyclic symmetric pipes ...... 81 4.4.2. Film forming ...... 82 4.4.3. Push-out-test ...... 87

5. Extension to the fully dynamical problem 91 5.1. Discretization in time: the α-method ...... 92 5.2. Embedding of the contact conditions ...... 95 5.3. Numerical examples ...... 98 5.3.1. Sphere against brick ...... 98 5.3.2. Rubber ball ...... 101 5.3.3. Resonator ...... 103

6. Summary and outlook 109

Appendix 115

A. Calculation of the mortar integrals - cutting algorithm 115

VIII Contents

B. A short comparison of contact algorithms 119 B.1. Patch test ...... 119 B.2. Visco-elastic ironing problem ...... 121 B.3. Hertz ...... 123 B.4. Shroud ...... 124

List of Symbols

A Variable for exponential regularization in normal direction 3 a Accelerations in continuum R , discretized vector ∈ of node values A0 Nominal contact area A Coefﬁcient matrix for dual basis functions for one slave surface αf Variable for α-method αi Variable for Iwan element i αm Variable for α-method αp Scalar from semi-smooth Newton in tangential direction, slip case an Normal contact stiffness Ar Real contact area aτ Tangential contact stiffness also called stick slope b friction bound in continuum description b¯ Given body forces B/BD Slave-master mortar coupling matrix, primal-dual coupling BP Slave-master coupling matrix for Petrov-Galerkin method, primal-primal coupling β Variable for exponential regularization in normal direction; variable for α-method βp Scalar from semi-smooth Newton in tangential direction, slip case bi Friction bound for Iwan element i in continuum bp Friction bound in discrete setting for point p C Right Cauchy–Green tensor c Numerical constant for semi-smooth Newton P Cn Non-linear complementary function for normal contact P Cτ Non-linear complementary function for tangential contact with linear regularization

XI List of Symbols

P,i Cτ Non-linear complementary function for tangential contact for Iwan element i P DCn Derivative of non-linear complementary function for normal contact P DCτ Derivative of non-linear complementary function for tangential contact with linear regularization P,i DCτ Derivative of non-linear complementary function for tangential contact for Iwan element i D/DD Slave-slave mortar coupling matrix, primal-dual coupling DP Slave-slave coupling matrix for Petrov-Galerkin method, primal-primal coupling D D˜ /D˜ Transformed slave-slave coupling matrix for quadratic elements, primal-dual coupling P D˜ Transformed slave-slave coupling matrix for quadratic elements and Petrov-Galerkin method, primal-primal coupling δGn Variation of the normal gap δλn Variation of the Lagrange multiplier in normal direction δΠc Weak contact forces δΠint,ext Weak internal and external forces δλ Variation of Lagrange multiplier δu Variation of displacements δu˜q Transformed variation of displacements for quadratic elements E Young’s modulus E Green–Lagrange strain tensor ep Scalar from semi-smooth Newton in tangential direction, slip case η local coordinate in reference element F p 2 2 matrix from semi-smooth Newton in tangential × direction, slip case γ Variable for Coulomb friction law; variable for α- method γ¯ Solution of last load increment for variable of Coulomb friction law Γc Contact boundary in reference conﬁguration γc Contact boundary in current conﬁguration A γc Active part of slave surface in continuum setting

XII List of Symbols

I γc Inactive part of slave surface in continuum setting slip γc Slipping part of the active slave surface in continuum description stick γc Sticking part of the active slave surface in continuum description ΓD Dirichlet boundary with given displacements in reference configuration γD Dirichlet boundary with given displacements in current configuration ΓN Neumann boundary with given stresses in reference configuration γN Neumann boundary with given stresses in current configuration γp Scalar from semi-smooth Newton in tangential direction, slip case g Real initial gap between two microscopic rough surfaces g vector of weak normal gaps gp in slave nodes Gn Normal gap functional g0 Normal gap at the start of the load increment c Gn(λn) Regularization operator depending on function λn of normal contact law c gn(λn) Scalar regularization function depending on discrete λn of normal contact law cmpl gn (λn) Piece-wise linear scalar regularization function depending on discrete λn of normal contact law cnpl gn (λn) Exponential scalar regularization function depending on discrete λn of normal contact law cpl gn (λn) Linear scalar regularization function depending on discrete λn of normal contact law gn(u) Scalar gap in normal direction depending on displacements R gp weak normal gap in slave point p, ψpg0 dS c,niwan Gτ (λτ ) Regularization operator for Iwan model c Gτ (λτ ) Regularization operator for tangential contact law F Deformation gradient H Displacements gradient Hp 2 2 matrix from semi-smooth Newton in tangential × direction, slip case

XIII List of Symbols

sl hp Vector from semi-smooth Newton in tangential direction, slip case st hp Vector from semi-smooth Newton in tangential direction, stick case hpτ Vector from semi-smooth Newton in tangential direction I Identity tensor, Identity matrix J = det(F ) K Stiffness matrix in Newton–Raphson iteration K˜ Transformed stiffness matrix for quadratic elements λ Lagrange multiplier λ Lame´ constant λ¯ Lagrange multiplier for last load increment λn Contact pressure λ¯n Fixed normal Lagrange multiplier for augmented La- grange formulation λs Scaled Lagrange multiplier vector in discrete setting (= Ddλ) 3 λsp scaled Lagrange multiplier vector R for node p T ∈ λspn n λsp T λspτ τ λsp i λτ Shear stress for Iwan Element i λτ Shear stress λ˜spτ = λspτ λ¯spτ − Lp 2 2 matrix from semi-smooth Newton in tangential × direction, slip case Lpτ 2 3 matrix from semi-smooth Newton in tangential × direction M Dual space of W M Mass matrix M ∗ Combined mass/stiffness matrix for α-method M(λ) Space for dual Lagrange multiplier including friction Set of master nodes M M p 2 2 matrix from semi-smooth Newton in tangential × direction, slip case µ Lame´ constant, friction coefﬁcient n Normal vector α nall Total number of ﬁnite Element nodes in body α N Matrix of transposed slave normals

XIV List of Symbols

niwan Number of Iwan elements used in serial-parallel Iwan model nm Number of master nodes Set of all other nodes except slave and master nodes N ns Number of slave nodes ν Poisson’s ratio Ω Domain of body in reference configuration ∂Ω Boundary of the body in reference configuration Ωt Domain of body in current configuration for time t P First Piola–Kichhoff stress tensor p Normal pressure Φ Function for Coulomb friction law analogous to plasticity formulation φ Vector of standard basis functions for one surface element ϕ Material point mapping from reference to current configuration φ˜p Altered dual basis function for node p in case of SPC/MPcs or overlap φp Standard basis function for node p φ˜ Vector of transformed standard basis function for quadratic elements φ˜p Transformed standard basis function for quadratic elements for node p Π Total energy Πc Contact potential Πint,ext Potential of internal and external forces ψ Vector of dual basis functions for one surface element ψp Dual basis function for node p Pt Projection operator for contact mapping r∗ residuum for α-method ρ Density Rpτ 2 3 matrix from semi-smooth Newton in tangential × direction S Second Piola–Kichhoff stress tensor σ Cauchy stress tensor Set of slave nodes S

XV List of Symbols

T Transformation for reference element to coordinates in finite Element mesh; Global Transformation matrix for for quadratic elements t Traction vector τ Tangent vectors (τ 1, τ 2) ¯t Given traction tc Contact traction dt time increment for current load increment l T e Transformation matrix for for quadratic element and quad-lin method q T e Transformation matrix for for quadratic element and quad-quad method 3 u Displacements in continuum R , discretized vector ∈ of node values u¯ Given displacements; Displacements of the last load increment uˆ weak relative displacements between slave and master side in discrete setting 3 up Displacement vector R for node p T ∈ upn = n up T upτ = τ up u˜pτ = uˆpτ u¯ˆ pτ − u˜q Transformed displacements for quadratic elements q u˜p Transformed displacements for quadratic elements and node p 3 v Velocities in continuum R , discretized vector of ∈ node values α V0 Sobolev-space for displacements and its test function of body α s, m ∈ { } vp Vector from semi-smooth Newton in tangential direction, slip case W Trace space for displacements W Weight matrix in construction of dual basis functions W (C) Strain-energy density function X material point in reference configuration x material point in current configuration ξ local coordinate in reference element Z Mass matrix in construction of dual basis functions

XVI Chapter 1. Introduction

1.1. Motivation and state of the art

A correct and robust modelling of contact interactions is crucial for many engineering applications. Here an application from the field of structural mechanics is given exemplary: Having a look at the assembly of a jet turbine, see Figure 1.1, a lot of different parts are in contact using bolted connections, hooks, fir-tree contacts, shrinks and other types of contacts. For correct stress calculations and thus living calculations during the construction of the separate parts of the turbine, the contacts to the surrounding parts have to be taken into account. Also the surface roughness plays an important role in the contact modelling leading to more complex interface conditions. These are crucial for the physically correct calculation of structural damping for example. In most cases, calculations of this type are solved using the finite element method. Although a lot of progress has been made in the implementation of contact algorithms in commercial codes, solving non-linear contact problems within the FEM framework is still a challenging task when facing large plastic deformations, non-linear material laws and several parts in contact in industrial applications. This led to a lot of research in this field in the past decades. An overview over analytical solutions of contact problems in general can be found in [76, 135, 33] and computational aspects focusing on the FEM method are discussed in [139, 90, 148]. An exhaustive description of contact of rough surfaces can be found in [116, 138, 14, 101]. Covariant contact descriptions for shell elements can be found in [118, 80]. Thus not in focus of this thesis, numerical aspects using isogeometric analysis instead of FEM are presented in [24, 72]. The most popular methods for contact enforcement are the penalty method and the Lagrange multiplier method. Within the Lagrange multiplier method, the contact traction is modelled as an extra variable, the Lagrange Multiplier, fulfilling the contact inequality conditions. Thus being able to model even hard contact in normal direction and a perfect Coulomb law in tangential direction, this method is usually connected with higher computational cost stemming from the additional variables in the system and the solution of a saddle point problem in the discrete setting. The penalty method on the other hand formulates the contact traction as a function of the displacements using a

1 Chapter 1. Introduction

Figure 1.1.: Schematic drawing of a jet turbine as an example for contact interactions in industry regularized version of the contact conditions. Although easy to implement, the penalty method has a strong dependency of the penalty parameters used in the regularization and results in an ill-conditioned matrix systems for high penalty parameters. Apart from these two methods, there exist also combinations of both like the perturbed Lagrange method or the augmented Lagrange method. Early contributions combined the contact enforcement methods with a simple node- to-node discretization in case of small deformations [41, 73] leading to a one-to-one connection between a slave node and a master node. But this discretization method is not favourable for more complex examples including non-matching meshes. The more sophisticated node-to-surface discretization enforces the contact conditions between a slave node and master surface [5, 11, 51, 91, 123, 149]. Thus suitable for non- matching meshes, the node-to-surface formulation has in general a strong dependency on the choice of the master and slave surface and does not satisfy the patch test. The tightest connection between the contacting surfaces is gained with a surface-to-surface discretization enforcing the contact constraints between a slave surface and master surface. Early contributions can be found in [104, 125]. Another more recent method is the Gauss point-to-surface discretization [36, 37, 117] enforcing the contact conditions between an integration point on the slave surface and a master surface usually combined with a penalty method. As the state of the art method for contact enforcement within the FEM framework

2 1.2. Aim of this work mortar methods are preferable. Mortar methods were originally introduced in the context of domain decomposition [13, 12, 6] but were later adapted to small deformational contact problems [7]. They beneﬁt from a strong mathematical background and are strongly connected the surface-to-surface discretization in the discrete setting. Exam- ples of the mortar method in combination with the penalty formulation can be found in [97, 151, 111, 112, 113, 92] and in combination with the Lagrange multiplier formulation in [57, 60, 137, 31]. Among the mortar methods, the dual mortar method [143] is of special interest, since it employs a consistent weak formulation of the contact conditions and models the contact traction via a dual Lagrange multiplier allowing for static condensation of the extra variables of the system before solving. The dual mortar formulation for small deformation quasi-static contact can be found in [70]. The dynamic case is considered in [53, 55] and [50, 48] including the dual mortar method for plasticity. The consistent linearization for large deformations was treated in [107, 108, 44, 110] and the extension to thermo-mechanical contact in [71, 43]. Another current area of research is the isogeometrical analysis. Here the same smooth and higher order basis functions are utilized for the representation of the CAD geometry as well as for the FEA solution ﬁelds. Contact simulation using a Gauss point-to-surface discretization can be found in [133, 26, 32] or a mortar formulation in [134, 27, 78].

1.2. Aim of this work

This work aims to derive a dual mortar method for contact problems modelling constitutive contact laws due to surface roughness on the micro scale using the finite element method. This surface roughness affects significantly the global contact behaviour. Since it is not intended to model this roughness in the finite element model, the local behaviour on the micro scale is reflected with constitutive contact laws leading to regularized contact conditions. As a method of choice for the contact enforcement method the Lagrange multiplier method is chosen including a consistent weak formulation of the regularized contact laws. This results in a surface-to-surface discretization having major advantages over more simple node-to-surface formulations. Generally, Lagrange multiplier methods lead to saddle point problems in the discrete setting or in case of regularized contact conditions to regularized saddle point problems. Here, the space of the Lagrange multiplier is set to be the dual space of the trace space of the displacements leading to the well known dual mortar method. This has the advantage, that the additional variables of the system can be condensed before solving and thus produce a purely displacement dependent system. Nevertheless the optimality of the solution is guaranteed. The discretized weak contact conditions are reformulated as non-linear complementary function leading to a semi-smooth Newton method handling all system

3 Chapter 1. Introduction non-linearities within one iteration loop. First, a dual mortar method for quasi-static contact problems with constitutive contact laws in normal and tangential direction using linear finite elements is derived. Then, the algorithm is extended to quadratic finite elements. For the quadratic case, also a Petrov–Galerkin dual mortar method is derived. This method avoids non-physical evaluations of the weak gap occurring for special mesh set-ups. In order to apply the derived algorithms to complex examples of industrial relevance, a workaround for single or multi point constraints like cyclic symmetry or directional blocking on the contact interface as well as overlap has to be found. Also the algorithm is extended to the dynamical case in order to simulate frictional damping correctly. Throughout this work, some comparisons of numerically obtained results with actual measurements are desired. The academic findings presented in this work are based on the following papers:

S. Sitzmann, K. Willner, and B. I. Wohlmuth. A dual Lagrange method for contact • problems with regularized contact conditions. International Journal for Numerical Methods in Engineering, 99(3):221–238, 2014.

S. Sitzmann, K. Willner, and B. I. Wohlmuth. A dual Lagrange method for contact • problems with regularized frictional contact conditions: Modelling micro slip. Com- puter Methods in Applied Mechanics and Engineering, 285(0):468–487, 2015.

S. Sitzmann, K. Willner, and B. I. Wohlmuth. Variationally consistent quadratic ﬁ- • nite element contact formulations for ﬁnite deformation contact problems on rough surfaces . Finite Elements in Analysis and Design, 109:37–53, 2015.

1.3. Outline

The outline of this thesis is as follows. In Chapter 2 the fundamentals of continuum solid mechanics involving contact are presented. Special focus is set on contact laws in normal and tangential direction taking the surface roughness on the micro scale into account which leads to constitutive interface laws. Also an overview over the most common contact enforcement and discretization methods for the ﬁnite element framework is given. The dual mortar method with constitutive interface laws is presented in detail in Chap- ter 3 for quasi-static calculations using linear ﬁnite elements. After the introduction of the dual Lagrange multiplier, the weak formulation of the problem is derived and discretized. The discrete regularized contact inequalities are treated with a semi-smooth Newton method and the whole algebraic representation is shown. Finally the dual La- grange multiplier is condensed from the system. Numerical examples demonstrate the wide applicability of the dual mortar method with constitutive contact laws including comparisons with real measurements.

4 1.3. Outline

In Chapter 4, the algorithm is extended to quadratic finite elements. Also a modification of the algorithm in case of non-physical evaluations of the weak gap due to the dual basis functions using a Petrov–Galerkin method is presented. In case of complex industrial applications including multi-point-constraints like cyclic symmetry and directional blocking as well as hanging nodes, a consistent alteration of the dual mortar method is shown. All modifications in this Chapter are tested with a variety of complex examples including large deformations, various potential contact zones and plasticity. The fully dynamic contact problem is covered in Chapter 5. The generalized α- method is employed for the discretization of the problem in time. Afterwards the modifications of the algorithm switching from displacements to accelerations as primary state variable is shown embedding the contact conditions in the system. Finally several numerical examples demonstrate robustness of the algorithm. Special focus is set to the simulation of frictional damping shown by a resonator example. Finally some conclusions as well an outlook is given in Chapter 6. In Appendix A a remark on the calculations of the mortar integrals and on the cutting algorithm needed for slave-master intersection is given. Finally a short comparison of different contact enforcement algorithms is presented in Appendix B including the patch test and a Hertz example. Here the derived dual mortar method with constitutive contact laws is compared with a node-to-face penalty method as well as a penalty mortar method modelling the same contact interaction laws. The numerical examples include linear and quadratic finite elements as well as visco-elastic materials and overlap of the bodies.

Chapter 2. Continuum solid mechanics

In this chapter the fundamentals of continuum solid mechanics involving contact are presented. It starts with an overview of the underlying kinematics and balance laws including the utilized material laws. Afterwards the contact conditions are described focusing on contact of microscopic rough surfaces leading to constitutive contact laws on the interfaces. Then a summary of the full problem description in its strong form is given and the weak form is derived with a method of virtual displacements. To solve the problem, an outline of the most popular method of contact constraint enforcement in the ﬁnite element framework is given using the principle of the minimal potential energy. At the end, an overview of different discretization methods for contact problems is given. Special focus is set on the dual mortar method employed in this work. In total, this chapter represents a brief overview of the case of two contacting bodies treated with the ﬁnite element method. For a more general description, the interested reader is referred to [77, 122, 64] for an exhaustive continuum description and to [139, 90, 148] for computational aspects. Here the following notions are used: scalar variables are described by normal letters, vector valued variables by small bold letters and tensors by capital bold letters.

2.1. Kinematics and balance laws

Let’s start with an overview of the 3D two-body contact problem in a non-linear elasticity setting. As one can see in Figure 2.1, two contacting bodies are considered, a slave α 3 body, denoted by s, and a master body, denoted by m, with domains Ω R , α = s, m ∈ in the reference conﬁguration. During the time interval [0,T ] the bodies are undergoing motion described by the mapping

α α 3 ϕ :Ω [0,T ] R × → (2.1) Xα xα = ϕα(Xα, t), α = s, m. 7→ With this mapping, any material point Xα of the reference conﬁguration is mapped to its counterpart in the current conﬁguration xα. With this nomenclature the displacements

7 Chapter 2. Continuum solid mechanics

Figure 2.1.: Two-body contact problem can be defined as u(X, t) = ϕ(X, t) X (2.2) − with u = (us, um) defined on Ω = Ωs Ωm. The boundary of the bodies ∂Ωα is × α divided into three disjoint sets, the Dirichlet boundary ΓD with given displacements, the α α Neumann boundary ΓN with given traction and the potential contact boundary Γc . Their spatial counterparts on ∂Ωα are denoted by γα , γα and γα. It holds Γ¯α Γ¯α Γ¯α = ∂Ωα. t D N c D ∪ N ∪ c In terms of the displacements u, the displacement gradient ∂u H = = u (2.3) ∂X ∇ and the unsymmetrical deformation gradient ∂x F := = [X + u(X, t)] = I + u = I + H (2.4) ∂X ∇ ∇ can be defined. With the deformation gradient F , an infinitesimal line element dX is mapped to its spatial counterpart dx dx = F dX (2.5) and vice versa dX = F −1dx (2.6) using it’s inverse F −1 = ∂X/∂x. Since the deformation gradient describes the whole deformation process including rigid body motion, it is not a desirable candidate for strain measurement. Symmetric strain measurements referred to the initial configuration are the right Cauchy–Green tensor C = F T F (2.7)

8 2.1. Kinematics and balance laws and the Green–Lagrange strain tensor 1 1 E = (F T F I) = (C I) 2 − 2 − (2.8) 1 = u + uT + u uT . 2 ∇ ∇ ∇ ∇ The stress can be calculated from the strain via the employed material model. The physical stress is described by the Cauchy stress tensor σ in the current conﬁguration while the energetic conjugate measurement to the Green–Lagrange strain tensor is the second Piola–Kirchhoff stress tensor S. This symmetric stress tensor is formulated completely in the reference conﬁguration and it holds

S = JF −1σF −T (2.9) with J = det(F ) > 0. (2.10) In contrast to S, the unsymmetrical ﬁrst Piola–Kirchhoff tensor P is a two ﬁeld tensor

P = JσF −T (2.11) where one base vector lies in Ω and the other in ϕ(Ω, t). It holds:

S = F −1P . (2.12)

Now that the underling kinematics are described, the balance laws will be studied in detail. First the balance of mass holds for all materials and for all time points t [0,T ] ∈ Z Z m = ρ0 dV = ρ dV = const (2.13) Ω ϕ(Ω,t)

−1 with ρ0 being the density of the reference configuration and ρ = J ρ0 the density in the current configuration. Next, the balance of linear momentum in the current configuration

div σ + ρb¯ = ρv˙ (2.14) is derived from Newton’s second law of motion using Cauchy’s-theorem

t = σT n. (2.15)

It states, that the divergence of Cauchy stress plus the given body forces (ρb¯) equals the inertia forces. Its counterpart in the reference conﬁguration reads as

DIV P + ρ0b¯ = ρ0v˙ 0. (2.16)

Adding initial conditions and boundary conditions to the balance of momentum, one can deﬁne the ﬁrst part of the two-body contact problem without contact conditions.

9 Chapter 2. Continuum solid mechanics

α α 3 Deﬁnition 1 (The two-body contact problem - part 1). Find u :Ω [0,T ] R satisfying × →

balance α DIV P α + ραb¯ = ραv˙ α, in (0,T ] Ωα, (2.17a) equation 0 0 ×  α α α u = u¯ , in (0,T ] ΓD, boundary  α × P αn = ¯t , in (0,T ] Γα , (2.17b) conditions × N  P αn = tα, in (0,T ] Γα, c × c initial uα(0, ) = uα, · 0 in Ωα, (2.17c) conditions vα(0, ) = vα, · 0 On the Dirichlet boundary, it is assumed u¯α = 0 and on the Neumann boundary α α α α P n = ¯t , where ¯t is a given surface force. The contact traction tc and the actual contact zone are not known a priori and have to be determined numerically. Since the α contact traction tc is not known, this system is under-constrained. Adding the contact conditions will complete the system, see Section 2.4.

2.2. Material laws

Material non-linearities are considered assuming the existence of a strain-energy function W with ∂W ∂W = 2 = S (2.18) ∂E ∂C relating the Green–Lagrange strain tensor E to the second Piola–Kirchhoff stress tensor S. Here three material laws are presented representatively. For a broader overview, the interested reader is referred to [64, 122]. Deﬁnition 2 (Saint Venant-Kirchhoff material law). This isotropic, hyper-elastic material is based on the quadratic strain energy function λ W = (tr(E))2 + µE : E (2.19) 2 setting S to S = λtr(E)I + 2µE. (2.20) The Lame´ constants λ and µ are deﬁned as Eν E λ = , µ = (2.21) (1 + ν)(1 2ν) 2(1 + ν) − using the Young’s modulus E and Poisson’s ratio ν. The Saint Venant-Kirchhoff material law describes a linear relationship between Green-Lagrange strain E and second Piola–Kirchhoff stress S.

10 2.3. Contact kinematics

Deﬁnition 3 (Compressible Neo–Hookean material). This isotropic, hyper-elastic material is based on the strain energy function µ λ W = (tr(C) 3) µln(J) + (ln (J))2 (2.22) 2 − − 2 leading to the second Piola–Kichhoff stress tensor

S = λln(J)C−1 + µ(I C−1). (2.23) − This describes a non-linear relationship between Green-Lagrange strain E and second Piola–Kirchhoff stress S. Deﬁnition 4 (Visco-elastic Kelvin–Voigt material). This isotropic, visco-elastic material is based on the strain energy function

λ η1 η2 W = (tr(E))2 + µE : E + (tr(E˙ ))2 + E˙ : E˙ (2.24) 2 2 2 with the viscosity parameters η1, η2 deﬁned as

η1 = τ λ, η2 = 2τ µ (2.25) · · using the retardation time τ [47]. This sets S to

S = λtr(E)I + 2µE + η1tr(E˙ )I + η2E˙ . (2.26)

2.3. Contact kinematics

In this section the contact kinematics are described. Though starting with the mathematically perfect contact conditions in normal and tangential direction, special focus is set to the constitutive contact laws taking the surface roughness on the micro scale into account. For a more detailed overview, the interested reader is referred to [142, 116, 138]. In this work, the contact constraints are enforced on the slave body and transferred to the master body via the contact traction. Thus for each point on the slave surface the orthogonal system (n, τ 1, τ 2) is deﬁned, with n = n(X, t) being the current normal on the slave surface in the point X at time t and τ = (τ 1, τ 2) the corresponding tangents. In terms of this nomenclature the projection operator can be stated as s m Pt :Γc Γc → (2.27) X Pt(X). 7−→ It takes a point x of the slave surface in the current conﬁguration and projects it onto the master surface in direction of the actual normal n(X, t), see Figure 2.1. With the

11 Chapter 2. Continuum solid mechanics

projection operator Pt the gap function in the normal direction can be deﬁned for the current conﬁguration:

s s s T s s m s Gn(X , t) := [X + u(X , t)]n = n (ϕ (X , t) ϕ (Pt(X ), t)). (2.28) · − It has to be noted that with this deﬁnition, the gap is negative as long as there is a gap between the two bodies and it is positive if there is overlap of the bodies.

