Nonsmooth Mechanics and Convex Optimization

This book presents a comprehensive methodology for dealing with nonsmooth behaviors in mechanics, in accordance with contemporary theory and algorithms of optimization. Delving into both theoretical studies and numerical algorithms to provide a new perspective on the subject, this book addresses how linear, semidefinite, and second-order cone optimizations are applied to real-world problems in nonsmooth mechanics. It also covers complementarity problems, optimality conditions, Fenchel and Lagrangian dualities, and algorithms in optimization. The author also explores working with many practical applications in structural engineering, including cable networks, membranes, masonry structures, contact problems, and plasticity. This is an ideal guide to nonsmooth mechanics for graduate students and researchers in computational and applied mechanics, applied mathematics, and operations research.

"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