Linear Computational Structural Mechanics for Wind Energy Systems

Module Linear Structural Computational Mechanics for Wind Energy Systems i Lecture Notes Linear Computational Structural Mechanics for Wind Energy Systems Prof. Dr.-Ing. habil. Detlef Kuhl Online M.Sc. Wind Energy Systems University of Kassel and Fraunhofer IWES www.uni-kassel.de/wes

ii D. Kuhl, Online M.Sc. Wind Energy Systems, University of Kassel Lecture Notes Linear Computational Structural Mechanics for Wind Energy Systems Prof. Dr.-Ing. habil. Detlef Kuhl 1 st Edition, December 2013 Online M.Sc. Wind Energy Systems (wes.online) University of Kassel Department of Civil and Environmental Engineering Institute of Mechanics and Dynamics Prof. Dr.-Ing. habil. Detlef Kuhl Mönchebergstraße 7 34109 Kassel, Germany www.uni-kassel.de/fb14/mechanics Prof. Dr.-Ing. habil. Detlef Kuhl, 3. November 2015 All rights reserved. In particular, the right to translate the text of this document into another language is reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any other information storage and retrieval system, without written permission of the author.

Module Linear Structural Computational Mechanics for Wind Energy Systems iii Lecture Linear Computational Structural Mechanics for Wind Energy Systems Abbreviation LCSM Abstract The present set of lecture notes is designed to assist the students of the online master s study Wind Energy Systems with their learning in linear finite element methods and linear structural dynamics of wind energy systems. For this reason it includes sections on the theory development, application of methods in selected examples and program flowcharts, as well as coding instructions supporting the homework and the final case study of the course. After an introduction to numerical methods for the static and dynamics simulation of structures, a brief review of the history and a first course classification of the applied models and methods the finite element method will be newly invented for the simple case of one dimensional continua. This all ows for an artless but also completed representation of the main ideas of the finite element method as well as the comparison of numerical and analytical solutions. Afterwards advanced topics of the one-dimensional finite element method will be extended in order to enable calculation of space frameworks, to obtain higher order accurate p finite element methods and also residual based error estimates, to include inhomogeneous DIRICHLET boundary, to have a first idea about static and dynamic solution procedures and, finally, to prepare the development of the finite element method for the simulation of general three dimensional structures. The development of the general n dimensional p finite element method starts with a brief repetition of linear continuum mechanics, then the finite element representations of virtual work term are realized and afte r- wards specialized for a family of three and two dimensional finite elements. The following chapter 'eigenvalue analysis' will provide methods the analyze the dynamic characteristics of stru c- tures and to provide a analytical solution of simple structural dynamics with con later on co m- pared with numerical results to validate the program development of time integrations schemes of the central difference and NEWMARK-type discussed in the following two chapters.

iv D. Kuhl, Online M.Sc. Wind Energy Systems, University of Kassel Online M.Sc. Wind Energy Systems (wes.online) Since the foundation of the University of Kassel in 1971, an awareness of the environment has always been an important part of science and education. More than 60 professors and their scientific employees work on environmental questions within departments, interdisciplinary research centers and institutes. The number of environmental research projects and environmental study programs has been increasing continuously over the years. Bicultural and international on-campus study programs with innovative teaching concepts are part of this portfolio. The Online M.Sc. Wind Energy Systems is another milestone in this story of a green university. The specific expertise of the University in the fields of computational science and engineering with regards to renewable energy systems is incorporated into the learning content of this wind energy study program. These competencies are extended by the Fraunhofer Institute of Wind Energy and Energy System Technology (IWES). The Fraunhofer IWES is one the largest institutes for wind energy and energy system technology in Europe. Lecturers from the Fraunhofer IWES introduce further aspects into the study program, such as the economic integration of a large amount of wind energy into the energy supplier system. Students also gain knowledge about how to design and develop innovative concepts for individual components of the wind energy converter systems, like the nacelle systems, rotor blades or support structures. Beside lecturers from Fraunhofer IWES and University Kassel leading experts from industry and cooperating universities enrich the team of Online M.Sc. Wind Energy Systems. The teaching methods of the Online M.Sc. Wind Energy Systems are new and innovative. The program is a part-time, extra-occupational Master's program. It is explicitly developed for students who would like to study alongside their job or family responsibilities. Our aim is also to provide a worldwide global student body with the knowledge of the se two institutions in the area of renewable and wind energy. We would like to extend the knowledge of male and female engineers from regions where this technology is not easily accessed. For this reason we teach the 28 modules of our program 100% online. We welcome you in our program. Partners offering the Online M.Sc. Wind Energy Systems (wes.online)