2.3.1. Normal contact with microscopic rough surfaces

Now, the Karush–Kuhn–Tucker (KKT) conditions for normal contact can be deﬁned on s the slave surface Γc:

s s Gn 0 t 0 t Gn = 0, (2.29) ≤ ∧ − cn ≥ ∧ − cn · with ts = nT ts = p, (2.30) cn c − see Figure 2.2. Here the ﬁrst equation of (2.29) states the impenetrability condition. The second one guarantees that only pressure is transmitted from the slave to the master surface using s s m s t (X , t) = t (Pt(X ), t). (2.31) c − c The last equation is known as the complementary condition and guarantees that pressure is only transmitted if the gap between the two bodies is closed. This KKT conditions divide the slave surface into two disjunct sets, the active part with a closed gap

A s s γ = x γ : Gn = 0 t > 0 (2.32) c { ∈ c ∧ − cn } and the inactive part

I s s γ = x γ : Gn < 0 t = 0 (2.33) c { ∈ c ∧ − cn } with vanishing contact forces for all t [0,T ]. ∈ In the last paragraph it was assumed that all surfaces are ideally smooth, but in reality all surfaces are rough on the micro-scale as one can see in Figure 2.3 showing a milled steel surface. This roughness affects significantly the normal contact behaviour as can be seen from the contact of two rough surfaces exposed to an increasing normal pressure P in Figure 2.4a [45, 140]. One notices that in the beginning only a few asperities are in contact. Therefore the real contact area Ar is significantly smaller than the nominal contact area A0. This leads to high contact pressure in these contacting asperities causing them to deform and flatten down. One gets a soft contact behaviour at the beginning of the process. With increasing pressure P , more and more asperities

12 2.3. Contact kinematics

Figure 2.2.: Visualization of the Karush–Kuhn–Tucker (KKT) conditions for hard contact in normal direction

Figure 2.3.: Milled steel surface on the micro scale

3.0

] 2.5 MP a [ 2.0

1.5

1.0

Contact pressure 0.5

0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 3 Penetration [ 10− mm] × (a) Contact of rough surfaces (b) Resulting constitutive contact law

Figure 2.4.: Normal contact for rough surfaces

13 Chapter 2. Continuum solid mechanics are getting into contact. So the real contact area increases and redistributes the pressure across a larger area causing a stiffer contact behaviour. Increasing the normal pressure again will close the initial gap g between the two rough surfaces leading to Ar = A0. From this point on, the asperities are totally flattened and the contact now acts like a hard contact. Thus in total there is a transition from soft to hard contact taking the surface roughness on the micro-scale into account as one can see in Figure 2.4b. One has to distinguish two cases: The initial loading and the cyclic loading. The initial loading occurs directly after the manufacturing process and during it the high asperities with small radius will be exposed to extremely high contact pressure, deform plastically and are flattened. This can be modelled numerically for example by the ABBOTT-model [76], but will be neglected in this work. After the initial loading the asperities are flattened and under cyclic loading they will now deform elastically. This behaviour can be realized numerically by the GREENWOOD–WILLIAMSON-model [46] or the model by BUSH,GIBBSON and THOMAS [16]. Here the focus lies on the cyclic loading after initial loading assuming a purely elastic deformations of the asperities. In order to model this behaviour numerically, the hard contact conditions in normal direction (2.29) will be regularized in Chapter 3 and combined with a perturbed Lagrange formulation to enforce the regularized contact conditions.

2.3.2. Tangential contact with microscopic rough surfaces

There exists a wide variety of possible friction models for contact problems, but this work will focus on the Coulomb friction law. It states, that transmitted shear stress of two contacting bodies is bounded by a friction bound deﬁned as the normal contact pressure multiplied by the friction coefﬁcient µ

ts µts , (2.34) k cτ k ≤ − cn

s T s with t = (t1, t2), ti = τ t . The friction coefﬁcient µ, 0 < µ 1 depends on cτ i · c ≤ the material properties and can be derived experimentally. Furthermore the Coulomb friction law distinguishes two cases for every point of the contact area: If the shear stress is smaller than the friction bound, no relative velocity between the bodies is allowed and thus the point sticks. If the shear stress reaches the friction bound, the bodies start to slide over one another and the point is slipping. This is visualized in Figure 2.5. In terms of KKT conditions, the perfect Coulomb law can be stated as

14 2.3. Contact kinematics follows [90]: s s s s Φ(tcn, tcτ ) := tcτ + µtcn 0, k ks ≤ tcτ [u˙ ]τ = γ˙ s , − tcτ (2.35) k k γ˙ 0, ≥ γ˙ Φ = 0, · with T s s m s [u˙ ] = τ (u˙ (X , t) u˙ (Pt(X ), t)). (2.36) τ · − The KKT conditions in tangential direction are formulated in u˙ in order to model the path dependency of the frictional problem. The first equation states that the shear stress is bounded by the friction bound. The second equation is the evolution law for the displacements and implicitly formulates the stick case. The last equation states the complementary condition. The case of no friction is included setting µ = 0 . This KKT conditions divide the active part of the slave surface into two disjunct sets, the sticking part γstick = x γA : ts < µts [u˙ ] = 0 (2.37) c ∈ c k cτ k − cn ∧ k τ k and the slipping part γslip = x γA : ts = µts [u˙ ] > 0 (2.38) c ∈ c k cτ k − cn ∧ k τ k for all t [0,T ]. In case of no friction, all points are sliding ts 0 ts = 0. ∈ k cτ k ≤ ⇒ cτ Again, this was an ideal description of the friction kinematics. Having again a closer look at the contact of two rough surfaces on the micro scale, a slightly different behaviour can be observed, see Figure 2.6. Applying a normal pressure, the asperities on the micro scale are pressed against each other, deform elastically and get caught-up with each other. For small shear stresses, these caught up asperities deform elastically in tangential direction and lead to an elastic stick region. With increasing shear stress, parts of the micro contacts are starting to slip while other micro contacts are still sticking. This effect is called micro slip [142]. Increasing the shear stress further leads to macro slip, where all micro contacts are slipping. This smooth transition from sticking to slipping can be observed locally on the micro scale but also globally on the macro scale when the contact zone is considered as a whole and it significantly influences the dynamical behaviour of the contact [119, 42]. For a more detailed description of tangential contact laws with microscopic rough surfaces, the interested reader is referred to [14, 101]. In this work special focus is set to the tangential behaviour under cyclic loading. This can be visualized by a hysteresis plot, that means a displacements-force plot for a given displacement history u(t). Therefore the Masing rules for a hysteresis of rough surfaces [95, 96] are briefly recalled here. Having a given displacement history u(t) and a resulting Force F (t), one defines r(u, F ) as r(u, F ) = F g(u). It holds: −

15 Chapter 2. Continuum solid mechanics

1. The initial loading curve is symmetric

r(u, F ) = r( u, F ) = 0. (2.39) − −

2. In case of load reversal, it holds

u∗ u F ∗ F r − , − = 0 (2.40) 2 2

with (u∗, F ∗) being the point of load reversal.

3. The tangent in the point of load reversal is the same as the initial tangent.

4. If a inner curve crosses the curve of a former load cycle, the curve follows the old path.

See Figure 2.7 for a hysteresis following the Masing rules. In order to model the smooth transition between sticking and slipping numerically, the perfect Coulomb law (2.35) will be regularized in Chapter 3 and enforced by a perturbed Lagrange formulation. The non-linear regularization will be achieved via a serial-parallel Iwan model satisfying the Masing rules for hysteresis by construction.

2.4. Problem deﬁnition

Now that the basic kinematics, balance laws and contact conditions have been deﬁned, the full two body contact problem in 3D can be deﬁned in its strong form:

α α 3 Deﬁnition 5 (The two-body contact problem - part 2). Find u :Ω [0,T ] R × →

16 2.4. Problem deﬁnition

Figure 2.5.: Perfect Coulomb law

macro slip

micro slip elastic stick elastic

Figure 2.6.: Constitutive contact law in tangential direction for rough surfaces

1.0 Masing model initial loading load reversal 0.5

τmax 0.0 /F τ F

0.5 −

1.0 − 1.5 1.0 0.5 0.0 0.5 1.0 1.5 − − − uτ /uτmax

Figure 2.7.: Hysteresis following the Masing rules

17 Chapter 2. Continuum solid mechanics satisfying

balance α DIV P α + ραb¯ = ραv˙ α, in (0,T ] Ωα, (2.41a) equation 0 0 ×  α α α u = u¯ , in (0,T ] ΓD, boundary  α × P αn = ¯t , in (0,T ] Γα , (2.41b) conditions × N  P αn = tα, in (0,T ] Γα, c × c α α initial u (0, ) = u0 , α α · α in Ω , (2.41c) conditions v (0, ) = v0 ,  · normal Gn 0  s ≤ s contact tcn 0 in (0,T ] Γc, (2.41d) −s ≥ × conditions  t Gn = 0 cn ·  s s Φ(tcn, tcτ ) 0  ≤ s tangential  tcτ  u s [ ˙ ]τ = γ˙ k k , s contact − tcτ in (0,T ] Γc, (2.41e)  γ˙ 0 × conditions  ≥  γ˙ Φ(ts , ts ) = 0, · cn cτ For simplicity here the unregularized versions of the contact conditions are employed. In Chapter 3 the contact conditions will be regularized. Again, it is assumed u¯α = 0 on the Dirichlet boundary. In order to ﬁnd the solution of this problem, the ﬁnite element method is used. One starts with the balance of momentum (2.41a) and multiplys it with the test func- α α α α tions δu satisfying δu = 0 on ΓD and integrates over the domain Ω . This leads to Z Z X α α α ¯α α α DIVP δu dV + ρ0 (b v˙ ) δu dV = 0. (2.42) α · α − · α∈{s,m} Ω Ω Performing a partial integration and inserting the boundary conditions yields   Z Z X  α α α α α P δu dV + ρ0 v˙ δu dV  Ωα · ∇ Ωα · α∈{s,m}  | {zα } | {zα } δΠint δΠkin    (2.43)   Z Z  Z   α¯α α α α  α α  ρ0 b δu dV + ¯t δu dS tc δu dS = 0. −  Ωα · Γα  − Γα N c    | {z } | {zα } α  δΠext δΠc

Equation (2.43) is equivalent to the weak balance of forces (internal, inertia, external,

18 2.5. Methods for contact constraint enforcement body plus contact forces equals null):

s δΠkin(v˙ , δu) + δΠint(u, δu) δΠext(δu) + δΠc(t , δu) = 0, (2.44) − c with Z s δΠc = tc[δu] dS (2.45) s − Γc using Equation (2.31) and the jump in the displacements deﬁned implicitly in Equation (2.28). This system is now solvable in a weak sense adding the contact conditions and applying one of the contact constraint enforcement methods deﬁned in Section 2.5. The problem will be discretized in Chapter 3 for the quasi-static case neglecting the inertia forces and in Chapter 5 for the dynamic case.

2.5. Methods for contact constraint enforcement

Here a brief overview of the most common methods for contact constraint enforcement is given. For simplicity this section is restricted to the linear-elastic frictionless two-body contact problem deﬁned in (2.41) setting µ = 0. Instead of using the methods of virtual displacements, the problem can also be solved with constrained minimization

Π(x) min −→ (2.46) subject tof(x) 0. ≥ More precisely the fact is utilized that for hyper-elastic materials there exists a strain- energy function W = W (E) as described in Section 2.2 and minimize the total potential energy of the two-body system:

Π(ϕ) = Πint,ext,kin(ϕ) + Πc(ϕ) min s −→s subject to Gn 0 tcn 0 tcnGn = 0 (2.47) s ≤ ∧ − ≥ ∧ tcτ = 0 with Πint,ext,kin(ϕ) deﬁned as

X α Πint,ext,kin(ϕ) = Πint,ext,kin(ϕ) α∈{s,m} Z Z (2.48) = W (E) ρ0(b¯ v˙ ) ϕ dV ¯t ϕ dS. Ω − − · − ΓN · The minimum can be found by variation

d δΠ = D Π(ϕ) η = Π(ϕ + αη) . (2.49) · dα α=0

19 Chapter 2. Continuum solid mechanics

It holds for ϕ∗ minimizing Π:

∗ D Π(ϕ ) η = δΠint,ext,kin + δΠc = 0 (2.50) · The variation of the ﬁrst term of (2.50) yields Z ∂W Z δΠint,ext,kin = δC ρ0(¯b v˙) η dV t¯ η dS (2.51) Ω ∂C − − · − ΓN · which is equivalent to the weak formulation of the balance of momentum (2.43). The most common choices for the contact potential Πc are discussed in this Section. The s simplest case where the whole contact zone Γc is active is studied avoiding the need s for a method to determine the subset of Γc that is currently in contact like for example an active-set strategy. For other choices of the contact potential and the more general case involving friction, the interested reader is referred to [90, 148].

2.5.1. Lagrange method

The ﬁrst option to incorporate the contact conditions is the Lagrange multiplier method. Here an extra variable λ, the Lagrange multiplier, is introduced modelling the unknown negative contact traction ts. The Lagrange multiplier fulﬁlls the contact conditions: − c

Gn 0 λn 0 λn Gn = 0 λτ = 0. (2.52) ≤ ∧ ≥ ∧ · ∧ This is equivalent to the contact potential Z Z LM Πc = λ G dS = λn Gn dS (2.53) s s Γc Γc in the minimization problem (2.47). Note, that the tangential part of the contact potential LM vanishes, since frictionless contact is assumed. The variation of Πc yields Z Z LM δΠc = λn δGn dS + δλn Gn dS (2.54) s s Γc Γc | {z } | {z } δΠc contact condition splitting the weak contact potential in the weak contact forces δΠc and the weak contact conditions, that means the impenetrability condition, in normal direction. This leads to the saddle point problem Z δΠint,ext,kin(u, δu) + λn δGn dS = 0, s Γc Z (2.55) δλn Gn dS = 0. s Γc

20 2.5. Methods for contact constraint enforcement

Thus with the Lagrange multiplier method, the KKT conditions can be fulﬁlled without regularization in a weak sense. This is also the basis for the mortar methods. The drawback, however, of the Lagrange multiplier method is that the number of variables is increased and one has to solve a saddle-point system in the discrete setting in general.

2.5.2. Penalty method

Another very popular method for constraint enforcement is the penalty method. Here one wants to avoid extra variables in the system and regularizes the KKT conditions instead with a penalty parameter an:

1 s Gn + t 0 an cn s ≤ s anGn if Gn > 0 tcn 0 t = p(Gn) = . (2.56) − ≥ ⇒ − cn 0 otherwise ts G + 1 ts = 0 cn n an cn

Here the simplest possible penalty function is utilized leading to the contact potential Z P 1 2 Πc = anGn dS. (2.57) s Γc 2

P The variation of Πc yields Z P δΠc = anGn δGn dS (2.58) s Γc leading to the problem Z δΠint,ext,kin(u, δu) + anGn δGn dS = 0. (2.59) s Γc

Thus introducing no additional unknowns, the system depends in this case on the penalty parameter. On the downside one can get an ill-conditioned matrix system in the discrete setting for high penalty parameters and cannot enforce hard contact conditions.

2.5.3. Perturbed Lagrange method

The perturbed Lagrange method also introduces an extra variable for the contact traction like the Lagrange multiplier method, but combines it with regularized contact conditions [125]: 1 1 Gn λn 0 λn 0 λn(Gn λn) = 0 (2.60) − an ≤ ∧ ≥ ∧ − an

21 Chapter 2. Continuum solid mechanics

This is achieved by the contact potential Z Z PL 1 2 Πc = λn Gn dS λn dS (2.61) s s Γc − Γc 2an and the variation Z Z Z PL 1 δΠc = λn δGn dS + δλn Gn dS δλn λn dS. (2.62) s s s Γc Γc − Γc a

This leads to the regularized saddle point problem Z δΠint,ext,kin(u, δu) + λn δGn dS = 0, Γs c (2.63) Z Z 1 δλn Gn dS δλn λn dS = 0. s s Γc − Γc an

Again, the contact condition is fulﬁlled in a weak sense. In contrast to the penalty method, the regularization parameter an can be arbitrary high and the case of hard contact is included by setting an = . Later, this approach will be the method of choice ∞ for the dual mortar formulation with microscopic rough surfaces.

2.5.4. Augmented Lagrange method

Another method more related to the penalty method is the augmented Lagrange method [123]. Here the drawbacks of the penalty methods should be compensated and combined with all advantages of the Lagrange multiplier method. Therefore the following contact potential is used Z Z AL ¯ 1 2 Πc = λn Gn dS + anGn dS (2.64) s s Γc Γc 2 introducing a Lagrange multiplier λ¯ that is ﬁxed during the solution of the system and therefore not varied Z AL ¯ δΠc = (λn + anGn) δGn dS. (2.65) s Γc The ﬁxed λ¯ is be found iteratively via the Uzawa-algorithm starting with ¯1 λn = 0 (2.66) and an successive augmentation of the Lagrange multiplier ¯i+1 ¯i λn = λn + anGn. (2.67)

22 2.6. Discretization methods

Thus this method starts with a penalty solution and converges after some iterations to the solution of the Lagrange method. This has the advantage, that there is no dependency on a regularization parameter an and that the contact constraints are fulﬁlled in its hard form. But on the other side an additional outer loop is added to the discrete algorithm.

2.6. Discretization methods

Next an overview over possible discretization methods for contact problems is given once the contact enforcement method is chosen. Theoretically all above mentioned techniques can be combined with the following discretization techniques, but there are some more natural combinations.

Slave Slave

Master Master

(a) NTS (b) STS

Slave

Slave Master

Master

Triangulation (c) GPTS (d) Mortar

Figure 2.8.: Discretization methods

2.6.1. Node-to-surface (NTS)

The simplest method is the node-to-surface discretization. Here the contact constraints are enforced between a slave node and a master face, see Figure 2.8a. This method is often combined with the penalty method for constraint enforcement and needs an active set strategy to choose the active slave nodes out of all slave nodes. Examples

23 Chapter 2. Continuum solid mechanics can be found in [51, 11, 149, 104] in combination with the penalty method and in [4] in combination with the augmented Lagrange method. A more recent review can be found in [154]. Unlike other methods, the node-to-surface discretization is unable to satisfy the patch test [153] and has a strong dependency on the ﬁnite element mesh as well as on the choice of the slave body. Also it is restricted to linear elements in most formulations [25]. But on the other hand this discretization technique is preferable for modelling edge or vertex contact.

2.6.2. Surface-to-surface (STS)

In order to overcome the drawbacks of the NTS formulations, the surface-to-surface disretization method (also called segment-to-segment discretization method in 2D) was developed. Here the contact constraints are enforced over the slave surface in an integral manner and not collocated at slave nodes, see Figure 2.8b. This approach typically employs so-called intermediate contact surfaces and a segmentation of the contacting surfaces. An early implementation can by found in [125] in combination with a perturbed Lagrange method and a piecewise constant approximation of the contact pressure. Other STS formulations were proposed in [155, 105] in combination with the penalty method. In contrast to the NTS formulation the STS discretization satisﬁes the patch test but does not fulﬁll the inf-sup condition also called LBB stability condition [34]. See [3] for details on the inf-sup condition.

2.6.3. Gauss Point-to-surface (GPTS)

Another possibility to overcome the drawbacks of the NTS formulation is the Gauss point-to-surface formulation. Here one enforces the contact conditions at an arbitrary number of contact quadrature points in the slave surface. In most cases, a ﬁxed number of integration points is chosen for every slave surface and only contributions of quadrature points with a closed gap will be included in the evaluation of the contact integral. Since no segmentation of the slave surface is performed, this discretization method is simple to implement. But it satisﬁes the patch test only up to the integration error and depends on the choice of the slave surface. An example of a GPTS formulation in combination with a penalty method can be found in [36, 37, 117].

2.6.4. Mortar method

A recent discretization method is the mortar method. It was originally developed in the context of domain decomposition [13, 12, 6] but was later adapted to contact treatment [7]. In contrast to other methods, it is characterized by a weak formulation of the contact constraints over the whole slave surface. The mortar method is strongly connected

24 2.6. Discretization methods to the STS formulation in the discrete setting, but beneﬁts from a strong mathematical background. Thus a variationally consistent treatment of the contact constraints is achieved while guaranteeing optimal convergences rates. A main advantage of this method is that it satisﬁes the patch test as well as the LBB stability requirements [113]. As a drawback to other methods the higher computational costs have to be mentioned. These stem mainly from the evaluation of the mortar integrals requiring a segmentation of the contact surface, see Figure 2.8d. Examples of the mortar method in combination with the penalty formulation can be found in [97, 151, 111, 112, 113, 92] and in combination with the standard Lagrange multiplier formulation in [57, 60, 137, 31]. This work is focused on the dual mortar method [143] combining the mortar method with the Lagrange multiplier formulation and a smart choice of the Lagrange multiplier space. Here so-called dual shape functions are constructed for the discrete Lagrange multiplier space allowing for condensation of the Lagrange multiplier before solving the linear system without loosing the optimality of the solution. It was applied to small deformation contact in [70] and expanded to dynamic contact in [53, 55] and [50, 48]. A consistent linearization of the contact terms was presented in [107, 108, 44, 110] while a consistent treatment of boundaries was proposed in [22]. In contrast to other work on the dual mortar method, here the dual mortar method is combined with a perturbed Lagrange formulation enabling the treatment of a variety of constitutive contact laws as described in Section 2.3.

Chapter 3. Dual mortar method for quasi-static contact problems with microscopic rough surfaces

In this chapter the dual mortar formulation for quasi-static contact problems with microscopic rough surfaces leading to constitutive contact laws is presented on the basis of [126, 127]. As mentioned in Section 2.6.4, the dual mortar method is a Lagrange multiplier method combined with a surface-to-surface discretization using a consistent weak formulation of the contact conditions [146]. Like all other mortar methods the dual mortar method was originally introduced in the context of domain decomposition, see [82, 144, 38, 39, 40]. Later the ideas were expanded to small deformation contact and thermal coupling [65], dynamical contact [52] and plasticity [48] as well as finite deformational contact with consistent linearization including wear [106, 43]. Here one focuses on a combined fix-point and Newton approach and employs the linearized version of the normal contact law as well as a linearized version of the weak contact forces based on [65]. This is based on the assumption that the load step has to be split up in small load increments to resolve path dependencies in case of friction or plasticity. In the beginning, the classical contact problem using hard contact and a perfect Coulomb law is presented in Section 3.1. Then the weak problem for the balance equation is recalled and one can have a closer look at the contact conditions in Sec- tion 3.2. Here, one switches from the ideal mathematical setting to regularized contact conditions in order to capture the physical effects on the micro scale. Thus the dual mortar method is combined with a perturbed Lagrange formulation [125]. After the full weak problem is described, the continuum is discretized with linear finite elements in Section 3.3. The extension to quadratic elements will be treated separately in Chapter 4. In Section 3.4 a semi-smooth Newton [1, 62] is used as an active set iteration for the discrete contact conditions. At the end, the full algebraic linear system is shown and the Lagrange multiplier is condensed from the global system in Section 3.5. In Section 3.6, several numerical examples demonstrate the robustness and wide applicability of the derived algorithm.

27 Chapter 3. Quasi-static dual mortar with CCL

3.1. Introduction of the dual Lagrange multiplier

As mentioned in Section 2.5.1, the Lagrange multiplier λ is deﬁned as the negative contact traction on the slave side

λ := λ(X, t) = ts(X, t), (X, t) Γs [0,T ] (3.1) − c ∈ c × leading to an alternation of Equation (2.41). With the deﬁnition of λ, the normal contact conditions read as Gn 0 λn 0 λn Gn = 0, (3.2) ≤ ∧ ≥ ∧ · and the tangential contact conditions become

λτ Φ(λn, λτ ) 0 [u˙ ]τ =γ ˙ γ˙ 0 γ˙ Φ(λn, λτ ) = 0. (3.3) ≤ ∧ λτ ∧ ≥ ∧ · k k One recalls the weak formulation of the momentum equation in the quasi-static mortar framework δΠint,ext(u, δu) + δΠc(λ, δu) = 0, (3.4) with Z Z Z ) X α α α¯α α α α δΠint,ext = S δE dV ρ0 b δu dV ¯t δu dS , (3.5) Ωα · − Ωα − Γα α∈{s,m} N as a reformulation of Equation 2.43 and the weak contact forces Z δΠc = λ [δu] dS. (3.6) s Γc

Since only quasi-static calculations are considered in this section, the term modelling the inertia forces is zero, and the time interval [0,T ] is split into a sequence of small load increments [ti, ti+1]. It follows a short functional analytical classiﬁcation of the utilized function spaces, for further considerations see [143, 65]. One recalls that the displacements uα and the test functions for the displacements δuα are lying in the Sobolev-space

α n α 1 α 3 α o V0 := δu H (Ω ) : u Γα = 0 , α = s, m (3.7) ∈ | D { } On the boundary ∂Ωs the displacements us are lying in the trace space of [H1(Ωs)]3, namely 3 h 1/2 s i W := H (ΓC ) . (3.8)

28 3.2. Regularization of the contact conditions

Since the dual Lagrange framework is utilized, the Lagrange multiplier λ is set to be in the dual space of W , M := W −1, in order to employ the duality pairing Z η, ν = η ν dS, η W ν M (3.9) h i s ∈ ∈ ΓC in the discrete weak formulation. More precisely, the space for the dual Lagrange multiplier λ considering friction is a closed convex cone of M, namely

M(λ) := ν M :< η, ν > < η , µλn > η W with ηn 0 , (3.10) { ∈ ≤ k τ k ∈ ≤ } see [49]. Using the duality pairing (3.9), Equation (3.6) can be written as

δΠC = [δu] , λ . (3.11) h i For comments on a priori error estimator of the used method, the interested reader is referred to [69, 66, 147]. Since in general a non-linear geometrical setting is assumed, the weak internal and α external forces Πint,ext depend non-linearly on the displacements u and a Newton– Raphson iteration has to be performed in the discrete setting using the linearization

α α α α α α α α L δΠ α = δΠ (u¯ , δu ) + DδΠ (u¯ , δu ) ∆u . (3.12) int,ext uα=u¯ int,ext int,ext · In this work a detailed descriptions of the Newton-Raphson procedure and the derivation of the linearization are skipped. For details see [148, 139, 30].

