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2016 | Book

Motivation and Framework Read chapter Read first chapter

Optimal Design through the Sub-Relaxation Method

Understanding the Basic Principles

Author: Pablo Pedregal

Publisher: Springer International Publishing

Book Series : SEMA SIMAI Springer Series

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This book provides a comprehensive guide to analyzing and solving optimal design problems in continuous media by means of the so-called sub-relaxation method. Though the underlying ideas are borrowed from other, more classical approaches, here they are used and organized in a novel way, yielding a distinct perspective on how to approach this kind of optimization problems. Starting with a discussion of the background motivation, the book broadly explains the sub-relaxation method in general terms, helping readers to grasp, from the very beginning, the driving idea and where the text is heading. In addition to the analytical content of the method, it examines practical issues like optimality and numerical approximation. Though the primary focus is on the development of the method for the conductivity context, the book’s final two chapters explore several extensions of the method to other problems, as well as formal proofs. The text can be used for a graduate course in optimal design, even if the method would require some familiarity with the main analytical issues associated with this type of problems. This can be addressed with the help of the provided bibliography.

Table of Contents

Frontmatter
Chapter 1. Motivation and Framework
Abstract
It is not difficult to motivate, from a practical point of view, the kind of situations we would like to deal with and analyze. We have selected a typical example in heat conduction, but many other examples are as valid as this one. Suppose we have two very different materials at our disposal: the first, with conductivity α 1 = 1, is a good and expensive conductor; the other is a cheap material, almost an insulator with conductivity coefficient α 0 = 0. 001. These two materials are to be used to fill up a given design domain Q, which we assume to be a unit square for simplicity (Fig. 1.1), in given proportions t 1, t 0, with t 1 + t 0 = 1. Typically, t 1 < t 0 given that the first material is much more expensive than the second. We will take, for definiteness, t 1 = 0. 4, t 0 = 0. 6. The thermal device is isolated all over ∂ Q, except for a small sink Γ 0 at the middle of the left side where we normalize temperature to vanish, and there is a uniform source of heat all over Q of size unity. The mixture of the two materials is to be decided so that the dissipated energy is as small as possible.
Pablo Pedregal
Chapter 2. Our Approach
Abstract
To introduce our analytical strategy, let us focus on the particular situation described in Sect. 1.1, but changed in a way to avoid any distraction from our main objective:
Pablo Pedregal
Chapter 3. Relaxation Through Moments
Abstract
We would like to reflect on the (sub)relaxation we found at the end of Chap. 2:
$$\displaystyle{\mbox{ Minimize in }\nu =\{\nu _{\mathbf{x}}\}_{\mathbf{x}\in \varOmega }:\quad I(\nu ) =\int _{\varOmega }\mathbf{F}(\mathbf{x}) \cdot \nabla u(\mathbf{x})\,d\mathbf{x}}$$
subject to
$$\displaystyle\begin{array}{rcl} & \nabla u(\mathbf{x}) =\int _{\mathbb{R}^{2}\times \mathbb{R}^{2}}\lambda \,d\nu _{\mathbf{x}}(\lambda,\rho ),\quad u = u_{0}\mbox{ on }\partial \varOmega, & {}\\ & \mathbf{V}(\mathbf{x}) =\int _{\mathbb{R}^{2}\times \mathbb{R}^{2}}\rho \,d\nu _{\mathbf{x}}(\lambda,\rho ),\quad \mbox{ div}\mathbf{V} = 0\mbox{ in }\varOmega, & {}\\ & \nabla u(\mathbf{x}) \cdot \mathbf{V}(\mathbf{x}) =\int _{\mathbb{R}^{2}\times \mathbb{R}^{2}}\lambda \cdot \rho \, d\nu _{\mathbf{x}}(\lambda,\rho )\mbox{ for a.e. }\mathbf{x} \in \varOmega,& {}\\ & \,\mbox{ supp}\,(\nu _{\mathbf{x}}) \subset \varLambda _{1} \cup \varLambda _{0},\mbox{ for a.e. }\mathbf{x} \in \varOmega, & {}\\ & \int _{\varOmega }\nu _{\mathbf{x}}(\varLambda _{1})\,d\mathbf{x} = t_{1}. & {}\\ \end{array}$$
Pablo Pedregal
Chapter 4. Optimality
Abstract
All of our previous analysis had the final goal of getting to the relaxation
$$\displaystyle{\mbox{ Minimize in }(t,\mathbf{s}):\quad \overline{I}(t,\mathbf{s}) =\int _{\varOmega }f(\mathbf{x})u(\mathbf{x})\,d\mathbf{x}}$$
subject to
$$\displaystyle\begin{array}{rcl} & t(\mathbf{x}) \in [0,1],\int _{\varOmega }t(\mathbf{x})\,d\mathbf{x} = r,\quad \vert \mathbf{s}(\mathbf{x})\vert \leq 1, & {}\\ & -\mbox{ div}\left [a(\mathbf{x})\nabla u(\mathbf{x}) + b(\mathbf{x})\vert \nabla u(\mathbf{x})\vert \mathbf{s}(\mathbf{x})\right ] = f(\mathbf{x})\mbox{ in }\varOmega,\quad u = u_{0}\mbox{ on }\partial \varOmega,& {}\\ \end{array}$$
Pablo Pedregal
Chapter 5. Simulation
Abstract
Once optimality conditions, for the relaxed version of the problem, have been worked out and understood, the next natural step is to deal with numerical simulation in order to test to what extent our analysis can be helpful to produce, at least in some examples, true minimizing sequences. This chapter focuses on this important task. Our contribution here is rather modest. A full numerical and computational treatment would be required to fully exploit this point of view for numerical simulation in more realistic circumstances. We just set to ourselves the task to illustrate, in some simplified situations, how the analytical ideas behind the sub relaxation method may be applied for the numerical approximation. A much more rigorous and systematic study ought to be performed in this regard by someone with a firm background on computational issues that the author of this text cannot claim to have.
Pablo Pedregal
Chapter 6. Some Extensions
Abstract
From the extensions enumerated in Sect. 1.2, we would like to examine two important ones. Some of them are so hard to deal with that they are fully open problems, and whatever new development would have a tremendous impact. For some others, only partial (or even very partial) results are known. Yet for some other situations, extension is almost immediate. The significance of non-linear situations (either in the cost functional or in the state law) for Engineering is not covered here, and should be explored in the bibliography provided. Our motivation is purely analytical.
Pablo Pedregal
Chapter 7. Some Technical Proofs
Abstract
We include in this final chapter those topics which have been deliberately deferred in our main exposition.
Pablo Pedregal
Metadata
Title
Optimal Design through the Sub-Relaxation Method
Author
Pablo Pedregal
Copyright Year
2016
Electronic ISBN
978-3-319-41159-0
Print ISBN
978-3-319-41158-3
DOI
https://doi.org/10.1007/978-3-319-41159-0

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