Module Linear Structural Computational Mechanics for Wind Energy Systems v Online M.Sc. wind Energy Systems Accredited by Reputational partners from research and industry Project founding and project alliance for the development of premium online master s courses

vi D. Kuhl, Online M.Sc. Wind Energy Systems, University of Kassel Curriculum Vitae Prof. Dr.-Ing. habil. Detlef Kuhl University of Kassel Faculty of Civil and Environmental Engineering Institute of Mechanics and Dynamics Moenchebergstrasse 7 34109 Kassel Email: kuhl@uni-kassel.de Webpage: www.uni-kassel.de/fb14/mechanics Scientific Vitae Prof. Dr.-Ing. habil. Detlef Kuhl has studied aerospace engineering at the University of Stuttgart with the main focus on regenerative energy systems and wind energy. 1992 he has finished his master s thesis at M.A.N Technology in Munich about the mechanical analysis and experimental verification of the wind turbine WKA 60 on the island Helgoland. His professional career has started as designing engineer at the wind turbine manufacturer Enercon in Aurich. Knowing the demands of wind engineering he has decided to devote his life to the investigation, application and teaching of methods of computational mechanics, in particular of structures, wind turbines and general multifield problems. The scientific qualification of Prof. Dr.-Ing. habil. Detlef Kuhl has started with the PhD study at the University of Stuttgart about dynamics of shell structures, finished in 1996. The years 1996 to 1998 he has spend his post doc as head of the research group Thermomechanical Modeling Group at the German Aerospace Center in Lampoldshausen, as postdoctoral fellow at the Department of Aeronautics, Imperial College of Science, Technology and Medicine in London (1997) and as senior scientist and lecturer at the Institute of Structural Mechanics, Ruhr University Bochum. He finished his Habilitation about the simulation of time dependent multi eld problems in 2004. Beside the thesis work he has researched about the thermo mechanical modeling and simulation of rocket combustion chambers and the computational analysis of tensegrity structures. Since 2007 Detlef Kuhl is Professor for Mechanics and Dynamics at the Faculty of Civil and Env i- ronmental Engineering, University of Kassel. He is teaching bachelor courses on soli d mechanics, master s courses on computational solid mechanics and is member of the teaching team of the lecture series Simulation of Wind Energy Systems. His research is related to the computational analysis of dynamics of structures, the modeling and simulation of thermo mechanical, electro magneto thermo mechanical and the fluid structure interaction, the simulation of tensegrity structures and wind turbines and didactical concepts of online university teaching. Furthermore, Prof. Dr.-Ing. habil. Detlef Kuhl is the Academic Director of Online M.Sc. Wind Energy Systems (wes.online), Dean of Students of Faculty of Civil and Environmental Engineering, University of Kassel, Head of Chair of Mechanics and Dynamics, University of Kassel and Guest Professor at Vietnamese German University (VGU), Binh Duong New City, Vietnam. In

Module Linear Structural Computational Mechanics for Wind Energy Systems vii 2015 he was as visiting Professor at the Department of Mathematics, University of Auckland. Lectures held in Online M.Sc. Wind Energy Systems Solid Mechanics of Wind Energy Systems Linear Computational Structural Mechanics Nonlinear Computational Structural Mechanics Research Interests Computational structural dynamics Non-linear multi-field finite element methods Adaptive time stepping schemes Computational tensegrity mechanics Simulation of wind turbines Projects Modeling and simulation of electro magneto dynamics coupled with heat conduction problems Modeling and simulation of electro magneto mechanical interactions Simulation of thermomechanical fluid structure interaction Form finding of tensegrity structures Higher order accurate integration of multifield elastoplasticity Recent Publications D. Kuhl, G. Meschke: Numerical Analysis of Dissolution Processes in Cementitious Materials Using Discontinuous and Continuous Galerkin Time Integration Schemes. International Journal for Numerical Methods in Engineering, Vol. 69, No. 9, 1775-1803, 2007 S. Carstens, D. Kuhl: Higher Order Accurate Implicit Time Integration Schemes for Transport Problems. Archive of Applied Mechanics, Vol. 82, 1007{1039, 2012 T. Gleim, D. Kuhl: Higher Order Accurate Discontinuous and Continuous p-galerkin Methods for Linear Elastodynamics. Zeitschrift f ur Angewandte Mathematik und Mechanik, Vol. 93, 177-194, 2013 P. Birken, T. Gleim, D. Kuhl, A. Meister: Fast Solvers for Unsteady Thermal Fluid Structure Interaction. International Journal for Numerical Methods in Fluids, DOI: 10.1002/d.4040, 2015 B. Schröder, D. Kuhl: Small Strain Plasticity: Classical Versus Multifield Formulation. Archive of Applied Mechanics, DOI 10.1007/s00419-015-0984-9, 2015 T. Gleim, B. Schröder, D. Kuhl: Nonlinear Thermo-Electromagnetic Analysis of Inductive Heating Processes. Archive of Applied Mechanics, DOI 10.1007/s00419-014-0968-1, 2015 Die Scientific Vitae kann in Stichworten dargestellt werden (siehe nächste Seite).