3.2. Regularization of the contact conditions for microscopic rough surfaces

3.2.1. Normal contact conditions

Now one can have a closer look at the normal contact condition employed for the dual mortar formulation. In most publications on mortar methods for contact problems hard contact is assumed [68, 113, 57, 36]. In contrast to that the dual mortar formulation is adapted here in order to use constitutive contact laws on the contact interfaces. Motivated by Section 2.3.1, one introduces a normal contact law with a data driven regularization of the normal contact switching from a classical dual Lagrange method to a perturbed Lagrange method:

c c Gn(u) G (λn) 0 λn 0 λn (Gn(u) G (λn)) = 0, (3.13) − n ≤ ∧ ≥ ∧ · − n c see Figure 3.1. The regularization term Gn(λn) models the transition from soft to c hard contact as described in Section 2.3.1. In this context capital Gn(λn) denotes the

29 Chapter 3. Quasi-static dual mortar with CCL

clp cmlp cnlp (a) Linear gn (b) Piecewise linear gn (c) Exponential gn

Figure 3.1.: Contact condition for normal contact with regularizations

−1/2 s operator taking an argument of the dual function space H (Γc) and mapping it to a 1/2 s c regularization term for the displacements lying in H (Γc). In contrast to that gn(λn) denotes a scalar function taking a discrete value for the contact pressure in one point and mapping it to a scalar regularization term for the displacements. In the discrete c setting Gn(λn) is deﬁned point-wise as   ns ns c X X c Gn  λpn ψp = gn(λpn) φp, (3.14) · · p=1 p=1 with φp being the standard basis function for the displacements and ψp being the dual basis function utilized for the Lagrange multiplier, see Section 3.3. c In this work, three different choices for the regularization function gn(λn) are considered, a simple linear, a piecewise linear and a non-linear regularization. The starting point for the linear regularization is the perturbed Lagrange formulation (Section 2.5.3) using the regularization term

clp 1 gn (λn) = λn (3.15) an see Figure 3.1a. This is the complementary term to the penalty formulation (Section 2.5.2, see Equation (2.56)) and the easiest possible regularization. As a model for the transition from soft to hard contact caused by the asperities on the micro-scale, the perturbed Lagrange formulation is adapted and a piecewise linear regularization function cmlp 1 gn (λn) = min λn + ti (3.16) i=1,..,np ani is introduced, see Figure 3.1b. It is assumed that the derivative of the function is 1 1 1 1 monotonically decreasing a a a a 0 since the deformation is purely n1 ≥ n2 ≥ n3 ≥ n4 ≥

30 3.2. Regularization of the contact conditions elastic. With this piecewise linear regularization one can either approximate experimentally measured contact behaviour or numerical calculated contact behaviour based on roughness measurements as with the modiﬁed BGT-model by WILLNER [141]. In terms of displacements, the derived constitutive contact law for the BGT-model model reads as 2 1 (gr0 gn(u)) p(u) = C1 exp − (3.17) gr gn(u) · − C2 0 − with gr0 being the mean distance between the two contacting rough surfaces without loading. C1 and C2 are constants depending on the measured surface roughness and the material parameters of the two contacting bodies. Since this function has no trivial closed inverse, it is approximated here with discrete data points and a piecewise linear interpolation. One could also use splines of higher order, but these weren’t considered in this work. For the sake of completeness also an exponential regularization that is often utilized in the literature is presented

gn(u) p(u) = A exp A, (3.18) · β − see e.g. [76]. This function can be considered as a simpliﬁcation of Equation (3.17). Reformulating (3.18) in terms of Lagrange multipliers, one ends up with

λn+A cnlp β ln( A ) λn > 0 gn (λn) = · β , (3.19) λn λn 0 A · ≤ for the discrete regularization, see Figure 3.1c. It has to be noted, that a constant term of A was added to the classical exponential approach to get a continuous transition − between inactive and active nodes. Additionally, the logarithm function was replaced by λn+A it’s linearization around the origin for negative values of λn, since β ln( ) is only · A deﬁned for λn ] A, [. ∈ − ∞ As a next step one switches to the incremental form of the non-penetration condition, which is a linearization around the current conﬁguration at the start of the load increment, assuming small deformations per load increment:

c c [u] G (λn) g0 λn 0 λn ([u] G (λn) g0) = 0, (3.20) n − n ≤ ∧ ≥ ∧ · n − n − with g0 being the normal gap between the two bodies at the start of the load increment and recalling the deﬁnition of the jump in the displacement [u] := us(t, X) m − u (t, Pt(X)).

31 Chapter 3. Quasi-static dual mortar with CCL

3.2.2. Tangential contact conditions

Now that the normal contact conditions are described in case of rough surfaces, one can have a closer look at the used tangential contact laws. So far a perfect Coulomb law has been considered in most publications using mortar formulation [68, 67, 81, 44]. More complicated friction models like thermomechanically coupled friction were explored in [71, 132]. The full Reynolds approximation for lubricated contact has been considered in [152]. But for use in structural dynamics it is important to model the transition between sticking and slipping, also called micro slip, in a physical correct way [116, 138]. With this smooth friction model one can reproduce measured frictional damping [142], like for example in the bolted connection in the casing of a jet turbine. Here one starts with a simple linear regularization modelling the elastic stick region and then employ the Iwan model for the micro slip, see Figure 2.6.

Figure 3.2.: Coulomb with linear regularization

(a) Micro slip model (b) Iwan model

Figure 3.3.: Scale dependent friction model realized by a serial-parallel Iwan model

32 3.2. Regularization of the contact conditions

Linear regularization

One starts with the perfect Coulomb law (2.35) and adds a regularization term depending on λτ to the stick branch, see Figure 3.2:

1 ˙ λτ Φ(λn, λτ ) := λτ b 0 [u˙ ]τ λτ =γ ˙ γ˙ 0 γ˙ Φ = 0, (3.21) k k − ≤ ∧ − aτ λτ ∧ ≥ ∧ · k k with aτ being the tangential stiffness also called stick slope and the friction bound b = µ λn. · u−u¯ Using a backward Euler method for the displacements u˙ = dt and the Lagrange ¯ ˙ λ−λ ¯ multiplier λ = dt with (u¯, λ) being the solution of the last load-increment and dt being the load increment, one gets

1 ¯ λτ Φ(λn, λτ ) := λτ b 0 ([u]τ [u¯]τ ) (λτ λτ ) = (γ γ¯) , k k − ≤ ∧ − − aτ − − λτ (3.22) k k (γ γ¯) 0 (γ γ¯) Φ = 0. − ≥ ∧ − · It has to be noted, that the term 1/dt can be neglected due to the linear relation in the stick branch.

Iwan regularization

Now the concepts from the last paragraph are expanded in order to construct a Coulomb law with elastic stick and micro slip (Figure 3.3a). The main idea is that the Coulomb law with a linear regularization can be regarded point-wise as an elastic–slip element consisting of a spring with spring constant k = aτ and a slider element with sliding bound b = µ λn in series. This elastic–slip element is also called Iwan element in liter- · ature, see e.g. [75]. The smooth transition from sticking to slipping is then realized by parallel arrangement of niwan of these elastic-slip elements, see Figure 3.3b, resulting in a serial-parallel Iwan model. In the dual Mortar framework, a separate local Lagrange i multiplier λτ is introduced for every Iwan element i satisfying i i i i i niwan i ¯i i i λτ Φ (λn, λτ ) := λτ b 0 ([u]τ [u¯]τ ) (λτ λτ ) = (γ γ¯ ) i , k k − ≤ ∧ − − aτ − − λ k τ k (γi γ¯i) 0 (γi γ¯i) Φi = 0. − ≥ ∧ − · (3.23)

The same aτ is used for every spring, but different friction bounds are chosen, satisfying Pniwan i i i i i b = i=1 b . In this work, one sets b = α b = α µλn with α = 2i/niwan(niwan + 1). The global Lagrange multiplier is ﬁnally assembled by

niwan ! X i λ = λn n + λ τ . (3.24) · τ · i=1

33 Chapter 3. Quasi-static dual mortar with CCL

It has to be noted, that the serial-parallel Iwan model is not the only possibility to model the smooth transition between sticking and slipping numerically. ODEN and co- workers [102, 103, 116] for example suggested a square-root regularization

[u˙ ]τ λτ = µλn q (3.25) T 2 [u˙ ]τ [u˙ ]τ + with > 0. This regularization works ﬁne for the initial tangential loading. Unfortunately it is quite complex to guarantee the Masing rules in case of load reversal keeping track of all points of load reversal. Using the serial-parallel Iwan model, the Masing rules are automatically guaranteed in every point even for complex loading scenarios. Now that all contact conditions have been described, the weak conditions can be stated.

3.2.3. Weak formulation of the contact conditions and summary of the weak problem

The weak formulation for the regularized contact conditions in normal and tangential direction is now derived from the quasi-variational inequality formulated in [79]: Find λ M(λ) such that ∈ c [u]n Gn(λn), δλn λn + h c − − i (3.26) ([u]τ [u¯]τ ) G (λτ λ¯τ ), δλτ λτ g0, δλn λn , δλ M(λ). h − − τ − − i ≤ h − i ∈ Here the linearized and regularized version of the normal contact condition is utilized c 1 as well as the linear regularized tangential contact law with G (λτ λ¯τ ) = (λτ λ¯τ ). τ − aτ − Following the same arguments as in [65], it can be shown that the weak form (3.26) fulﬁls (3.20) and (3.22) in a weak sense. In case of the Iwan model with niwan Iwan elements for the tangential contact law, (3.26) reads as ﬁnd λ M(λ) such that ∈ c [u]n G (λn), δλn λn + h − n − i niwan X c,niwan i i i (3.27) ([u]τ [u¯]τ ) G (λ λ¯ ), δλτ λ g0, δλn λn , h − − τ τ − τ − τ i ≤ h − i i=1

c,niwan niwan δλ M(λ) with Gτ ( ) = ( ). The full weak problem can then be summarized ∈ · aτ · as ﬁnd (u, λ) satisfying

δΠint,ext(u, δu) + [δu] , λ = 0 c h i [u]n G (λn), δλn λn + h − n − i niwan (3.28) X c,niwan i i i ([u]τ [u¯]τ ) G (λ λ¯ ), δλτ λ g0, δλn λn . h − − τ τ − τ − τ i ≤ h − i i=1

34 3.3. Discretization

Figure 3.4.: Cuboid surface element in reference square and in spatial coordinates

3.3. Discretization

Now that the full weak problem is known, the domain Ωα is dicretized by a finite element α mesh Ωh and the unknown functions are discretized by a finite basis. Here the standard isogeometric finite elements are employed. Thus one considers meshes consisting of Hex8, T et4 and W edge6 elements. In Chapter 4 also Hex20, T et10 and W edge15 elements are considered. The displacements uα as well as their test functions δuα are discretized by the standard basis functions φp

α α nall nall α X α α X α u = φp u , δu = φp δu , (3.29) h · p h · p p=1 p=1

α 3 with n being the number of finite Element nodes in body α s, m and up R all ∈ { } ∈ being the displacement vector for node p. Since the main focus of this work lies in the contact interaction, one has a closer look on the discretization on the boundary of the domain. The basis functions are defined as φp ∂Ω = φp(ξ, η) with (ξ, η) being | the local coordinates in the reference element inheriting its definition from its volumetric counterpart φp = φp(ξ, η, ζ). As an example the standard basis functions for one Q4 surface element stemming from a Hex8 volume element defined on the reference square ( 1, 1) ( 1, 1) are shown, see Figure 3.4: − × − Q4 Q4 φ1 (ξ, η) = (1 ξ)(1 η)/4, φ2 (ξ, η) = (1 + ξ)(1 η)/4, − − − (3.30) φQ4(ξ, η) = (1 + ξ)(1 + η)/4, φQ4(ξ, η) = (1 ξ)(1 + η)/4. 3 4 − The nodes of the finite Element mesh can be split up into three disjoint sets, the Slave nodes = p Γs , the master nodes = p Γm and all other nodes . It S { ∈ c,h} M { ∈ c,h} N holds = ns and = nm. |S| |M| The Lagrange multiplier λ and the test function for the Lagrange multiplier δλ are

35 Chapter 3. Quasi-static dual mortar with CCL discretized with dual basis functions n n Xs Xs λh = ψp λp, δλh = ψp δλp. (3.31) · · p=1 p=1 As defined in the previous section, the space for the Lagrange multiplier is defined to be the dual space of the trace space for the displacement. Therefore the dual basis functions ψp are constructed in the discrete setting as a linear combination of the standard basis functions satisfying the biorthogonality condition Z Z ψq φp dS = δpq φp dS, p, q (3.32) s s γc · γc ∈ S in the current configuration. The construction of the dual basis functions is shown exemplary for one Q4 surface element. It holds 4 Q4 X Q4 Q4 Q4 Q4 ψ = aij φ , i = 1, ..., 4 ψ = A φ . (3.33) i · j ⇒ j=1 Inserting (3.33) in (3.32) results in the linear system AQ4 ZQ4 = W Q4, (3.34) · Q4 R Q4 Q4 Q4 R Q4 with Z [i, j] = T (Q4) φi φj dS and W [i, j] = δij T (Q4) φi dS, i, j = 1, ..., 4. For the undeformed reference element, AQ4 results in  4 2 1 2 − − Q4  2 4 2 1  A = − −  . (3.35)  1 2 4 2 − − 2 1 2 4 − − Q4 See Figure 3.5 for a visualization of ψ1 in the reference element.

3.3.1. Balance equation

h s,h s s m,h m m The term δΠint/ext(uh, δuh) := (δΠint/ext(uh, δuh), δΠint/ext(uh , δuh )) for the virtual internal and external work can be discretized with the standard basis functions. One recalls that there is no coupling between the master and the slave side for this term except if one considers self contact. To solve the non-linear discrete equation due to material non-linearities or non-linear strain calculation, a Newton–Raphson algorithm is used, resulting in a linear system h k k DδΠint/ext ∆u = r , · (3.36) uk+1 = uk + ∆uk,

36 3.3. Discretization

Ψ1(ξ, η ) 4

2 Φ1(ξ, η ) 1

1 − 2 1.0 − 0.5 1.0 0.0 0.5 0.0 0.5 − η − 1.0 1.0 0.5 − − ξ

Figure 3.5.: Standard basis function in blue and dual basis function in red for one node of a Q4 surface

k in every Newton iteration with DδΠint/ext = DδΠint/ext(u ) also known as the tangent matrix K and the residuum rk. For details, the interrested reader is referred to the monographs [139, 90, 148]. To get to the discretized term for the contact forces, the test functions for the displacements δu are discretized again with standard basis functions and the Lagrange multipliers λ with dual basis functions:

h δΠc = [δuh] , λh h  i  Z ns ms ns ! X s s X m m X (3.37) =  φj δuj φk δuk  ψi λi dS. s · − · · γc j=1 k=1 i=1

For a point p this can be reformulated using (3.32): ∈ S Z ! ( nm Z ! ) h s T X m T m δΠc p = (δup) ψp φp dS I3 λp (δuk ) ψp φk dS I3 λp . | s · − s · Γc k=1 Γc (3.38) Deﬁning the Mortar matrices as Z Z D[p, q] = ψp φq dS I3 = δpq φp dS I3 p, q , s s Γc · Γc ∈ S Z (3.39) B[p, q] = ψp (φq Pt) dS I3 p S, q , s − Γc · ◦ ∈ ∈ M the discrete version of the virtual contact work can be stated as

h T T T T δΠc = (δuS ) D λ + (δuM) B λ. (3.40)

37 Chapter 3. Quasi-static dual mortar with CCL

Hence the virtual contact work couples the slave body with the master body. In total one solves DδΠh ∆uk + δΠh λk+1 = rk (3.41) int/ext · c · | {z } |{z} K [BD]T in every Newton–Raphson iteration. It has to be noted that the coupling matrices are deformation dependent in the fully non-linear setting D = D(u) and B = B(u). In order to avoid the complex derivation of the fully deformation dependent solution here, a combined ﬁx-point and Newton approach is employed setting D = D(u¯) and B = B(u¯) with u¯ being the solution of the last load increment. From the numerical point of view, this is suboptimal, i.e. the super-linear convergence rate of a global semi- smooth Newton method is lost. However other system non-linearities (plasticity and friction) need small load increments to resolve the path dependencies and thus one has a good start iterate in each iteration step. Moreover, this allows more easily to exploit the full advantage of the dual Lagrange multiplier concept since only then the matrix D is diagonal and no Jacobian has to be taken into account. The same approach is followed for the calculation of the normals and tangents at the slave nodes n = n(u¯) and τ = τ (u¯). This formulation has the advantage of having to calculate the coupling of the master and slave surface and evaluating the discrete gap only once per increment. In the fully deformation dependent formulation, one would get

∂ ∂ ∂ δΠh ∆uk + δΠh ∆uk + δΠh λk+1 = rk, (3.42) ∂u int/ext · ∂u c · ∂λ c · see [108, 44].

3.3.2. Normal contact conditions

Now can have a closer look at the discretization of the weak contact conditions in normal direction. Again, the displacements u are discretized with standard basis functions and Lagrange multiplier λ as well as its test function with dual basis functions. The regularization term for the contact law is discretized with standard basis functions. This choice is motivated by the fact that one wants to preserve biorthogonality in the alge- −1/2 s braic system. In the variational setting it holds λn H (Γ ). Consequently one ∈ c needs a regularization operator Gc : H−1/2(Γs) H1/2(Γs) to utilize the duality pairing n c → c < , > in the variation. In the continuous setting, this is employed with mollifiers like in · · c 1/2 s the monograph by ECK [33] in order to get G (λn) H (Γ ). In the discrete setting, n ∈ c the action of the mollifier on λn are mimicked by a change in the basis. More precisely, c Pns Pns c one defines G λpn ψp = g (λpn) φp and thus increases the regularity. n p=1 · p=1 n · This step can be seen as a mass lumping similar to [71].

38 3.3. Discretization

Here one focuses on the discrete version of the non-penetration condition included in Equation (3.26):   Z ns ! ns nm ns X X s s X m m X c ψi δλin  φj ujn φk ukn gn (λjn) φj dS s · · · − · − · ≤ Γc i=1 j=1 j=1 k=1 (3.43) n ! Z Xs ψi δλin g0 dS. s · Γc i=1

For a point p S this can be expressed as ∈   nm s X m m  δλpn < ψp, φp > upn δλpn < ψp, φq > ukn −   q=1 (3.44) c δλpn < ψp, φp > gn(λpn) δλpn < ψp, g0 >, − ≤ | {z } gp using the assumption, that the normal does not change much during the load increment i i+1 t t , n = n0. Due to the duality pairing, the regularization term reduces to a → diagonal matrix. With the mortar matrices D and B (3.39) the non-penetration condition in the discrete setting can be expressed as

c N (D uS + B uM) D g (λ) g (3.45) − n ≤ c c T with (g (λ))p = g (λnp) and N[p, p] = n . Introducing the transformation uS uˆS , n n →

uˆS := D uS + B uM (3.46)

[145], switching from the displacements to the relative displacements between master and slave side for all slave nodes and deﬁning λs = Dλ, one gets point-wise decoupled contact constraints:

c c uˆpn g (λspn) gp λspn 0 λspn (ˆupn g (λspn) gp) = 0. (3.47) − n ≤ ∧ ≥ ∧ · − n − Equations (3.47) can be reformulated as the projection operator

c λspn = max 0, λspn + c (ˆupn g (λspn) gp) . (3.48) { − n − } with the numerical constant c > 0 similar to [62]. It has to be noted, that for a non-linear cnlp cnlp c c regularization D[p, p]gn (λpn) = gn (λspn). Thus gn(λspn) is deﬁned as gn(λspn) := c 6 D[p, p]gn(λpn).

39 Chapter 3. Quasi-static dual mortar with CCL

3.3.3. Tangential contact conditions

Linear regularization

Performing the same discretization for the weak tangential contact conditions with a linear regularization, this leads to point-wise decoupled contact constraints in tangential direction for each slave node p S: ∈  λspτ bp 0 k k − ≤ λ b < 0 u˜ 1 λ˜ = 0 spτ p pτ aτ spτ (3.49) k k − ⇒ k − 1 · k  λspτ bp = 0 u˜pτ λ˜spτ > 0 k k − ⇒ k − aτ · k with the point-wise friction bound bp = µλspn, deﬁning u˜pτ := uˆpτ u¯ˆ pτ , λ˜spτ := λspτ − − λ¯spτ and using the well known basis transformation u uˆ (3.46). Equation (3.49) can → be reformulated as a projection operator :

1 λspτ + c u˜pτ λ˜spτ · − aτ λspτ = bp (3.50) n 1 o max bp, λspτ + c u˜pτ λ˜spτ · − aτ similar to [20, 19]. Here one can use different numerical constants for the normal and the tangential contact laws. For a low tangent stiffness aτ , c should be set to aτ . In this case the projection operator looses its non-linearity in λ. The same is achieved by choosing the simple radial return mapping

λ¯spτ + aτ u˜pτ λspτ = bp (3.51) λ¯spτ + aτ u˜pτ as the projection operator for the slip case.

Iwan regularization

Performing the discretization analogue to the linear regularization, but using the sum over all Iwan elements for the tangential part instead of one Iwan element, one ends up with point-wise decoupled contact constraints in tangential direction for every Iwan element i:  i i λspτ bp 0 k k − ≤ i λi bi < 0 u˜ niwan λ˜ = 0 spτ p pτ aτ spτ (3.52) k k − ⇒ k − i k  i i niwan  λ b = 0 u˜pτ λ˜ > 0 k spτ k − p ⇒ k − aτ spτ k i i with b = α bp. This is equivalent to applying the projection operator p · i aτ λ¯ + u˜pτ i i spτ niwan λspτ = α bp n · o (3.53) i i aτ max α bp, λ¯ + u˜pτ spτ niwan ·

40 3.4. Semi-smooth Newton for contact equations

for every Iwan element i setting c = aτ in (3.50), see (3.51). The simpler projection operator depending only linearly on λ is utilized here deliberately for the Iwan elements since the case of aτ = makes no sense when modelling micro slip. ∞

3.4. Semi-smooth Newton for contact equations

Now that the whole weak problem is discretized, one has to deal with the discretized contact inequalities. In combination with the dual mortar method, a natural choice is using a semi-smooth Newton method [1, 62]. Here the contact inequalities are reformulated point-wise as a, in general semi-smooth, projection operator

λ = P (u, λ).

The projection operator can than be reformulated as a non-linear complementary function C(u, λ) = 0 for which a Newton method can be derived using the concept of the generalized derivative. The non-linear complementary function then separates the nodes into active and inactive nodes in every Newton iteration. Thus the contact conditions will be fulﬁlled in a weak sense only in the limit case and not in every Newton–Raphson iteration like it would be the case for an active-set strategy. But one ends up with one iteration loop handling all system non-linearities (geometrical, material and contact). In contrast to the Newton–Raphson for the balance equation, the contact Newton has only a super-linear convergence rate. For further details on semi-smooth Newton methods, the interested reader is referred to [74, 114, 115].

3.4.1. Normal direction

Starting with the projection operator (3.48), one can easily construct a non-linear complementary function and derive a semi-smooth Newton similar to [65]. For the data driven regularized contact law in normal direction, one obtains the non-linear complementary function

P c C (uˆp, λsp) := λspn max 0, λspn + c (ˆupn g (λspn) gp) . (3.54) n − { − n − } P In order to derive the semi-smooth Newton, Cn has to be differentiated

P DCn (uˆp, λsp)(∆uˆp, ∆λsp) = ∆λspn c (3.55) A (uˆp, λsp) (∆λspn + c (∆ˆupn g (λspn)∆λspn)) − X n − ∇ n

41 Chapter 3. Quasi-static dual mortar with CCL

with A being the characteristic function based on the set X n c n := p S : λspn + c (ûpn g (λspn) gp) > 0 (3.56) A { ∈ − n − } defined as c 1 if λspn + c (ûpn gn(λspn) gp) > 0 An (uˆp, λsp) := − c − . (3.57) X 0 if λspn + c (ûpn g (λspn) gp) 0 − n − ≤ k The function A separates the nodes p S into the active set and the inactive set X n ∈ An k in every Newton iteration k, namely In k n k k c k o n := p S : λspn + c uˆpn gn(λspn) gp > 0 , A ∈ − − (3.58) k n k k c k o := p S : λ + c uˆ g (λ ) gp 0 . In ∈ spn pn − n spn − ≤ k k Given the previous iterate (up, λp), one obtains the new iterate k+1 k+1 k k k k (uˆp , λsp ) = (uˆp, λsp) + (∆uˆp, ∆λsp) by solving the Newton-equation DCP (uˆk, λk )(∆uˆk, ∆λk ) = CP (uˆk, λk ). (3.59) n p sp p sp − n p sp k k+1 A straightforward computation of (3.59) for (∆up, λp ) results in: Active nodes: • k c k k+1 k c k c k k ∆û g (λ ) λ = gn uˆ + g (λ ) g (λ ) λ . (3.60) pn − ∇ n spn · spn − pn n spn − ∇ n spn · spn Instead of a Dirichlet boundary condition as in the case of hard contact, one ends up with a Robin boundary condition in the normal direction. Writing (3.60) in a short form, one gets T k k+1 n uˆ g λ = hpn (3.61) p − np sp with g = gc (λk )nT (3.62) np ∇ n spn and k c k c k k hpn = gn uˆ + g (λ ) g (λ ) λ (3.63) − pn n spn − ∇ n spn · spn Inactive nodes: • k+1 λspn = 0. (3.64) For inactive nodes one has the same Neumann condition as in the case of hard contact. The equations resulting from the tangential contact constraints completing the algebraic system are treated in the next section.

42 3.4. Semi-smooth Newton for contact equations

3.4.2. Tangential direction

Linear regularization

Analogous to the normal contact conditions, one can easily construct a non-linear complementary function from the projection operator (3.50) and derive a semi-smooth New- ton similar to that in [68]. For the Coulomb law with elastic stick one obtains: P 1 ˜ Cτ (λsp, uˆp) = max bp, λspτ + c u˜pτ λspτ λspτ · − aτ · (3.65) 1 bp λspτ + c u˜pτ λ˜spτ − · · − aτ with cpl/mpl bp = µ max 0, λspn + c(ûpn gp gn (λspn)) . (3.66) · { − − } Defining bp in the semi-smooth Newton as above, the non-linear complementary function in normal direction is reflected and a better numerical convergence is achieved. cpl/mpl With gn one denotes again the regularization functions for the normal contact defined in Section 3.2.1. Here we restrict our considerations to a linear or a piece-wise linear regularization of the normal contact, since this covers most of the relevant cases. The derived Newton for an exponential regularization of the normal contact would be c slightly different, since the linearity of gn is employed in the derivation. Differentiating (3.65), one gets

P DCτ (uˆp, λsp)(∆uˆp, ∆λsp) = T ! λspτ (dp) −1 sl Iτ ∪A (uˆp, λsp) dp ∆λspτ + ∆λspτ + c ∆uˆpτ aτ ∆λspτ X τ k k · dp − k k −1 st (uˆp, λsp) cbp ∆uˆpτ a ∆λspτ + −XAτ − τ −1 c +cµ(u˜pτ a λ˜spτ ) (∆λspn + c (∆ûpn g (λspn) ∆λspn)) − τ − ∇ n −1 + sl (uˆp, λsp) bp ∆λspτ + c ∆uˆpτ a ∆λspτ + XAτ − τ c +µdp (∆λspn + c (∆ûpn g (λspn)∆λspn))) , − ∇ n (3.67) with 1 dp = λspτ + c u˜pτ λ˜spτ , (3.68) · − aτ and the characteristic functions defined on the sets

Inactive nodes τ = p n , • I { ∈ I }

43 Chapter 3. Quasi-static dual mortar with CCL

st n 1 o Sticking nodes = p n : λspτ + c u˜pτ λ˜spτ < bp , • Aτ ∈ A · − aτ

sl n 1 o Slipping nodes = p n : λspτ + c u˜pτ λ˜spτ bp 0 . • Aτ ∈ A · − aτ ≥ ≥ Like in the complementary function for the normal constraints, the max function in (3.65) deﬁnes a characteristic function separating the sticking nodes from the slipping nodes, while the max function in (3.66) reﬂects the active and inactive sets in normal direction. k k k+1 k+1 Given the previous iterate (uˆp, λsp), one obtains the new iterate (uˆp , λsp ) by solving the Newton-equation

DCP (uˆk, λk )(∆uˆk, ∆λk ) = CP (uˆk, λk ) (3.69) τ p sp p sp − τ p sp using the three disjoint sets: n o k = p k , Iτ ∈ In st,k k k k 1 ˜k k τ = p n : λspτ + c u˜pτ λspτ < bp , (3.70) A ∈ A · − aτ sl,k k k k 1 ˜k k τ = p n : λspτ + c u˜pτ λspτ bp 0 . A ∈ A · − aτ ≥ ≥

k k+1 Again a computation of (3.69) for (∆uˆp, λsp ) results in the equality condition:

Inactive nodes: • k+1 λspτ = 0. (3.71)

The choice for λspτ is not a rigorous consequence of the Newton iteration, but di- P 1 rectly derived from C (uˆp, λsp) = λspτ + c u˜spτ λ˜spτ λspτ = 0. Another τ · − aτ possibility to derive the equality constraint for the inactive nodes is to take the projection operator from [20],     bp 1 ˜ λspτ = min , 1 λspτ + c u˜pτ λspτ , 1 · · − a  λspτ + c u˜spτ λ˜spτ  τ · − aτ and construct the alternative non-linear complementary function P ∗ 1 ˜ Cτ (uˆp, λsp) = λspτ min bp/ λspτ + c u˜pτ λspτ , 1 − · − aτ 1 λspτ + c u˜pτ λ˜spτ . · · − aτ

44 3.4. Semi-smooth Newton for contact equations

Sticking nodes: • k 1 k+1 k 1 ˜k k k+1 1 k ∆uˆpτ λspτ + µ u˜pτ λspτ /bp λspn = λspτ . − aτ · − aτ · −aτ (3.72) | {z } st hpτ

Slipping nodes: • k+1 k k+1 sl λ + Lp ∆uˆ + µvp λ = h , (3.73) − spτ · pτ · spn pτ with

k bp ep = , dp k k k T λspτ (dp) F p = k , M p = ep (I2 F p), bp dp · − k k c −1 Hp = I2 1 M p, Lp = cHp M p, − − aτ sl 1 1 k −1 dp hpτ = Lp λspτ , vp = Hp . c − aτ · dp k k sl Here vp and hpτ are two-dimensional vectors, ep is a scalar and F p, M p, Hp and Lp are 2 2 matrices. Since all of these quantities can be built point-wise, the × computational costs can be neglected.

