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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Serie: Wiley Series in Computational Mechanics
- Sprache: Englisch
Introduction to Computational Contact Mechanics: A Geometrical Approach covers the fundamentals of computational contact mechanics and focuses on its practical implementation. Part one of this textbook focuses on the underlying theory and covers essential information about differential geometry and mathematical methods which are necessary to build the computational algorithm independently from other courses in mechanics. The geometrically exact theory for the computational contact mechanics is described in step-by-step manner, using examples of strict derivation from a mathematical point of view. The final goal of the theory is to construct in the independent approximation form /so-called covariant form, including application to high-order and isogeometric finite elements.
The second part of a book is a practical guide for programming of contact elements and is written in such a way that makes it easy for a programmer to implement using any programming language. All programming examples are accompanied by a set of verification examples allowing the user to learn the research verification technique, essential for the computational contact analysis.
Key features:
- Covers the fundamentals of computational contact mechanics
- Covers practical programming, verification and analysis of contact problems
- Presents the geometrically exact theory for computational contact mechanics
- Describes algorithms used in well-known finite element software packages
- Describes modeling of forces as an inverse contact algorithm
- Includes practical exercises
- Contains unique verification examples such as the generalized Euler formula for a rope on a surface, and the impact problem and verification of thå percussion center
- Accompanied by a website hosting software
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Seitenzahl: 250
Veröffentlichungsjahr: 2015
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Table of Contents
Cover
Wiley Series in Computational Mechanics
Title Page
Copyright
Series Preface
Preface
Acknowledgments
Part I: Theory
Chapter 1: Introduction with a Spring-Mass Frictionless Contact System
1.1 Structural Part—Deflection of Spring-Mass System
1.2 Contact Part—Non-Penetration into Rigid Plane
1.3 Contact Formulations
Chapter 2: General Formulation of a Contact Problem
2.1 Structural Part—Formulation of a Problem in Linear Elasticity
2.2 Formulation of the Contact Part (Signorini's problem)
Chapter 3: Differential Geometry
3.1 Curve and its Properties
3.2 Frenet Formulas in 2D
3.3 Description of Surfaces by Gauss Coordinates
3.4 Differential Properties of Surfaces
Chapter 4: Geometry and Kinematics for an Arbitrary Two Body Contact Problem
4.1 Local Coordinate System
4.2 Closest Point Projection (CPP) Procedure—Analysis
4.3 Contact Kinematics
Chapter 5: Abstract Form of Formulations in Computational Mechanics
5.1 Operator Necessary for the Abstract Formulation
5.2 Abstract Form of the Iterative Method
5.3 Fixed Point Theorem (Banach)
5.4 Newton Iterative Solution Method
5.5 Abstract Form for Contact Formulations
Chapter 6: Weak Formulation and Consistent Linearization
6.1 Weak Formulation in the Local Coordinate System
6.2 Regularization with Penalty Method
6.3 Consistent Linearization
6.4 Application to Lagrange Multipliers and to Following Forces
6.5 Linearization of the Convective Variation
6.6 Nitsche Method
Chapter 7: Finite Element Discretization