viii D. Kuhl, Online M.Sc. Wind Energy Systems, University of Kassel Curriculum Vitae Prof. Dr.-Ing. habil. Detlef Kuhl University of Kassel Faculty of Civil and Environmental Engineering Institute of Mechanics and Dynamics Mönchebergstraße 7 34109 Kassel Email: kuhl@uni-kassel.de Webpage: www.uni-kassel.de/fb14/mechanics Current Positions Academic Director of International Online Master s Course Wind Energy Systems Dean of Students of Faculty of Civil and Environmental Engineering Head of Chair of Mechanics and Dynamics Guest Professor at Vietnamese German University (VGU), Binh Duong New City, Vietnam Scientific Vitae 1985-1992 Study of Aerospace Engineering, University Stuttgart 1992 Mechanical Engineer, Enercon, Aurich 1992-1996 PhD student, PhD degree 1996, Department of Civil Engineering, Institute of Structural Mechanics, University of Stuttgart 1996-1998 Head of Thermomechanical Modelling Group, Institute of Space Propulsion, German Aerospace Center, Lampoldshausen 1997 Postdoctoral fellow at Department of Aeronautics, Imperial College of Science, Technology and Medicine, London 1998-2007 Senior scientist and lecturer, Habilitation 2004, Institute of Structural Mechanics, Ruhr University Bochum since 2007 Professor at Chair of Mechanics and Dynamics, University of Kassel Lectures held in Online M.Sc. Wind Energy Systems Solid Mechanics of Wind Energy Systems Linear Computational Structural Mechanics Nonlinear Computational Structural Mechanics Research Interests Computational structural dynamics Non-linear multifield finite element methods Adaptive time stepping schemes Computational tensegrity mechanics Simulation of wind turbines

Module Linear Structural Computational Mechanics for Wind Energy Systems ix Projects Modeling and simulation of electro magneto dynamics coupled with heat conduction problems Modeling and simulation of electro magneto mechanical interactions Simulation of thermomechanical fluid structure interaction Form finding of tensegrity structures Higher order accurate integration of multifield elastoplasticity Recent Publications D. Kuhl, G. Meschke: Numerical Analysis of Dissolution Processes in Cementitious Materials Using Discontinuous and Continuous Galerkin Time Integration Schemes. International Journal for Numerical Methods in Engineering, Vol. 69, No. 9, 1775-1803, 2007 S. Carstens, D. Kuhl: Higher Order Accurate Implicit Time Integration Schemes for Transport Problems. Archive of Applied Mechanics, Vol. 82, 1007{1039, 2012 T. Gleim, D. Kuhl: Higher Order Accurate Discontinuous and Continuous p-galerkin Methods for Linear Elastodynamics. Zeitschrift f ur Angewandte Mathematik und Mechanik, Vol. 93, 177-194, 2013 P. Birken, T. Gleim, D. Kuhl, A. Meister: Fast Solvers for Unsteady Thermal Fluid Structure Interaction. International Journal for Numerical Methods in Fluids, DOI: 10.1002/d.4040, 2015 B. Schröder, D. Kuhl: Small Strain Plasticity: Classical Versus Multifield Formulation. Archive of Applied Mechanics, DOI 10.1007/s00419-015-0984-9, 2015 T. Gleim, B. Schröder, D. Kuhl: Nonlinear Thermo-Electromagnetic Analysis of Inductive Heating Processes. Archive of Applied Mechanics, DOI 10.1007/s00419-014-0968-1, 2015

x D. Kuhl, Online M.Sc. Wind Energy Systems, University of Kassel Contents Introduction to Linear Structural Computational Mechanics for Wind Energy Systems 1 1 Finite Element Method for One Dimensional Continua and Truss Elements... 7 1.1 Learning Goals... 8 1.2 Required Prior Knowledge (Empfehlung)... 8 1.3 Section 2... 8 1.3.1 Section 3... 8 1.3.2 Formula... 9 1.3.3 Essenz (Empfehlung)... 9 References... 10 2 Finite Element Method for One Dimensional Continua and Truss Elements... 13 2.1 Introduction... 14 2.1.1 Learning goals... 14 2.1.2 Section 3... 14 2.1.3 Sections 3... 14 2.1.4 Formula... 15 2.2 Essenz... 15 References... 15 Bibliography (Möglichkeit)... 17 Appendix... 18 Glossary (Möglichkeit)... 18

Module Linear Structural Computational Mechanics for Wind Energy Systems xi Index (Möglichkeit)... 19 Nomenclature (Möglichkeit)... 20

xii D. Kuhl, Online M.Sc. Wind Energy Systems, University of Kassel List of Figures Figure 1.1: Tension of a truss: Geometry and loading cases... 11 Figure 1.3: Wind power Plant... 10 Figure 2.1: Wind power Plant... 15

