• Curriculum Vitae
• Forschungsschwerpunkte
• Lehrveranstaltungen
• Betreuung: Studien- und Diplomarbeiten, Bachelor- und Masterarbeiten
• Veröffentlichungen
• Vorträge 

Forschungsschwerpunkte:

Young Investigator Group Computer Aided Material Modeling (CAMM) [YIG]
(YIG Computerbasierte Materialmodellierung)


‎Ordnungsreduktionsverfahren zur nicht-linearen Homogenisierung heterogener Materialien
(Order reduction methods for the homogenization of non-linear heterogeneous materials)

Modellierung der Thermoplastizität metallischer Werkstoffe
(Modeling of thermoplasticity of metallic materials)

Numerische Implementierung von Mehrfeldproblemen mit der Methode der Finiten Elemente
(Numerical implementation of multifield problems using the Finite-Element-Method)


Netzgenierung für mikrostrukturierte Werkstoffe:

Räumliche Diskretisierung eines partikelverstärkten Materials



 

Deformierte polykristalline Mikrozugprobe

Sommersemester 2013

• Vorlesung Technische Mechanik II
• Vorlesung Finite-Elemente-Methode (FEM)

Wintersemester 2012/13

• Vorlesung Mikrostrukturcharakterisierung und –modellierung (MCM)

Sommersemester 2012

• Vorlesung Finite-Elemente-Methode (FEM)

Wintersemester 2011/12

• Vorlesung Mikrostrukturcharakterisierung und –modellierung (MCM)

Sommersemester 2011

• Vorlesung Finite-Elemente-Methode (FEM)
• Übungen zu Rechnerunterstützte Mechanik II (RUM II)
• Sprechstunde zu Rechnerunterstützte Mechanik II (RUM II)

Wintersemester 2010/11

• Übungen zu Rechnerunterstützte Mechanik I (RUM I)
• Übungen zu Technische Mechanik I (TM I)

Sommersemester 2010

• Übungen zu Rechnerunterstützte Mechanik II (RUM II)
• Übungen zu Technische Mechanik II (TM II)

Wintersemester 2009/2010

• Vorlesung Rechnerunterstützte Mechanik I (RUM I)
• Übungen zu Rechnerunterstützte Mechanik I (RUM I)

Sommersemester 2009

• Übungen zu Rechnerunterstützte Mechanik II (RUM II)

Wintersemester 2008/09

• Übungen zu Rechnerunterstützte Mechanik I (RUM I)
• Höhere Technische Festigkeitslehre (HTF)
• Block-Veranstaltung: Implementierung von Stoffgesetzen in FE-Software

2013

Diplomarbeit von Max Hodapp
Implementierung ordnungsreduktionsbasierte Homogenisierungsmethoden auf Nvidia GPUs
Betreuer:
Dipl.-Ing. Dipl.-Math. techn. Felix Fritzen
Prof. Dr.-Ing. habil. Thomas Böhlke

2012
‎
Diplomarbeit von Matthias Leuschner
Ordnungsreduktionsbasierte Homogenisierung visko-plastischer Materialien
Betreuer:‎
Dipl.-Ing. Dipl.-Math. techn. Felix Fritzen
Prof. Dr.-Ing habil. Thomas Böhlke

2011

Bachelorarbeit von Johanna Beer
Numerische Charakterisierung des inelastischen Grenzflächenverhaltens von Faserausdrückversuchen zur Erstellung einer Datenbasis für inverse Optimierungsverfahren
Betreuer:
Prof. Dr.-Ing. habil. Thomas Böhlke
Dipl.-Ing. Dipl.-Math. techn. Felix Fritzen

Bachelorarbeit von Benedikt Dorschner
Inverse Bestimmung der Materialparameter von Kohäsivzonenmodellen aus Faserausdrückversuchen
Betreuer:
Prof. Dr.-Ing. habil. Thomas Böhlke
Dipl.-Ing. Dipl.-Math. techn. Felix Fritzen

Bachelorarbeitvon Michael Pecher
Generierung randomisierter Strukturen für poröse Materialien
Betreuer:
Prof. Dr.-Ing. habil. Thomas Böhlke
Dipl.-Ing. Dipl.-Math. techn. Felix Fritzen

