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 Introduction to Computational Contact Mechanics: A Geometrical Approach is an ideal textbook for graduates and senior undergraduates, and is also a useful reference for researchers and practitioners working in computational mechanics. 

Table of Contents:
Preface viii Acknowledgements xi Part One Theory 1 1 Introduction with a spring-mass frictionless contact system  2 1.1 Structural part - deflection of spring-mass system 3 1.2 Contact part – non-penetration into rigid plane 3 1.3 Contact formulations 4 2 General formulation of a contact problem 11 2.1 Structural part – formulation of a problem in linear elasticity 11 2.2 Formulation of the contact part (Signorini’s problem) 14 3 Differential geometry 19 3.1 Curve and its properties 19 3.2 Frenet formulas in 2D 23 3.3 Description of surfaces by Gauss coordinates 24 3.4 Differential properties of surfaces 32 4 Geometry and kinematics for arbitrary two body contact problem 40 4.1 Local coordinate system 41 4.2 Closest Point Projection (CPP) procedure – Analysis 43 4.3 Contact kinematics. 50 5 Abstract form of formulations in computational mechanics 54 5.1 Operator necessary for the abstract formulation 54 5.2 Abstract form of iterative method 55 5.3 Fixed point theorem (Banach) 56 5.4 Newton iterative solution method 58 5.5 Abstract form for contact formulations 61 6 Weak formulation and consistent linearization 65 6.1 Weak formulation in the local coordinate system 66 6.2 Regularization with penalty method 67 6.3 Consistent linearization 67 6.4 Application to the Lagrange multipliers and to the following forces 71 6.5 Linearization of the convective variation δξ 73 6.6 Nitsche method 73 7 Finite element discretization 76 7.1 Computation of the contact integral for various contact approaches 76 7.2 Node-To-Node (NTN) contact element 78 7.3 Nitsche Node-To-Node (NTN) contact element 80 7.4 Node-To-Segment (NTS) contact element 81 7.5 Segment-To-Analytical-Surface (STAS) approach 88 7.6 Segment-To-Segment (STS) Mortar approach 94 8 Verification with analytical solution 99 8.1 Hertz problem 99 8.2 Rigid flat punch problem 104 8.3 Impact on moving pendulum – center of percussion 106 8.4 Generalized Euler-Eytelwein problem 108 9 Frictional contact problems 111 9.1 Measures of contact interactions – sticking and sliding case. Friction law. 111 9.2 Regularization of tangential force and return mapping algorithm 112 9.3 Weak form and its consistent linearization 118 9.4 Frictional Node-To-Node (NTN) contact element 119 9.5 Frictional Node-To-Segment (NTS) contact element 123 9.6 NTS frictional contact element 125 Part Two Programming and Verification Tasks 127 10 Introduction into programming and verification tasks 128 11 Lesson 1 Nonlinear structural truss – elmt1.f 132 11.1 Implementation 134 11.2 Examples 138 12 Lesson 2 Nonlinear structural plane – elmt2.f 144 12.1 Implementation 145 12.2 Examples 150 13 Lesson 3 Penalty Node-To-Node (NTN) – elmt100.f 154 13.1 Implementation 156 13.2 Examples 158 14 Lesson 4 Lagrange multiplier Node-To-Node (NTN) – elmt101.f 161 14.1 Implementation 163 14.2 Examples 165 15 Lesson 5 Nitsche Node-To-Node (NTN) – elmt102.f 167 15.1 Implementation 169 15.2 Examples 171 16 Lesson 6 Node-To-Segment (NTS) – elmt103.f 173 16.1 Implementation 175 16.2 Examples 178 16.3 Inverted contact algorithm – following force 182 17 Lesson 7 Segment-To-Analytical-Segment (STAS) – elmt104.f 186 17.1 Implementation 188 17.2 Examples 191 17.3 Inverted contact algorithm – general case of following forces 194 18 Lesson 8 Mortar / Segment-To-Segment (STS) – elmt105.f 202 18.1 Implementation 204 18.2 Examples 207 18.3 Inverted contact algorithm – following force 209 19 Lesson 9 Higher order Mortar / STS – elmt106.f 213 19.1 Implementation 215 19.2 Examples 219 20 Lesson 10 3D Node-To-Segment (NTS) elmt107.f 223 20.1 Implementation 225 20.2 Examples 229 21 Lesson 11 Frictional Node-To-Node (NTN) – elmt108.f 233 21.1 Implementation 235 21.2 Examples 237 22 Lesson 12 Frictional Node-To-Segment (NTS) – elmt109.f 239 22.1 Implementation 241 22.2 Examples 245 23 Lesson 13 Frictional higher order NTS – elmt110.f 250 23.1 Implementation 251 23.2 Examples 256 24 Lesson 14 Transient contact problems 259 24.1 Implementation 260 24.2 Examples 262 A Numerical integration 264 A.1 Gauss quadrature 266 B Higher order shape functions of different classes 268 B.1 General 268 B.2 Lobatto class 268 B.3 Bezier class 271