Module Linear Structural Computational Mechanics for Wind Energy Systems xiii List of Tables Table 0.1: Embedding of the Module in Online M.Sc. Wind Energy Systems... 4 Table 1.1: Nomenclature for one dimensional linear continuum mechanics and linear truss mechanics... 10

Module Linear Structural Computational Mechanics for Wind Energy Systems 1 Introduction to Linear Structural Computational Mechanics for Wind Energy Systems Jedes Kapitel beginnt mit einem Deckblatt, auf welchem die Kapitelüberschrift, eine kurze Zusammenfassung des Kapitels sowie die Key Words zu finden sind. Das erste Kapitel des Skripts bei n- haltet eine Zusammenfassung des Moduls: Learning Goals of the Module, Motivation, Prior Knowledge Required for the Module, Embedding in Online M.Sc. Wind Energy Systems, Learning Schedule Abstract In the present chapter continuum mechanical for the simulation of wind turbine components are briefly reviewed. In particular, linear, physically nonlinear and geometrically nonlinear models are characterized. Furthermore, the significance of dynamical effects on the deformation of wind turbines is demonstrated and, consequently, continuum mechanical models for stationary and transient analysis of wind turbines are distinguished. Since wind power plants are using components made of different kind of materials also the modeling of isotropic and transversal isotropic as well as elastic and inelastic material models are briefly discussed. Above reviewed continuum mechanical models of wind turbine components constitute time dependent or time independent partial differential equations. In general these model can nor be solved analytically. Therefore, sequences of mathematical reformulations and numerical methods for the solution of linear dynamics, linear statics and non-linear dynamics are sketched. Linear dynamics is numerically solved by the spatial weak formulation, the finite element method and time integration schemes. These principal solution steps can also used for non-linear dynamics. Only the linearization and an iterative solution procedure must be used additionally. For the particular reason to become familiar with continuum mechanics and later also with the lin e- ar finite element method also the differential equation of one dimensional continuum mechanics is presented, the solution procedures for dynamics and statics are shown and, finally, the analytical solution of static one dimensional continua is deviated. Beside the technical aspects of computational mechanics for wind turbines also the history of mechanics, the finite element method and also the time integration method is briefly reviewed. Key Words linear and non-linear elasticity, finite element method, computational mechanics, time integration, history of mechanics and computational mechanics

2 D. Kuhl, Online M.Sc. Wind Energy Systems, University of Kassel Motivation (Empfehlung) Modern engineering structures as vehicles, air planes and wind turbines are subjected to m e- chanical as well as non-mechanical external actions. Due to external loading deformations and internal stress states, which may lead to the failure of the structure, are observed. In order to design wind turbines with a high level of safety and a long life time, the deformations and stresses in the structure should be known in advance for standard operation and extremely co n- ditions. The prognosis of the mechanical behavior, including collapse, low and high cycle fatigue, of wind turbines and their components is based on their adequate mechanical models, considering for the applied materials, dynamic and static actions, wind and death loads and also temperature changes. Only very simple models, far away from a realistic description, but, nevertheless, valid as basis for a first design of estimation of mechanical behavior of wind turbines or interactions of components, can be solved analytically. More realistic mechanical models taking into account realistic geometries, materials and mechanical effects require numerical solution procedures for an approximated solution of these highly sophisticated models. During the last six de c- ades the finite element method has been developed to a powerful tool for the mechanical analysis of structures of civil and environmental, mechanical, aerospace, electrical and wind energy engineering. A broad range of strong commercial tools have been developed for the linear mechanical analysis of structures using a more or less automated procedure for the meshing, the calculation of deformations and stresses and the post-processing of engineering relevant results. Beside these basic calculation competences several commercial finite element programs have strong capabilities on selected advanced simulation methods. For example for advanced material modeling, dynamic analysis, contact problems, soil and structural analysis, stability and multifield analyses. However, a general tool for the adequate mechanical analysis of wind turbines is not available. It is worth to mention that not only the highly sophisticated mechanical models of wind turbines needs real experts for the application of commercial programs, but also the parameter identification, the decision of adequate algorithms and finite elements and the interpretation of results and errors. Already linear models requires a deep knowledge of the underlying numerical methods for not only the reason of watching colorful pictures but also providing a serious and tough prognosis of expected deformations and stresses. It is self evident that more advanced mechanical models and computational methods require a strong knowledge for the educated decision for problem specific software packages and of course to overcome the limitations of commercial finite element programs for special applications in wind turbine mechanics. Beside the classical engineering prognosis of the mechanical behavior of wind turbines the n u- merical solution procedure consisting of spatial and temporal discretization methods are powe r- ful tools as basis for the classical engineering design and optimization method. Within these process simulations of wind turbines or components are used to study the influence of design modifications. Obviously, the applied numerical methods can also be used together with a sens i- tivity analysis and gradient based of evolutionary optimization algorithms for the systematic and computer oriented improvement of the design. Furthermore, computational wind turbine mechanics can be used as ingredient for the operation control and the damage detection of wind turbines using simplified or reduced models and inverse analysis, respectively. Above sketched applications, requirements and limitations of computational wind turbine mechanics motivate to reach a strong knowledge of numerical methods for the simulation, optimization and control of high tech wind power plants. In order to be able to achieve this goal, two