Diplomarbeit von Roman Stierhof
Finite Elemente Implementierung thermo-mechanisch gekoppelter Probleme unter Einbeziehung von Operatorsplittingmethoden
Betreuer:
Prof. Dr.-Ing. habil. Thomas Böhlke
Dipl.-Ing. Dipl.-Math. techn. Felix Fritzen

Diplomarbeit von Florian Rieger
Finite Elemente Implementierung thermo-mechanisch gekoppelter Probleme
Betreuer:
Prof. Dr.-Ing. habil. Thomas Böhlke
Dipl.-Ing. Dipl.-Math. techn. Felix Fritzen

Studienarbeit von Stefan Liebenstein
Untersuchung des konstitutiven Grenzflächenverhaltens im Rahmen generalisierter Standard Materialien und deren Implementierung in ABAQUS
Betreuer:
Prof. Dr.-Ing. habil. Thomas Böhlke
Dipl.-Ing. Dipl.-Math. techn. Felix Fritzen

2010

Diplomarbeit von Christoph Engelhardt
Mikrostrukturelle Modellierung kurzfaserverstärkter Werkstoffe mit Faserclustern und deren Einfluss auf die effektiven elastischen Eigenschaften
Betreuer:
Prof. Dr.-Ing. habil. Thomas Böhlke
Dipl.-Ing. Dipl.-Math. techn. Felix Fritzen

2009

Diplomarbeit von Eugen Oswald

Belastungsoptimierte Gusswanddimensionierung bei Umformwerkzeugen unter Berücksichtigung von statischen und dynamischen Lastfällte mit Hilfe der Finiten-Elemente-Methode

Betreuer:

Dipl.-Ing. Dipl.-Math. techn. Felix Fritzen

Prof. Dr.-Ing. habil. Thomas Böhlke

2008
 

Studienarbeit von Johannes Wippler
Implementierung von elasto-plastischen Materialgesetzen
Betreuer:
Dipl.-Ing. Dipl.-Math. techn. Felix Fritzen
Prof. Dr.-Ing. habil. Thomas Böhlke

(Bold Roman: Refereed Journal Articles)

2015

Leuschner, M., Fritzen, F., van Dommelen, J.A.W., Hoefnagels, J.P.M.:
Potential-based constitutive models for cohesive interfaces: Theory, implementation and examples.
Composites: Part B 68, 38-50 (2015)

2014

Fritzen, F., Hodapp, M., Leuschner, M.:
GPU accelerated computational homogenization based on a variational approach in a reduced basis framework.
Computer Methods in Applied Mechanics and Engineering 278, 186-217 (2014)

2013

Fritzen, F., Forest, S., Kondo, D., Böhlke, T.:
Computational homogenization of porous materials of Green type.
Comput Mech 52, 121-134 (2013)
DOI 10.1007/s00466-012-0801-z
Fritzen, F., Leuschner, M.:
Reduced basis hybrid computational homogenization based on a mixed incremental formulation.
Computer Methods in Applied Mechanics and Engineering 260, 143–154 (2013)

Fritzen, F., Thomas Böhlke:
Reduced basis homogenization of viscoelastic composites.
Composites Science and Technology 76, 84-91 (2013)

2012

Fritzen, F., Forest, S., Böhlke, T., Kondo, D., Kanit, T.:
Computational homogenization of elasto-plastic porous metals.
International Journal of Plasticity 29, 102-119 (2012) DOI: 10.1016/j.ijplas.2011.08.005

2011

Fritzen, F.:
Microstructural modeling and computational homogenization of the physically linear and nonlinear constitutive behavior of micro-heterogeneous materials
Doctoral dissertation, Schriftenreihe Kontinuumsmechanik im Maschinenbau Nr. 1, KIT Scientific Publishing, Karlsruhe (2011)
https://publikationen.bibliothek.kit.edu/1000023534

Fritzen, F., Böhlke, T.:
Nonlinear homogenization using the nonuniform transformation field analysis
PAMM • Proc. Appl. Math. Mech. 11, 519-522 (2011)
DOI: 10.1002/pamm.201110250

Brylka, B., Fritzen, F., Böhlke, T., Weidenmann, K.A.:
Influence of micro-structure on fibre push-out tests.
PAMM • Proc. Appl. Math. Mech. 11, 141-142 (2011)
DOI: 10.1002/pamm.201110062

