Shape Optimization Problems

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Shape Optimization Problems

Hideyuki Azegami
Shape Optimization Problems Book PDF

Author : Hideyuki Azegami
Publisher : Springer Nature
Page : 646 pages
File Size : 49,8 Mb
Release : 2020-09-30
Category : Mathematics
ISBN : 9789811576188

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Shape Optimization Problems by Hideyuki Azegami Pdf

This book provides theories on non-parametric shape optimization problems, systematically keeping in mind readers with an engineering background. Non-parametric shape optimization problems are defined as problems of finding the shapes of domains in which boundary value problems of partial differential equations are defined. In these problems, optimum shapes are obtained from an arbitrary form without any geometrical parameters previously assigned. In particular, problems in which the optimum shape is sought by making a hole in domain are called topology optimization problems. Moreover, a problem in which the optimum shape is obtained based on domain variation is referred to as a shape optimization problem of domain variation type, or a shape optimization problem in a limited sense. Software has been developed to solve these problems, and it is being used to seek practical optimum shapes. However, there are no books explaining such theories beginning with their foundations. The structure of the book is shown in the Preface. The theorems are built up using mathematical results. Therefore, a mathematical style is introduced, consisting of definitions and theorems to summarize the key points. This method of expression is advanced as provable facts are clearly shown. If something to be investigated is contained in the framework of mathematics, setting up a theory using theorems prepared by great mathematicians is thought to be an extremely effective approach. However, mathematics attempts to heighten the level of abstraction in order to understand many things in a unified fashion. This characteristic may baffle readers with an engineering background. Hence in this book, an attempt has been made to provide explanations in engineering terms, with examples from mechanics, after accurately denoting the provable facts using definitions and theorems.

Variational Methods in Shape Optimization Problems

Dorin Bucur,Giuseppe Buttazzo
Variational Methods in Shape Optimization Problems Book PDF

Author : Dorin Bucur,Giuseppe Buttazzo
Publisher : Springer Science & Business Media
Page : 218 pages
File Size : 47,7 Mb
Release : 2006-09-13
Category : Mathematics
ISBN : 9780817644031

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Variational Methods in Shape Optimization Problems by Dorin Bucur,Giuseppe Buttazzo Pdf

Shape optimization problems are treated from the classical and modern perspectives Targets a broad audience of graduate students in pure and applied mathematics, as well as engineers requiring a solid mathematical basis for the solution of practical problems Requires only a standard knowledge in the calculus of variations, differential equations, and functional analysis Driven by several good examples and illustrations Poses some open questions.

Existence and Regularity Results for Some Shape Optimization Problems

Bozhidar Velichkov
Existence and Regularity Results for Some Shape Optimization Problems Book PDF

Author : Bozhidar Velichkov
Publisher : Springer
Page : 349 pages
File Size : 49,6 Mb
Release : 2015-03-21
Category : Mathematics
ISBN : 9788876425271

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Existence and Regularity Results for Some Shape Optimization Problems by Bozhidar Velichkov Pdf

​We study the existence and regularity of optimal domains for functionals depending on the spectrum of the Dirichlet Laplacian or of more general Schrödinger operators. The domains are subject to perimeter and volume constraints; we also take into account the possible presence of geometric obstacles. We investigate the properties of the optimal sets and of the optimal state functions. In particular, we prove that the eigenfunctions are Lipschitz continuous up to the boundary and that the optimal sets subject to the perimeter constraint have regular free boundary. We also consider spectral optimization problems in non-Euclidean settings and optimization problems for potentials and measures, as well as multiphase and optimal partition problems.

Introduction to Shape Optimization

Jan Sokolowski,Jean-Paul Zolesio
Introduction to Shape Optimization Book PDF

Author : Jan Sokolowski,Jean-Paul Zolesio
Publisher : Springer Science & Business Media
Page : 254 pages
File Size : 47,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642581069

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Introduction to Shape Optimization by Jan Sokolowski,Jean-Paul Zolesio Pdf

This book is motivated largely by a desire to solve shape optimization prob lems that arise in applications, particularly in structural mechanics and in the optimal control of distributed parameter systems. Many such problems can be formulated as the minimization of functionals defined over a class of admissible domains. Shape optimization is quite indispensable in the design and construction of industrial structures. For example, aircraft and spacecraft have to satisfy, at the same time, very strict criteria on mechanical performance while weighing as little as possible. The shape optimization problem for such a structure consists in finding a geometry of the structure which minimizes a given functional (e. g. such as the weight of the structure) and yet simultaneously satisfies specific constraints (like thickness, strain energy, or displacement bounds). The geometry of the structure can be considered as a given domain in the three-dimensional Euclidean space. The domain is an open, bounded set whose topology is given, e. g. it may be simply or doubly connected. The boundary is smooth or piecewise smooth, so boundary value problems that are defined in the domain and associated with the classical partial differential equations of mathematical physics are well posed. In general the cost functional takes the form of an integral over the domain or its boundary where the integrand depends smoothly on the solution of a boundary value problem.