Equations (3.73) and (3.73) can be summed up as

k+1 k Rpτ λ + Lpτ ∆uˆ = hpτ (3.74) − sp · p with ( 1 T k ˜k T st,k a τ µ(u˜pτ 1/aτ λspτ )/bp n p τ Rpτ = τ − − · ∈ A (3.75) T T sl,k τ µvpn p τ − ∈ A and ( T st,k ( st st,k τ p τ hpτ p τ Lpτ = T ∈ Asl,k , hpτ = sl ∈ Asl,k (3.76) Lpτ p τ h p τ ∈ A pτ ∈ A for all active slave nodes p k . In case of no friction, that means µ = 0, the system ∈ An reduces to T Rpτ = τ , Lpτ = 0, hpτ = 0. (3.77)

45 Chapter 3. Quasi-static dual mortar with CCL

Within the semi-smooth Newton for the slipping branch it has to be assured that the k k matrix Hp is invertible for each (uˆp, λsp). Therefore a modifications to the Robin system is made based on [68, 2, 17, 63]. Here the matrix F p is replaced by T λspτ d F˜ p = (3.78) max bp, λspτ d { k k}k k leading to M˜ p. Furthermore Hp is replaced by c H˜ p = I2 βp 1 M˜ p (3.79) − − aτ using λT d λ 1 α < 0 spτ spτ 1−αpγp if αp = , δp = min k k, 1 , βp = (3.80) λspτ d bp 1 otherwise k k · k k leading to L˜ p, h˜p and v˜p. This modification of the Robin system yields positive eigenvalues of H˜ p and L˜ p following the same argument as in [68]. In [68] two other possibilities for the modification of the system are suggested, but since these modifications performed worse from the numerical point of view, they are neglected here.

Iwan model

Starting with Equation (3.53), a non-linear complementary function is constructed for every Iwan Element i:

P,i i n i i o i C (λ , uˆp) = max α bp, λ¯ + aτ /niwan u˜pτ λ τ sp spτ · · spτ i i α bp λ¯ + aτ /niwan u˜pτ − spτ · cpl/mpl P,i with bp = µ max 0, λspn + c(ûpn gp gn (λspn)) . Differentiating Cτ , one gets · { − − } P,i i i DCτ (uˆp, λsp)(∆uˆp, ∆λsp) = i T ! i i λspτ dp aτ i sl,i uˆp, λsp dp λspτ + ∆uˆpτ Iτ ∪A i X tau · dp · niwan i i aτ Ast,i uˆp, λsp α bp ∆λspτ ∆uˆpτ + − X τ − niwan (3.81) i ˜ aτ c +α λspτ u˜pτ (∆λspn + c (∆ûpn gn(λspn)∆λspn)) − niwan · − ∇ i i aτ Asl,i uˆp, λsp α bp ∆uˆpτ + − X τ · niwan i i c + µα d (∆λspn + c (∆ûpn g (λspn)∆λspn)) p − ∇ n

46 3.4. Semi-smooth Newton for contact equations with i ¯i aτ dp = λspτ + u˜pτ (3.82) niwan decomposing the slave nodes into three sets i Inactive nodes = p n , • Iτ { ∈ I } st,i n i o Sticking nodes τ = p n : λ¯ + aτ /niwan u˜pτ < bp , • A ∈ A spτ · sl,i n i o Slipping nodes τ = p n : λ¯ + aτ /niwan u˜pτ bp 0 . • A ∈ A spτ · ≥ ≥ for each Iwan element i. As one can easily see, the choice whether a node is active depends on the normal component of the global Lagrange multiplier and the choice whether an active node is sticking or slipping depends on the local tangential part of the Lagrange multiplier for the Iwan element i. Now the computation of the equality constraints for the Iwan element i is straightforward: Inactive: • i,k+1 λspτ = 0

Sticking: • ˜i,k k λspτ aτ /niwan u˜pτ i,k+1 − · k+1 aτ k i,k λspτ µ λspn + ∆uˆpτ = λspτ (3.83) − − bp niwan −

Slipping: • i dp λi,k+1 + µαi λk+1 + Li ∆uˆk = 0 (3.84) − spτ di · spn p · pτ k pk with

i i,k i,T α bp λspτ d ei = , F i = , p i p i i d α bp d k k || || i i i i i aτ M p = ep(I2 F p), Lp = M p . − niwan In contrast to the previous case, here no modiﬁcation of the Robin system is necessary. Before one can condense the global Lagrange multiplier from the algebraic system, one has to calculate the equality constraints for the global Lagrange multiplier. Having i the equality constraints for every λspτ , the global constraint is calculated as the sum over all Iwan elements (3.24). In total one gets:

47 Chapter 3. Quasi-static dual mortar with CCL

Inactive nodes: • k+1 λspτ = 0. (3.85)

Active nodes (sum over all Iwan elements): • k+1 k Rpτ λ + Lpτ ∆uˆ = hpτ (3.86) − sp · p with

nslip niwan X i T X aτ T Lpτ = Lpτ + τ niwan i=1 i=nslip+1 n i,k slip i niwan ˜ k T X d i T X λspτ aτ /niwan u˜pτ T Rpτ = τ i µα n + µ − · n (3.87) − d bp i=1 k k i=nslip+1 niwan X i,k hpτ = λ − spτ i=nslip+1

with nslip being the number of slipping Iwan elements per node.

For the global Lagrange multiplier, the active nodes are no longer separated into sticking and slipping, because each node can be in niwan + 1 different states.

3.5. Full algebraic linear system and condensation of the Lagrange multiplier

Now that the problem is discretized and the the weak contact inequalities are treated with a semi-smooth Newton method, the problem can be summed up by: Find (uh, λh) satisfying

h h δΠint,ext(δuh, uh) + δΠc (δuh, λh) = 0 P Cn (uh, λh) = 0 (3.88) P Cτ (uh, λh) = 0

k k iteratively by a semi-smooth Newton/Newton–Raphson algorithm for (∆uh, λh) satisfying

K ∆uk + [DB]T λk+1 = rk · h h k k k k k k DCn(u , λ )(∆u , ∆λ ) = Cn(u , λ ) (3.89) h h h h − h h k k k k k k DCτ (u , λ )(∆u , ∆λ ) = Cτ (u , λ ), h h h h − h h

48 3.5. Full algebraic linear system and condensation of the Lagrange multiplier with rk = δΠh (uk). In the full algebraic version, (3.89) reads as − int,ext h     KNN KNM KNI KNA 0 0  k  rN T T ∆uN KMN KMM KMI KMA BI BA  k rM  T  ∆uM    KIN KIM KII KIA DII 0   k   rI   T   ∆uI     KAN KAM KAI KAA 0 DAA   k  =  rA  , (3.90)   ·  ∆uA     0 0 0 0 DII 0   k+1   0     λI     0 NBA 0 NDAA 0 GnDAA k+1  hn  − λA 0 Lτ BA 0 Lτ DAA 0 Rτ DAA hτ − with Lτ [p, p] = Lpτ , Rτ [p, p] = Rpτ and Gn[p, p] = gnp. Due to the regularization of the contact conditions, one ends up with a regularized saddle point problem. Before solving (3.90), the full advantage of the dual mortar formulation is exploited and the Lagrange multiplier is locally condensed from the global system

λk+1 = D−T (rk K ∆uk), (3.91) h − · h derived from the third and fourth row of (3.90). Inserting (3.91) in the remaining rows of (3.90) leads to   KNN KNM KNI KNA  ˜ T ˜ T ˜ T ˜ T  KMN + BAKAN KMM + BAKAM KMI + BAKAI KMA + BAKAA     KIN KIM KII KIA     GnKAN GnKAM + NBA GnKAI GnKAA + NDAA  Rτ KAN Rτ KAM + Lτ BA Rτ KAI Rτ KAA + Lτ DAA  k   r  ∆uN N k ˜ T ∆uM rM + BArA  k  =   ·  ∆uI   GnrA + hn  k ∆uA Rτ rA + hτ (3.92)

˜ −1 for the condensed system with BA = DAABA. Though internally assuming a full k k+1 Lagrange multiplier setting with (∆uh, λh ) in the k-th Newton–Raphson iteration, the linear system has to be solved only for the primary variable due to the smart choice of the basis functions for the Lagrange multiplier. The description of the whole algorithm can be found in Algorithm 1. One has to note that the condensation of λ is deliberately performed with the balance of forces and not with the contact condition to avoid an ill-conditioning of the matrix system due to the regularisation parameters an and aτ for high regularisation parameters. Although the focus of this work lies in the contact of rough surfaces, one does not want to renounce one of the main advantages of the Lagrange multiplier method, namely the

49 Chapter 3. Quasi-static dual mortar with CCL ability to model a mathematically ideal contact, hard contact in normal direction and a perfect Coulomb law in tangential direction. Doing so, one ends up with a multi-scale algorithm being able to model ideal contact for micro- or nano-scale models and and constitutive contact laws for larger scales.

Algorithm 1 Combined ﬁx-point semi-smooth Newton for contact problems loop over all steps s T = Ts j = 1 while t < Tend do loop over all increments j { } Update n and τ Cutting of slave-master surface and evaluate gap 1 1 1 st,1 sl,1 Set , , , τ and τ In An Iτ A A Calculation of residuum r1 k = 1 while rk > ε CP > ε CP > ε do || n || τ Newton-Raphson iteration k { } Calculation of stiffness matrix Kk Embedding of contact conditions and condensing λk+1 via (3.92) k Solving K˜ ∆uk =r ˜k · Calculation of λk+1 via (3.91) Calculation of uk+1 = u + ∆uk k+1 k+1 k+1 st,k+1 sl,k+1 Update , , , τ and τ In An Iτ A A Calculation of residuum rk+1 k = k + 1 end while j = j + 1 t = t + ∆t end while end loop

3.6. Numerical examples

In this Section, different numerical examples demonstrate the robustness and wide applicability of the derived algorithm. Special focus is set here to microscopic rough surfaces leading to constitutive contact laws. More general standard examples like the patch test and Hertz problems as well as a comparison of different contact enforcement

50 3.6. Numerical examples methods can be found in Appendix B. The ﬁrst two examples concentrate on the constitutive contact laws for normal contact without friction while the last two examples focus on the frictional contact.

3.6.1. Lamination stack of an electric motor

The first example is a quite simple numerical set-up without friction. Here numerically obtained solutions for different normal contact laws in the dual mortar framework are compared with experimental data [94]. The experimental data were obtained from com- pression measurements of a stack of steel sheets similar to the lamination stack in an electrical motor. In this scenario, a correct constitutive description of the interface behaviour is necessary to identify a homogenized model of the rotor and stator stiffness. For the experiment, a stack of nine circular sheets cut from a typical lamination sheet with radius R = 15.0 mm and height h = 0.65 mm with the material characteristics of steel E = 2.1 105 MPa, ν = 0.3 is put between two plungers of an electro-mechanical · universal testing machine, see Figure 3.6a. Then a packing load of 3 MPa is applied slowly at the top of the model to avoid dynamical effects. Since a constant load is applied and all surfaces are flat on the macro scale, a constant pressure distributions is obtained on all interfaces, see Figure 3.6c. Thus it is sufficient to model a cut-out of 3 3 mm of the circular sheets, see Figure 3.6b, with adequate boundary conditions to × avoid shear stresses. The FEM-model consists of linear Hex8 Elements. In the numerical calculation, the load is applied in one quasi-static non-linear geometrical step in 20 load increments. The sheets of the lamination stack have a rough surface on the micro scale as one can see in Figure 3.7a leading to the measured contact law in Figure 3.7b in red. As a naive approach one neglects these asperities and models hard contact on all interfaces(see Figure 3.7b hard). Then the solution is compared with the regularized contact formulation. Here one uses a piecewise linear regularization obtained from the experimental data (see Figure 3.7b MLP). Figure 3.8a shows the measured displacements for increasing loads from 0.0 MPa to 3.0 MPa. As one can see clearly, the naive assumption of hard contact on all interfaces results in a load-displacement relation far away from the measured one. In this case the displacements simply result from the elastic deformation of the material and the lamination stack acts like one solid block, see Figure 3.8b. In contrast to that, the dual mortar formulation with regularized contact condition is able to reproduce the measured load-displacement relation without increasing the computational costs, see Figure 3.8a.

3.6.2. Simpliﬁed disk blade foot contact

The second example is more complex: As one can see in Figure 3.9a, the rotating part of a high pressure turbine exposed to centrifugal force without friction is considered.

51 Chapter 3. Quasi-static dual mortar with CCL

Plunger

9 Sheets

Plunger

(a) Test set-up (b) FEM mesh (c) Contact pressure

Figure 3.6.: Lamination stack example, real set-up, FEM mesh and resulting contact pressure on the interfaces

Approximation of constitutive contact law 3.0 hard contact 2.5 Measurements ] MLP

MP a 2.0 [

1.5

1.0 Contact pressure 0.5

0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Penetration [ 102mm] × (a) Surface of sheets on micro-scale (b) Simulated contact laws

Figure 3.7.: Lamination stack example, surface roughness on the micro-scale and used normal contact laws

Total displacement vs. pressure elastic displacement vs. pressure 3.0 3.0 Measurements 2.5 Hard contact 2.5 MLP ] 2.0 ] 2.0 MP a MP a [ [ 1.5 1.5

Pressure 1.0 Pressure 1.0

0.5 0.5 El.displacement Hard contact 0.0 0.0 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.00 0.05 0.10 0.15 0.20 − 2 3 Displacement [ 10− mm] Displacement [ 10− mm] × × (a) Hard and regularized contact (b) Elastic displacement

Figure 3.8.: Lamination stack example, displacements versus applied load at the top of the model for different normal contact laws

52 3.6. Numerical examples

Blade centrifugalforce

Bladefoot 200mm Disk 300mm fixed tangential direction

fixed all directions (a) Set-up hole disk (b) Set-up one blade (c) Hex20 mesh

Figure 3.9.: Simpliﬁed disk blade foot contact, set-up

Contact laws 1000

800 ] MP a [ 600

400 LP an = 1.e6

Contact pressure MLP 200 NLP expo Hard contact 0 0.0 0.2 0.4 0.6 0.8 3 Penetration [ 10− mm] × Figure 3.10.: Simpliﬁed disk blade foot contact, simulated contact laws

Contact law turbine 1000

800 ] MP a [ 600

400

Contact pressure 200

0 0.0 0.2 0.4 0.6 0.8 1.0 3 Penetration [ 10− mm] × (a) Example blade (b) Surface on the micro scale (c) Calculated contact law

Figure 3.11.: Simpliﬁed disk blade foot contact, basis for piecewise linear regularization

53 Chapter 3. Quasi-static dual mortar with CCL

∅ number it. total it. hard 3.9 39 6 LP an = 10 3.9 39 MLP 4.6 46 NLP expo 4.6 46

Table 3.1.: Average number of iterations per increment (and total number of iterations) for simpliﬁed disk blade foot contact example

More precisely, a disk of radius 300 mm assembled with 20 blades is examined. For 1 simplicity, only 20 of the whole set-up is modelled and everything except the contact zone is meshed with Hex20 elements, see Figure 3.9c. The contact zone is re-meshed to linear Hex8 elements and linked to Hex20 elements with multi-point constraints. All parts are modelled as steel with E = 2.1 105 MPa and ν = 0.3, and a non-linear · geometrical calculation in one quasi-static step is performed. A centrifugal force of 7000 turns is applied to the whole model in 10 load increments (dt = 0.1 ω2 with ω = min · 2π 7000 1 ). The bottom of the disk is fixed in all direction, the left and right side of · · 60s the disk is fixed in tangential direction, see Figure 3.9b. In the contact zone, the disk and the blade foot are modelled with different curvatures leading to an overlap at the beginning of the calculation. For the numerical simulation the following contact laws and their numerical performance are compared: Hard contact (Figure 3.10 hard), a linear regularization with 6 an = 10 MPa/mm (Figure 3.10 LP an = 1.e6), a piecewise linear regularization gained from roughness measurements (Figure 3.10 MLP) and an exponential regularization as in equation (3.19) with A = 9.072 and β = 8.689 10−5 (Figure 3.10 NLP expo). · The values for A and β were obtained via linear data fitting according to the discrete data points of the piecewise linear regularization. The piecewise linear regularization is obtained from the roughness measurement of a test blade seen in Figure 3.11b. Here a gas turbine blade is chosen, see Figure 3.11a, having a similar surface roughness than the blade foot in a high pressure jet turbine. The contact law is then calculated numerically based on the measured roughness parameters with a modified BGT-model [141], see Figure 3.11c. turns The resulting contact pressure in the blade foot for hard contact and 7000 min can be seen in Figure 3.12. One can see nicely the different contact pressures at the top and the bottom because of the different convexities of the blade foot and the disk, and between the right and the left contact zones because of the asymmetric deformation of the blade. Now one has a closer look at the red marked zone on the top left for different contact laws. As one can see in Figure 3.13, the solution of the variationally consistent weak formulations of the contact condition matches the strong contact laws quite well. It has

54 3.6. Numerical examples

top left

singularities bottom left

(a) Contact pressure left side (b) Contact pressure right side

turns Figure 3.12.: Simpliﬁed disk blade foot contact, contact pressure for t = 1.0 (7000 min ) and hard contact (Hex20 mesh displayed)

Contact Pressure t=1.0 (left top) Contact Penetration t=1.0 (left top) 800 0.8

700 ] 0.7 ] mm 600 3 0.6 − 10 MP a [ × 500 [ 0.5

400 0.4

300 0.3 LP an = 1.e6 LP an = 1.e6

Contact pressure 200 MLP 0.2 MLP

NLP expo Contact penetration NLP expo 100 0.1 Hard Hard 0 0.0 280.0 280.5 281.0 281.5 282.0 282.5 283.0 283.5 280.0 280.5 281.0 281.5 282.0 282.5 283.0 283.5 Z-coordinate [mm] Z-coordinate [mm] (a) Contact pressure (b) Contact penetration

Figure 3.13.: Simpliﬁed disk blade foot contact: contact pressure and contact penetration for dual mortar formulation and different contact laws

55 Chapter 3. Quasi-static dual mortar with CCL to be noted here, that this numerical example is force driven, therefore there are only minor differences in the resulting contact pressure for different contact laws, see Figure 3.13a, but signiﬁcant differences in the contact penetration, see Figure 3.13b. In case of a displacement driven example, like in a crimp connection for example, one would get similar displacements for all contact laws, but signiﬁcant differences in the contact pressure. It has to be pointed out, that the regularization term has only minor impact on the numerical performance of the algorithm, see Table 3.1. One obtains the same number of iterations per load increment for hard contact and a linear regularization term and only one iteration per increment more for the piecewise linear and for the exponential regularization on average.

3.6.3. Menq example

To test the impact of the regularized tangential contact law, a quite simple numerical benchmark example is considered which was studied analytically by MENQ [98] in 2D. A steel block of 10.0 1.0 0.5 mm (L x B x H) with the material characteristics of steel × × E = 2 105 MPa and ν = 0.0 is pressed against a rigid surface with p = 4 MPa, see · Figure 3.14. Here the Poisson’s ratio is set to zero to be able to compare the FEM calculations with the analytically obtained results. A friction coefficient of µ = 0.15 is assumed and the bar is meshed with 40 linear Hex8 elements over the length of the bar. Then one applies the following steps in a linear geometrical quasi-static calculation: In the first step the bar is pressed against the surface. Afterwards one starts to pull at one side of the bar denoted by A (initial loading), see Figure 3.14. In the third step one pushes at side A in tangential direction (load reversal) and finally one pulls again (second load reversal). The shear stress in y-direction over the length of the bar for initial loading denoted by tτ can be calculated analytically for aτ < by 1 ∞

( cosh(κxL) tmax cosh(κ(1−a)) 0 xl < 1 a tτ1 = ≤ − , (3.93) tmax 1 a xl 1 − ≤ ≤ with xL = x/L being the normalized length of the bar and tmax the friction bound. The ratio a describes the portion of the contact zone, which is already slipping and the parameter κ is set to be r 2 aτ L b κ := · · . E A · Here the perfect Coulomb law (aτ = MPa/mm) is compared with a linear regularized 5 ∞ Coulomb law (aτ = 10 MPa/mm) and an Iwan model with 10 elements per node with the same tangential stiffness. One recalls that the tangential stiffness aτ /niwan is used i for every Iwan element and the friction bound for Iwan element i is deﬁned as bp = bp 2i/(niwan (niwan + 1)). In normal direction hard contact is assumed. · ·

56 3.6. Numerical examples

2 3 4 A

Figure 3.14.: Menq example, set-up

5 Shear stress initial loading p=4MPa, perfect Coulomb Shear stress initial loading p=4MPa, aτ = 10

1.0 1.0

0.5 0.5 n n

/µλ 0.0 /µλ 0.0 τ τ λ λ

0.5 0.5 − − t=1.10 t=1.10

1.0 t=1.60 1.0 t=1.60 − t=2.00 − t=2.00 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 xL xL

(a) Perfect Coulomb (b) Linear regularization

5 Shear stress initial loading p=4MPa,Iwan aτ = 10

1.0

0.5 n

/µλ 0.0 τ λ

0.5 − t=1.10

1.0 t=1.60 − t=2.00 0.0 0.2 0.4 0.6 0.8 1.0 xL

Figure 3.15.: Menq example, initial tangential loading, normalized shear stress over normalized length of the bar for different frictional contact laws

57 Chapter 3. Quasi-static dual mortar with CCL

5 Shear stress reversal p=4MPa, perfect Coulomb Shear stress reversal p=4MPa, aτ = 10 t=2.20 t=2.20 1.0 1.0 t=2.80 t=2.80 t=3.50 t=3.50 0.5 t=4.0 0.5 t=4.0 n n

/µλ 0.0 /µλ 0.0 τ τ λ λ

0.5 0.5 − −

1.0 1.0 − −

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 xL xL

(a) Perfect Coulomb (b) Linear regularization

5 Shear stress reversal p=4MPa, Iwan aτ = 10

t=2.20 1.0 t=2.80 t=3.50 0.5 t=4.0 n

/µλ 0.0 τ λ

0.5 −

1.0 −

0.0 0.2 0.4 0.6 0.8 1.0 xL

Figure 3.16.: Menq example, tangential load reversal, normalized shear stress over normalized length of the bar for different frictional contact laws

58 3.6. Numerical examples

In Figure 3.15, the normalized shear stress over the normalized length of the bar is plotted for different load increments of the initial loading step (from t = 1 to t = 2) and different contact laws: The perfect Coulomb law in Figure 3.15a, the linear regularization in Figure 3.15b and the Iwan regularization in Figure 3.15c. One can easily see the differences in the transition between sticking and slipping for the different tangential contact laws and how the ratio a of the slipping region is increasing over time. At the end of step 2 (t = 2.0) around 80% of the contact zone is slipping for the perfect Coulomb law , for the linear regularization around 70% and for the Iwan model around 65%. After side A was pulled to a maximal displacement of 2 10−4mm, macro slip is not · reached in the whole contact zone. When the tangential displacement in now reversed, one can observe the transition from slipping back to sticking and to slipping again but in the inverse direction, see Figure 3.16. Also the doubling of the shear stress function due to the load reversal can be seen nicely at xL = 0.0. For the Iwan model, the smooth transition between sticking and slipping can be seen in Figure 3.16c.

Now one can have a closer look at the local hysteresis for this simple example. Figure 3.17 shows the resulting local hysteresis in one node in the middle of the contact zone. Here, the tangential displacement is plotted versus the normalized shear stress. For all tangential contact laws, the contact conditions are fulﬁlled and the transition from sticking to slipping is clearly visible. Especially the smooth transition from sticking to slipping in the case of the Iwan regularization can be observed.

This behaviour changes if one looks at the global behaviour. In Figure 3.18a, the total reaction force on side A is plotted versus the displacements. One sees that with fine divisions in the contact zone one can reproduce globally a smooth transition from sticking to slipping even in the case of a locally linear regularized Coulomb law or even a perfect Coulomb law. Differences in the maximal force values can be observed because one is still in the state of partial slip at the points of load reversal. The hysteresis for the Iwan regularization exactly matches the solution of a linear regularized Coulomb law, because they are both based on the same tangential stiffness, and thus the transmitted tangential contact force is the same for both regularizations in this simple example. When one reduces the division over the length of the bar to only two elements, the numerical results change significantly. Figure 3.18b shows, that there is no smooth transition from sticking to slipping any more in case of a perfect Coulomb law and a linear regularization. Only the Iwan regularization can still reproduce globally a smooth behaviour. But it has to be noted that the tangential stiffness had to be reduced in this 4 case in order to approximate the solution for the fine mesh (aτ = 2 10 ). In summary · one can observe that in the case of large scales and consequently coarse meshes in the contact zone a more complex friction model like the Iwan regularization is needed in order to reproduce a physically correct behaviour.