7.1 Computation of the Contact Integral for Various Contact Approaches
7.2 Node-To-Node (NTN) Contact Element
7.3 Nitsche Node-To-Node (NTN) Contact Element
7.4 Node-To-Segment (NTS) Contact Element
7.5 Segment-To-Analytical-Surface (STAS) Approach
7.6 Segment-To-Segment (STS) Mortar Approach
Chapter 8: Verification with Analytical Solutions
8.1 Hertz Problem
8.2 Rigid Flat Punch Problem
8.3 Impact on Moving Pendulum: Center of Percussion
8.4 Generalized Euler–Eytelwein Problem
Chapter 9: Frictional Contact Problems
9.1 Measures of Contact Interactions—Sticking and Sliding Case: Friction Law
9.2 Regularization of Tangential Force and Return Mapping Algorithm
9.3 Weak Form and its Consistent Linearization
9.4 Frictional Node-To-Node (NTN) Contact Element
9.5 Frictional Node-To-Segment (NTS) Contact Element
9.6 NTS Frictional Contact Element
Part II: Programming and Verification Tasks
Chapter 10: Introduction to Programming and Verification Tasks
Chapter 11: Lesson 1 Nonlinear Structural Truss—elmt1.f
11.1 Implementation
Setup of tangent matrix and residual (
isw = 3
)
Description of subroutines
Global
FEAP
-arrays
Hints for implementation
11.2 Examples
Chapter 12: Lesson 2 Nonlinear Structural Plane—elmt2.f
12.1 Implementation
Setup of tangent matrix and residual (
isw = 3
)
Setup of mass matrix (
isw = 5
)
Description of subroutines
Global
FEAP
-arrays
Hints for implementation
12.2 Examples
Chapter 13: Lesson 3 Penalty Node-To-Node (NTN)—elmt100.f
13.1 Implementation
Setup of tangent matrix and residual (
isw = 3
)
Description of subroutines
Global
FEAP
-arrays
Hints for implementation
13.2 Examples
Chapter 14: Lesson 4 Lagrange Multiplier Node-To-Node (NTN)—elmt101.f
14.1 Implementation
Setup of tangent matrix and residual (
isw = 3
)
Description of subroutines
Global
FEAP
-arrays
Hints for implementation
14.2 Examples
Chapter 15: Lesson 5 Nitsche Node-To-Node (NTN)—elmt102.f
15.1 Implementation
Setup of tangent matrix and residual (
isw = 3
)
Description of subroutines
Global
FEAP
-arrays
Hints for implementation
15.2 Examples
Chapter 16: Lesson 6 Node-To-Segment (NTS)—elmt103.f
16.1 Implementation
Setup of tangent matrix and residual (
isw = 3
)
Description of subroutines
Global
FEAP
-arrays
Hints for implementation
16.2 Examples
16.3 Inverted Contact Algorithm—Following Force
Chapter 17: Lesson 7 Segment-To-Analytical-Segment (STAS)—elmt104.f
17.1 Implementation
Setup of tangent matrix and residual (
isw = 3
)
Description of subroutines
Global
FEAP
-arrays
Hints for implementation
17.2 Examples
17.3 Inverted Contact Algorithm—General Case of Following Forces
Chapter 18: Lesson 8 Mortar/Segment-To-Segment (STS)—elmt105.f
18.1 Implementation
Setup of tangent matrix and residual (
isw = 3
)
Description of subroutines
Global
FEAP
-arrays
Hints for implementation
18.2 Examples
18.3 Inverted Contact Algorithm—Following Force
Implementation
Setup of the tangent matrix and residual (
isw = 3
)
Chapter 19: Lesson 9 Higher Order Mortar/STS—elmt106.f
19.1 Implementation
Setup of tangent matrix and residual (
isw = 3
)
Description of subroutines
Global
FEAP
-arrays
Hints for implementation
19.2 Examples
Chapter 20: Lesson 10 3D Node-To-Segment (NTS)—elmt107.f
20.1 Implementation
Setup of tangent matrix and residual (
isw = 3
)
Description of subroutines
Global
FEAP
-arrays
Hints for implementation
20.2 Examples
Chapter 21: Lesson 11 Frictional Node-To-Node (NTN)—elmt108.f
21.1 Implementation
Setup of tangent matrix and residual (
isw = 3
)
Description of subroutines
Global
FEAP
-arrays
Hints for implementation
21.2 Examples
Chapter 22: Lesson 12 Frictional Node-To-Segment (NTS)—elmt109.f