Fritzen, F., Forest, S., Böhlke, T., Kondo, D., Kanit, T.:
Computational homogenization of elasto-plastic porous metals.
International Journal of Plasticity, in press (2011)
DOI: 10.1016/j.ijplas.2011.08.005

Fritzen, F., Böhlke, T.:
Nonuniform transformation field analysis of materials with morphological anisotropy.
Composite Science and Technology 71, 433-442 (2011)

Fritzen, F., Böhlke, T.:
Periodic three-dimensional mesh generation for particle reinforced composites with application to metal matrix composites.
Int. J. Solids ‎Struct. 48, 706-718 (2011)

Wippler, J., Fünfschilling, S., Fritzen, F., Böhlke, T., Hoffmann, M.J.:
Homogenization of the thermoelastic properties of silicon nitride.
Acta Materialia 59, 6029-6038 (2011)
DOI: 10.1016/j.actamat.2011.06.011

2010

Brylka, B., Fritzen, F., Böhlke, T., Weidenmann, K.:
Study of Experimental Methods for Interface Problems Based on Virtual Testing.
PAMM • Proc. Appl. Math. Mech. 10, 109-110 (2010)
DOI: 10.1002/pamm.201010047

Fritzen, F., Böhlke, T.:
Analytical inversion of the Jacobian for a class of generalized standard materials.
PAMM • Proc. Appl. Math. Mech. 9, 407-408 (2010)
DOI: 10.1002/pamm.200910177

Fritzen, F., Böhlke, T.:
Three-dimensional finite element implementation of the nonuniform transformation field analysis.
Int. J. Numer. Meth. Engng. 84, 803-829 (2010)
DOI: 10.1002/nme.2920

Fritzen, F., Böhlke, T.:
Influence of the type of boundary conditions on the numerical properties of unit cell problems.
Technische Mechanik, 30(4), 354-363 (2010)

Jöchen, K., Böhlke, T., Fritzen, F.:
Influence of the crystallographic and the morphological texture on the elastic properties of fcc crystal aggregates.
Solid State Phenomena, Vol. 160, 83-86 (2010)
DOI: 10.4028/www.scientific.net/SSP.160.83

2009

Böhlke, T., Fritzen, F., Jöchen, K., Tsotsova, R.:
Numerical methods for the quantification of the mechanical properties of crystal aggregates with morphologic and crystallographic texture.
International Journal of Material Forming 2, 915–917, Suppl. 1 (2009)
DOI: 10.1007/s12289-009-0470-4

Fritzen, F., Böhlke, T.:
Homogenization of three-dimensional micro-heterogeneous materials using nonuniform transformation fields.
7th EUROMECH Solid Mechanics Conference, J. Ambrosio et.al. (eds.), 7–11 September 2009, Lisbon, Portugal (2009)

Fritzen, F., Böhlke, T., Schnack, E.:
Periodic three-dimensional mesh generation for crystalline aggregates based on ‎Voronoi tessellations.
Computational Mechanics 43(5), 701-713 (2009)
DOI: 10.1007/s00466-008-0339-2

Fritzen, F., Böhlke, T.:
Homogenization of the physically nonlinear properties of three-dimensional metal matrix composites using the nonuniform transformation field analysis.
Proceedings of the 17th International Conference on Composite Materials, 27-31 July 2009, Edinburgh, Scotland (2009)
[pdf]

2008

Jöchen, K., Böhlke, T., Fritzen, F.:
On estimates for the effective shear modulus of cubic crystal aggregates.
PAMM, 8(1), 10551-10552 (2008)
DOI: 10.1002/pamm.200810551

Fritzen, F., Böhlke, T., Schnack, E.:
Periodic three-dimensional mesh-generation for Voronoi tessellations with application to cubic crystal aggregates.
Proc. Appl. Math. Mech. 8, 10545-10546 (2008)
DOI: 10.1002/pamm.200810545

2007

Fritzen, F., Böhlke, T., Schnack, E.:
Modeling of latent energy storage effects in thermoplasticity of metals.
PAMM • Proc. Appl. Math. Mech. 7, 4080017–4080018 (2007)
DOI: 10.1002/pamm.200700449

Piat R. , Roser M., Fritzen F., Schnack E.:
Numerische Modellierung des Rissfortschritts in porösen CVI-CFC Verbundwerkstoffen.
MP Materialprüfung 49(4), 170-176 (2007)