59 Chapter 3. Quasi-static dual mortar with CCL

Local hysteresis p=4MPa 1.5 a = τ ∞ 5 aτ = 10 1.0 Iwan

0.5 n /µλ τ λ 0.0

0.5 −

1.0 − 0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3 − − − − 4 u [ 10− mm] τ × Figure 3.17.: Menq example, local hysteresis at a point in the middle of the contact zone, normalized shear stress over relative tangential displacement

Hysteresis Menq example p=4MPa, div40 Hysteresis Menq example p=4MPa, Div 2

5 perf. Coulomb lin. reg. aτ = 10 4 4 lin. reg. Iwan 10 a = 2 104 τ · Iwan 5 div40 aτ = 10 2 2 ] ] N N [ 0 [ 0 tot tot F F

2 2 − −

4 4 − −

2 1 0 1 2 2 1 0 1 2 − − A 4 − − A 4 u [ 10− mm] u [ 10− mm] x × x × (a) Fine mesh (b) Coarse mesh

Figure 3.18.: Menq example, global hysteresis, total tangential contact force over applied displacement at side A, ﬁne mesh and coarse mesh

60 3.6. Numerical examples

A B

(a) Experimental set-up (b) FEM set-up

Global hysteresis resonator LTM 1500 measurements 1000

500 ] N [ 0 Rx F

500 −

1000 −

1500 − 4 2 0 2 4 − − 3 u [ 10− mm] τ × (c) Measured global hysteresis

Figure 3.19.: Resonator example, experimental set-up, FEM mesh and measured global hysteresis, relative displacement between measured points vs. total tangential contact force

3.6.4. Resonator

The second frictional example is an rather academic scenario for which measurements exist [130]. One considers a resonator consisting of two disks with a bolted connection under pretension with a M10 bolt in it, see Figure 3.19a. The resonator is excited at one end with an excitation force of 100 N, and the accelerations of the two disks are measured at points A and B, see Figure 3.19b. The relative displacements and the transmitted tangential force are subsequently calculated with a reduced 1D model leading to the measured global hysteresis seen in Figure 3.19c. In order to reproduce the measured behaviour, the linear Hex8 mesh from Figure 3.19b is used and the measured relative displacement between the points A and B is applied in a quasi-static calculation as a ﬁrst approximation. Therefore one has to add quasi-static mounting in order to avoid singularities during the calculation. All parts of the resonator are modelled with the material properties of steel E = 2.1 105 MPa, ν = 0.3. A friction ·

61 Chapter 3. Quasi-static dual mortar with CCL

(a) Step 1 (b) Step 2

Figure 3.20.: Resonator example, steps in FEM calculation, step 1 pretension of bolt, step 2 tangential movement, step 3 inverse tangential movement, step 4 tangential movement (displacements [mm] multiplied by 103 for better visualization)

coefﬁcient of µ = 0.8 is deduced from the measurements. Then the following steps are simulated: First a pretension force Fn = 1588 N is applied in the bolt (Figure 3.20a). Then a tangential displacement of 4.8 10−3mm between A and B is added in 20 load · increments (Figure 3.20b). Afterwards the inverse tangential displacement is applied in 40 load increments (Figure 3.20c). At the and the tangential displacement is reversed in 40 load increments (Figure 3.20d) As a naive approach, one models a perfect Coulomb law in tangential direction. Then the solution is compared to the regularized contact formulation, a linear regularization with tangential stiffness aτ = 550 MPa/mm and a Iwan model with 5 elements per node and the same tangential stiffness. Again hard contact in normal direction is assumed.

62 3.6. Numerical examples

Global hysteresis resonator LTM Global hysteresis resonator LTM 1500 1500 perfect Coulomb lin reg., a = 5.5 102 τ · 1000 measurements 1000 Iwan 5 a = 5.5 102 τ ·

500 500 ] ] N N [ 0 [ 0 Rx Rx F F

500 500 − −

1000 1000 − −

1500 1500 − 4 2 0 2 4 − 4 2 0 2 4 − − 3 − − 3 u [ 10− mm] u [ 10− mm] τ × τ × (a) Perfect Coulomb (b) Regularization

Figure 3.21.: Resonator example: global hysteresis, relative displacement between measured points vs. total tangential contact force

Reg. #it per cycle ∅ #it per inc min #it max #it Perfect Coulomb 363 4.53 2 13 linear reg. 284 3.55 2 6 Iwan with 5 el. 253 3.16 2 5

Table 3.2.: Total number of iterations for one cycle, average number of iterations and minimum and maximum number of iterations per increment for resonator example

(a) Step 2 (b) Step 3

Figure 3.22.: Resonator example, contact pressure for step 2 and step 3

63 Chapter 3. Quasi-static dual mortar with CCL

Figure 3.21a indicates clearly that the perfect Coulomb law does not catch the hysteresis effects properly and fails to reproduce the measurements. From this fact it can be concluded that the set-up is dominated by the micro slip part of the physical tangential contact law (Figure 2.6). Figure 3.21b shows that with a tangential stiffness 2 of aτ = 5.5 10 , the hysteresis for the linear regularized Coulomb law and the Iwan · model with ﬁve elements can approximate the measured behaviour. Of course there are still some differences in the solutions since a quasi-static calculation with quasi- static mounting is performed leading to some unsymmetrical behaviour not observed in the real model (see Figure 3.22 which shows the distribution of the normal contact pressure for different steps). Another observation made during the FEM calculation was that the model with linear regularization and the Iwan model needed one more cycle (tangential displacement and inverse tangential displacement) in order to reach the limit hysteresis shown in Figure 3.21b even in the case of a quasi-static calculation. However, from the numerical point of view the regularization in the tangential contact law leads to a faster convergence as one can see in Table 3.2. Averaged over all increments of a cycle, the linear regularization needed one iteration less per increment and the Iwan model needed even 1.4 increments per iteration less than the perfect Coulomb law in order to reach convergence. The maximal number of iterations per increment also reduced signiﬁcantly using the regularized versions of the tangtential contact laws. This better numerical performance can be deduced from the decreased non-linearity in the semi-smooth Newton iteration due to the simpler projection operator (3.51).

64 Chapter 4. Dual mortar methods facing industrial applications

The algorithms derived in Chapter 3 were all restricted to linear finite elements. But for use in industrial applications, linear finite elements have several drawbacks. For example, they do not approximate curvilinear interfaces accurately and possibly show numerical artefacts like shear locking, volumetric locking and hour-glassing. There are several methods to overcome these issues from numerical point of view. One possibility shows the isogeometric analysis for contact problems to enable a tighter connection between CAD and FEA [24, 72]. Here the same smooth and higher order basis functions are used for the representation of the CAD geometry as well as for the FEA solution fields. Although showing very promising results in the field of domain decomposition [61] or in contact simulation combined with a Gauss point-to- surface [133, 26, 32] or a mortar formulation [134, 27, 78], the isogeometric methods complicate the local problem treatment due to the broader support. This possibly results in inaccuracies in the boundary between contact and no-contact regions [93] and can be seen as a limitation to industrial applications. Therefore another path is chosen here and the in Chapter 3 well studied dual mortar method is employed in combination with quadratic finite elements as a reasonable compromise between linear finite element formulations and the full isogeometrical approach. Quadratic finite elements approximate curvilinear interfaces more accurately and lack the above mentioned numerical artefacts while at the same time avoiding the costly step switching to the isogeometrical approach. Thus they are quite attractive from the point of accuracy and computational complexity. As an alternative solution hybrid strategies combining finite element formulations with isogeometric analysis were proposed in [10, 89, 23]. Quadratic finite element formulations have been studied for the standard mortar formulation by Puso et. al. [113] and for the dual mortar formulation by Popp et. al. [110] expanding the ideas from Lamichhane, Wohlmuth [86]. Here these ideas are picked up and combined with constitutive contact laws due to rough surfaces on the micro scale presented in Chapter 3 based on [128]. Another critical point dealing with the dual mortar method can arise in set-ups with

65 Chapter 4. Dual mortar methods facing industrial applications curved interfaces leading to a possible non-physical evaluation of the weighted dual gap. This problem can be avoided using a Petrov–Galerkin dual mortar formulation combining the beneﬁts of standard and the dual Lagrange multiplier method in the linear setting [109]. Here this idea is picked up and extended to quadratic ﬁnite elements with microscopic rough surfaces. The third criteria bridging the gap to industrial applications of the dual mortar method is the ability to handle hanging nodes on the slave surface or single and multi-point constraints like directional blocking and cyclic symmetry in the contact zone. For the latter case, this results in an over-constrained system for these slave nodes carrying both the Lagrange multiplier and a multi-point constraint. Here an idea presented in [144] for mesh tying is extended to contact problems. This also solves the problem of hanging nodes on the slave surface. The outline of this Chapter is as follows: One starts with the extension of the dual mortar method with regularized contact conditions to quadratic elements in Section 4.1. Two different approaches are studied: a quadratic-quadratic method and a quadratic-linear method. In Section 4.2, the quadratic-linear dual mortar formulation is combined with a Petrov–Galerkin approach. The adoption of the algorithm solving over-constraints or hanging nodes is presented in Section 4.3. In Section 4.4, various numerical examples demonstrate the robustness of the derived algorithms. The examples involve plastic effects as well as rough surfaces on the micro scale.

4.1. Quadratic dual mortar

The main feature of the dual mortar method for linear ﬁnite elements is that the La- grange multiplier can be condensed locally from the global linear system before solving. This can be done since the dual basis functions ψp are constructed satisfying the biorthogonality condition (3.32) leading to a diagonal coupling matrix Dd on the slave side in the term of the virtual contact work. In this construction one uses the integral R positivity of the linear standard basis functions s φi dS > 0, i = 1, .., n needed for γc ∀ the contact condition. For quadratic Hex20, T et10 and W edge15 elements, this condition is violated. This is shown here exemplary for one Q8 surface. The basis functions of one Q8 surface read as follows: φQ8(ξ, η) = (1 ξ)(1 η)( ξ η 1)/4, φQ8(ξ, η) = (1 ξ2)(1 η)/2, 1 − − − − − 5 − − Q8 Q8 2 φ2 (ξ, η) = (1 + ξ)(1 η)(+ξ η 1)/4, φ6 (ξ, η) = (1 + ξ)(1 η )/2, − − − − (4.1) φQ8(ξ, η) = (1 + ξ)(1 + η)(+ξ + η 1)/4, φQ8(ξ, η) = (1 ξ2)(1 + η)/2, 3 − 7 − φQ8(ξ, η) = (1 ξ)(1 + η)( ξ + η 1)/4, φQ8(ξ, η) = (1 ξ)(1 η2)/2. 4 − − − 8 − − Indeed, it holds R φQ8 dS = 1 while R φQ8 dS = 4 for the undeformed refer- T (Q8) 1 − 3 T (Q8) 5 3 ence element (Figure 3.4).

66 4.1. Quadratic dual mortar

Following the ideas presented in [110, 86], a linear basis transformation T e is introduced for every slave surface transforming the standard basis function φi to the trans- ˜ R ˜ formed standard function φi satisfying s φi dS > 0, for all i = 1, .., n. Then the dual γc basis functions ψi can be constructed satisfying the alternative biorthognality condition Z Z ψi φ˜j dS = δij φ˜j dS (4.2) s s γc · γc in the quadratic setting. In theory, there are plenty of possibilities for the choice of T e, but this work is restricted to two different T e: A quadratic-quadratic method and a quadratic-linear method. First one has a closer look at the quadratic-quadratic method referred to as quad- quad method. Here locally quadratic basis functions are employed for the displacements as well as the Lagrange multiplier and every node of the quadratic slave surface carries a Lagrange multiplier contribution. In order to guarantee integral positivity for all transformed basis functions, parts of the edge basis functions are redistributed to the corner basis functions. For a Q8 surface the basis transformation has the following form:

 ˜Q8T  Q8T   φ1 φ1 1 0 0 0 0 0 0 0  ˜Q8  Q8   φ2  φ2  0 1 0 0 0 0 0 0   ˜Q8  Q8   φ3  φ3  0 0 1 0 0 0 0 0   ˜Q8  Q8   φ4  φ4  0 0 0 1 0 0 0 0    =     . (4.3) φ˜Q8 φQ8 · α α 0 0 1 2α 0 0 0   5   5   −  φ˜Q8 φQ8 0 α α 0 0 1 2α 0 0   6   6   −  φ˜Q8 φQ8 0 0 α α 0 0 1 2α 0   7   7   −  φ˜Q8 φQ8 α 0 0 α 0 0 0 1 2α 8 8 − | {zq } T e

P8 ˜Q8 It still holds i=1 φi = 1 and choosing α = 1/5, integral positivity is achieved for the undeformed as well as for the deformed element, see [110]. For a visualization of ˜Q8 ˜Q8 φ1 and φ5 see Figure 4.1. This choice of α also guarantees integral positivity for q T et10 and W edge15 elements. At the same time the transformation matrix T e can be inverted trivially. Using the transformed basis in the construction of the dual basis (4.2) q analogous to Section 3.3 , one gets the dual basis functions ψi as in Figure 4.2 for the undeformed reference element. The second possibility is the quadratic-linear method referred to as the quad-lin method in the following. In this case the displacements are discretized with quadratic basis functions, but the Lagrange multiplier is approximated with a linear combination of the linear basis functions. Additionally only the corner nodes carry a Lagrange multiplier distribution. In terms of the basis transformation, parts of the edge basis functions

67 Chapter 4. Dual mortar methods facing industrial applications

1.0 φ˜q 1.0 1 0.8 0.8 0.6 0.6 φ5 0.4 0.4 φ1 0.2 0.2 0.0 0.0 ˜q 1.0 φ5 0.2 − 0.5 1.0 0.5 η 0.0 1.0 0.0 − 0.5 η 0.5 − 0.0 0.5 0.5 1.0 ξ 0.5 1.0− − 0.0 1.0 1.0 0.5 − 1.0 − ξ − − Q8 ˜Q8 Q8 ˜Q8 (a) φ1 and φ1 (b) φ5 and φ5

q Figure 4.1.: Transformed standard basis functions using T e with α = 1/5; standard basis function in blue and transformed basis function in green

q ψ1 2.5 q ψ5 4 2.0 2 1.5 ˜q 1.0 0 φ1 0.5 2 0.0 ˜q − φ5 0.5 4 − − 1.0 1.0 0.5 0.5 . 0.0 1.0 1 0 0.5 0.5 η − 0.5 0.0 − 0.0 − η − ξ 0.5 1.0 1.−0 0.0 0.5 − ξ 0.5 − 1.0 1.0 − ˜Q8 q ˜Q8 q (a) φ1 and ψ1 (b) φ5 and ψ5

q Figure 4.2.: Dual basis functions using T e with α = 1/5; transformed basis functions in green and dual basis functions in red

68 4.1. Quadratic dual mortar are added to the corner basis functions while at the same time leaving the edge basis functions unchanged. For a Q8 surface, i.e. a Hex20 element, the basis transformation has the following form:

 ˜Q8T  Q8T   φ1 φ1 1 0 0 0 0 0 0 0  ˜Q8  Q8   φ2  φ2  0 1 0 0 0 0 0 0  ˜Q8  Q8   φ3  φ3  0 0 1 0 0 0 0 0  ˜Q8  Q8   φ4  φ4  0 0 0 1 0 0 0 0   =     (4.4) φ˜Q8 φQ8 · α α 0 0 1 0 0 0  5   5     ˜Q8  Q8   φ6  φ6  0 α α 0 0 1 0 0  ˜Q8  Q8   φ7  φ7  0 0 α α 0 0 1 0 ˜Q8 Q8 φ8 φ8 α 0 0 α 0 0 0 1 | {z } l T e

Choosing α = 1/2, it holds R φ˜Q8 dS > 0, i = 1, ..8 and P4 φ˜Q8 = 1. With T (Q8) i ∀ i=1 1 this choice of α a hierarchical basis is built and even point-wise positivity is guaranteed for Hex20, T et10 and W edge15 elements. A visualization of the transformed basis functions can be seen in Figure 4.3 and of the resulting dual basis functions in Figure 4.4. Again the inverse of the transformation can be stated explicitly, see [110]. Once a basis transformation has been chosen, the displacements on the slave side can now be written in two different ways:

ns ns s X X q ˜ uh = upφp = u˜pφp (4.5) p=1 p=1 and it holds u = T u˜q, u˜q = T −1u. Here T is the global transformation matrix assem- l,q bled with T e over all slave faces. Having a closer look at the contact virtual work on the slave side and applying the basis transformation in the discretization of this term, one ends up with

s,h T −T ˜ T q T ˜ T δΠc (δu, λ) = δuS T D λ = (δu˜S ) D λ (4.6) with the two different coupling matrices on the slave side. The ﬁrst one refers to the transformed system Z Z D˜ [i, j] = ψi φ˜j dS I3 = δij φ˜i dS I3, i, j S (4.7) s s γc · γc ∈ and is diagonal, while the second one refers to the non-transformed system Z D[i, j] = ψi φj dS I3, i, j S (4.8) s γc · ∈

69 Chapter 4. Dual mortar methods facing industrial applications

φ˜l 1 1.0 1.0 0.8 0.8 φ5 0.6 0.6 φ 1 0.4 0.4 0.2 0.2 ˜l 0.0 φ5 0.0 0.2 1.0 − 1.0 0.5 0.00.5 1.0 0.5 0.0 0.5 η 1.0 0.0 − − ξ 0.5 1.0 1.−0 − 0.5 η − − 0.0 0.5 ξ 0.5 − 1.0 1.0 − Q8 ˜Q8 Q8 ˜Q8 (a) φ1 and φ1 (b) φ5 and φ5

l Figure 4.3.: Local basis transformation using T e with α = 1/2; standard basis function in blue and transformed basis function in green

l ψ1 1.0 4 3 0.8 φ5 2 0.6 φ˜l 1 1 0.4 0 0.2 ˜l φ5 1 0.0 − 1.0 1.0 0.5 0.00.5 1.0 0.5 0.0 0.5 1.0 1.00.5 η 1.0 0.0 − − ξ − − − 0.5 η − 0.0 0.5 ξ 0.5 − 1.0 1.0 − ˜Q8 q ˜Q8 q (a) φ1 and ψ1 (b) φ5 and ψ5

q Figure 4.4.: Dual basis functions using T e with α = 1/5; transformed basis functions in green and dual basis functions in red

70 4.1. Quadratic dual mortar and is a rectangular one. The discretization of the master side stays the same as in (3.40). The discretization of the weak contact conditions is next. For simplicity a linear regularization of the tangential contact is assumed, but the derivation for the Iwan model is analogous. As in Section 3.3.2 and 3.3.3, one wants to ﬁnd λh M h such that ∈

[u]h Gc (λh), νh λh + h n − n n n − ni h h c h h h h h h h h h ([u] [u¯] ) G (λ λ¯ ), ν λ gn, ν λ , ν M (λ ) h τ − τ − τ τ − τ τ − τ i ≤ h n − ni ∈

h with (u¯h, λ¯ ) being the solution of the last load increment. Here a mass lumping similar c h to Chapter 3 is performed and the regularization terms Gn,τ (λ ) are approximated in the discrete setting with the transformed standard basis functions

c X X c G ( ψi λi) = φ˜i g (λi) (4.9) n,τ · · n,τ in order to guarantee diagonality of the contact law in λ, otherwise the semi-smooth Newton method looses its locality. Then the weak contact condition in normal direction for point p S reads as follows: ∈

c c uˆpn g (λsp) gp λspn 0 (ˆupn g (λsp) gp) λspn = 0 (4.10) − n ≤ ∧ ≥ ∧ − n − · with uˆS = D uS + B uM and λs = Dλ˜ . Analogously, one gets the discretized weak · · tangential contact conditions:

 λspτ bp 0 k k − ≤ λ b < 0 u˜ 1 λ˜ = 0 spτ p pτ aτ spτ , (4.11) k k − ⇒ k − 1 · k  λspτ bp = 0 u˜pτ λ˜spτ > 0 k k − ⇒ k − aτ · k with the friction bound bp = µλspn and deﬁning again u˜ := uˆ u¯, λ˜s := λs λ¯s. That − − means that the weak contact conditions decouple point-wise for the pair (uˆ, λs) and the semi-smooth Newton method is derived with respect to this pair. The same non-linear complementary functions as in the linear case are utilized, namely (3.54) for the normal direction and (3.65) for the tangential direction analogously to [126, 127]. Thus the basis transformation is not needed in the construction of the semi-smooth Newton method, but later in the condensation of the Lagrange multiplier, since only D˜ is a diagonal and thus a trivial invertible matrix. The general approach can be summed up as follows: The dual basis functions ψi are constructed with φ˜i and the matrix representation of the weak discretized and linearized

71 Chapter 4. Dual mortar methods facing industrial applications two-body contact problem is derived in (∆uk, δu, λk+1, δλ) analogous to Chapter 3:

    KNN KNM KNI KNA 0 0  k  rN T T ∆uN KMN KMM KMI KMA BI BA  k rM  T T  ∆uM    KIN KIM KII KIA DII DAI   k   rI   T T   ∆uI     KAN KAM KAI KAA DIA DAA   k  =  rA  .   ·  ∆uA     0 0 0 0 D˜ II 0     0    λk+1  ˜   I     0 NBA NDAI NDAA 0 GnDAA k+1  hn  − λA 0 Lτ BA Lτ DAI Lτ DAA 0 Rτ D˜ AA hτ − (4.12)

In this matrix system, the Lagrange multiplier cannot be condensed without numerically inverting D. Therefore the system is transformed to (∆u˜q,k, δu˜q, λk+1, δλ):   K K K˜ K˜ 0 0   NN NM NI NA  k  rN T T ∆uN KMN KMM K˜ MI K˜ MA B B   I A  ∆uk rM  ˜ ˜ ˜ ˜ ˜ T   M    KIN KIM KII KIA DII 0   q,k  r˜I   T  ∆u˜I     K˜ K˜ K˜ K˜ D˜   q,k =  r˜A  . (4.13)  AN AM AI AA 0 AA  · ∆u˜     ˜   A   0   0 0 0 0 DII 0   λk+1     ˜ ˜   I   hn   0 NBA 0 NDAA 0 GnDAA λk+1 − A hτ 0 Lτ BA 0 Lτ D˜ AA 0 Rτ D˜ AA − Now the full advantage of the dual mortar formulation is exploited and the Lagrange multiplier is locally condensed from the global system using

−T k λk+1 = D˜ (r˜k K˜ ∆u˜q,k). (4.14) − This leads to   KNN KNM K˜ NI K˜ NA  ˜ T ˜ ˜ T ˜ ˜ ˜ T ˜ ˜ ˜ T ˜  KMN + BAKAN KMM + BAKAM KMI + BAKAI KA + BAKAA   ˜ ˜ ˜ ˜   KIN KIM KII KIA   ˜ ˜ ˜ ˜ ˜   GnKAN GnKAM + NBA GnKAI GnKAA + NDAA  Rτ K˜ AN Rτ K˜ AM + Lτ BA Rτ K˜ AI Rτ K˜ AA + Lτ D˜ AA  r   ∆uk  N N  ˜ T  ∆uk rM + BArÃ  M     q,k =  r˜I  · ∆u˜I    q,k  GnrÃ + hn  ∆uÃ Rτ rÃ + hτ (4.15)

72 4.1. Quadratic dual mortar

Algorithm 2 Combined ﬁx-point semi-smooth Newton for quadratic ﬁnite elements loop over all steps s T = Ts j = 1 while t < Tend do loop over all increments j { } Update n and τ Cutting of slave-master surface and evaluate gap 1 1 1 st,1 sl,1 Set , , , τ and τ In An Iτ A A Calculation of residuum r1 k = 1 while rk > ε CP > ε CP > ε do || n || τ Newton-Raphson iteration k { } Calculation of stiffness matrix Kk k Transformation Kk K˜ and rk r˜k → → Embedding of contact conditions and condensing λk+1 via (4.15) k Solving K˜ ∆u˜q,k =r ˜k · Calculation of λk+1 via (4.14) Transformation ∆u˜q,k ∆uk → Calculation of uk+1 = u + ∆uk k+1 k+1 k+1 st,k+1 sl,k+1 Update , , , τ and τ In An Iτ A A Calculation of residuum rk+1 k = k + 1 end while j = j + 1 t = t + ∆t end while end loop

73 Chapter 4. Dual mortar methods facing industrial applications

˜ ˜ −1 for the condensed transformed system with BA = DAABA. The full combined fix-point Newton can be found in Algorithm 2. In contrast to [110], the linear algebraic system is transformed to u˜q in every Newton-Raphson iteration for both, the quad-quad and the quad-lin method, since in the general setting with hanging nodes and multi-point constraints like cyclic symmetry, the coupling matrix D˜ may loose its diagonality and thus D can not be inverted trivially like proposed in [110] for the quad- quad method, see Section 4.3. Nevertheless, the transformation of the linear system can be achieved in negligible time, since the transformation is applied element-wise in the construction of K˜ e and r˜e. Assuming a linear Euler-Lagrange strain tensor, one ends up with ˜ T Ke = T e KeT e, (4.16) and ˜ T f int,e = T e f int,e (4.17) k ˜ k q ˜ k q leading to r˜ = f ext(u , δu˜ ) f int(u , δu˜ ). Here one has to note that for the external ˜ k q − k s forces it holds f ext(u , δu˜ ) = f ext(u , δu) since ΓN Γc = ∅. The transformation ∩ matrices T e are defined on a surface and they have to be complemented by the identity matrix for all other nodes of the element not included in the slave surface definition. For non-linear geometrical calculations the transformation will be shown in more detail. In this case it cannot be applied after the assembly of the untransformed element matrix and right hand side, but during the construction due to the non-linear terms. For one element e having nnodes nodes one gets the 3 3 transformed sub-matrix for row × node i and column node j by

nintpoints X n o K˜ e,ij = Jm wm λtr(W˜ ij,m)I + 2µW˜ ij,m , (4.18) · m=1 i, j = 1, ..., nnodes with

W˜ ij,m[k, l] = φ˜i,k(ξm, ηm, ζm) φ˜j,l(ξm, ηm, ζm), k, l = 1, 2, 3, (4.19) · integration weight wm and the Jacobian determinant Jm with respect to the untransformed basis functions and assuming an Staint Venat-Kirchhoff material law with Lame constants λ and µ. This deﬁnition is straight-forward. Calculating the transformed internal forces for degree of freedom k and node i of element e in the non-linear geometrical situation requires more attention:

nintpoints X n f˜ [i, k] = Jm wm Sm[k, :] φ˜ + int,e · · m,i m=1 (4.20) 3 3 ) X X ∂um,k ∂um,k Sm[kk, ll] φ˜m,i,ll + φ˜m,i,kk 0.5 , ∂Xkk · ∂Xll · · kk=1 ll=1

74 4.2. Petrov–Galerkin quadratic dual mortar

3×3 i = 1, ..., nnodes, k = 1, 2, 3. In Equation 4.20, Sm R defines the second Piola- ∂u ∈ Kirchhoff stress tensor and m,k the derivative of the k-th component of the displace- ∂Xl ments with respect to the l-th component evaluated in integration point m on basis of the untransformed current nodal displacements uk. Therefore the transformation is only applied in the redistribution of the integration point values to the nodes and not in the integration point values itself, since only δu is transformed for this term and not u. Thus there is no need of defining a certain metric between the contact kinematic and the calculated stress field.