22.1 Implementation
Setup of tangent matrix and residual (
isw = 3
)
Description of subroutines
Global
FEAP
-arrays
Hints for implementation
22.2 Examples
Chapter 23: Lesson 13 Frictional Higher Order NTS—elmt110.f
23.1 Implementation
Setup of tangent matrix and residual (
isw = 3
)
Description of subroutines
Global
FEAP
-arrays
Hints for implementation
23.2 Examples
Chapter 24: Lesson 14 Transient Contact Problems
24.1 Implementation
Description of subroutines
24.2 Examples
Appendix A: Numerical integration
A.1 Gauss Quadrature
Appendix B: Higher Order Shape Functions of Different Classes
B.1 General
B.2 Lobatto Class
B.3 Bezier Class
References
Index
End User License Agreement
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Guide
Cover
Table of Contents
Preface
Begin Reading
List of Illustrations
Chapter 1: Introduction with a Spring-Mass Frictionless Contact System
Figure 1.1 Spring-mass system in contact with a rigid plane
Figure 1.2 Mechanical interpretation of the Penalty method
Chapter 2: General Formulation of a Contact Problem
Figure 2.1 Source of nonlinearity of contact problem: neither contact stresses, nor contact surface are known
Figure 2.2 Contact with a rigid obstacle: Signorini's problem
Figure 2.3 Split of the stress vector into normal and tangential parts
Chapter 3: Differential Geometry
Figure 3.1 Definition of the Frenet frame and the center of curvature
Figure 3.2 A circle as a curve with constant curvature and its Frenet frame
Figure 3.3 Surface in space: definition of a surface coordinate system (). A curve on the surface is defined as
Figure 3.4 Calculation of a surface area . Infinitesimal area
Figure 3.5 Covariant , and contravariant , coordinate vectors. Representation of a vector via co- and contravariant components
Figure 3.6 Cylinder and its local (surface) coordinates system (), ()
Chapter 4: Geometry and Kinematics for an Arbitrary Two Body Contact Problem
Figure 4.1 Local coordinate systems in 2D: (a) basis vectors , for arbitrary parametrization of the curve; (b) unit basis vectors and (or ) for natural parametrization
Figure 4.2 Allowable projection domains for the circular arch: (a) inner part , (b) outer part
Figure 4.3 Projection domain for the outer part of the circle:
Figure 4.4 Projection domain for the inner part of the circle:
Figure 4.5 Linear FE representation of the boundary. Discontinuity of normals: , , and
Figure 4.6 Violation of -continuity leads to non-allowable domains with regard to the projection onto master curve . Both Point-To-Curve and Point-To-Point contact pairs are necessary to describe geometrically exact kinematics of contact with the master curve
Figure 4.7 Contact kinematics for the 2D case
Figure 4.8 Contact kinematics in the 3D case
Chapter 5: Abstract Form of Formulations in Computational Mechanics
Figure 5.1 Geometrical interpretation of the Newton Iterative Solution Method:, while
Figure 5.2 Convex function and its convex domain
Chapter 6: Weak Formulation and Consistent Linearization
Figure 6.1 Equilibrium during contact. Infinitesimally small contact lines: ;
Chapter 7: Finite Element Discretization
Figure 7.1 Geometry and kinematics of the Node-To-Segment contact element
Figure 7.2 Node-To-Segment (NTS) contact element. The lower part is the inner part of the body. Coordinate system , and coordinates ,
Figure 7.3 Geometry and kinematics of the Segment-To-Analytical-Surface (STAS)
Figure 7.4 Closed form solution for STAS penetration: Contact with a rigid line. Coordinate system for computation of the penetration