Fritzen, F., Böhlke, T., Schnack, E.:
Periodic three-dimensional mesh-generation for Voronoi tessellations with application to cubic crystal aggregates.
Proc. Appl. Math. Mech. 8, 10545-10546 (2008)
DOI: 10.1002/pamm.200810545

Fritzen, F., Böhlke, T., Schnack, E.:
Thermoplasticity with nonlinear self-heating.
Preprint Kontinuumsmechanik Nr. 2008-6

Fritzen, F., Böhlke, T., Schnack, E.:
Periodic three-dimensional mesh generation for crystalline aggregates based on Voronoi tessellations.
Preprint Kontinuumsmechanik Nr. 2008-4

2014

27.-29.08.2014
Leuschner, M., Fritzen, F.:
Towards reduced-order homogenization for materials with interfaces
2nd International Workshop on Order-Reduction Methods for application to Mechanics of ‎Materials, Bad Herrenalb, Germany

20.-25.07.2014
Leuschner, M., Fritzen, F.:
A potential-based constitutive interface model for application in reduced order nonlinear homogenization
WCCM XI - ECCM V - ECFD VI, Barcelona, Spain

25.06.2014
Leuschner, M., Fritzen, F.:
A potential-based framework for the modeling of cohesive interfaces with examples for damaging and mixed-mode behaviour
STW Workshop Interfacial Mechanics, Eindhoven, The Netherlands

10.-14.03.2014
Leuschner, M., Fritzen, F.:
A potential-based framework for cohesive zone models
85th annual meeting of GAMM, Erlangen-Nuremberg, Germany

11.03.2014
Fritzen, F., Forest, S., Böhlke, T., Kondo, D.:
Computational limit analysis of incompressible and compressible porous materials
GAMM Conference 2014, Erlangen-Nürnberg, Germany

19.03.2014
Fritzen, F., Forest, S.:
Voided materials with internal length scale: a computational study
EUROMECH563 colloquium, Cisterna di Latina, 17.-21.03.2014, Italy

2013

06.11.2013
Leuschner, M., Fritzen, F.:
Reduced order homogenization for viscoplastic materials with cohesive interfaces
Seminar presentation, Department of Mechanical Engineering, Eindhoven University of ‎Technology, Netherlands

30.09.-02.10.2013
Leuschner, M., Fritzen, F.:
Reduced order nonlinear homogenization
‎5th GACM Colloquium on Computational Mechanics, Hamburg, Germany

25.-29.08.2013
Leuschner, M., Fritzen, F.:
Reduced order homogenization for viscoplastic materials
‎1st International Workshop on Order-Reduction Methods for application to Mechanics of ‎Materials, Bad Herrenalb, Germany

04.-07.03.2013
Fritzen, F.:
Scale bridging techniques based on incremental energy minimization in a reduced basis setting
Workshop über gekoppelte Deskriptorsysteme, Geseke (Westfalen), Germany

07.02.2013
Fritzen, F., Böhlke, T.:
Efficient reduced basis homogenization of visco-elastic composite materials
Invited presentation, Laboratoire Modelisation et Simulation Multiéchelles (MSME), Université Paris-Est, Marne-la-Vallée, France

03.-08.01.2013
Fritzen, F., Forest, S., Kondo, D., Böhlke, T.:
Computational Homogenization of Plastically Compressible Porous Materials
International Symposium Plastiticity, Nassau, Bahamas

2012

06.-07.12.2012
Fritzen, F.:
Mixed variational formulation of a reduced basis approach to nonlinear homogenization
25th Workshop Research in Mechanics of Composites, Bad Herrenalb, Germany

08.-12.10.2012
Fritzen, F.:
Computational Approaches to Linear and Nonlinear Two-scale Homogenization
COMMAS Summer School 2012 ’Computational Mechanics of Materials and Structures’, Stuttgart, Germany

10.-14.09.2012
Fritzen, F., T. Böhlke, T.:
Order-reduction based computational homogenization of three-dimensional visco-elastic properties of composites ECCOMAS 2012, Vienna, Austria

03.-07.09.2012
Fritzen, F.:
Model order reduction in nonlinear homogenization: conceptual ideas and numerical examples
Workshop Multiscale Material Modeling, Bad Herrenalb, Germany