4.2. Petrov–Galerkin quadratic dual mortar

Besides the many advantages of the dual mortar method for contact treatment, the method has one major drawback: Since the dual basis functions are only integral pos- R itive and not point-wise positive, the evaluation of the weak gap gp = s ψpg0 dS may γc produce some non-physical effects in the weak formulation of the non-penetration condition (3.44) in the discrete setting for point p S. Due to the construction of ψp it holds R ∈ that ψpg0 dS < 0 if g0 = const < 0. But for a non-constant gap function g0 there exist R set-ups where g0(ξ, η) < 0 1 ξ, η, 1 but ψpg0 dS > 0. That means that a ∀ − ≤ ≤ slave node can be set active, even if there is still a gap between the bodies and some non-physical pushing apart occurs. An example of this behaviour is shown in Section 4.4.2. Of course this is a very rare h-dependent phenomenon and in most of the cases the problem with the dual gap can be avoided by simply switching the master and the slave side in the discrete setting. Nevertheless a more sophisticated workaround is presented here. Adapting ideas from [71] a smoothing operator I is introduced. Switching from the dual space to the trace space of the displacements for selected terms in the discretization of the weak contact conditions, the problem can be avoided. The same is achieved using a Petrov–Galerkin approach discretizing the variation of the Lagrange multiplier δλ with different basis functions than the Lagrange multiplier itself. For linear ﬁnite elements this method was employed by Popp et. al [109]. The Lagrange multiplier Pns is discretized with the dual basis functions λ = p=1 ψp λp but the variation of the · Pns Lagrange Multiplier is discretized by the standard basis functions δλ = φp δλp p=1 · leading to the weak non-penetrability condition

ns Z ! nm Z ! Z X s X m m φpφk dS ukn φpφk dS ukn φpg0 dS (4.21) s − s ≤ s k=1 γc k=1 γc γc for the unregularized problem. Since the standard basis functions for linear elements are point-wise positive, the weak gap gp using the Petrov–Galerkin approach is always negative if g0(ξ, η) < 0 1 ξ, η, 1 and no non-physical pushing apart occurs. ∀ − ≤ ≤

75 Chapter 4. Dual mortar methods facing industrial applications

This idea can be extended to quadratic elements for the quad-lin mortar method using the transformed standard basis function φ˜p for the discretization of the variation of the Lagrange multiplier n Xs δλ = φ˜p δλp (4.22) · p=1 since the transformed basis functions for the corner nodes coincide with standard basis functions for linear elements. It has to be noted that the same idea cannot be utilized for the quad-quad mortar method since the transformed basis functions are not point- wise positive in this case, since only setting α = 0.5 leads to a point-wise positive transformed basis functions coinciding with the quad-lin method. Using the Petrov–Galerkin quadratic-linear mortar approach, from now on referred as PG quad-lin method one ends up with six different coupling matrices in the discrete set- up instead of three for the quad-lin method. This leads to three different coupling matrices in the term for the contact forces, the slave-slave coupling in the non-transformed as well as in the transformed system Z Z Z D D D [p, q] = ψp φq dS I3, D˜ [p, q] = ψp φ˜q dS I3 = δpq φ˜q dS I3 (4.23) s s s γc · γc · γc and the master-slave coupling Z D Bd [p, q] = ψp (φq Pt) dS I3. (4.24) s γc · ◦

Here the same primal-dual coupling as in Section 4.1 is employed but the nomenclature is slightly different. For the weak contact condition one obtains the non-transformed and transformed slave-slave coupling Z Z P P D [p, q] = φ˜p φq dS Id, D˜ [p, q] = φ˜p φ˜q dS Id (4.25) · · as well as the slave-master coupling Z P B [p, q] = φ˜p φq dS Id. (4.26) d ·

Due to the Petrov–Galerkin approach, a primal-primal coupling of the displacements is P s achieved for this part of the linear system. Thus the weighted pair (uˆ, λs) = (D u + D BP um, D˜ λ) is used in the construction of the semi-smooth Newton. In contrast to Section 4.1 and Chapter 3, no further smoothing is needed for the regularization terms of the contact laws.

76 4.2. Petrov–Galerkin quadratic dual mortar

In total the following discretized problem is obtained for (∆u, δuk):   KNN KNM KNI KNA 0 0 r  D T D T ∆uk  N KMN KMM KMI KMA (BI ) (BA)  N r  D T D T  uk  M  KIN KIM KII KIA (D ) (D )  ∆ M   II DAI rI  D T D T   ∆uk     KAN KAM KAI KAA (D ) (D )   I     DIA DAA   k  =  rA   ˜ D  ·  ∆uA     0 0 0 0 DII 0     0    λk+1   P P P ˜ D  I  h  0 NBA NDAI NDAA 0 GnDAA k+1  n   −  λA P P P D hτ 0 Lτ B Lτ D Lτ D 0 Rτ D˜ A AI AA − AA (4.27) As in Section 4.1, the system is transformed to (∆u˜q, δu˜q,k):   KNN KNM K˜ NI K˜ NA 0 0  k  r  K K K˜ K˜ BD T BD T  ∆u N  MN MM MI MA ( I ) ( A)  N D uk rM  K˜ K˜ K˜ K˜ (D˜ )T 0  ∆ M    IN IM II IA II   q,k r˜I  D      ˜ ˜ ˜ ˜ ˜ T ∆u˜I    KAN KAM KAI KAA 0 (DAA)   q,k =  rÃ     u   ˜ D  · ∆Ã     0 0 0 0 DII 0   k+1   0   P P D  λI   P ˜ ˜ ˜    hn   0 NBA NDAI NDAA 0 GnDAA k+1   λA P P P − D hτ 0 Lτ B Lτ D˜ Lτ D˜ 0 Rτ D˜ A AI AA − AA (4.28) Finally the Lagrange multiplier is condensed from the global system using λk+1 = D k (D˜ )−1(r˜k K˜ ∆u˜q,k) leading to d −   KNN KNM K˜ NI K˜ NA  ˜ T ˜ ˜ T ˜ ˜ ˜ T ˜ ˜ ˜ T ˜  KMN + BAKAN KMM + BAKAM KMI + BAKAI KA + BAKAA   ˜ ˜ ˜ ˜   KIN KIM KII KIA   P P   G K˜ G K˜ + NBP G K˜ + ND˜ G K˜ + ND˜   n AN n AM A n AI AI n AA AA  ˜ ˜ P ˜ ˜ P ˜ ˜ P Rτ KAN Rτ KAM + Lτ BA Rτ KAI + Lτ DAI Rτ KAA + Lτ DAA   rN ∆uk  N  ˜ T  uk rM + BArÃ ∆ M     k  =  r˜I  ·  ∆u˜I    k  GnrÃ + hn  ∆uÃ Rτ rÃ + hτ (4.29) ˜ ˜ D−1 D with BA = DAA BA . One has to note that in contrast to the previous section, one has to consider twice as much coupling matrices. Additionally the coupling of the transformed displacements in the contact condition is denser than the coupling in the contact

77 Chapter 4. Dual mortar methods facing industrial applications forces. This yields a asymmetric matrix for the linear system even in the frictionless case. One also has to have a closer look into the implications of the used Petrov-Galerkin approach, see Figure 4.5. Here one considers a very simple set-up with linear elements, where a rigid form with sharp edges (in red) is pressed into a soft block assuming mathematically ideal contact. One can see clearly in Figure 4.5b, that one can resolve the sharp edge with the dual mortar method resulting in point-wise hard contact as also achieved for example with an augmented Lagrange method. Using the Petrov-Galerkin approach one introduces a mesh-dependent point-wise overlap of the two bodies due to the weak contact condition as also induced by the standard Lagrange method, see Figure 4.5c for the solution using a coarse mesh and see Figure 4.5d for the solution using a ﬁne mesh. Especially in complex forming processes with edged tools, this mesh-dependent effect may be undesirable. Another point is the extension of the method to dynamical problems. Here, the dual Petrov–Galerkin method may corrupt the conservation of energy due to the inherent asymmetry of the approach. Using the standard dual mortar method, the contact or mesh tying constraints do not generate any energy. Let’s clarify this behaviour with a simple mesh tying example leading to a static matrix system T T T KC1 u T T T T (δu , δλ ) = δu Ku + δu C1 λ + δλ C2u, (4.30) · C2 0 · λ with the coupling matrices C1, C2. For the standard dual mortar method it holds C1 = T T T C2 and thus u C1 λ + λ C2u = 0 since C2u = 0. If C1 = C2 as it is the case for T T T T 6 T the Petrov–Galerkin method, then u C λ + λ C2u = u C λ = 0. This potentially 1 1 6 causes a non-physical energy growth by the constraint and may lead to instability in the time integrator.

4.3. Modiﬁcation for SPCs/MPCs and overlap

Now that suitable algorithms for quadratic finite element problems in the dual mortar setting have been derived in Section 4.1 and Section 4.2, there is one piece missing limiting the application of the algorithm to industrial problems. When considering complex industrial applications with several parts in contact, typically not the whole set-up is modelled numerically, but only a sub model with multi-point constraints like cyclic symmetry or directional blocking applied to the boundaries of the finite element model. In this context, a slave node carrying a Lagrange multiplier distribution can have additionally a multi-point constraint defined on it leading to an over constraint linear system in total. There exist different strategies to overcome this problem. In some cases, the Lagrange multiplier on the slave node can be kept under the assumption that the equivalent multi-point constraint for the Lagrange multiplier is known.

78 4.3. Modiﬁcation for SPCs/MPCs and overlap

8 mm

master

slave

(a) Simple set-up

(b) Dual mortar (c) PG dual mortar (d) PG dual mortar

Figure 4.5.: Simple Hex8 example comparing the Petrov-Galerkin dual mortar method with the standard dual mortar method for linear elements in case of ideal contact,(a) set- up, (b) displacements for a cut along the symmetry axis for the dual mortar method, (c) the Petrov-Galerkin dual mortar method and (d) the Petrov-Galerkin dual mortar method using a ﬁner mesh

master 2 6 5 0.5 3 slave

1 7 8 0.5 4

(a) Set-up (b) one Hex8 element

Figure 4.6.: Example of a hanging node for a Hex8 elements and redistribution of the Lagrange multiplier distribution

79 Chapter 4. Dual mortar methods facing industrial applications

Then both dependent degrees of freedom, from the displacements and the Lagrange multiplier, can be condensed from the global system matrix before solving. But this can lead to a non-diagonal condensed coupling matrix Dd and increases the computational costs for inverting it. Also for most multi-point constraints in industrial applications, the equivalent constraint for the Lagrange multiplier cannot be derived trivially, see [30, p.96]. Therefore this solution won’t be shown in detail here. The simpler solution is removing the Lagrange multiplier distribution for this slave node and redistributing it to the neighbouring nodes during the construction of the dual basis functions ψp. Thus the nodes with multi-point constraints are treated as master nodes by the algorithm and are solely forced by the multi-point constraint. This is very similar to the classical mortar situation of cross-points in case of domain decomposition [144]. Here, a redistribution of the nodal basis function to the other nodes is required, otherwise a constant traction force cannot be preserved. The same can be done for hanging slave nodes like in Figure 4.6a, when parts of the slave surface overlap the master surface. Here it is not favourable to keep the hanging nodes active, since it would result in non-physical displacements and stresses in the numerical simulation or divergence of the algorithm. This divergence problem can be seen even in the simplest test scenarios like in Figure 4.6a, were two steel cubes are pressed again each other and the lower cube is ﬁxed at the bottom. Applying the redistribution of the Lagrange multiplier, the problem converges without difﬁculties and all hanging nodes can move freely. In order to clarify the procedure, the redistribution of the Lagrange multiplier distribution of one Q8 surface with one hanging node using the quad-lin method introduced in Section 4.1 will be shown in detail, see Figure 4.6b. The other cases can be derived P4 Q8 Q8 analogously. It holds λ face = ψi λp with ψ = A φ˜ . Matrix A is calculated | i=1 · · using the biorthogonality condition on the common area (light blue in Figure 4.6b with triangulation). Since node one is a hanging node, the Lagrange multiplier contribution is redistributed to the nodes two and four, λ1 = 0.5 λ2 + 0.5 λ4. The edge nodes do · · P4 Q8 not carry Lagrange multiplier degrees of freedom. This leads to λ face = ψˆ λp | i=1 i · with

 0 0 0 0  aˆ21 aˆ22 aˆ23 aˆ24 Aˆ =   , aˆji = aji + 0.5 a1i. (4.31) a31 a32 a33 a34 · aˆ41 aˆ42 aˆ43 aˆ44

P4 ˆQ8 ˜ It still holds i=1 ψi = 1. The coupling matrix Dd restricted on the current slave

80 4.4. Numerical examples surface using the alternated dual basis functions becomes:   0 0 0 0  ˆQ8 Q8 I ˆQ8 Q8 I  ˜ < ψ2 , φ1 > 3 < ψ2 , φ2 > 3 0 0  Dd =  Q8 Q8   0 0 < ψ3 , φ3 > I3 0  ˆQ8 Q8 ˆQ8 Q8 < ψ4 , φ1 > I3 0 0 < ψ4 , φ4 > I3 (4.32)

Thus D˜ d looses its diagonality, but the Lagrange multiplier can nevertheless be condensed without extra cost, since D˜ d restricted to all slave nodes carrying Lagrange multipliers is still a diagonal one and can be inverted trivially. All hanging nodes with no Lagrange multiplier contributions are treated as master nodes in the linear system as one can see here for the discretized transformed problem restricted to the Lagrange multiplier nodes of the considered slave surface:   ∆u˜1  ˜ ˜ ˜ ˜ ˜ ˜  ∆u˜2   K11 K12 K13 K14 D21 0 D41   r˜1 ∆u˜3 K˜ 21 K˜ 22 K˜ 23 K˜ 24 D˜ 22 0 0    r˜2   ∆u˜4 =   . K˜ 31 K˜ 32 K˜ 33 K˜ 34 0 D˜ 33 0  ·   r˜3  λ2  K˜ 41 K˜ 42 K˜ 43 K˜ 44 0 0 D˜ 44   r˜4  λ3  λ4

4.4. Numerical examples

In this section, a variety of industrial motivated examples with microscopic rough surfaces and different contact zones are studied including large plastic deformations. The surface roughness resulting in regularized contact laws are considered in all examples but is not the main focus in this Chapter. For a detailed study see Chapter 3. Here a comparison between the different numerical approaches for quadratic elements is made.

4.4.1. Cyclic symmetric pipes

The first numerical example is a simple set-up demonstrating the advantages of the modification of the dual mortar method for multi-point constraints in the contact zone. One considers two nested steel pipes (E = 2.1 105 MPa) as one can see in Figure · 4.7a. Since the model is axial symmetric, only 30◦ of the pipes are modelled with Hex20 elements and a cyclic symmetry tie is applied to the sides of the model, see blue marks in Figure 4.7b. For simplicity, all nodes are fixed in axial direction and

81 Chapter 4. Dual mortar methods facing industrial applications one sets ν = 0.0. Furthermore two nodes of each pipe belonging to the symmetry tie are fixed in circumferential direction to guarantee static mounting. Then a pressure of 100 MPa is applied to the inside of the inner pipe (marked with green) to expand both pipes in one linear-geometrical quasi-static step. In this set-up the slave surface touches the cyclic symmetry tie and would lead to an over-constraint system. As a naive approach one removes all slave faces with nodes in the cyclic symmetry tie from the slave surface formulation. Afterwards the results are compared to the modification introduced in Section 4.3 keeping all slave nodes and simply removing the Lagrange multiplier contribution from the over-constraints nodes. In both cases the quad-quad mortar method is employed and for the contact interaction one sets a normal stiffness 7 3 of an = 5 10 MPa/mm, µ = 0.2 and the tangential stiffness aτ = 5 10 MPa/mm. · · Looking at the norm of the displacements, one would expect a linear distribution from the outer to the inner pipe in this simple example. As one can see clearly in Figure 4.8a, the naive approach fails to reproduce the physical behaviour of the model and violates the contact condition near the cyclic symmetry tie (Figure 4.8b). The modification introduced in Section 4.3 on the other hand reproduces the physically correct displacements and even satisfies the contact condition in the nodes of the cyclic symmetry due to the active Lagrange multiplier distributions of the surrounding slave nodes, see Figure 4.8c. The same can be observed looking at the worst principle stress defined as the maximum of the absolute value of the principal stresses times its original sign (Figure 4.9). Here one can see the correct solution for the modified algorithm in Figure 4.9b compared to the naive approach leading to extreme non-physical results in Figure 4.9a.

4.4.2. Film forming

The second numerical example is inspired by the pharmaceutical industry and shows clearly the benefit of the Petrov–Galerkin dual mortar formulation for quadratic elements. The model consists of a cylindrical hollow matrix, a thin plastic film and a cylindrical indenter with rounded edges pressing out a pill form of the plastic film. In Figure 4.10a, one can see a visualization of the set-up without the plastic film. Since the model is axial symmetric, only 30◦ are modelled and the two sides of the model are blocked in circumferential direction (Figure 4.10b). It has to be noted that therefore the slave nodes at the boundary of the contact zone loose their Lagrange multiplier contributions. Also the thin plastic film is fixed to the indenter at one end and to the matrix at the other end. The plastic film is set to be the slave body and the matrix and the indenter are the master bodies. The matrix and the indenter are modelled as rigid. For the thin film the elastic material properties E = 5000 MPa and ν = 0.3 are used. Since the model shows a forming process, incremental plasticity with multiplicative split is utilized due to [120, 121] with the following data pairs, von Mises stress versus the equivalent plastic strain, for the isotropic hardening curve: (50.0 MPa, 0.0),(100 MPa, 0.3), see

82 4.4. Numerical examples

15 mm pipe 1 (slave) 20 mm pipe 2 (master) 10 mm load

(a) Full set-up (b) Set-up

Figure 4.7.: Cyclic symmetric pipes, full set-up and FEM model with cyclic symmetry tie

(a) Naive approach (b) Naive approach zoom (c) With modiﬁcation

Figure 4.8.: Cyclic symmetric nested pipes, resulting displacements 100 for naive ap- × proach and new method with modiﬁcation

(a) Naive approach (b) With modiﬁcation

Figure 4.9.: Cyclic symmetric nested pipes, resulting worst principle stress for naive approach and new method with modiﬁcation

83 Chapter 4. Dual mortar methods facing industrial applications

10mm mm 6 10mm

master indenter indenter slave 0.25 mm film 16.5 mm

matrix master matrix

fixed 5 mm

(a) Full set-up (b) Set-up (c) Hex20 mesh

100

80 ]

60 MP a [

p vM 40 σ

0 0.0 0.2 0.4 0.6 0.8 1.0 pl (d) Zoom contact zone (e) Plasticity law

Figure 4.10.: Film forming example, setup, mesh, zoom to contact zone and used plasticity law with isotropic bi-linear hardening

Figure 4.10e. The thin film consists of Hex20 elements and a few W edge15 elements at the pointy end of the model as one can see in Figure 4.10c. Using this mesh, the problem with the dual gap occurs at point p S as one can see in Figure 4.10d, since R ∈ g ψp dS > 0 while g 0 for the two standard dual mortar formulations (quad-quad · ≤ and quad-lin method). One performs a quasi-static non-linear geometrical calculation in 600 increments and move the indenter down by 6 mm. For the contact interaction a nor- 4 mal stiffness of an = 5 10 MPa/mm, the friction coefficient µ = 0.2 and the tangential 2 · stiffness aτ = 10 MPa/mm are used. The plastic forming process is illustrated in Figure 4.11. In Figure 4.11a-4.11c, three positions of the indenter are shown. Here the von Mises stress is plotted while the displacement of the intender is applied. In Figure 4.11d, one can observe large plastic deformations of the film due to the high von Mises stress leading to rupture of the thin film in practice. A special focus is set to the first increment of the calculation, since it illustrates per-

84 4.4. Numerical examples

(a) 2 mm (b) 4 mm

Figure 4.11.: Film forming example, von Mises stress during forming process

fectly the problem with the dual gap. Here, the intender moves down 0.01 mm and all deformations remain elastic. Figure 4.12 shows the resulting displacements after the first increment for the three different methods presented in this paper amplified by 100. One can see clearly the non-physical pushing apart between the matrix and the film due to the use of the dual basis functions in the evaluation of the weighted gap for the quad-quad method in Figure 4.12a and for the quad-lin method in Figure 4.12c. Only the PG quad-lin method reproduces the correct physical behaviour, see Figure 4.12e. The non-physical pushing apart also results in a higher worst principle stress within the film as one can see in Figure 4.12b for the quad-quad method and in Figure 4.12d for the quad-lin method and compared to the PG quad-lin method in Figure 4.12f.

85 Chapter 4. Dual mortar methods facing industrial applications

(a) Disp. quad-quad (b) WPS Quad-quad

(e) Disp. PG quad-lin (f) WPS PG quad-lin

Figure 4.12.: Film forming example, displacements 100 and worst principle stress for × the ﬁrst increment of the step showing spurious effects due to the construction of the dual basis functions for the quad-quad and the quad-lin method

86 4.4. Numerical examples

4.4.3. Push-out-test

Bladefoot

Disk

(a) General retainer (b) Set-up front

Bladefoot

Retainer

Disk

Figure 4.13.: Push-out-test example, used retainer, set-up front, set-up side and applied displacements

An even more complex example of industrial relevance is considered in the following. A blade foot disk contact is modelled with a retainer holding the blade foot within the disk, see Figure 4.13b. Here the retainer is removed from the model for better visibility of the ﬁr tree contact between disk and blade foot. In practise, the retainer (Figure 4.13a) has to withstand a certain minimum force. The numerical simulation can help in the design of prototypes satisfying this criteria. Therefore the blade foot is pushed out of the disk by a rectangular block like in Figure 4.13c and Figure 4.13d. The model consists of Hex20 and T et10 elements with around 180.000 nodes in total and all parts are modelled 5 as steel with E = 2.1 10 MPa and ν = 0.3. For the contact simulation one sets an = 6 · 5 10 MPa/mm, µ = 0.2 and aτ = 5 10 MPa/mm. A quasi-static non-linear geometrical · calculation is performed in 20 load increments with friction but without plasticity. In reality, the large displacements in the retainer will result in plastic deformations, but in contrast to example 4.4.2 here the focus lies in the dissolving of the various contact

87 Chapter 4. Dual mortar methods facing industrial applications

Method #it total ∅ #it per inc max #it min #it Quad-quad 170 8.5 15 6 Quad-lin 161 8.05 22 5 PG quad-lin 176 8.8 33 5

Table 4.1.: Total number of iterations for step, average number of iterations and maximum and minimum number of iterations per increment for push-out-test example zones. Indeed, this is a challenging example, since there are 21 different potential contact zones (disk-blade foot, disk-retainer and blade foot-retainer) and because of the high non-linearities in the system due to friction and large deformations. Also no additional static mounting is used for the blade foot and the retainer being kept in place only via the contact deﬁnitions. The calculation converged for all three different methods. Here one has a closer look at the results for the retainer. In Figure 4.14 and Figure 4.15, one can see the scaled worst principle stress at the top and at the bottom of the retainer for the three different numerical methods. We restrict our observations to the qualitative behaviour of the retainer and show no absolute values since the resulting stresses without plasticity are very high and do not mirror the correct physical behaviour. Nevertheless the transition from areas under pressure (in blue) to areas of tension (in red) can be seen clearly. One can observe clearly the contact to the disk at the lower part of the retainer indicated with red in Figure 4.14d and the contact to the blade foot and to the disk at the upper part of the retainer in Figure 4.15d. Only small differences in the solution of the worst principle stress for the different methods are obtained, mainly in the contact to the disk at the lower part of the retainer, see Figure 4.14a to 4.14c and in the contact to the blade foot at the upper part of the retainer, see Figure 4.15a to 4.15c. Looking at the numerical behaviour, one sees that the PG quad-lin method needed the most iterations in total, the quad-lin method needed 15 iterations less, since it uses less slave nodes carrying Lagrange multipliers than the quad-quad method, see Table 4.1. Also the maximum number of iterations per increment is highest for the PG quad-lin method.