Figure 7.5 STAS: Contact with a rigid circle. Coordinate system for computation of the penetration
Figure 7.6 Geometry and kinematics of the Segment-To-Segment contact element
Chapter 8: Verification with Analytical Solutions
Figure 8.1 Geometry of contacting surfaces for the Hertz problem
Figure 8.2 Geometry of contact and non-penetration condition for the Hertz problem
Figure 8.3 Rigid punch (stamping) problem
Figure 8.4 Impact on a moving pendulum—definition of the center of percussion
Figure 8.5 Plane 2D curve: Interpretation of the curvature
Figure 8.6 Comparison between a rope wrapped over a circular cylinder and a rope on an ellipse in 2D: The same ratio of forces for the same angle :
Chapter 9: Frictional Contact Problems
Figure 9.1 Kinematics of frictional contact: Computation of the tangential measure
Figure 9.2 Coulomb friction. Updating of sliding displacements in convective coordinates. Motion of the elastic region and update of the center as initial sticking point
Chapter 10: Introduction to Programming and Verification Tasks
Figure 10.1 Global setup of a finite element analysis program
Chapter 11: Lesson 1 Nonlinear Structural Truss—elmt1.f
Figure 11.1 Geometry and parameters of the nonlinear truss finite element
Figure 11.2 Compression of truss element. Comparison: (a) physically and geometrically linear theory, (b) Neo-Hooke and (c) St. Venant and geometrically nonlinear theory
Figure 11.3 Tension of truss element. Comparison: (a) physically and geometrically linear theory, (b) Neo-Hooke and (c) St. Venant and geometrically nonlinear theory
Figure 11.4 Force displacement diagram for snap-through buckling (Neo-Hooke)
Chapter 12: Lesson 2 Nonlinear Structural Plane—elmt2.f
Figure 12.1 Geometry and parameters of the nonlinear plane finite element
Figure 12.2 Compression of plane element. Comparison: (a) physically and geometrically linear theory, (b) Neo-Hooke and geometrically nonlinear theory
Figure 12.3 Tension of plane element. Comparison: (a) physically and geometrically linear theory, (b) Neo-Hooke and geometrically nonlinear theory
Chapter 13: Lesson 3 Penalty Node-To-Node (NTN)—elmt100.f
Figure 13.2 Influence of the penalty parameter on the value of penetration
Chapter 14: Lesson 4 Lagrange Multiplier Node-To-Node (NTN)—elmt101.f
Figure 14.1 Geometry and parameters of the Lagrange multiplier NTN contact element
Chapter 15: Lesson 5 Nitsche Node-To-Node (NTN)—elmt102.f
Figure 15.1 Geometry and parameters of the Nitsche NTN contact element
Figure 15.2 Influence of the stabilization parameter on the value of penetration
Chapter 16: Lesson 6 Node-To-Segment (NTS)—elmt103.f
Figure 16.1 Geometry and parameters of NTS contact element
Figure 16.2 Distribution of the contact stresses along the contact radius
Figure 16.3 Bending of a beam into a circle
Chapter 17: Lesson 7 Segment-To-Analytical-Segment (STAS)—elmt104.f
Figure 17.1 Geometry and parameters of STAS contact element: (1) contact with a rigid line; (2) contact with a rigid circle
Figure 17.2 Bending of a beam into a circle: moment is modeled as a pair of single forces
Figure 17.3 Bending of a beam into a circle: moment is modeled as exact distributed forces for pure bending
Figure 17.4 Inflating a bar. A bar deforms into a circular shaped bar with constant curvature, which is correlated with an analytical result
Chapter 18: Lesson 8 Mortar/Segment-To-Segment (STS)—elmt105.f
Figure 18.2 Within the STS-type inverted contact algorithm the forces follow the master normal only. The following master segment forces act on the slave segment. This is by no means inflating pressure for both sides, but a test for the rotational part!