24.-27.04.2012
Fritzen, F., T. Böhlke, T.:
Order-reduction based homogenization – theory, properties and examples
Young Investigator Conference, Phd Olympiad, Apr. 24-27, 2012, Aveiro, Portugal

26.-30.03.2013
Fritzen, F., Böhlke, T., Forest, S.:
Efficient simulation of non-linear micro-heterogeneous structures using an order-reduction approach
GAMM Jahrestagung 2012, YR-Minisymposium, Darmstadt, Germany

20.-21.01.2012
Fritzen, F., Böhlke, T.:
Structural simulation of composites with morphological anisotropy based on the NTFA
GAMM Seminar on Microstructures 2012, Essen, Germany

2011

07.-09.09.2011
Fritzen, F., Forest, S., Böhlke, T.:
Computational homogenization of porous metals
International Conference on Computational Plasticity, Barcelona, Spanien

31.08.-02.09.2011
Fritzen, F., Böhlke, T.:
Nonuniform transformation field analysis of composite materials: Theory and structural application
International Conference on Material Modeling 2, Paris, Frankreich

14-17.06.2011
Fritzen, F., Böhlke, T.:
Nonuniform Transformation Field Analysis of composites with high morphological anisotropy
SimTECH 2011, Stuttgart

01.06.2011 - Invited Lecture
Fritzen, F.:
Microstructural modeling and computational homogenization of the physically linear and nonlinear constitutive behavior of micro-heterogeneous materials
Seminar des Lehrstuhls für Kontinuumsmechanik, Ruhr-Universität Bochum

18.-21.04.2011 - Keynote lecture
Fritzen, F., Böhlke, T.:
Nonlinear homogenization using the nonuniform transformation field analysis
GAMM Jahrestagung 2011, Graz, Österreich

29.03-01.04.2011
Fritzen, F., Böhlke, T.:
Nonuniform Transformation Field Analysis of composites with anisotropic morphology
DGM Tagung Verbundwerkstoffe/Werkstoffverbunde 2011, Chemnitz

2010

29.11.-01.12.2010
Fritzen, F., Engelhardt, C., Böhlke, T.: 
Influence of the fiber clustering on the apparent and effective elastic properties of short fiber reinforced composites
23rd International Workshop Research in Mechanics of Composites, Bad Herrenalb

29.11.-01.12.2010
Brylka, B., Fritzen, F., Böhlke, T., Weidenmann, K.: 
Influence of microstructure on fibre delamination test
23rd International Workshop Research in Mechanics of Composites, Bad Herrenalb

13.10.2010
Fritzen, F., Böhlke, T.:
Nonuniform transformation field analysis of particulate metal matrix composites with anisotropic microstructure
Seminaire à ’ONERA Chatillon, France

14.09.2010
Fritzen, F., Forest, S., Kondo, D., Böhlke, T., Kanit, T.:
Homogenization of porous metal plasticity: computational investigations
In collaboration with Université Lille 1 - Sciences and Technologies, Grand Cocas, Evry, France

Fritzen, F., Böhlke, T.:
Nonuniform transformation field analysis of particulate metal matrix composites with anisotropic microstructure
ECCM 2010, IV European Conference on Computational Mechanics, Palais des Congrès, May 16-21, 2010, Paris, France

Fritzen, F., Böhlke, T.:
Nonuniform Transformation Field Analysis of Composites with Anisotropic Morphology
GAMM 2010, 81st Annual Meeting of the International Association of Applied Mathematics and Mechanics, March 22-26, 2010, Karlsruhe, Germany

2009

02.12.2009
F. Fritzen, T. Böhlke:
Nonuniform Transformation Field Analysis of materials with anisotropic microstructure
22. Workshop Composite Forschung in der Mechanik, 02.12.2009, Paderborn, Germany

01.-02.12.2009
F. Fritzen, T. Böhlke:
Nonuniform Transformation Field Analysis: Theoretical investigation and numerical examples
22.Workshop Composite Forschung in der Mechanik, 3. Forum Metallplastizität, 01.-02.12.2009, Paderborn, Germany

04.11.2009
F. Fritzen, T. Böhlke (invited):
Computational homogenization of materials with microstructure: Microstructural modelling and numerical implementation of the Nonuniform Transformation Field Analysis
Seminar "Mathematische Materialwissenschaften", Zentrum für Technomathematik, Universität Bremen, 04.11.2009, Bremen, Germany

