Vladislav A. Yastrebov

Numerical Methods in Contact Mechanics (eBook, ePUB)

• Format: ePub

Produktbeschreibung

Computational contact mechanics is a broad topic which brings together algorithmic, geometrical, optimization and numerical aspects for a robust, fast and accurate treatment of contact problems. This book covers all the basic ingredients of contact and computational contact mechanics: from efficient contact detection algorithms and classical optimization methods to new developments in contact kinematics and resolution schemes for both sequential and parallel computer architectures. The book is self-contained and intended for people working on the implementation and improvement of contact algorithms in a finite element software. Using a new tensor algebra, the authors introduce some original notions in contact kinematics and extend the classical formulation of contact elements. Some classical and new resolution methods for contact problems and associated ready-to-implement expressions are provided. Contents: 1. Introduction to Computational Contact. 2. Geometry in Contact Mechanics. 3. Contact Detection. 4. Formulation of Contact Problems. 5. Numerical Procedures. 6. Numerical Examples. About the Authors Vladislav A. Yastrebov is a postdoctoral-fellow in Computational Solid Mechanics at MINES ParisTech in France. His work in computational contact mechanics was recognized by the CSMA award and by the Prix Paul Caseau of the French Academy of Technology and Electricité de France.

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Produktdetails

• Verlag: John Wiley & Sons
• Seitenzahl: 416
• Erscheinungstermin: 13. Februar 2013
• Englisch
• ISBN-13: 9781118648056
• Artikelnr.: 38254455

Autorenporträt

Vladislav A. Yastrebov, Postdoctoral-fellow in Computational Solid Mechanics.