88 4.4. Numerical examples

Tension Tension Tension

Null Null Null

Pressure Pressure Pressure

(a) Quad-quad (b) Quad-lin (c) PG quad-lin

Active

Inactive

(d) Contact to disk

Figure 4.14.: Push-out-test example, worst principle stress and active nodes at the bottom of the retainer for retainer-disk contact with different numerical methods

Tension Tension Tension

Null Null Null

Pressure Pressure Pressure

(a) Quad-quad (b) Quad-lin (c) PG quad-lin

contact to bladefoot side Active contact to bladefoot bottom

pushing direction pushing contact to disk side

Inactive

(d) Contact to blade foot

Figure 4.15.: Push-out-test example, worst principle stress and active nodes at the top of the retainer for retainer-blade contact with different numerical methods

Chapter 5. Extension to the fully dynamical problem

In this Chapter the dynamical contact problem will be solved numerically using the dual mortar method. There are basically two different approaches to solve the contact problem in the time domain, implicit methods where the solution at time tj+1 is found by solving a equation of type uj+1 = f(tj+1) and explicit methods where the solution at time tj+1 is directly extrapolated from the previous solution uj+1 = f(tj). Explicit methods are simple to implement since no linear system has to be solved while implicit methods are in general unconditionally stable and allow for larger time steps compared to explicit methods. For a more general overview to time integration methods in the finite element framework, the interested reader is referred to [8]. This work will focus on implicit methods resolving the time dependency. Within the implicit time integration methods, the most popular ones are the generalized α-method [21] and the energy momentum method [124, 84, 85] as enhancements of the classical Newmark method [100]. Both algorithms are unconditionally stable [9]. The generalized α-method is a numerical dissipative method with a stable numerical integration while the energy momentum method guarantees conservation of the total energy as well as the momentum and angular momentum within a time step [83]. Using the dual mortar method for dynamical contact problems, both time integration methods have been studied in [52, 53, 55, 54] building a basis for this Chapter. The ideas were extended in [48, 15] proposing a modification of the mass matrix via a modified integration [50] in order to overcome the problems of high frequency oscillations stemming from the inertia terms of the system. Other mass modification strategies were presented in [136]. Another contact stabilization can be found in [29]. Extensions to the fully deformation dependent case were presented in [106]. Within the time integration, all implicit algorithms assume a continuous velocity field. In case of contact, this condition is violated and one gets jumps in the velocity fields leading to a loss of energy in general. Thus not in focus of this work, there are several opportunities to achieve the conservation of the total energy in case of contact. In [88] a velocity update algorithm was presented updating the velocity field at the end of every time iteration to guarantee energy conservation. Other contributions dealing with the energy conservation in case of contact problems can be found in [87, 18]. Apart from the dynamical contact problem with the dual mortar method, energy conservation

91 Chapter 5. Extension to the fully dynamical problem using a node-to-surface method or the standard mortar method was studied in detail in [59, 57, 60] (for the case of domain decomposition [58]). The outline of this chapter is as follows: In in Section 5.1 a special conﬁguration of the generalized α-method [99] will be introduced as a time integration method improving the high-frequency dissipation. Combined with the dual mortar method with regularized contact laws presented in Section 5.2, one bypasses the problem of the oscillating Lagrange multiplier and the need of a modiﬁcation of the mass matrix for the contact nodes. In Section 5.3, several numerical examples demonstrate the wide applicability of the derived algorithm. Here the resonator example introduced in Section 3.6.4 is revisited for the dynamical case.

5.1. Discretization in time: the α-method

In contrast to the previous chapter were the inertia terms were neglected, the discretized problem stemming from Equation 2.44 can be stated as

δΠh (a) + δΠh (u) δΠh + δΠh(λ) = 0. (5.1) kin int − ext c h h h Again, δΠkin represents the body forces, δΠint the internal forces, δΠext the external h forces and δΠc the contact forces. Using the deﬁnition of the mass matrix M[p, q] = R φpφq dV I3 and the well-known mortar matrices (3.39), one can rewrite (5.1) as

h T h Ma + Πint(u) + [BD] λ = Πext. (5.2)

Here the displacements u, the velocities v, the accelerations a as well as the Lagrange multiplier λ and the coupling matrices D and B are time dependent. Since Equation (5.2) has to be valid for t [0,T ], the time interval is split into several time increments ∈ [tj, tj+1] and the solution for tj+1 is found on the basis of tj. It holds:

Z tj+1 uj+1 = uj + v(η) dη (5.3) tj and Z tj+1 vj+1 = vj + a(ξ) dξ. (5.4) tj Using the generalized α-method [73], the accelerations are approximated with the constant function a(t) = (1 γ)aj + γaj+1, t [tj, tj+1] (5.5) − ∈ with a ﬁxed γ ]0, 1[. Inserting(5.5) in (5.4) and (5.3) yields ∈

vj+1 = vj + ∆t[(1 γ)aj + γaj+1], t [tj, tj+1] (5.6) − ∈

92 5.1. Discretization in time: the α-method and 1 2 uj+1 = uj + ∆tvj + ∆t [(1 2β)aj + 2βaj+1] t [tj, tj+1] (5.7) 2 − ∈ with ∆t = tj+1 tj. It has to be pointed out, that two different linear combinations of aj − and aj+1 are used for vj+1 and uj+1. Then equation (5.2) is evaluated at the generalized midpoint tj+1+α, α [ 1, 0] ∈ − Ma h u h λ h j+1+αm + Πint( j+1+αf ) + Πc ( j+1+αf ) = Πext,j+1+αf . (5.8) with

aj+1+α = (1 + αm)aj+1 αmaj, m − vj+1+αf = (1 + αf )vj+1 αf vj, − (5.9) uj+1+α = (1 + αf )uj+1 αf uj, f − λj+1+α = (1 + αf )λj+1 αf λj, f − in order to guarantee second-order accuracy. It has to be noted, that in general two different generalized midpoints are used for the body forces and all other forces. Special focus has to be set on the evaluation of the internal forces at the generalized h midpoint. There are two possibilities of evaluating Πint(uj+1+αf ) in the discrete setting, either with the midpoint rule

h h Π (uj+1+α ) = Π ((1 + αf )uj+1 αf uj) (5.10) int f int − or with the trapezoid rule

h h h Π (uj+1+α ) = (1 + αf )Π (uj+1) αf Π (uj+1). (5.11) int f int − int For linear-geometrical calculation, both methods coincide, whereas the two possibilities yield different numerical algorithms for the non-linear geometrical case [83]. This thesis is restricted to the latter approach (5.11). Also, special attention has to be paid to the h evaluation of Πc (λj+1+αf ). Using again the trapezoid rule, one gets

h h h Π (λj+1+α ) = (1 + αf )Π (λj+1) αf Π (λj+1) c f c − c T T (5.12) = (1 + αf )[Bj+1 Dj+1] λj+1 αf [Bj Dj] λj. − The parameter of the generalized α-method are deﬁned as

2p∞ 1 p∞ 1 2 1 αm = − , αf = , β = (1 + αm αf ) , γ = + αm αf . (5.13) −2p∞ 1 −p∞ + 1 2 − 2 − − For the special choice of αm = αf = 0, one ends up with the standard Newmark method. In this thesis, a special variant of the generalized α-method is employed,

93 Chapter 5. Extension to the fully dynamical problem

setting αm = 0 and αf ] 1/3, 0[ achieving second-order accuracy and unconditional ∈ − stability [99]. At the same time, this choice of the α-parameters avoid high frequency oscillations stemming from the inertia terms. Since the internal forces in (5.8) depend non-linearly on uj+1 = uj+1(aj+1), a lin- k k 2 k earization of the problem around aj+1 is performed by substituting ∆uj+1 = β∆t aj+1:

k k 2 k T k+1 M∆aj+1 + (1 + αf )K(uj+1)β∆t ∆aj+1 + (1 + αf )[Bj+1 Dj+1] λj+1 = k k Ma (1 + αf )Πint(u ) + αf Πint(uj) (5.14) − j+1 − j+1 T +αf [Bj Dj] λj + (1 + αf )Πext,j+1 αf Πext,j. − 1 1 In order to get a good initial guess for uj+1 utilized in the calculation of Πint(uj+1) and 1 K(uj+1), a prediction step is introduced at the start of every time increment

1 aj+1 = 0 1 v = vj + (1 γ)∆taj j+1 − (5.15) 1 1 2 u = uj + ∆tvj + ∆t (1 2β)aj j+1 2 − derived from (5.6) and (5.7). The Newton–Raphson algorithm in iteration k + 1 can then be summed up by: Solve

∗ k T k+1 ∗ M ∆aj+1 + (1 + αf )[Bj+1 Dj+1] λj+1 = r (5.16) with ∗ k 2 M = M + (1 + αf )K(uj+1)β∆t (5.17) and ∗ k k r = Ma (1 + αf )Πint(u ) + αf Πint(uj) − j+1 − j+1 T (5.18) + αf [Bj Dj] λj + (1 + αf )Πext,j+1 αf Πext,j, − calculated with the solution of iteration k. Afterwards, the variables are updated for the next Newton iteration via k+1 k k aj+1 = aj+1 + ∆aj+1, k+1 k k vj+1 = vj+1 + γ∆t∆aj+1, (5.19) k+1 k 2 k uj+1 = uj+1 + β∆t ∆aj+1. It has to be noted here, that in contrast to [52, 48, 106] the linear system is solved for the accelerations a instead of the displacements u. Nevertheless the same combined ﬁx-point Newton algorithm for the mortar matrices is used as in the quasi-static case (see Chapter 3). For further details on the employed α-method, the interested reader is referred to [30].

94 5.2. Embedding of the contact conditions

5.2. Embedding of the contact conditions

Analogous to the quasi-static case the weak discretized contact condition in normal direction for a point p S reads as follows: ∈ c uˆpn,j+1 gn(λsp,j+1) gp,j+1 λspn,j+1 0 − ≤c ∧ ≥ ∧ (5.20) (ˆupn,j+1 g (λsp,j+1) gp,j+1) λspn,j+1 = 0 − n − · with uˆS,j+1 = Dj+1 uS,j+1 +Bj+1 uM,j+1 and λs = D˜ j+1λj+1. The tangential contact · · conditions for point p S reads as : ∈  λspτ,j+1 bp 0 k k − ≤ λ b < 0 u˜ 1 λ˜ = 0 spτ,j+1 p pτ,j+1 aτ spτ,j+1 (5.21) k k − ⇒ k − 1 · k  λspτ,j+1 bp = 0 u˜pτ,j+1 λ˜spτ,j+1 > 0 k k − ⇒ k − aτ · k with the friction bound bp = µλspn,j+1 and deﬁning again u˜j+1 := uˆj+1 u¯j+1, λ˜s,j+1 := − λs,j+1 λ¯s,j+1. It has to be noted, that the stick part of the friction law is formulated in − u and not v. But due to the linear regularization, both formulations are equivalent. As in the quasi-static case, the weak contact conditions decouple point-wise for the pair (uˆj+1, λs,j+1) and the semi-smooth Newton is derived with respect to this pair. For the semi-smooth Newton the same non-linear complementary functions as in the quasi-static case are utilized, (3.54) for the normal direction and (3.65) for the tangential k 2 k direction leading the linear equations (3.61) and (3.74). Substituting ∆uj+1 = β∆t aj+1 one ends up with the full algebraic system

 ∗ ∗ ∗ ∗  M NN M NM M NI M NA 0 0  ∗ ∗ ∗ ∗ T T  M MN M MM M MI M MA (1 + αf )BI (1 + αf )BA   ∗ ∗ ∗ ∗ T T   M IN M IM M II M IA (1 + αf )DII (1 + αf )DAI   ∗ ∗ ∗ ∗ T T   M M M M (1 + αf )D (1 + αf )D   AN AM AI AA IA AA  0 0 0 0 D˜ 0   II   2 2 2 ˜   0 NBAβ∆t NDAI β∆t NDAAβ∆t 0 GnDAA  2 2 2 − 0 Lτ BAβ∆t Lτ DAI β∆t Lτ DAAβ∆t 0 Rτ D˜ AA −   (5.22) r∗ ∆ak  N N r∗  k  M ∆aM  ∗   k   rI   ∆a     I  =  r∗  . ·  ∆ak   A   A   0   λk+1     I    k+1  hn  λA hτ

95 Chapter 5. Extension to the fully dynamical problem

Again the system is transformed to (∆a˜q,k, δa˜, λk+1, δλ):  ∗ ∗ ˜ ∗ ˜ ∗  M NN M NM M NI M NA 0 0 ∗ ∗ M ∗ M ∗ M˜ M˜ (1 + α )BT (1 + α )BT   MN MM MI MA f I f A   ˜ ∗ ˜ ∗ ˜ ∗ ˜ ∗ ˜ T   M IN M IM M II M IA (1 + αf )DII 0   ∗ ∗ ∗ ∗ T   ˜ ˜ ˜ ˜ ˜   M AN M AM M AI M AA 0 (1 + αf )DAA  ˜   0 0 0 0 DII 0   2 2   0 NBAβ∆t 0 ND˜ AAβ∆t 0 GnD˜ AA  − 0 L B β∆t2 0 L D˜ β∆t2 0 R D˜ τ A τ AA τ AA (5.23) −   r∗  ∆ak  N N  ∗  k rM ∆aM   ∗   q,k  r˜I  ∆a˜   ∗   I  =  r˜  · ∆a˜q,k  A   A   0   k+1     λI    k+1  hn  λA hτ and ﬁnally the full advantage of the dual mortar formulation is exploited condensing the Lagrange multiplier from the global system via

1 −T ∗ λk+1 = D˜ (r∗ M˜ ∆a˜q,k). (5.24) (1 + αf ) − This leads to  ∗ ∗ ˜ ∗ M NN M NM M NI  ∗ ˜ T ˜ ∗ ∗ ˜ T ˜ ∗ ˜ ∗ ˜ T ˜ ∗ M MN + BAM AN M MM + BAM AM M MI + BAM AI  ˜ ∗ ˜ ∗ ˜ ∗  M IN M IM M II ...  1 ˜ ∗ 1 ˜ ∗ 2 1 ˜ ∗  GnM AN GnM AM + NBAβ∆t GnM AI  (1+αf ) (1+αf ) (1+αf ) 1 ∗ 1 ∗ 2 1 ∗ Rτ M˜ Rτ M˜ + Lτ BAβ∆t Rτ M˜ (1+αf ) AN (1+αf ) AM (1+αf ) AI ∗ (5.25) ˜   ∗  M NA rN ∗ T ∗  k  ˜ ˜ ˜  ∆aN ∗ ˜ T ∗ M A + BAM AA k  r + B r˜  ∗  ∆a  M A A  M˜   M   r∗  ... IA   q,k =  I  1 ˜ ∗ ˜ 2  · ∆a˜I   1 ∗  GnM AA + NDAAβ∆t  q,k (1+α ) GnrÃ + hn (1+αf )  ∆a˜  f  1 ∗ 2 A 1 ∗ Rτ M˜ + Lτ D˜ AAβ∆t (1+α ) Rτ rÃ + hτ (1+αf ) AA f ˜ ˜ −1 for the condensed transformed system with BA = DAABA. The whole algorithm for dynamical contact problems is summarized in Algorithm 3. Here the algorithm was derived for the more general case of quadratic finite elements. The linear case is included in the derivation, since the transformation matrix for the standard basis functions reduces to the identity.

96 5.2. Embedding of the contact conditions

Algorithm 3 Combined ﬁx-point semi-smooth Newton for dynamic contact problems loop over all steps s T = Ts j = 1 while t < Tend do loop over all increments j { 1 } 1 1 Perform prediction of aj+1,vj+1 and uj+1 via (5.15) Update n and τ Cutting of slave-master surface and evaluate gap 1 1 1 st,1 sl,1 Set , , , τ and τ In An Iτ A A Calculation of residuum r∗,1 (5.18) k = 1 while rk > ε CP > ε CP > ε do || n || τ Newton-Raphson iteration k { } Calculation of stiffness matrix M ∗,k (5.17) ∗,k Transformation M ∗,k M˜ and r∗,k r˜∗,k → → k+1 Embedding of contact conditions and condensing λj+1 via (5.25) ∗,k Solving M˜ ∆a˜q,k =r ˜∗,k · j+1 Calculation of λk+1 via (5.24) q,k k Transformation ∆a˜j+1 ∆aj+1 k+1 k+1 → k+1 Update aj+1 , vj+1 and uj+1 via (5.19) k+1 k+1 k+1 st,k+1 sl,k+1 Update , , , τ and τ In An Iτ A A Calculation of residuum r∗,k+1 k = k + 1 end while k aj+1 = aj+1 k vj+1 = vj+1 k uj+1 = uj+1 j = j + 1 t = t + ∆t end while end loop

97 Chapter 5. Extension to the fully dynamical problem

5.3. Numerical examples

In this chapter the algorithm derived in the previous section is applied to a variety of numerical examples. Special focus is set here to the inﬂuence of the inertia terms in the problem formulation. It has to be noted, that the algorithm derived in Section 5.2 does not guarantee the conservation of energy during contact also conﬁrmed by [52, 106]. Also numerical dissipation of energy is introduced using the α-method.

5.3.1. Sphere against brick

The ﬁrst example is an academical dimensionless one validating the derived algorithm and can also be found in [48]. Here a sphere of radius 0.6 with constant initial velocity T v0 = (0, 0, 5) hits a brick of size 0.6 1.4 1.4 (H B L) being at rest, see − × × × × Figure 5.1a. The sphere has the elastic properties E = 60000 and ν = 0.3 with the density % = 0.8 while the brick has the elastic properties E = 30000 and ν = 0.3 with the density % = 1.0. The initial gap between the two bodies at time t = 0.0 is 0.1. For this example the sphere is meshed with Hex8 elements while the brick is meshed with Hex20 elements. The sphere is set to be the slave body and the brick is set to be the master body. The mesh is visualized in Figure 5.1b. In contrast to [48] a normal 7 6 stiffness of an = 10 is employed and friction is included setting µ = 0.5 and aτ = 5 10 . · A time period of T = 7.0 10−2 time units is simulated with ∆t = 2.0 10−4 in a dynamic · · linear-geometrical calculation using αf = 0.05 in the generalized α-method. In order − to stabilize the problem, additional points on the symmetry axes x and y of the sphere are blocked with single-point constraints. During the simulation the impulse of the sphere is transferred to the brick as visualized in Figure 5.2. In the upper part of Figure 5.2, the displacement uz is shown for a cut along the x-axes through the whole model. In the lower part of Figure 5.2 the corresponding contact pressure on the sphere is shown. The contact pressure in the middle of the contact zone over time is visualized in Figure 5.3. It has to be noted that no oscillations of the contact pressure can be observed for the derived algorithm even though no extra modiﬁcation of the mass matrix was implemented as suggested in [50]. This is partly owed to the special α-method used for the discretization in time avoiding high frequency dissipation [99] and partly to the chosen regularized contact laws. max The utilized regularized contact also results in a lower maximum value λn and larger contact time compared to [50]. Now one has a closer look at the total energy of the system

Etot = Ekin + Eint, consisting of the internal enery Z Z t Eint = σ dε dV Ω 0

98 5.3. Numerical examples

(a) Set-up (b) Mesh

Figure 5.1.: Sphere-brick example, set-up and ﬁnite element mesh

Figure 5.2.: Sphere-brick example, displacement uz and contact pressure λn over time

99 Chapter 5. Extension to the fully dynamical problem

6000

5000

4000 n

λ 3000

2000

1000

0 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 Time

Figure 5.3.: Sphere-brick example, contact pressure over time

10 9.00

8.99 total with friction 8 total without friction 8.98

6 8.97

Energy 4 Energy 8.96 internal 8.95 2 kinetic 8.94 total 0 8.93 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 Time Time (a) Energy over time (b) Zoom contact period

Figure 5.4.: Sphere-brick example, energy over time and zoom contact period

100 5.3. Numerical examples and the kinetic energy Z 1 T 1 T Ekin = %v v dV = v Mv, Ω 2 2 see Figure 5.4a. One can see clearly, how the kinematic energy of the ball is converted to internal energy in case of contact. As one can see in Figure 5.4b, a small amount of energy gets lost due to the frictional dissipation and a small part of the total energy is stored in the contact zone due the used regularized contact laws acting like springs. Without friction, also a small amount of energy gets lost, since no extra corrections of the algorithm in case of contact are made. Also the total energy is slightly smaller than in [50] due to the different meshes leading to small differences in the total mass in the discrete setting.

5.3.2. Rubber ball

The second example is a quite coarsely meshed example of a small rubber ball rolling down a steel plate, see Figure 5.5a. The rubber ball with radius 20 mm has the material properties E = 103 MPa and ν = 0.3 with the density % = 2.5 10−9 Ns2/mm4. Everything · is meshed with linear Hex8 elements as one can see in Figure 5.5b. The rubber ball is placed right above a steel plate (E = 2.1 105 MPa, ν = 0.3) with a slope of 20◦ · that is ﬁxed in all directions at the lower side. For the contact interaction a normal 6 stiffness of an = 10 MPa/mm, the friction coefﬁcient µ = 0.2 and the tangential stiffness 6 3 aτ = 5 10 MPa/mm are utilized. Applying a gravity force of 9.81 10 N to the ball, × × the ball should start to roll over the plate. Here a time period of t = 0.3 s is simulated with the increment size ∆t = 5 10−6s. × As one can see in Figure 5.6, the ball rotates around a constant point during the simulation due to the friction as indicated by the movement of the black dot. It can be observed, that the ball gains velocity during the simulation and starts to hop on the plate. As indicated in Figure 5.5b, there is a small gap between the ball and the plate at the beginning of the calculation leading to small bounces on the plate. Switching the friction off in the numerical calculation, the ball slides down the plate without rolling, see Figure 5.7. During the simulation, the potential Energy

Epot = m g h · · is transformed to kinematic energy using a starting height of h = 100 mm, see Figure 5.8a. For this test set-up, the internal energy can be neglected, since almost no deformation occurs in the rubber ball. One can see nicely in Figure 5.8b, that the total energy of the system decreases over time due to the frictional dissipation. Again, the downward peaks indicate the contact, since parts of the total energy are stored in the regularized contact laws.

101 Chapter 5. Extension to the fully dynamical problem

rubber ball

(a) Set-up (b) Mesh

Figure 5.5.: Rubber ball example, set-up and ﬁnite element mesh

Figure 5.6.: Rubber ball example, v for different time points k k

Figure 5.7.: Rubber ball example, v for different time points without friction k k

102 5.3. Numerical examples

f [Hz] C3D8RC3D8 C3D8R with mod. el. params. Mode 1 240.9 255.3 253.5 Mode 2 302.7 328.7 318.5 Mode 3 1089.79 1122.3 1145.1

Table 5.1.: Resonator example, frequencies (eigenvalues) corresponding to the ﬁrst three modes (eigenvectors) of undamped dynamical system for a model using C3D8R elements, C3D8 elements or C3D8R elements with modiﬁed elastic material parameters

5.3.3. Resonator

As a last dynamical example the resonator from Section 3.6.4 is revisited for the fully dynamic case as considered in [130]. The set-up of two disks with a bending spring and a bolted connection under pretension stays the same, but now ﬁner meshes for the disks are employed to catch the dynamic effects properly, see Figure 5.9a. Here around 34.000 Hex8 elements with reduced integration are used. Again a pretension force of 1588 N is applied in the bolt and the resonator is excited at one end with a point force of 100 N using an excitation frequency between f = 290 316 Hz near the experimentally − obtained resonance frequency. Thus the point load over time can be written as

Fx(t) = sin(t 2π f). (5.26) · · The material and contact parameters are set according to [130]. More precisely, a steel material is modelled with E = 1.822 105, ν = 0.28 and a density of % = 7.884 10−9 2 4 · 6 · Ns /mm . For the contact interaction a normal stiffness of an = 10 MPa/mm is utilized 2 and friction is included setting µ = 0.84 and aτ = 5.51 10 MPa/mm. As in Section 3.6.4, · either a linear regularization of the tangential contact or 5 Iwan elements are simulated. The aim is to reproduce the measured hysteresis obtained at an excitation frequency of 306 Hz numerically, see Figure 5.9b. Here the relative displacements between the point A and B are plotted against the total transmitted force in the bolted connection in tangential direction. A time period of t = 0.05 s is simulated setting dt = 4.084967 10−5 · s. This is equivalent to using around 80 increments per cycle for an excitation frequency of 306 Hz. Now no mounting is applied to the model and only a few points on the symmetry axis are ﬁxed in the y-direction to stabilize the problem. The resulting displacements ux[mm] multiplied by 100 for the limit cycle using a frequency of f = 290 Hz and 5 Iwan elements can be seen in Figure 5.10. One can see clearly how the spring bends from one side to the other transmitting the excitation force ampliﬁed due to resonance to the bolted connection. The small gap between the bolt and the block stems from a small overlap at the beginning of the calculation in order to have a good starting active set in the pretension step.