Introduction to Shape Optimization

J. Haslinger,R. A. E. Makinen
Introduction to Shape Optimization Book PDF

Author : J. Haslinger,R. A. E. Makinen
Publisher : SIAM
Page : 276 pages
File Size : 53,7 Mb
Release : 2003-01-01
Category : Mathematics
ISBN : 9780898715361

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Introduction to Shape Optimization by J. Haslinger,R. A. E. Makinen Pdf

Treats sizing and shape optimization in a comprehensive way, covering everything from mathematical theory through computational aspects to industrial applications.

Topological Derivatives in Shape Optimization

Antonio André Novotny,Jan Sokołowski
Topological Derivatives in Shape Optimization Book PDF

Author : Antonio André Novotny,Jan Sokołowski
Publisher : Springer Science & Business Media
Page : 423 pages
File Size : 45,5 Mb
Release : 2012-12-14
Category : Technology & Engineering
ISBN : 9783642352454

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Topological Derivatives in Shape Optimization by Antonio André Novotny,Jan Sokołowski Pdf

The topological derivative is defined as the first term (correction) of the asymptotic expansion of a given shape functional with respect to a small parameter that measures the size of singular domain perturbations, such as holes, inclusions, defects, source-terms and cracks. Over the last decade, topological asymptotic analysis has become a broad, rich and fascinating research area from both theoretical and numerical standpoints. It has applications in many different fields such as shape and topology optimization, inverse problems, imaging processing and mechanical modeling including synthesis and/or optimal design of microstructures, fracture mechanics sensitivity analysis and damage evolution modeling. Since there is no monograph on the subject at present, the authors provide here the first account of the theory which combines classical sensitivity analysis in shape optimization with asymptotic analysis by means of compound asymptotic expansions for elliptic boundary value problems. This book is intended for researchers and graduate students in applied mathematics and computational mechanics interested in any aspect of topological asymptotic analysis. In particular, it can be adopted as a textbook in advanced courses on the subject and shall be useful for readers interested on the mathematical aspects of topological asymptotic analysis as well as on applications of topological derivatives in computation mechanics.

Optimal Shape Design

B. Kawohl
Optimal Shape Design Book PDF

Author : B. Kawohl
Publisher : Springer Science & Business Media
Page : 404 pages
File Size : 53,9 Mb
Release : 2000-11-16
Category : Mathematics
ISBN : 3540679715

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Optimal Shape Design by B. Kawohl Pdf

Optimal Shape Design is concerned with the optimization of some performance criterion dependent (besides the constraints of the problem) on the "shape" of some region. The main topics covered are: the optimal design of a geometrical object, for instance a wing, moving in a fluid; the optimal shape of a region (a harbor), given suitable constraints on the size of the entrance to the harbor, subject to incoming waves; the optimal design of some electrical device subject to constraints on the performance. The aim is to show that Optimal Shape Design, besides its interesting industrial applications, possesses nontrivial mathematical aspects. The main theoretical tools developed here are the homogenization method and domain variations in PDE. The style is mathematically rigorous, but specifically oriented towards applications, and it is intended for both pure and applied mathematicians. The reader is required to know classical PDE theory and basic functional analysis.

Shape Optimization by the Homogenization Method

Gregoire Allaire
Shape Optimization by the Homogenization Method Book PDF

Author : Gregoire Allaire
Publisher : Springer Science & Business Media
Page : 470 pages
File Size : 41,7 Mb
Release : 2012-12-06
Category : Technology & Engineering
ISBN : 9781468492866

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Shape Optimization by the Homogenization Method by Gregoire Allaire Pdf

This book provides an introduction to the theory and numerical developments of the homogenization method. It's main features are: a comprehensive presentation of homogenization theory; an introduction to the theory of two-phase composite materials; a detailed treatment of structural optimization by using homogenization; a complete discussion of the resulting numerical algorithms with many documented test problems. It will be of interest to researchers, engineers, and advanced graduate students in applied mathematics, mechanical engineering, and structural optimization.