Inhaltsangabe

Foreword xi Preface xiii Notations xv Chapter 1. Introduction to
Computational Contact 1 1.1. Historical remark 5 1.1.1. The augmented
Lagrangian method 7 1.2. Basics of the numerical treatment of contact
problems 9 1.2.1.Contact detection 9 1.2.2.Contact discretization 10
1.2.3.Contact resolution 13 Chapter 2. Geometry in ContactMechanics 15 2.1.
Introduction 15 2.2. Interaction between contacting surfaces 19 2.2.1.Some
notations 19 2.2.2.Normal gap 21 2.2.3.Closest point on a surface 26
2.2.4.Closest point on a curve 28 2.2.5.Shadow-projectionmethod 32
2.2.6.Tangential relative sliding 35 2.3. Variations of geometrical
quantities 38 2.3.1.First-ordervariations 38 2.3.2. Second-order variations
40 2.4. Numericalvalidation 42 2.5. Discretized geometry 44 2.5.1. Shape
functions andfinite elements 44 2.5.2. Geometryof contact elements 45 2.6.
Enrichmentof contactgeometry 51 2.6.1. Derivation of enriched quantities 53
2.6.2. Variations of geometrical quantities 58 2.6.3.Exampleof enrichment
65 2.6.4.Concludingremarks 68 Chapter 3. Contact Detection 71 3.1.
Introduction 71 3.2.All-to-all detection 76 3.2.1.Preliminaryphase 76
3.2.2.Detection phase 79 3.3.Bucket sort detection 84
3.3.1.Preliminaryphase 86 3.3.2.Numerical tests 87 3.3.3.Detection phase 90
3.3.4. Multi-face contact elements 91 3.3.5. Improvements 92 3.4. Case of
unknown master-slave 93 3.5.Parallel contactdetection 97 3.5.1.General
presentation 97 3.5.2. Single detection, multiple resolution approach 97
3.5.3. Multiple detection, multiple resolution approach 99 3.5.4.
Scalability test 100 3.6.Conclusion 101 Chapter 4. Formulation of Contact
Problems 103 4.1. Contact of a deformable solid with a rigid plane 103
4.1.1.Unilateral contactwith a rigid plane 104 4.1.2. Interpretation of
contact conditions 109 4.1.3.Friction 111 4.1.4.An analogywith plastic flow
117 4.1.5. Interpretation of frictional conditions 121 4.2. Contact of a
deformable solid with an arbitrary rigid surface 124 4.2.1. Non-penetration
condition 125 4.2.2. Hertz-Signorini-Moreau's contact conditions 129 4.2.3.
Interpretation of contact conditions 130 4.2.4. Frictional conditions and
their interpretation 132 4.2.5. Example: rheology of a one-dimensional
frictional system on a sinusoidal rigid substrate 133 4.3. Contact between
deformable solids 135 4.3.1. General formulation and variational inequality
135 4.3.2. Remarks on Coulomb's frictional law 142 4.4. Variational
equality and resolution methods 144 4.5. Penaltymethod 145
4.5.1.Frictionless case 145 4.5.2. Example 148 4.5.3.
Nonlinearpenaltyfunctions 151 4.5.4. Frictional case 153 4.6. Method of
Lagrange multipliers 157 4.6.1.Frictionless case 158 4.6.2. Frictional case
161 4.6.3. Example 164 4.7. AugmentedLagrangianMethod 170 4.7.1.
Introduction 170 4.7.2.Applicationto contact problems 174 4.7.3.Example 183
Chapter 5. Numerical Procedures 189 5.1.Newton'smethod 189 5.1.1.
One-dimensional Newton's method 190 5.1.2. Multidimensional Newton's method
193 5.1.3. Application to non-differentiable functions 195 5.1.4.
Subdifferentials and subgradients 196 5.1.5 GeneralizedNewtonmethod 200
5.2. Returnmappingalgorithm 203 5.3. Finite elementmethod 210 5.3.1.
Introduction 211 5.3.2.Contact elements 216 5.3.3. Discretization of the
contact interface 219 5.3.4. Virtual work for discretized contact interface
220 5.3.5.Linearizationof equations 223 5.3.6.Example 225 5.4. Residual
vectors and tangent matrices for contact elements 225 5.4.1. Penalty
method: frictionless case 226 5.4.2. Penalty method: frictional case 228
5.4.3. Augmented Lagrangian method: frictionless case 237 5.4.4. Augmented
Lagrangian method: frictional case 240 5.5. Method of partial
Dirichlet-Neumann boundary conditions 248 5.5.1. Description of the
numerical technique 248 5.5.2.Frictionless case 250 5.5.3.Frictional case
254 5.5.4.Remarks 255 5.6. Technicaldetails 255 5.6.1. Rigidmaster surface
256 5.6.2. Multi-face contact elements and smoothing techniques 257
5.6.3.Heterogeneous friction 260 5.6.4.Short remarks 261 Chapter 6.
Numerical Examples 265 6.1.Two dimensionalproblems 265 6.1.1. Indentation
by a rigid flat punch 265 6.1.2. Elastic disk embedded in an elastic bored
plane 269 6.1.3. Indentation of an elastic rectangle by a circular indenter
272 6.1.4. Axisymmetricdeepcup drawing 274 6.1.5. Shallowironing 278 6.1.6.
Axisymmetric post-buckling of a thin-walled cylinder 279 6.2.
Three-dimensionalproblems 286 6.2.1. Accordion post-buckling folding of a
thin-walled tube 286 6.2.2. Hydrostatic extrusion of a square plate through
a circular hole 288 6.2.3. Frictional sliding of a cube on a rigid plane
292 Appendix 1. Vectors, Tensors and s-Structures 297 A1.1. Fundamentals
298 A1.2.Vector space basis 303 A1.2.1. Transformation matrices, covariant
and contravariant objects 306 A1.2.2. Gradient operator or Hamilton's
operator 308 A1.3. Sub-basis, vector function of v-scalar argument 311
A1.4.Tensors 314 A1.5.Tensor as a linear operatoron vector space 322
A1.6.S-structures 325 A1.6.1. Formal definition, notations and types 327
A1.6.2.Simple operations 331 A1.6.3. Invariant s-structures 333 A1.6.4.
Scalar products of v-vectors 336 A1.6.5. Inversev-vector 341 A1.6.6.
Isomorphism of s-space and tensor space 343 A1.6.7. Tensor product of
v-vectors 348 A1.7.Reducedformof s-structures 349 Appendix 2. Variations of
Geometrical Quantities 353 A2.1.First-ordervariations 353 A2.1.1.Normal
projectioncase 354 A2.1.2. Shadow-projection case: infinitely remote
emitter 356 A2.1.3. Shadow-projection case: close emitter 361 A2.2.
Second-order variations 362 A2.2.1.Normal projectioncase 362 A2.2.2.
Shadow-projection case: infinitely remote emitter 369 A2.2.3.
Shadow-projection case: close emitter 370 Bibliography 375 Index 387