103 Chapter 5. Extension to the fully dynamical problem

100 83.0 total 82.5 80 J] J]

6 6 82.0 − 60 pot − 10 10

× kinetic × 81.5 40 total 81.0 Energy[ Energy[ 20 80.5

0 80.0 0.00 0.05 0.10 0.15 0.20 0.25 0.00 0.05 0.10 0.15 0.20 0.25 Time [s] Time [s] (a) Energy over time (b) Zoom total energy

Figure 5.8.: Rubber ball example, energy over time and zoom contact period

Global hysteresis resonator LTM 1500 measurements Load 1000 A 500

B ] N [ 0 Rx F

500 −

1000 −

1500 − 4 2 0 2 4 − − 3 u [ 10− mm] τ × (a) Fine mesh (b) Measured hysteresis

Figure 5.9.: Resonator example, ﬁne mesh for dynamic calculation and measured global hysteresis between points A and B for an excitation frequency of 306 Hz

104 5.3. Numerical examples

(a) Turning point 1 (b) Turning point 2

Figure 5.10.: Resonator example, dynamic calculation, resulting displacements for turning points of limit cycle for excitation frequency of 290 Hz and 5 Iwan elements

The dynamic behaviour for the linear regularization is visualized more detailed in Figure 5.11 for excitation frequencies between 290 and 306 Hz. With an increasing excitation frequency, the resonance force increases switching from micro slip to macro slip in the bolted connection damping the dynamical system. As one can see clearly, the resonance frequency of the FEM model is lower compared to the actual measurements. In the FEM model the critical frequency resulting in microslip is 291 Hz, see Figure 5.11f, showing good agreement with the measured hysteresis at 306 Hz in red. For an excitation of 306 Hz the FEM model already is in the state of macro slip as one can see in Figure 5.11l. Here the linear regularization was chosen, since the Iwan model resulted in a lower slope, see Figure 5.12. For the Iwan model one would have needed additional tuning of the tangential stiffness aτ in order to get a better match with the measurements. For this FEM model using Hex elements with reduced integration (C3D8R), the numerically calculated eigenfrequency of the undamped model corresponding to the bending of the spring lies around 302.7 Hz, using regular Hex elements with full integration (C3D8) the eigenfrequency it lies around 328.7 Hz, see Table 5.1. The eigenfrequency of the real model deduced from the measurements lies around 315 Hz [130]. Thus this ﬁnite element model using the reduced integration underestimates the real resonance frequency and the full integration overestimates the real eigenfrequency. In order to tune the numerically calculated eigenfrequency near the measured one, one can change the elastic material parameters of the model to E = 2.0 105 and · ν = 0.28. This results in a eigenfrequency of 318 Hz using reduced integration (see Table 5.1). Exciting the model now with a frequency of 306 Hz, one gets the dynamical

105 Chapter 5. Extension to the fully dynamical problem

resonator LTM resonator LTM Global hysteresis resonator LTM 10 1500 1500 FEM 1000 1000 meas. at 306 Hz 5 500 500 ] ] ] mm 3 N N [ [ − 0 0 0 Rx Rx 10 F F × [ τ u 500 500 − − 5 − 1000 1000 − −

10 1500 1500 − 0.00 0.01 0.02 0.03 0.04 0.05 − 0.00 0.01 0.02 0.03 0.04 0.05 − 10 5 0 5 10 − − 3 t[s] t[s] u [ 10− mm] τ ×

(a) ux 290 Hz (b) Fx 290 Hz (c) Hysteresis 290 Hz

10 1500 1500 − 0.00 0.01 0.02 0.03 0.04 0.05 − 0.00 0.01 0.02 0.03 0.04 0.05 − 10 5 0 5 10 − − 3 t[s] t[s] u [ 10− mm] τ ×

(d) ux 291 Hz (e) Fx 291 Hz (f) Hysteresis 291 Hz

10 1500 1500 − 0.00 0.01 0.02 0.03 0.04 0.05 − 0.00 0.01 0.02 0.03 0.04 0.05 − 10 5 0 5 10 − − 3 t[s] t[s] u [ 10− mm] τ ×

(g) ux 292 Hz (h) Fx 292 Hz (i) Hysteresis 292 Hz

10 1500 1500 − 0.00 0.01 0.02 0.03 0.04 0.05 − 0.00 0.01 0.02 0.03 0.04 0.05 − 10 5 0 5 10 − − 3 t[s] t[s] u [ 10− mm] τ ×

(j) ux 306 Hz (k) Fx 306 Hz (l) Hysteresis 306 Hz

Figure 5.11.: Resonator example, dynamic calculation, resulting relative displacements between node A and B , total force Fx and limit hysteresis for excitation frequencies of 290, 291, 292 and 306 Hz , material E = 1.822 105, ν = 0.28 2 · density % = 7.884 10−9 Ns /mm4 and a linear regularization in tangential · direction

106 5.3. Numerical examples

Global hysteresis resonator LTM Global hysteresis resonator LTM 1500 1500 FEM FEM 1000 meas. at 306 Hz 1000 measurements

500 500 ] ] N N [ 0 [ 0 Rx Rx F F

500 500 − −

1000 1000 − −

1500 1500 − 10 5 0 5 10 − 10 5 0 5 10 − − 3 − − 3 u [ 10− mm] u [ 10− mm] τ × τ × (a) Lin. regularization (b) 5 Iwan elements

Figure 5.12.: Resonator example, dynamic calculation, resulting hysteresis for an excitation frequency of 291 Hz of Iwan model with 5 elements compared to a linear regularization, material E = 1.822 105, ν = 0.28 density 2 · % = 7.884 10−9 Ns /mm4 ·

Global hysteresis resonator LTM Global hysteresis resonator LTM 1500 1500 FEM FEM 1000 meas. at 306 Hz 1000 meas. at 306 Hz

500 500 ] ] N N [ 0 [ 0 Rx Rx F F

500 500 − −

1000 1000 − −

1500 1500 − 10 5 0 5 10 − 10 5 0 5 10 − − 3 − − 3 u [ 10− mm] u [ 10− mm] τ × τ × (a) Lin. regularization (b) 5 Iwan elements

Figure 5.13.: Resonator example, dynamic calculation, resulting limit hysteresis for an excitation frequency of 306 Hz using a linear regularization or 5 Iwan elements for the tangential contact, material E = 2.02 105, ν = 0.28 density 2 · % = 7.884 10−9 Ns /mm4 ·

107 Chapter 5. Extension to the fully dynamical problem behaviour visualized in Figure 5.13a using a linear regularization and in Figure 5.13b using 5 Iwan elements for the tangential contact. For the modiﬁed material parameters, the Iwan models yields a better approximation of the measured hysteresis without further tuning of the used contact parameters. Now the FEM model is in the state of micro slip for a frequency of 306 Hz and the hysteresis matches the measured one ad- equately. For a frequency under 306 Hz the model is in the state of elastic stick and for a frequency of over 306 Hz in the state of macro slip before it switches back to elastic stick using a frequency over 320 Hz. In total this example clearly demonstrates, that it is possible to reproduce real frictional damping with the perturbed Lagrange dual mortar method in the fully dynamic case. A further tuning of the model and the employed material and contact parameters would be possible in order to get an even better match with the measurements, but the focus of this work lies in the numerical contact treatment.

108 Chapter 6. Summary and outlook

In this thesis a perturbed Lagrange dual mortar method for the finite element framework using constitutive contact laws was presented. Thus the contact traction was realized via a Lagrange multiplier as an additional unknown of the system leading to a saddle point problem in the discrete setting. Due to the weak formulation of the balance equation and the contact conditions a consistent force transmission between the contacting bodies is guaranteed by construction. The space of the Lagrange multiplier is chosen to be the dual space of the displacements trace space enabling static condensation of the additional unknown of the system before solving without loosing the optimality of the solution. In the discrete setting, the dual basis functions are constructed fulfilling the biorthogonality condition using a full segmentation of the slave and master surface. Having a closer look at the constitutive contact laws taking the surface roughness on the micro scale into account, in normal direction the transition from soft to hard contact was considered. Numerically this was modelled via a perturbed Lagrange formulation adding a linear, a piece-wise linear or a non-linear regularization operator to the contact inequalities in the continuum formulation. In tangential direction the smooth transition from sticking to slipping modelling elastic stick, micro slip and macro slip using a Coulomb type friction law was considered. The elastic stick was modelled again via a perturbed Lagrange formulation using a linear regularization operator for the tangential contact inequalities. The smooth transition between sticking and slipping is achieved by combining the perturbed Lagrange formulation with a serial-parallel Iwan model in the discrete setting. This method has the advantage of fulfilling the Masing rules for arbitrary load paths. The case of hard contact in normal direction and a perfect Coulomb law in tangential direction was included setting the regularization parameters to infinity. In the first part of the work, the consistent weak formulation as well as the discretization is derived for the quasi-static case and linear finite elements. Thus considering large deformations in the balance equation, the contact interaction is modelled with a combined fix-point Newton approach avoiding complex Jacobian computations stemming from the deformation dependent dual basis functions. However other system non-linearities like plasticity and friction need small load increments to resolve the path dependencies and thus a good start iterate is obtained in each iteration step. More- over a smoothing operator is employed in the weak discrete contact conditions for the

109 Chapter 6. Summary and outlook regularization operator in order to exploit the full advantages of the duality pairing leading to a point-wise decoupling of the contact conditions with respect to the Lagrange multiplier. Reformulating the discrete contact inequalities in normal and tangential direction as non-linear complementary functions, a semi-smooth Newton method can be derived for the more general case of constitutive contact laws leading to a single iteration loop for all system non-linearities. Nevertheless, the dual Lagrange multiplier can be condensed from the global system before solving. The wide applicability and the robustness of the derived perturbed Lagrange dual mortar method is shown by various numerical examples including comparisons to measurements. In the second part of the work, the numerical algorithm for linear finite elements is extended to the quadratic case using a transformation of the standard basis functions. Here two cases are presented: a quadratic-quadratic approach and a quadratic-linear approach. Furthermore a solution to the problem of the evaluation of the dual gap in the discrete setting leading to non-physical effects in some set-ups is studied for the quadratic case. Here a Petrov–Galerkin approach is utilized leading to a primal-dual coupling of the slave and master side for the contact forces guaranteeing a static condensation of the Lagrange multiplier. At the same time the Petrov–Galerkin approach results in a primal-primal coupling of the displacements in the weak contact conditions avoiding non-physical numerical effects in the discrete setting. Also a modification of the algorithm suitable for single or multi-point constraints like cyclic symmetry or directional blocking on the contact interface is proposed. This modification can also be used in case of hanging nodes or cross-points of different contact zones. At the end the perturbed Lagrange dual mortar method with constitutive contact laws is considered for the fully dynamic set-up. The generalized α-method is employed for the discretization in time leading to a numerical dissipative algorithm. Due to a special tuning of the used parameters of the generalized α-method and the employed regularized contact conditions, oscillations of the Lagrange multiplier can be avoided without further modifications of the algorithm. Here special focus is set to the frictional dissipation. Having a look at a resonator example that is dynamically exited, good agreements of the numerically obtained results with the measurements can be obtained using constitutive interface laws. In total the presented perturbed Lagrange dual mortar method represents a robust and powerful tool for complex contact problems using constitutive contact laws. This was confirmed by a comparison of different contact enforcement methods for constitutive contact laws in the Appendix. On the one hand, the comparison shows clearly the superiority of the surface-to-surface discretization induced by the mortar formulation over a simple node-to-surface formulation for all selected examples. On the other hand the dual Lagrange mortar method benefits from the Lagrange multiplier formulation leading to a more robust algorithm compared to a penalty mortar formulation as clearly shown by the last three examples. This benefit in robustness can be seen most dominant in case of overlap in the model as for example in crimped connections

110 used in various industrial applications and indicated with the shroud example. Although modelling the same constitutive interface laws as it is possible with a penalty method, the perturbed Lagrange dual mortar method does not suffer from induced high contact forces in case of overlap and thus has no convergence problems in this case. Also the utilized semi-smooth Newton for friction generally yields a more robust numerical behaviour than the standard radial-return-mapping employed for penalty contact. As an outlook for further studies, one could for example consider the fully consistent linearization of the weak formulation of the constitutive contact laws in the discrete setting. This would allow for even larger load increments for complex simulations including overlap and large deformations. On the other hand one could have a look at even more complex interface laws like for example normal contact with viscous asperities like in lacquer coatings or tangential contact with viscous damping. Also a temperature dependent friction coefﬁcient solving the fully coupled thermo-mechanical contact problem could be considered. Another possibility would be to study the dynamical contact behaviour in the frequency domain using the perturbed Lagrange dual mortar method as needed in frequency calculations or one could focus further on the energy conservation for the dynamic case in the time domain, since this was not focus of this work.

111

Appendix

113

Appendix A. Calculation of the mortar integrals - cutting algorithm

This Section will focus on the calculation of mortar matrices (3.39) facing non-matching meshes in 3D. The calculation of the entries of the mortar matrices can be seen more abstract as the integration of a function f = f(x) in which x is parametrized by the local coordinates ξ and η, x = T (ξ, η). To evaluate the integral in the discrete setting, it is split up to the slave surfaces

nfaces Z X Z f dS = f(xj) dxj (A.1) s,h j γc j=1 T (Q8 )

j Pnnodes j with x = T (ξ, η) = φi(ξ, η) x . Here (A.1) is evaluated performing a seg- k=1 · k mentation of the slave surfaces in triangles based on [111, 112] using the Sutherland– Hodgman polygon clipping algorithm [131], see Firgure A.1. An other possibility was presented in [137] using no segmentation of the surface and simply integrating over a ﬁxed number of Gauss points for each slave surface. For a comparison of the two methods and a hybrid strategy, the interested reader is referred to [35]. Nevertheless, the from numerical point of view expensive segmentation is used here, since only this integration guarantees a consistent traction transmission independent of the meshes and chosen slave/master side and satisﬁes the patch test. Here the integration of f over one slave surface is shown, representatively for a Q8 slave face. The integration for other types of surfaces can be derived analogously. The integral of f can be approximated as a sum over all triangles of the segmentation Z Z 1 Z 1 ∂T ∂T f(x) dx = f(T (ξ, η)) (ξ, η) (ξ, η) dξdη T (Q8) −1 −1 ∂ξ × ∂η ntria ngp (A.2) X X ∂T ∂T wm f(T (ξm , ηm )) (ξm , ηm ) (ξm , ηm ) ≈ k k k ∂ξ k k × ∂η k k k=1 mk=1 performing a transformation from the spatial coordinates to the local coordinates in the reference surface.

115 Appendix A. Calculation of the mortar integrals - cutting algorithm

Slave

Master

Triangulation

Figure A.1.: Triangulation of the slave surface needed for the calculation of the mortar matrices

Slave Slave

Master Master Mean plane

(a) (b) Slave

Master

Mean plane Reference surface (c) (d) Slave

Master

(e)

Figure A.2.: Triangulation of the slave surface

116 Q4 Q8

2 5 1 1 3 4

T3 T6

4 1 2 1

Figure A.3.: Division of surfaces in convex sub-elements

The segmentation in triangles is calculated for each slave surface in the following steps, see Figure A.2:

1. Calculate the normal of the slave surface using the transformation T (Figure A.2a)

nnodes ∂T ∂T X ∂ξ (ξk, ηk) ∂η (ξk, ηk) n = × (A.3) ∂T ∂T (ξk, ηk) (ξk, ηk) k=1 ∂ξ × ∂η

2. Construct a mean plane and project all slave nodes to the mean plane in direction of n switching to piece-wise linear line segments (Figure A.2b)

3. Find a starting set of corresponding master surfaces for an active line search. Therefore for every node of slave surface, a corresponding master surface in direction n is searched and added to the stack of master surfaces.

4. While there is a master surface on the stack: for master surface j a) Divide the master surface into convex sub-surfaces according to Figure A.3 [113] b) For each sub-surface k i. Project the sub-surface master nodes to the mean surface switching to piece-wise linear line segments (Figure A.2b) ii. Generate the clipped polygon with the Sutherland-Hodgman algorithm using the master sub-surface as the convex clipping plane(Figure A.2c). Within the Sutherland-Hodgman algorithm, the master element stack is updated using an active line search.

117 Appendix A. Calculation of the mortar integrals - cutting algorithm

iii. Project clipped polygon back to the slave surface (x, y, z) (ξ, η) (Fig- → ure A.2d) iv. Now that the clipped polygon is described in (ξ, η), the center of gravity is determined and the triangles are generated v. For each triangle A. Generate the integration points in (ξ, η) (Figure A.2d) B. Project integration points to master surface (Figure A.2e) C. Evaluate the normal gap

5. The whole slave surface is triangulated

In the derivation of the algorithm, the center of gravity is used in the calculation of the triangles for every clipping polygon. This assumes that the slave surfaces are not degenerated possibly leading to a center of gravity outside the clipping polygon. Al- ternatively, one could calculate the triangulation with the Delaunay triangulation [28] as used in [106]. It follows a short comment on the used Sutherland-Hodgman algorithm. This algorithm was originally introduced for polygon clipping in computer graphics in order to cut objects with the visual field during the rendering of an image. With this algorithm, a convex polygon can be clipped with an arbitrary polygon in 2D. Therefore all edges of the clipping polygon will be extended to clipping planes and the polygon is successively cut against the clipping planes. Every time the polygon crosses the current clipping plane, this line will be added to the active line search in order to find neighbouring master surfaces. These neighbouring master faces are added to the stack of master surfaces keeping track of all master surfaces, that have already been treated. For a more detailed description of the Sutherland–Hodgman algorithm, the interested reader is referred to [129]. In contrast to the segmentation algorithm described in [111, 112], the integration points are generated directly on the slave surface in the local coordinates instead of generating them in the mean slave plane and projecting them back to slave surface see [31]. This reduces the numerical error in case of quadratic finite elements and significantly reduces the number of projection operations. Even more sophisticated algorithms calculating the slave-master surface segmentation including search trees can be found in [150].

118 Appendix B. A short comparison of contact algorithms

In this Chapter a comparison between three different contact algorithms modelling constitutive contact laws is made. Aside from the perturbed Lagrange dual mortar formulation introduced in this work a simple NTS penalty formulation and a penalty mortar formulation is studied. The penalty mortar method uses the same slave surface segmentation as the mortar method and enforces the contact conditions in the generated Gauss points (see the CalculiX manual version 2.8p2 for further details on the penalty formulations). All three contact algorithms are implemented in the FEM code CalculiX used in this work. But it has to be noted, that only the NTS penalty method and the penalty mortar method are available in the ofﬁcial version of CalculiX. The ﬁrst numerical example is a simple contact patch test with linear elements. A visco-elastic ironing example is studied next followed by a hertz problem for the quadratic case. At the end, an industrially motivated quadratic shroud example is considered.

B.1. Patch test

The patch test is a main tool to test the accuracy of a contact algorithm facing non- matching meshes [25, 34, 153] and was originally introduced in the context of domain decomposition methods. Here two blocks of equal size (1 1 1 mm) and material × × (E = 2.1 105 MPa, ν = 0.0) are placed on top of each other. The lower block is × ﬁxed at the bottom in all direction and a pressure of 100 MPa is applied to the top of the upper block in a quasi-static non-linear geometrical step, see Figure B.1. For this simple set-up, an analytical solution exists and one should get a linear distribution of of the displacements uz and a constant Cauchy stress σzz. For the ﬁnite element approximation, the upper block is meshed with linear Hex8 elements and the lower block is meshed with T et4 elements. The coarser Hex8-mesh is chosen as the slave side and the T et4-mesh is set to be the master side. Now the perturbed Lagrange dual mortar method is compared with the NTS penalty method and the penalty mortar 7 method. For all three contact algorithms the contact constants an = 5 10 MPa/mm, · µ = 0.0 and aτ = are used modelling a linear regularization for the normal and ∞ tangential contact.

119 Appendix B. A short comparison of contact algorithms

Load

Slave

Master

Fixed

Figure B.1.: Patch test, set-up

(a) Penalty NTS (b) Penalty GPTS (c) Mortar

Figure B.2.: Patch test, displacements for different contact enforcement methods

(a) Penalty NTS (b) Penalty GPTS (c) Mortar

Figure B.3.: Patch test, stress σzz for different contact enforcement methods

120 B.2. Visco-elastic ironing problem

As one can see in Figure B.2, one gets a linear distribution of the displacements uz for the perturbed Lagrange dual mortar method and the penalty mortar formulation but not for the NTS penalty method. The effect can be seen even more dominant looking a the Cauchy stress σzz in Figure B.3. The perturbed Lagrange dual mortar method and the penalty method both achieve a consistent load transmission from the slave to the master side using the same segmentation of the slave surface while the NTS penalty formulation suffers from the coarser slave side. Thus this example clearly demonstrates the beneﬁt of a surface-to-surface discretization compared to a node-to- surface discretization.

B.2. Visco-elastic ironing problem

Here a punch that is pressed into a visco-elastic block is considered and then moved across it. The punch, a 30 30mm block with a rounded end has the material properties × of steel, E = 2.1 105 MPa and ν = 0.3 and is set to be the slave body. The visco- × elastic block of size 120 120 60mm has the elastic parameters E = 1.0 MPa, ν = 0.3 × × and is set to be the master body. The whole set-up can be seen in Figure B.4a. Furthermore Hex8 elements are employed with division 64 64 16 for the block and × × division 16 16 16 for the punch, see Figure B.4b. A normal stiffness of an = × × 107 MPa/mm is used for the contact interaction and friction is included setting µ = 0.2 4 and aτ = 5 10 MPa/mm. The viscosity of the block is modelled using the visco-elastic · Kelvin-Voigt material law. Using (2.25) for the calculation of the viscosity parameters 2 2 with τ = 15 s, one ends up with η1 = 2.64 Ns/mm and η2 = 23 Ns/mm in our set-up. Now the following quasi-static linear geometrical steps are performed: Since there is an initial gap of 5.4 mm between the two bodies, the punch is first moved down by 8 mm in one second in 20 increments. Then the punch is moved by a vector of (60, 60, 0) mm − in tangential direction in 10 seconds using 1000 increments. This movement is applied in a quarter cycle as indicated by Figure B.4a. At the end the punch is again moved upwards 8 mm in 3 seconds in 60 increments. One can see the resulting displacements in Figure B.5 for the perturbed Lagrange dual mortar method. Due to viscous material, the path of the punch is memorized by the block as indicated in Figure B.5b. During the third step, the block slowly relaxes to its undeformed configuration after contact, see Figure B.5c. This effect can be seen even better looking at the elastic strain εzz for a 2D cut along the symmetry axis of the punch at time t = 11 s compared to t = 14 s, see Figure B.6. Although very small increments are used for the numerical simulation, only the perturbed Lagrange dual mortar method converged without adaptive time-stepping. The NTS penalty method as well as the penalty mortar method diverged in the first step. Switching to adaptive time stepping, the penalty mortar method converged using three cutbacks in the first step. The NTS penalty method still diverged in the first step after

121 Appendix B. A short comparison of contact algorithms

step 3 step 1

step 2

(a) Set-up (b) Mesh

Figure B.4.: Ironing example, set-up and mesh

(a) mortar t = 1 s (b) mortar t = 11 s (c) mortar t = 14 s

Figure B.5.: Ironing example, displacements uz over time for dual mortar method

(a) mortar t = 11s (b) mortar t = 14s

Figure B.6.: Ironing example, elastic strain εzz for dual mortar method and a 2D cut along the symmetry axis of the punch

122 B.3. Hertz

Method #it total ∅ #it per inc min #it max #it PL dual mortar 2814 2.6 2 15 NTS penalty ( ) ( ) ( ) ( ) − − − − − − − − Penalty mortar (7474) (6.9) (2) (60) − − − − Table B.1.: Ironing example, total number of iterations, average number of iterations and minimum and maximum number of iterations per increment for different contact enforcement methods. Values in brackets were calculated with adaptive time stepping and indicates divergence −

R=8 mm

slave

master

(a) Set-up half-sphere (b) Hex20 mesh (c) Displacements u || || Figure B.7.: Hertz example, half-sphere against rigid plane various cutbacks. As one can see in Table B.1, the dual mortar method had no problem with the soft visco-elastic material as the algorithm needed only an average of 2.6 iterations per load increment. The penalty mortar method needed almost three times more iterations to converge per load increment. Here the average number of iterations per load increment was obtained for the total number of increments of the problem without adaptive time stepping. Thus this example underlines the robustness of the perturbed Lagrange dual mortar method compared to other methods.

B.3. Hertz

As a ﬁrst quadratic example, a simple Hertz example of a half-sphere against a rigid plate is considered [110] , see Figure B.7a. The half-sphere has the radius R = 8 mm and the material properties E = 200 MPa and ν = 0.3. The whole set-up is meshed with Hex20 elements as one can see in Figure B.7b. In contrast to [110] a normal 7 stiffness of an = 10 MPa/mm is used for the contact interaction and friction is included 6 setting µ = 0.2 and aτ = 5 10 MPa/mm. Then a pressure of p = 0.2 MPa is applied ·

123 Appendix B. A short comparison of contact algorithms in a quasi-static non-linear geometrical calculation in 10 load increments. Here the quadratic-quadratic method is used for the perturbed Lagrange dual mortar method. For hard contact and a linear-geometrical set-up, there exists an analytical solutions for the maximum pressure p3 2 pmax = 0.388 4pπE and the contact radius r 3 3 pR π a = 1.109 2E leading to the pressure distribution

1/2 x2 + y2 p(x, y) = pmax 1 , − a2 see [56]. The resulting displacements u can be seen in Figure B.7c. For this set-up, only || || the penalty mortar method and the perturbed Lagange dual mortar method converged, the NTS penalty method diverged after 2 load increments. As one can see in Figure B.8, the resulting displacements uz are almost the same for the penalty mortar method and the perturbed Lagrange dual mortar method. This is also mirrored in the resulting Cauchy stress σzz visualized in Figure B.9. Having a closer look at σzz over the length on a 2D cut along the symmetry axis of the set-up, the solution of the two different methods almost coincide stemming from the same regularized contact laws in both calculations (Figure B.10). Here the red graph indicates the analytical solution for hard contact conditions and a linear-geometrical calculation. One can see clearly how the regularization of the contact laws lead to a smoothing of the stress σzz in the transition between contact and no contact and to a lower maximum value. Thus both surface-to-surface contact enforcement methods, perturbed Lagrange dual mortar and penalty mortar, converged for this example, the dual mortar method shows are more robust numerical behaviour as indicated by the number of iterations seen in Table B.2. The penalty mortar method needed over three times more iterations in total for the 10 load increments compared to the perturbed Lagrange dual mortar method. Also the minimum number of iterations is doubled for the penalty mortar method.

B.4. Shroud

The last example is an industrial motived one, more precisely the contact in the shroud of a rotating turbine part. A disk with 72 blades is considered, see Figure B.11a. Here the ﬁrtree contact between the disk and the blade is not modelled and the shroud contact of three blades is studied in detail as one can see in Figure B.11b. As in reality the shroud is twisted against the blade by a small angle leading to an interlock of the

124 B.4. Shroud

(a) PL dual mortar (b) Penalty mortar

Figure B.8.: Hertz example, displacement uz in the contact zone for different contact enforcement methods

(a) PL dual mortar (b) Penalty mortar

Figure B.9.: Hertz example, stress σzz in the contact zone for different contact enforcement methods

20 mortar 15 STS penalty analytical zz σ 10 Stress

0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 − − − Length

Figure B.10.: Hertz example, stress σzz in the contact zone for different contact enforcement methods, 2D cut and analytical solution for linear geometrical calculation with hard contact conditions

125 Appendix B. A short comparison of contact algorithms

Method #it total ∅ #it per inc min #it max #it PL dual mortar 59 5.9 5 8 NTS penalty − − − − Penalty mortar 194 19.4 11 30

Table B.2.: Hertz example, total number of iterations, average number of iterations and minimum and maximum number of iterations per increment for different contact enforcement methods

Shroud

30 mm

rigid

Blade 142 mm

Disk

fixed

(a) Full disk (b) Set-up

Shroud 1

Shroud 2

Figure B.11.: Shroud example

126 B.4. Shroud

(a) u (b) active nodes || || Figure B.12.: Shroud example, displacements u and contact zone || || blades after the assembling process, see Figure B.11d for a zoom onto one contact zone. For this abstracted example an angle of 1◦ is used leading to a small overlap of the different blades in the undeformed situation. Two untwisted rigid shroud parts are added to the finite element set-up to complete the reduced model. The whole set-up is meshed with Hex20 elements as one can see in Figure B.11c. In order to guarantee static-mounting, the support of the disk is fixed in all directions. The material properties are set to E = 7.5 104 and ν = 0.3. For the contact interaction a normal 7 · stiffness of an = 5 10 MPa/mm, a friction coefficient µ = 0.4 and a tangential stiffness 4 · aτ = 7.5 10 MPa/mm is used. Again the quadratic-quadratic method is used for the · perturbed Lagrange dual mortar method. The overlap is resolved in one quasi-static non-linear geometrical calculation and the perturbed Lagrange dual mortar method is compared to the performance of the NTS pentalty and the penalty mortar method for this example. The resulting displacements u for the dual mortar method are visualized in Figure || || B.12a showing clearly the twisting of the blades due to the overlap at the beginning. The contact zone is shown in Figure B.12b in red looking at the inside of the shrouds. For this example only the perturbed Lagrange dual mortar method was able to resolve the overlap in one increment. The NTS penalty method as well as the penalty mortar method diverged. Enabling the adaptive time stepping and resolving the overlap linearly over the step, the penalty mortar method managed to converge in two load increments while the NTS penalty method still diverged. Nevertheless, the penalty mortar method needed 6 times more iterations in total including the cutbacks than the dual mortar method, see Table B.3. Thus the dual mortar method is more robust facing overlap of the contacting bodies at the beginning of the calculation even though the same interface

127 Appendix B. A short comparison of contact algorithms

Method #it direct #it with adaptive time stepping PL dual Mortar 23 23 NTS penalty − − Penalty mortar 139 − Table B.3.: Shroud example, total number of iterations without and with adaptive time stepping for different contact enforcement methods laws are used for all three methods. The reason for that could be that in the penalty formulations, the contact conditions are enforced by a Neumann condition (force driven), while the perturbed Lagrange dual mortar method uses a Robin condition (combined displacement-force driven).

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