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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

Unive

Online M.Sc. Wind Energy Systems University of Kassel and Fraunhofer IWES

www.uni-kassel.de/wes

http://www.uni-kassel.de/uni/internationales/english-version/university/about-us.html

http://www.iwes.fraunhofer.de/en.html

ii D. Kuhl, Online M.Sc. Wind Energy Systems, University of Kassel

Lecture Notes

Linear Computational Structural Mechanics for Wind Energy Systems


1st 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


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 an-

other 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 sys-

tem, without written permission of the author.

http://www.uni-kassel.de/fb14/mechanics

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 develop-

ment, 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 introduc-

tion 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 chap-

ter 'eigenvalue analysis' will provide methods the analyze the dynamic characteristics of struc-

tures and to provide a analytical solution of simple structural dynamics with con later on com-

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 environ-

mental study programs has been increasing continuously over the years. Bicultural and interna-

tional 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 Sys-

tem 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 as-

pects into the study program, such as the economic integration of a large amount of wind ener-

gy into the energy supplier system. Students also gain knowledge about how to design and de-

velop 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 stu-

dents 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



Faculty of Civil and Environmental Engineering


Moenchebergstrasse 7

34109 Kassel

Email: [email protected]

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 De-

partment 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 Univer-

sity Bochum. He finished his Habilitation about the simulation of time dependent multi eld prob-

lems 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 Us-

ing 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 Multi field 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



Faculty of Civil and Environmental Engineering


Mönchebergstraße 7

34109 Kassel

Email: [email protected]

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 Me-

chanics, 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 Us-

ing 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 Multi field 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 Me-

chanics 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 bein-

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 brief ly 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 meth-

ods for the solution of linear dynamics, linear statics and non-linear dynamics are sketched. Lin-

ear 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 dy-

namics. 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 line-

ar finite element method also the differential equation of one dimensional continuum mechan-

ics 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 integra-

tion, 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 me-

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, consider-

ing for the applied materials, dynamic and static actions, wind and death loads and also temper-

ature 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 interac-

tions of components, can be solved analytically. More realistic mechanical models taking into

account realistic geometries, materials and mechanical effects require numerical solution proce-

dures 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 anal-

ysis 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 me-

chanical 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 materi-

al modeling, dynamic analysis, contact problems, soil and structural analysis, stability and mult i-

field 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 pa-

rameter identification, the decision of adequate algorithms and finite elements and the interpre-

tation 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 limita-

tions 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 nu-

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 me-

chanics 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 me-

chanics motivate to reach a strong knowledge of numerical methods for the simulation, optimi-

zation and control of high tech wind power plants. In order to be able to achieve this goal, two


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 se-

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, special-

ized and realistic computational models of wind turbines. In the second part 'Non-Linear Compu-

tational Structural Mechanics for Wind Energy Systems' non-linear continuum mechanical mod-

els, 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:

Module Mathematics, Module Solid Mechanics of Wind Energy

Systems

Recommended prior

learning:

Module Application of Software Tools

Modules:

Solid Mechanics for Wind Energy Systems

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


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 com-

ponents 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 mechan-

ics 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)

Specialization:

Energy System Technology


Additional Key Competencies:

Energy and Law


Rotor Aerody-

namics

Strength Dura-

bility and

Reliability

Rotor

Blades

Wind Energy

Meteorology Energy Storage

Contract

Law

Occupational

Safety On and

Offshore

Energy Law

Computational

Fluid Dynam-

ics

Nonlinear

Computational

Structural

Mechanics

Theoreti-

cal Fluid

Mechanics

Construction

and Design of

Nacelle

Systems

Control and

Operational

Management of

Wind Turbines

and Wind Farms

Project

Manage-

ment

Planning and

Constructions

of Wind

Farms

Business

Administration

and Manage-

ment of Wind

Turbines and

Wind Farms

On and Off-

shore Founda-

tions

Linear Compu-

tational Struc-

tural Mechan-

ics

Towers

Reliability,

Availability

Maintenance

Strategies

Technical and

Economic As-

pects of Grid

Integrations

Personal

Management

Fundamentals of Mathematics and Engineering for Wind Energy Systems (each 6 ECTS-Credits)