Module Linear Structural Computational Mechanics for Wind Energy Systems 3 lectures about the computational solid mechanics of wind turbines are included in the schedule of the master s course 'Wind Energy Systems'. The first one 'Linear Computational Structural Mechanics for Wind Energy Systems' is carefully limited to linear computational analysis of static and dynamic deformation of wind turbines. The main focus of this part is to understand the methods of spatial and temporal discretization, to know disadvantages and advantages of s e- lected numerical methods and to be able to select algorithms and finite elements and to capable interpret results of commercial software packages. Simultaneously the competence will be reached to overcome limitations of commercial codes and to develop more advanced, specialized and realistic computational models of wind turbines. In the second part 'Non-Linear Computational Structural Mechanics for Wind Energy Systems' non-linear continuum mechanical models, non-linear finite element methods and algorithms for non-linear statics and dynamics will be studied. Learning Goals for the Module Reviewing linear continuum mechanics Knowing different, also non-linear, models of continuum mechanics Having a idea of numerical methods applied for the solution of continuum mechanical models Having fun with the histories of the finite element method and the time integration schemes Prior Knowledge Required for the Module Requirements according to examination: Recommended prior learning: Modules: Solid Mechanics for Wind Energy Systems Module Mathematics, Module Solid Mechanics of Wind Energy Systems Module Application of Software Tools Mathematics for Wind Energy Systems Practice of Software Tools for Wind Energy Systems Design of Mechanical and Electrical Components of Wind Energy Systems Competencies: Vector and tensor analysis Basic knowledge on differential equations Integration and Differentiation in one to three spatial dimensions Linear systems of equations Mechanical forces and stress resultants Linear continuum mechanics in one to three spatial dimensions Beam and truss models of mechanics Programming in MATLAB

4 D. Kuhl, Online M.Sc. Wind Energy Systems, University of Kassel How this fits into the Online M.Sc. Wind Energy Systems The present module 'Linear Computational Structural Mechanics for Wind Energy Systems' is one six-credit module of the specialist study 'Simulation and Structural Technology for Wind Energy Systems'. This specialist study will enable the students to understand the structural components of wind energy systems, to permit prognoses of their life time for working and extreme conditions, and to design future wind turbines with an optimized use of foundations, materials, structural components and design concepts. As basis for this, a deep knowledge of the mechanics and technology of structural components is provided. Advanced fluid and solid mechanics and related novel simulation methods are thought as basis for studying the aerodynamic and mechanical behavior of wind turbines and their components. Together with the technological knowledge about on- and offshore foundations, towers, rotor blades and safer materials the generation of efficient and reliable wind turbines can be designed. Present course Strong basis Strong interaction Master s Thesis (in academia or industry) Specialization: Simulation and Structural Technology (each 6 ECTS-Credits) Rotor Aerodynamics On and Offshore Foundations Strength Durability and Reliability Nonlinear Computational Structural Mechanics Linear Computational Structural Mechanics Rotor Blades Computational Fluid Dynamics Theoretical Fluid Mechanics Towers Specialization: Energy System Technology (each 6 ECTS-Credits) Wind Energy Meteorology Construction and Design of Nacelle Systems Reliability, Availability Maintenance Strategies Energy Storage Control and Operational Management of Wind Turbines and Wind Farms Technical and Economic Aspects of Grid Integrations Contract Law Project Management Additional Key Competencies: Energy and Law (each 3 ECTS-Credits) Occupational Safety On and Offshore Planning and Constructions of Wind Farms Personal Management Energy Law Business Administration and Management of Wind Turbines and Wind Farms Design of Mechanical and Electrical Components Fundamentals of Mathematics and Engineering for Wind Energy Systems (each 6 ECTS-Credits) Electrical Engineering Mathematics Solid Mechanics Application of Software Tools Fluid Mechanics Table 0.1: How this Module fits into the Online M.Sc. Wind Energy Systems Table 0.1 shows the present module Linear Computational Structural Mechanics for Wind Energy Systems embedded in the specialist studies Simulation and Structural Technology for Wind Ene r- gy Systems and the master s course Wind Energy Systems. The present lecture is based on the knowledge of the modules of Fundamental Studies of Mathematics and Engineering. In particular, very good knowledge of Mathematics for Wind Energy Systems, Design of Mechanical and Electrical Components of Wind Energy Systems and Practice of Software Tools for Wind Ene rgy Systems is essential for the successful graduation of the present module. Since in the present module almost all continuum and structural mechanical problems, previously presented in module Solid Mechanics for Wind Energy Systems, are solved numerically, it is quite important to understand the topics of this fundamental module. The present module is extended to the nu-