Chapter 19: Lesson 9 Higher Order Mortar/STS—elmt106.f
Figure 19.1 Geometry and parameters of STS contact element with higher order approximations
Figure 19.2 Initial and final configuration for (a) Lagrange, (b) Lobatto and (c) Bezier classes of shape functions
Figure 19.3 Initial and final configuration for (a) Lagrange, (b) Lobatto and (c) Bezier classes of shape functions
Chapter 20: Lesson 10 3D Node-To-Segment (NTS)—elmt107.f
Figure 20.1 Geometry and parameters of the 3D NTS contact element
Chapter 21: Lesson 11 Frictional Node-To-Node (NTN)—elmt108.f
Figure 21.1 Geometry and parameters of the frictional NTN contact element
Chapter 22: Lesson 12 Frictional Node-To-Segment (NTS)—elmt109.f
Figure 22.1 Geometry and parameters of the frictional NTS contact element
Figure 22.2 Ratio between reaction forces of both rope ends for different frictional coefficients
Chapter 23: Lesson 13 Frictional Higher Order NTS—elmt110.f
Figure 23.1 Geometry and parameters of the frictional higher order NTS contact element
Figure 23.2 Initial and final configuration for sticking with (a) Lagrange, (b) Lobatto and (c) Bezier classes of shape functions, respectively
Figure 23.3 Initial, intermediate and final configuration for sliding with (a) Lagrange, (b) Lobatto and (c) Bezier classes of shape functions
Chapter 24: Lesson 14 Transient Contact Problems
Figure 24.1 Displacement of point at the first non-penetrating time step after impact vs. the position of the barrier
Appendix B: Higher Order Shape Functions of Different Classes
Figure B.1 1D quadratic shape functions
Figure B.2 Line representation with quadratic Lagrange and Lobatto shape functions using (square) nodal values
Figure B.5 Quadratic 2D additional Lobatto shape functions
Figure B.6 Used numbering for quadratic Lobatto plane element
Figure B.7 Line representation with quadratic Lagrange and Bezier shape functions using (square) nodal values
Figure B.9 Selection of quadratic Bezier shape functions
Figure B.10 Used numbering for quadratic Bezier plane element
Wiley Series in Computational Mechanics
Series Advisors:
René de BorstPerumal NithiarasuTayfun E. TezduyarGenki YagawaTarek Zohdi
Introduction to Computational Contact Mechanics: A Geometrical Approach
Konyukhov
April 2015
Extended Finite Element Method: Theory and Applications
Khoei
December 2014
Computational Fluid-Structure Interaction: Methods and Applications
Bazilevs, Takizawa and Tezduyar
January 2013
Introduction to Finite Strain Theory for Continuum Elasto-Plasticity
Hashiguchi and Yamakawa
November 2012
Nonlinear Finite Element Analysis of Solids and Structures, Second Edition
De Borst, Crisfield, Remmers and Verhoosel
August 2012
An Introduction to Mathematical Modeling: A Course in Mechanics
Oden
November 2011
Computational Mechanics of Discontinua
Munjiza, Knight and Rougier
November 2011
Introduction to Finite Element Analysis: Formulation, Verification and Validation
Szabó and Babuška
March 2011
Introduction to Computational Contact Mechanics
A Geometrical Approach
Alexander Konyukhov
Karlsruhe Institute of Technology (KIT), Germany
Ridvan Izi
Karlsruhe Institute of Technology (KIT), Germany
This edition first published 2015
© 2015 John Wiley & Sons Ltd
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Library of Congress Cataloging-in-Publication Data
Konyukhov, Alexander.
Introduction to computational contact mechanics : a geometrical approach / Alexander Konyukhov, Karlsruhe Institute of Technology (KIT), Germany, Ridvan Izi, Karlsruhe Institute of Technology (KIT), Germany.
pages cm. – (Wiley series in computational mechanics)
Includes bibliographical references and index.
ISBN 978-1-118-77065-8 (cloth : alk. paper) 1. Contact mechanics. 2. Mechanics, Applied. I. Izi, Ridvan. II. Title. III. Title: Computational contact mechanics.
TA353.K66 2015
620.1′05–dc23
2015005384
A catalogue record for this book is available from the British Library.
Series Preface
Since the publication of the seminal paper on contact mechanics by Heinrich Hertz in 1882, the field has grown into an important branch of mechanics, mainly due to the presence of a high number of applications in many branches of engineering. The advent of computational techniques to handle contact between deformable bodies has greatly enhanced the possibility of analyzing contact problems in detail, resulting, for instance, in an enhanced insight into wear problems. The numerical treatment of contact belongs to the hardest problems in computational engineering, and many publications and books have been written to date, marking progress in the field. An Introduction to Computational Contact Mechanics: A Geometrical Approach stands out in terms of the clear and geometric approach chosen by the authors. The book covers many aspects of computational contact mechanics and benefits from clear notation. It comes with detailed derivations and explanations, and an exhaustive number of programming and verification tasks, which will help the reader to master the subject.