Shape Optimization And Optimal Design

John Cagnol,Michael P. Polis,Jean-Paul Zolesio
Shape Optimization And Optimal Design Book PDF

Author : John Cagnol,Michael P. Polis,Jean-Paul Zolesio
Publisher : CRC Press
Page : 451 pages
File Size : 55,9 Mb
Release : 2017-08-02
Category : Mathematics
ISBN : 9780203904169

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Shape Optimization And Optimal Design by John Cagnol,Michael P. Polis,Jean-Paul Zolesio Pdf

This volume presents developments and advances in modelling passive and active control systems governed by partial differential equations. It emphasizes shape analysis, optimal shape design, controllability, nonlinear boundary control, and stabilization. The authors include essential data on exact boundary controllability of thermoelastic plates with variable transmission coefficients.

Applied Shape Optimization for Fluids

Bijan Mohammadi,Olivier Pironneau
Applied Shape Optimization for Fluids Book PDF

Author : Bijan Mohammadi,Olivier Pironneau
Publisher : OUP Oxford
Page : 296 pages
File Size : 55,9 Mb
Release : 2009-09-24
Category : Mathematics
ISBN : 9780191574214

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Applied Shape Optimization for Fluids by Bijan Mohammadi,Olivier Pironneau Pdf

The fields of computational fluid dynamics (CFD) and optimal shape design (OSD) have received considerable attention in the recent past, and are of practical importance for many engineering applications. This new edition of Applied Shape Optimization for Fluids deals with shape optimization problems for fluids, with the equations needed for their understanding (Euler and Navier Strokes, but also those for microfluids) and with the numerical simulation of these problems. It presents the state of the art in shape optimization for an extended range of applications involving fluid flows. Automatic differentiation, approximate gradients, unstructured mesh adaptation, multi-model configurations, and time-dependent problems are introduced, and their implementation into the industrial environments of aerospace and automobile equipment industry explained and illustrated. With the increases in the power of computers in industry since the first edition, methods which were previously unfeasible have begun giving results, namely evolutionary algorithms, topological optimization methods, and level set algortihms. In this edition, these methods have been treated in separate chapters, but the book remains primarily one on differential shape optimization. This book is essential reading for engineers interested in the implementation and solution of optimization problems using commercial packages or in-house solvers and graduates and researchers in applied mathematics, aerospace, or mechanical engineering, fluid dynamics, and CFD. More generally, anyone needing to understand and solve design problems or looking for new exciting areas for research and development in this area will find this book useful, especially in applying the methodology to practical problems.

Applied Shape Optimization for Fluids

Bijan Mohammadi,Olivier Pironneau
Applied Shape Optimization for Fluids Book PDF

Author : Bijan Mohammadi,Olivier Pironneau
Publisher : Oxford University Press
Page : 292 pages
File Size : 55,9 Mb
Release : 2010
Category : Mathematics
ISBN : 9780199546909

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Applied Shape Optimization for Fluids by Bijan Mohammadi,Olivier Pironneau Pdf

Contents: PREFACE; ACKNOWLEDGEMENTS; 1. Introduction; 2. Optimal shape design; 3. Partial differential equations for fluids; 4. Some numerical methods for fluids; 5. Sensitivity evaluation and automatic differentiation; 6. Parameterization and implementation issues; 7. Local and global optimization; 8. Incomplete sensitivities; 9. Consistent approximations and approximate gradients; 10. Numerical results on shape optimization; 11. Control of unsteady flows; 12. From airplane design to microfluidic; 13. Toplogical optimization for fluids; 14. Conclusion and perspectives; INDEX.

Variational methods in some shape optimization problems

Dorin Bucur,Giuseppe Buttazzo
Variational methods in some shape optimization problems Book PDF

Author : Dorin Bucur,Giuseppe Buttazzo
Publisher : Edizioni della Normale
Page : 217 pages
File Size : 44,9 Mb
Release : 2002-10-01
Category : Mathematics
ISBN : 8876422978

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Variational methods in some shape optimization problems by Dorin Bucur,Giuseppe Buttazzo Pdf

The study of shape optimization problems is a very wide field, both classical, as the isoperimetric problem and Newton problem of the best aerodynamical shape show, and modern, for all the recent results obtained in the last two, three decades. The fascinating feature is that the competing objects are shapes, i.e. domains of Rn, instead of functions, as usually occurs in problems of calculus of variations. This constraint often produces additional difficulties that lead to a lack of existence of a solution and the introduction of suitable relaxed formulations of the problem. However, in a few cases an optimal solution exists, due to the special form of the cost functional and to the geometrical restriction on the class of competing domains. This volume collects the lecture notes of two courses given in the academic year 2000/01 by the authors at the University of Pisa and at the Scuola Normale Superiore respectively. The courses were mainly addressed to Ph. D. students and required a background in the topics in functional analysis that are usually taught in undergraduate courses.