Design of Mechanical and

Electrical Components 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 particu-

lar, 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 mod-

ule 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-


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 tech-

nology 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 care-

fully 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

discreti-

zation

Dynamic

analysis

Tem-

poral

discreti-

zation

Chapter 1, page 1: Introduction to Linear Com-

putational Structural Mechanics

Chapter 2, page 25: Finite Element Method for

One Dimensional Continua

Chapter 3, page 35: Advanced Topics and Spa-

tial 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


thought and intensively studied by means of analytical analyses and representative and illustra-

tive examples. The simulation of wind turbines under real operation condition enforces the con-

sideration of time dependent loads and inertial forces. These simulations are performed by ap-

plying 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 inte-

grations 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 spa-

tial and temporal discretization methods are thought and intensively studied by means of analyt-

ical analyses and representative and illustrative examples.


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


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.

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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

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Example for citation:

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and soil pollution" (Wendland & Efendiev, 2003, S. 37).

Section 4 The section 4 will not be consecutively numbered.

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1 This is a footnote.







1.3.2 Formula

(𝒙 + 𝒂)𝒏 = ∑ (𝒏𝒌

)𝒙𝒌𝒂𝒏−𝒌𝒏

𝒌=𝟎 (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)


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Enumeration 1. level - Enumeration 2. level

* Enumeration 3. level

Memotechnic verse: Field shaded in gray to give short (!) memos or advices.


Figure 1.1: Wind power Plant

Nomenclature

Symbol Equivalent Uni Explanation

T s Time

𝚯 Κ temperature

𝚾𝟏 m position

𝝊𝟏 m displacement

Table 1.1: Nomenclature for one dimensional linear continuum mechanics and linear truss me-

chanics

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.


[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' 𝑘𝑒𝑖𝑗

and the consistent load 'tensors' 𝑟𝑒𝑖 for all load cases using the Gauss-Legendre integra-

tion 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 ten-

sors 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 el-

ements 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.


Extend your p finite element program by a post-processing procedure, calculating the

approximations of the displacement 𝑢1, stress 𝜎11 and residuum 𝜎11,1 + 𝑝𝑏1

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


2 Finite Element Method for One Di-

mensional Continua and Truss Ele-

ments

Abstract In the present chapter ... Kurze Zusammenfassung des Kapitels

Key Words linear elasticity, finite element method, history of mechanics


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

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2.1.3 Sections 3

Memotechnic verse: Field shaded in gray to give short (!) memos or advices.

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Enumeration 1. level - Enumeration 2. level

* Enumeration 3. level


2.1.4 Formula

(𝒙 + 𝒂)𝒏 = ∑ (𝒏𝒌

)𝒙𝒌𝒂𝒏−𝒌𝒏

𝒌=𝟎 (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)



Figure 2.1: Wind power Plant

2.2 Essenz Example for citation:



References Ehlers, W. & Bluhm, J. (2002). Porous Media. Theory, Experiments and Numerical Appli-

cations. Berlin: Springer.

Wendland, W. L. & Efendiev, M. (2003). Analysis and Simulation of Multield Problems.

Berlin: Springer.


Homework (Möglichkeit)

Hausaufgaben können auch in Moodle oder in anderer Form den

Studierenden zur Verfügung gestellt werden.


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.


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 of-

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 Poisson ratio

𝝂 Poisson ratio

𝝈𝟏𝟏 normal stress / normal stress component in direction 𝑒1



𝜺𝟏𝟏 normal stress / normal strain component in direction 𝑒1

𝜺𝟏𝟏 normal stress / normal strain component in direction 𝑒1


Online M.Sc. Wind Energy Systems www.uni-kassel.de/wes University of Kassel and Fraunhofer IWES

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 ele-ment methods and linear structural dynamics of wind energy systems. For this reason it includes various elements: the theoretical development, ap-plication of methods in selected examples and program flowcharts, as well as coding instructions supporting the homework and the final case study of the course.

http://www.uni-kassel.de/uni/internationales/english-version/university/about-us.html

http://www.iwes.fraunhofer.de/en.html

linear computational structural mechanics for wind energy ... › ... ›...

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