Preface
Computational contact mechanics within the last decade has developed into a separate branch of computational mechanics dealing exclusively with the numerical modeling of contact problems. Several monographs on computational contact mechanics summarize the study of computational algorithms used in the computational contact mechanics. The most famous, and subject to several editions, are the monographs by Wriggers (2002) and Laursen (2002). Most of the topics are explained at a high research level, which requires a very good knowledge of both numerical mathematics and continuum mechanics. Therefore, this book was the idea of Professor Dr. Ing. Schweizerhof back in 2006, who proposed to me to introduce a course in computational contact mechanics in such a manner that the prerequisite knowledge should be minimized. The main goal was to explain many algorithms used in well-known Finite Element Software packages (ANSYS, ABAQUS, LS-DYNA) in a simple manner and learn their finite element implementation. The starting point of the course was a reduction of the original 3D finite element algorithms into the 2D case and an introductory part to differential geometry. As field of the research has developed, the exploitation of the geometrical methods, the so-called covariant approach, after years of research has lead to the joint research monograph together with Professor Dr. Ing. Schweizerhof in Konyukhov and Schweizerhof (2012) Computational Contact Mechanics: Geometrically Exact Theory for Arbitrary Shaped Bodies. At this point, we would like to mention other monographs that we recommend for reading in computational contact mechanics Kikuchi (1988), Sofonea (2012), Yastrebov (2013), in friction and tribology Popov (2010) and also, the famous book on analytical methods in contact mechanics by Johnson (1987).
Ridvan Izi joined the computational contact mechanics course in 2009 and started to give assistance from 2011 to the exercise programming part, and made a lot of effort to make the exercises “easy going” for the students. Thus, the joint work started, leading to the current structure of the exercises in Part II. We were trying to keep this structure independent as much as possible from the programming language, although the course has been given in FEAP (Finite Element Analysis Program) written in FORTRAN.
The current book is based on the course being taught over several years at the Karlsruhe Institute of Technology and proved to be an effective guide for graduate and PhD students studying computational contact mechanics. The geometrically exact theory for contact interaction is delivered in a simple attractiveengineering manner available for undergraduate students starting from 1D geometry.
The book is subdivided into two parts:
Part I
contains the theoretical basis for the computational contact mechanics, including necessary material for lectures in computational contact mechanics.
Part II
includes the necessary material for the practical implementation of algorithms, including verification and numerical analysis of contact problems.
Part II
is consequently constructed following the theory considered in
Part I
.
In addition, the original FORTRAN programs, including all numerical examples considered in
Part II
, are available from the supporting Wiley website at
www.wiley.com/go/Konyukhov
The basis of the geometrically exact theory for contact interaction is to build the proper coordinate system to describe the contact interaction in all its geometrical detail. This results in the special structure of the computational mechanics course—study in applied differential geometry, kinematics of contact, formulation of a weak form and linearization in a special coordinate system in a covariant form. Afterward, most popular methods to enforce contact conditions – the penalty method, Lagrange multipliers, augmented Lagrange multipliers, Mortar method and the more seldomly used Nitsche method – are formulated consequently, first for 1D and then for 2D systems finally leading to examples in 3D. It then applies to finite element discretization. The structure of contact elements for these methods is studied in detail and all numerical algorithms are derived in a form ready for implementation. Thus, the structure of contact elements is carefully derived for various situations: Node-To-Node (NTN), Node-To-Segment (NTS) and Segment-To-Segment (STS) contact approaches. Special attention is given to the derivation of contact elements with rigid bodies of simple geometry such as the Segment-To-Analytical Segment (STAS) approach.