September 2009
F. Fritzen, T. Böhlke, K. Jöchen:
Homogenization Of Three-Dimensional Micro-Heterogeneous Materials Using Nonuniform Transformation Fields
7th Euromech, Sep 7th-11th, 2009, Lisboa, Portugal

September 2009
F. Fritzen, T. Böhlke:
Numerical Implementation of the Nonuniform Transformation Field Analysis for three-dimensional composites with structural applications
1st International Conference on Material Modeling, Dortmund, Sep 15th-17th, 2009, Germany

July 2009
F. Fritzen, T. Böhlke:
Homogenization of the physically nonlinear properties of three-dimensional metal matrix composites using the nonuniform transformation field analysis
17th International Conference on Composite Materials, 27-31 July 2009, Edinburgh, Scotland

26.06.2009
F. Fritzen, T. Böhlke:
Nonuniform Transformation Field Analysis: Theoretical investigation and numerical examples
GAMM Seminar "Multiscale Material Modelling", 26.06.2009, Karlsruhe, Germany

11.02.2009
F. Fritzen, T. Böhlke:
Homogenization of Non-linear Three-dimensional Metal Matrix Composites Using the Non-uniform Transformation Field Analysis
GAMM 2009, 09.-13.02.2009, Danzig, Polen

2008

02.12.2008
F. Fritzen, T. Böhlke:
Boundary condition induced effects on the homogenization of heterogeneous materials: Theoretical and numerical investigations
21. Workshop Composite-Forschung in der Mechanik
Bad Herrenalb, 01.12.-03.12.2008, Germany

July 2008
T. Böhlke, F. Fritzen, K. Jöchen, R. Tsotsova:
Continuum mechanical models for anisotropic polycrystalline materials: Phenomenological and micromechanical approaches
Second GAMM-Seminar on Multiscale Material Modellingm Stuttgart, Germany

July 2008
T. Böhlke, F. Fritzen, K. Jöchen:
Numerical Homogenization of Elastic and Plastic Properties of Polycrystalline Microcomponents
8th World Conference on Computational Mechanics, Venice, Italy

18.04.2008
F. Fritzen, K. Jöchen, T. Böhlke, E. Schnack:
Examination of boundary effects and sample sizes fore the numerical homogenization of physically non-linear metal-ceramic composites
IWRMM Statusseminar, Karlsruhe, Germany

03.04.2008
F. Fritzen, K. Jöchen, T. Böhlke, E. Schnack:
Examination of boundary effects and sample sizes fore the numerical homogenization of physically non-linear metal-ceramic composites
GAMM-Jahrestagung, Bremen, Germany

February 2008
T. Böhlke, F. Fritzen, K. Jöchen:
Numerical Homogenization of Elastic and Plastic Properties of Polycrystalline Microcomponents Computational Mechanics of Polycrystals
Bad Honnef, Germany

2007

06.12.2007
F. Fritzen, T. Böhlke, E. Schnack:
A model of latent energy storage processes with numerical examples
20. Workshop Composite Forschung in der Mechanik, Paderborn, Germany

12.10.2007
F. Fritzen:
Modeling of non-linear self-heating of metals in a thermodynamically consistent thermoplasticity model
2nd GACM Colloqium of computational mechanics, Technical University of Munich, Germany

19.07.2007
F. Fritzen:
A consistent thermo-plasticity model for the constitutive modeling of interpenetrated metal-ceramic-composites
6th International Congress on Industrial and Applied Mathematics, Zurich, Switzerland

27.04.2007
F. Fritzen, T. Böhlke:
Modellierung der temperaturabhängigen Selbsterwärmung bei der Thermoplastizität von Metallen
Arbeitskreis Stoffgesetze, Universität Paderborn, Paderborn, Germany

20.04.2007
F. Fritzen, T. Böhlke:
Modellierung der temperaturabhängigen Selbsterwärmung bei der Thermoplastizität von Metallen
IWRMM, Statusseminar, Universität Karlsruhe (TH), Karlsruhe, Germany

2006

06.10.2006
F. Fritzen:
Mathematical aspects of gradient-free shape optimization
International Conference on Multifield Problems, University of Stuttgart, Germany