New Trends in Shape Optimization

Aldo Pratelli,Günter Leugering
New Trends in Shape Optimization Book PDF

Author : Aldo Pratelli,Günter Leugering
Publisher : Birkhäuser
Page : 314 pages
File Size : 53,7 Mb
Release : 2015-12-01
Category : Mathematics
ISBN : 9783319175638

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New Trends in Shape Optimization by Aldo Pratelli,Günter Leugering Pdf

This volume reflects “New Trends in Shape Optimization” and is based on a workshop of the same name organized at the Friedrich-Alexander University Erlangen-Nürnberg in September 2013. During the workshop senior mathematicians and young scientists alike presented their latest findings. The format of the meeting allowed fruitful discussions on challenging open problems, and triggered a number of new and spontaneous collaborations. As such, the idea was born to produce this book, each chapter of which was written by a workshop participant, often with a collaborator. The content of the individual chapters ranges from survey papers to original articles; some focus on the topics discussed at the Workshop, while others involve arguments outside its scope but which are no less relevant for the field today. As such, the book offers readers a balanced introduction to the emerging field of shape optimization.

Numerical Methods in Sensitivity Analysis and Shape Optimization

Emmanuel Laporte,Patrick Le Tallec
Numerical Methods in Sensitivity Analysis and Shape Optimization Book PDF

Author : Emmanuel Laporte,Patrick Le Tallec
Publisher : Springer Science & Business Media
Page : 202 pages
File Size : 52,6 Mb
Release : 2012-12-06
Category : Technology & Engineering
ISBN : 9781461200697

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Numerical Methods in Sensitivity Analysis and Shape Optimization by Emmanuel Laporte,Patrick Le Tallec Pdf

Sensitivity analysis and optimal shape design are key issues in engineering that have been affected by advances in numerical tools currently available. This book, and its supplementary online files, presents basic optimization techniques that can be used to compute the sensitivity of a given design to local change, or to improve its performance by local optimization of these data. The relevance and scope of these techniques have improved dramatically in recent years because of progress in discretization strategies, optimization algorithms, automatic differentiation, software availability, and the power of personal computers. Numerical Methods in Sensitivity Analysis and Shape Optimization will be of interest to graduate students involved in mathematical modeling and simulation, as well as engineers and researchers in applied mathematics looking for an up-to-date introduction to optimization techniques, sensitivity analysis, and optimal design.

Shape Design Sensitivity Analysis and Optimization Using the Boundary Element Method

Zhiye Zhao
Shape Design Sensitivity Analysis and Optimization Using the Boundary Element Method Book PDF

Author : Zhiye Zhao
Publisher : Springer Science & Business Media
Page : 203 pages
File Size : 48,5 Mb
Release : 2012-12-06
Category : Technology & Engineering
ISBN : 9783642843822

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Shape Design Sensitivity Analysis and Optimization Using the Boundary Element Method by Zhiye Zhao Pdf

This book investigates the various aspects of shape optimization of two dimensional continuum structures, including shape design sensitivity analysis, structural analysis using the boundary element method (BEM), and shape optimization implementation. The book begins by reviewing the developments of shape optimization, followed by the presentation of the mathematical programming methods for solving optimization problems. The basic theory of the BEM is presented which will be employed later on as the numerical tool to provide the structural responses and the shape design sensitivities. The key issue of shape optimization, the shape design sensitivity analy sis, is fully investigated. A general formulation of stress sensitivity using the continuum approach is presented. The difficulty of the modelling of the ad joint problem is studied, and two approaches are presented for the modelling of the adjoint problem. The first approach uses distributed loads to smooth the concentrated adjoint loads, and the second approach employs the singu larity subtraction method to remove the singular boundary displacements and tractions from the BEM equation. A novel finite difference based approach to shape design sensitivity is pre sented, which overcomes the two drawbacks of the conventional finite difference method. This approach has the advantage of being simple in concept, and eas ier implementation. A shape optimization program for two-dimensional continuum structures is developed, including structural analysis using the BEM, shape design sensitiv ity analysis, mathematical programming, and the design boundary modelling.