Module Linear Structural Computational Mechanics for Wind Energy Systems 5 merical analysis of non-linear static and dynamic problems in module Non-Linear Computational Structural Mechanics for Wind Energy Systems (NCSM) and to the valuation of strength, failure, low and high cycle fatigue in lecture Strength Durability and Reliability for Wind Energy Systems. The present module can be combined with the fluid mechanics modules of specialist studies Simulation and Structural Technology for Wind Energy Systems in order to obtain the knowledge to overcome traditional borders between solid and fluid mechanics with study of both and finally with the analysis of fluid structure interaction. Furthermore, it can be combined with the technology modules of the specialist study in order to use numerical analysis of towers, foundations and rotor blades to improve or optimize these components of wind turbines. Learning Schedule (Beispiel) The simulation of wind turbines under real operating conditions enforces the consideration of time dependent loads and inertial forces. These simulations are performed by applying time integration schemes. Since these schemes are requiring a large numerical effort and significantly influencing the quality of the prognosis of the dynamic behavior of structures, it is worth to carefully develop these methods in Chapters 6 to 8 and to enrich the basic time integrations schemes by error measures and adaptive time stepping procedures. Methodologically oriented we will review continuum mechanics and we will discuss the dynamic characteristic and analytical sol u- tion of structural dynamics. Basics Static analysis Spatial Dynamic analysis discretization Temporal discretization Chapter 1, page 1: Introduction to Linear Computational Structural Mechanics Chapter 2, page 25: Finite Element Method for One Dimensional Continua Chapter 3, page 35: Advanced Topics and Spatial Truss Structures Chapter 4, page 7: Generalized Finite Element Method for n-dimensional Continua Chapter 5, page 13: Dynamic Characteristics and Analytical Solution of Dynamics Chapter 6, page 17: Central Difference Method Chapter 7, page 21: Newmark Time Integration Schemes Chapter 8, page 25: Galerkin Time Integration Schemes Figure 1: Learning Schedule of Linear Computational Structural Mechanics for Wind Energy Systems Afterwards, as main tasks of the present lecture, methods for the numerical solution of statics and dynamics are presented. In particular, the spatial and temporal discretization methods are

6 D. Kuhl, Online M.Sc. Wind Energy Systems, University of Kassel thought and intensively studied by means of analytical analyses and representative and illustrative examples. The simulation of wind turbines under real operation condition enforces the consideration of time dependent loads and inertial forces. These simulations are performed by applying time integration schemes. Since these schemes are requiring a large numerical effort and significantly influencing the quality of the prognosis of the dynamic behavior of structures, it is worth to carefully develop these methods in Chapters 6 to 8 and to enrich the basic time integrations schemes by error measures and adaptive time stepping procedures. Methodologically oriented we will review continuum mechanics and we will discuss the dynamic characteristic and analytical solution of structural dynamics. Afterwards, as main tasks of the present lecture, methods for the numerical solution of statics and dynamics are presented. In particular, the spatial and temporal discretization methods are thought and intensively studied by means of analy t- ical analyses and representative and illustrative examples.

Module Linear Structural Computational Mechanics for Wind Energy Systems 7 1 Finite Element Method for One Dimensional Continua and Truss Elements Abstract In the present chapter... Kurze Zusammenfassung des Kapitels. Jedes Kapitel beginnt mit einem Deckblatt, auf welchem die Kapitelüberschrift, eine kurze Zusammenfassung des Kapitels sowie die Key Words zu finden sind. Key Words linear elasticity, finite element method, history of mechanics

8 D. Kuhl, Online M.Sc. Wind Energy Systems, University of Kassel 1.1 Learning Goals reviewing linear continuum mechanics knowing different, also non-linear, models of continuum mechanics having a idea of numerical methods applied for the solution of continuum mechanical models having fun with the histories of the finite element method and the time integration schemes 1.2 Required Prior Knowledge (Empfehlung) Welche Voraussetzungen müssen erfüllt sein, um dieses Kapitel zu verstehen. 1.3 Section 2 Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invi d- unt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea taki mata sanctus est Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet. 1.3.1 Section 3 Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invi d- unt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea taki mata sanctus est Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet. Example for citation: "Fluid ows at and below the earth's surface are the cause and the cure for problems of water and soil pollution" (Wendland & Efendiev, 2003, S. 37). Section 4 The section 4 will not be consecutively numbered. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invi d- unt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea taki mata sanctus est Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua 1. At vero eos et 1 This is a footnote.

