Nonsmooth Mechanics and Convex Optimization

"This book concerns matter that is intrinsically difficult: convex optimization, complementarity and duality, nonsmooth analysis, linear and nonlinear programming, etc. The author has skillfully introduced these and many more concepts, and woven them into a seamless whole by retaining an easy and consistent style throughout. The book is not all theory: There are many real-life applications in structural engineering, cable networks, frictional contact problems, and plasticity… I recommend it to any reader who desires a modern, authoritative account of nonsmooth mechanics and convex optimization."
— Prof. Graham M.L. Gladwell, Distinguished Professor Emeritus, University of Waterloo, Fellow of the Royal Society of Canada

"… reads very well—the structure is good, the language and style are clear and fluent, and the material is rendered accessible by a careful presentation that contains many concrete examples. The range of applications, particularly to problems in mechanics, is admirable and a valuable complement to theoretical and computational investigations that are at the forefront of the areas concerned."
— Prof. B. Daya Reddy, Department of Mathematics and Applied Mathematics, Director of Centre for Research in Computational and Applied Mechanics, University of Cape Town, South Africa

"Many materials and structures (e.g., cable networks, membrane) involved in practical engineering applications have complex responses that cannot be described by smooth constitutive relations. … The author shows how these difficult problems can be tackled in the framework of convex analysis by arranging the carefully chosen materials in an elegant way. Most of the contents of the book are from the original contributions of the author. They are both mathematically rigorous and readable. This book is a must-read for anyone who intends to get an authoritative and state-of-art description for the analysis of nonsmooth mechanics problems with theory and tools from convex analysis."
— Prof. Xu Guo, State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Dalian University of Technology

Yoshihiro Kanno is an associate professor in the Department of Mathematical Informatics at the University of Tokyo, Japan. Dr. Kanno received his Ph.D in structural engineering from Kyoto University, Japan, in 2002. He received the Maeda Prize in Engineering in 2005 and CJK-OSM4 Award for Young Investigator in 2006.
The author and coauthor of numerous professional articles on applied mechanics and optimization, Dr. Kanno's research interest is in the interface between mechanics and mathematics. He is a member of the International Society for Structural and Multidisciplinary Optimization, the Japan Society of Mechanical Engineers, the Architectural Institute of Japan, and the Operations Research Society of Japan.

Part I: Convex Optimization Over Symmetric Cone

Cones, Complementarity, and Conic Optimization
Proper Cones and Conic Inequalities
Complementarity over Cones
Positive-Semidefinite Cone
Second-Order Cone
Conic Constraints and Their Relationship
Conic Optimization

Optimality and Duality
Fundamentals of Convex Analysis
Optimality and Duality
Application to Semidefinite Programming

Applications in Structural Engineering
Compliance Optimization
Eigenvalue Optimization
Set-Valued Constitutive Law


Part II: Cable Networks: An Example in Nonsmooth Mechanics

Principles of Potential Energy for Cable Networks
Constitutive law
Potential Energy Principles in Convex Optimization Forms
More on Cable Networks: Nonlinear Material Law

Duality in Cable Networks: Principles of Complementary Energy
Duality in Cable Networks (1): Large Strain
Duality in Cable Networks (2): Linear Strain
Duality in Cable Networks (3): Green-Lagrange Strain

Part III: Numerical Methods
Algorithms for Conic Optimization
Primal-Dual Interior-Point Method
Reformulation and Smoothing Method

Numerical Analysis of Cable Networks
Cable Networks with Pin-Joint
Cable Networks with Sliding Joints
Form-Finding of Cable Networks
Part IV: Problems in Nonsmooth Mechanics

Masonry Structures
Introduction
Principle of Potential Energy for Masonry Structures
Principle of Complementary Energy for Masonry Structures
Numerical Aspects

Planar Membranes
Analysis in Small Deformation
Principle of Potential Energy for Membranes
Principle of Complementary Energy for Membranes
Numerical Aspects

Frictional Contact Problems
Friction Law
Incremental Problem
Discussions on Various Complementarity Forms

Plasticity
Fundamentals of Plasticity
Perfect Plasticity
Plasticity with Isotropic Hardening
Plasticity with Kinematic Hardening