The efficiency and reliability of manufactured products depend on, among other things, geometrical aspects; it is therefore not surprising that optimal shape design problems have attracted the interest of applied mathematicians and engineers. This self-contained, elementary introduction to the mathematical and computational aspects of sizing and shape optimization enables readers to gain a firm understanding of the theoretical and practical aspects so they may confidently enter this field. In contrast to existing texts on structural optimization, Introduction to Shape Optimization: Theory, Approximation, and Computation treats sizing and shape optimization in a comprehensive way, covering everything from mathematical theory (existence analysis, discretizations, and convergence analysis for discretized problems) through computational aspects (sensitivity analysis, numerical minimization methods) to industrial applications. Some of the applications included are contact stress minimization for elasto-plastic bodies, multidisciplinary optimisation of an airfoil, and shape optimization of a dividing tube. By presenting sizing and shape optimization in an abstract way, the authors are able to use a unified approach in the mathematical analysis for a large class of optimization problems in various fields of physics. Audience; Introduction to Shape Optimization: Theory, Approximation, and Computation is written primarily for students of applied mathematics, scientific computing, and mechanics. Most of the material is directed toward graduate students, although a portion of it is suitable for senior undergraduate students. Readers are assumed to have some knowledge of partial differential equations and their numerical solution, as well as modern programming language such as C++ Fortran 90.
J. Haslinger is a Professor in the Department of Physics of Metals at Charles University in Prague. He is the author or co-author of five monographs and over 110 publications in scientific journals. His research interests include the approximation of elliptic equations and inequalities, contact mechanics, nonsmooth analysis, approximation of differentialinclusions, and shape and material optimization. R. A. E. Makinen is a Professor in the Department of Mathematical Information Technology at the University of Jyvaskyla in Finland. He has also acted as a modeling specialist at Numerola Oy (Inc.). Professor Makinen is the author or co-author of over 50 publications in scientific journals and conference proceedings. His research interests include structural optimization, especially shape optimization; numerical solution of partial differential equations; and numerical software.
Preface; Notation; Introduction; Part I. Mathematical aspects of sizing and shape optimization. 1. Why the mathematical analysis is important; 2. A mathematical introduction to sizing and shape optimization; Part II. Computational aspects of sizing and shape optimization. 3. Sensitivity analysis; 4. Numerical minimization methods; 5. On automatic differentiation of computer programs; 6. Fictitious domain methods in shape optimization; Part III. Applications. 7. Applications in elasticity; 8. Fluid mechanical and multidisciplinary applications; Appendix A. Weak formulations and approximations of elliptic equations and inequalities; Appendix B. On parametrizations of shapes and mesh generation; Bibliography; Index.
Treats sizing and shape optimization in a comprehensive way, covering everything from mathematical theory through computational aspects to industrial applications.