Module Linear Structural Computational Mechanics for Wind Energy Systems 9 accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet. Example for citation: "Fluid ows at and below the earth's surface are the cause and the cure for problems of water and soil pollution" (Wendland & Efendiev, 2003, S. 37). Memotechnic verse: Field shaded in gray to give short (!) memos or advices. Special texts like examples, excursions or tips are framed in a box: At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet. Enumeration 1. level - Enumeration 2. level * Enumeration 3. level 1.3.2 Formula (x + a) n = n k=0 ( n k )xk a n k (1.1) (1 + x) n = 1 + nx 1! + n(n 1)x2 2! + (1.2) x = b± b2 4ac 2a (1.3) 1.3.3 Essenz (Empfehlung) Chapter Checks 1. (Question/Task 1 of the paragraph 1.1) 2. (Question/Task 1 of the paragraph 1.1) 3. (Question/Task 1 of the paragraph 1.1)

10 D. Kuhl, Online M.Sc. Wind Energy Systems, University of Kassel Figure 1.1: Wind power Plant Nomenclature Symbol Equivalent Uni Explanation T s Time Θ Κ temperature Χ 1 m position υ 1 m displacement Table 1.1: Nomenclature for one dimensional linear continuum mechanics and linear truss mechanics References Ehlers, W. & Bluhm, J. (2002). Porous Media. Theory, Experiments and Numerical Applications. Berlin: Springer. Wendland, W. L. & Efendiev, M. (2003). Analysis and Simulation of Multield Problems. Berlin: Springer. Verschiedene Darstellungsweisen möglich! [1] M. Ameen. Computational Elasticity. Theory of Elasticity and Finite and Boundary Element Methods. Alpha Science International, Harrow, 2005. [2] T. L. Anderson. Fracture Mechnics. Fundamentals and Applications. Taylor & Francis Group, Broken, 3. edition, 2005.

Module Linear Structural Computational Mechanics for Wind Energy Systems 11 [3] Archimedes. De planorum aequilibriis. 285-212 v.chr. [4] J. Argyris. Dynamics of Structure. Elsevier, Amsterdam, 1991. [5] V. I. Arnold. Lectures on Partial Differential Equations. Springer & Phasis, Berlin & Moscow, 2004. [6] G. Galilei. Discorsi e dimostrazioni matematiche intorno a due nuove scienze. Leiden, 1638. Homework (Möglichkeit) Hausaufgaben können auch in Moodle oder in anderer Form den Studierenden zur Verfügung gestellt werden. Figure 1.2: Tension of a truss: Geometry and loading cases In the present homework your own finite element program for the static analysis of one dime n- sional continua should be extended in order to allow for the application of the p finite element method. Therefore, higher order (ρ = 1; 2; 3; 4; 5; 6), one dimensional continuum elements should be applied together with the Gauss-Legendre integration. The correct implementation of the finite element and finite element procedure on the structural level should be verifie d by means of above sketched model problems. These examples are described by a truss loaded by load cases i, ii and iii. They should be analyzed using ΝΕ = 1; 2; 4; 8; 16; 32 p finite elements for the discretization of the truss. For these reasons the following working stages are proposed: Develop a finite element routine for calculation of the element stiffness 'tensors' k eij and the consistent load 'tensors' r ei for all load cases using the Gauss-Legendre integration with GAUSS point coordinates and weights as given in the file gauss.f provided in the Moodle course. Chose the number of GAUSS points NG such that the stiffness tensors and the load tensors for load cases i and ii are exactly integrated. The load tensors according to load case iii cannot integrated exactly. For these integrations please use a integration rule with NG = p + 5. Develop finite element procedure for analyses with NE = 1; 2; 4; 8; 16; 32 finite elements of polynomial degrees p = 1; 2; 3; 4; 5; 6 and check your solutions for all load cases. Perform all forthcoming tasks only for load case iii, but for all implemented polynomial degrees p.

12 D. Kuhl, Online M.Sc. Wind Energy Systems, University of Kassel Extend your p finite element program by a post-processing procedure, calculating the approximations of the displacement u 1, stress σ 11 and residuum σ 11,1 + pb 1 Calculate the local (at position X1) and global (of the hole system) displacement errors with respect to the analytical solution. Plot diagrams of the displacements, stresses, the residuum and the local displacement error. Your homework submission should include a brief report documenting your results in form of diagrams your program code

Module Linear Structural Computational Mechanics for Wind Energy Systems 13 2 Finite Element Method for One Dimensional Continua and Truss Elements Abstract In the present chapter... Kurze Zusammenfassung des Kapitels Key Words linear elasticity, finite element method, history of mechanics

14 D. Kuhl, Online M.Sc. Wind Energy Systems, University of Kassel 2.1 Introduction 2.1.1 Learning goals reviewing linear continuum mechanics knowing different, also non-linear, models of continuum mechanics having a idea of numerical methods applied for the solution of continuum mechanical models having fun with the histories of the finite element method and the time integration schemes Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invi d- unt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea taki mata sanctus est Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet. 2.1.2 Section 3 Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invi d- unt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea taki mata sanctus est Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet. Example for citation: "Fluid ows at and below the earth's surface are the cause and the cure for problems of water and soil pollution" (Wendland & Efendiev, 2003, S. 37). 2.1.3 Sections 3 Memotechnic verse: Field shaded in gray to give short (!) memos or advices. Special texts like examples, excursions or tips could be framed in a box: At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet. Enumeration 1. level - Enumeration 2. level * Enumeration 3. level

