Nonsmooth Mechanics and Convex Optimization  book cover
1st Edition

Nonsmooth Mechanics and Convex Optimization

ISBN 9781138072787
Published March 29, 2017 by CRC Press
445 Pages 79 B/W Illustrations

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Book Description

"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

Table of Contents

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

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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