1st Edition

Nonsmooth Mechanics and Convex Optimization

By Yoshihiro Kanno Copyright 2011
    445 Pages 79 B/W Illustrations
    by CRC Press

    446 Pages 79 B/W Illustrations
    by CRC Press

    "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

    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


    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


    Fundamentals of Plasticity

    Perfect Plasticity

    Plasticity with Isotropic Hardening

    Plasticity with Kinematic Hardening


    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.

    "The text is rather self-contained with a clear structure, the material is presented nicely which makes it accessible to young researchers, the style is fluent and the examples are carefully selected. The monograph confirms the existence of strong interaction between mechanics and applied mathematics. It will certainly acquire an important position in everybody’s Mechanics and Applied Mathematics library."
    Mathematical Reviews, 2012