Module Linear Structural Computational Mechanics for Wind Energy Systems 15 2.1.4 Formula (x + a) n = n k=0 ( n k )xk a n k (2.1) (1 + x) n = 1 + nx 1! + n(n 1)x2 2! + (2.2) x = b± b2 4ac 2a (2.3) Chapter Checks 1. (Question/Task 1 of the paragraph 1.1) 2. (Question/Task 1 of the paragraph 1.1) 3. (Question/Task 1 of the paragraph 1.1) Figure 2.1: Wind power Plant 2.2 Essenz Example for citation: "Fluid ows at and below the earth's surface are the cause and the cure for problems of water and soil pollution" (Wendland & Efendiev, 2003, S. 37). References Ehlers, W. & Bluhm, J. (2002). Porous Media. Theory, Experiments and Numerical Applications. Berlin: Springer. Wendland, W. L. & Efendiev, M. (2003). Analysis and Simulation of Multield Problems. Berlin: Springer.

16 D. Kuhl, Online M.Sc. Wind Energy Systems, University of Kassel Homework (Möglichkeit) Hausaufgaben können auch in Moodle oder in anderer Form den Studierenden zur Verfügung gestellt werden.

Module Linear Structural Computational Mechanics for Wind Energy Systems 17 Bibliography (Möglichkeit) Beinhaltet die gesamte Literatur im Text. Die Literatur sollte jedoch in jedem Kapitel aufgelistet sein. Die Gesamtdarstellung stellt ein Zusatz dar. Capital 1 Ehlers, W. & Bluhm, J. (2002). Porous Media. Theory, Experiments and Numerical Applications. Berlin: Springer. Wendland, W. L. & Efendiev, M. (2003). Analysis and Simulation of Multifield Problems. Berlin: Springer. Verschiedene Darstellungsmöglichkeiten. Diese müssen einheitlich im Dokument sein! Capital 1 [1] M. Ameen. Computational Elasticity. Theory of Elasticity and Finite and Boundary Element Methods. Alpha Science International, Harrow, 2005. [2] T. L. Anderson. Fracture Mechnics. Fundamentals and Applications. Taylor & Francis Group, Broken, 3. edition, 2005. [3] Archimedes. De planorum aequilibriis. 285-212 v.chr.

Module Linear Structural Computational Mechanics for Wind Energy Systems 18 Appendix Glossary (Möglichkeit) Der Glossary stellt ein Zusatz dar. Actuator Actuator is a device to convert an electrical control signal to a physical action. Actuators may be used for flow-control valves, pumps, positioning drives, motors, switches, relays and meters. Floating-Point Operations Per Second (FLOPS) Floating-Point Operations Per Second (FLOPS) is a measurement of performance of capability assigned to a floating-point processor. It is usually noted as MFLOPS or Million FLOPS. Local Area Network A Local Area Network is a group of interconnected devices that share common processing and file management resources, usually within a specific physical area. An example would be an o f- fice computer network. Resolution Resolution is a measure of accuracy or dynamic range of an A/D or D/A converter.

D. Kuhl, Online M.Sc. Wind Energy Systems, University of Kassel 19 Index (Möglichkeit) Der Index stellt ein Zusatz dar. Actuator 8 Formula 5 Internet adress 7 Questions/Tasks 2, 5

D. Kuhl, Online M.Sc. Wind Energy Systems, University of Kassel 20 Nomenclature (Möglichkeit) Beinhaltet die gesamte Nomenklatur aller Kapitel. Diese sollte jedoch in jedem Kapitel aufgelistet sein. Die Gesamtdarstellung stellt ein Zusatz dar. ν ν Poisson ratio Poisson ratio σ 11 normal stress / normal stress component in direction e 1 σ 11 normal stress / normal stress component in direction e 1 σ 11 normal stress / normal stress component in direction e 1 ε 11 normal stress / normal strain component in direction e 1 ε 11 normal stress / normal strain component in direction e 1

Module Linear Structural Computational Mechanics for Wind Energy Systems 21 Lecture Notes Linear Computational Structural Mechanics for Wind Energy Systems Prof. Dr.-Ing. habil. Detlef Kuhl These lecture notes are designed to assist students of the online master s study wind energy systems with their learning process in linear finite element methods and linear structural dynamics of wind energy systems. For this reason it includes various elements: the theoretical development, application of methods in selected examples and program flowcharts, as well as coding instructions supporting the homework and the final case study of the course. Online M.Sc. Wind Energy Systems www.uni-kassel.de/wes University of Kassel and Fraunhofer IWES