Part II of the book contains programming schemes for the following finite elements: surface-to-analytical (rigid) surface, NTN for several methods: penalty, Lagrange, Nitsche methods; node-to-segment for both non-frictional and frictional cases, with Mortar type segment-to-segment and 3D node-to-segment contact elements. Through examples, special attention is given to the implementation of normal following forces, which is derived a particular case of implementation for the frictionless contact algorithm.
All examples are given in a sequential manner with increasing complexity, which allows the reader to program these elements easily. Though the course has been designed for the FEAP user using FORTRAN, the structure of all examples is given in a programming-block manner, which allows the user to program all elements using any, convenient programming language or just mathematical software such as MATLAB.
The examples and corresponding tests are conceptualized in order to study many numerical phenomena appearing in computational contact mechanics, such as influence of the penalty parameters, selection of meshes and element type for the contact patch test for non-frictional and frictional cases.
The original implementation of the derived contact elements was carried out in one of the earliest versions of FEAP originating from Professor Robert Taylor, University of California, Berkley. The Finite Element Analysis Program (FEAP) appeared at the Institute of Mechanics of the Karlsruhe Institute of Technology due to the joint collaboration between Robert Taylor and Karl Schweizerhof who further developed the FEAP code into FEAP-MeKa with the famous solid-shell finite element. The code used in the current course is a simplified student version without any finite elements used for research and is used for educational purposes. During private communications, Professor Taylor confirmed that a free updated version is available and is still supported at http://www.ce.berkeley.edu/projects/feap/feappv/. I am particularly thankful for his kind agreement to link the programming given in the current course to the updated version of FEAP. Though all originally implemented subroutines for contact elements are shown within the old version of FEAP (or FEAP-MeKa) together with all necessary specifications (geometry, loads, boundary conditions, etc.) of tasks, the subroutines can be easily rearranged for the updated version of FEAP. The code, together with numerical examples, is essential in order to work with examples given in Part II. Any reader familiar with FEAP can straightforwardly adopt this code to his/her needs. The code is written in FORTRAN, but the straightforward programming structure, without using any math library, is intentionally preserved in order that any user can easily adopt the code to any other programming language. Moreover, we do really hope that the flowcharts, provided for each contact element can be used for programming of computational contact mechanics exercises using symbolic mathematical software such as MATLAB, MATHEMATICA, and so on.
Alexander KonyukhovKarlsruhe Institute of TechnologyGermany
Acknowledgments
We are thankful to Professor Karl Schweizerhof for giving us the great opportunity to develop such a course for students.
We would like to thank Professor Robert Taylor for his kind agreement to link our course of computational contact mechanics to the current and updated version of FEAP, thus encouraging us to work with the current code. Professor Taylor confirmed that the free version is available and is still supported at http://www.ce.berkeley.edu/projects/feap/feappv/.
The group of excellent student assistants has been busy carefully testing all examples given in Part II in less than a year. We would particularly like to thank Christian Lorenz, Merita Haxhibeti, Isabelle Niesel and Oana Mrenes for careful testing of the contact mechanics examples and Marek Fassin for testing necessary structural finite elements. An additional thanks to Oana Mrenes for the many editing efforts made with contemporary LATEX packages.
Many thanks to Johann Bitzenbauer for the careful reading of the current manuscript version and his fruitful proposals that lead to improvement.
The book in its current version has been tested in a workshop for computational contact mechanics recently at the Bundeswehr Universität München—and we are thankful to Georgios Michaloudis for his careful reading and proposals.
The work on this book took us many weekends, sacrificing time spent with our families. At this last, but not least point, we would like to especially thank our families for their understanding and moral support during the work on this book.
Part I
Theory
Chapter 1Introduction with a Spring-Mass Frictionless Contact System
We start our introduction to contact mechanics from the simplest possible system: a mass point suspended on a spring, but free deformations of the system can be restricted by the additional plane. This chapter gives the general idea on how to handle contact by specifying contact constraints as well as illustrating numerical methods: Lagrange multipliers, Penalty and Augmented Lagrangian.
