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# Adaptive Systems Advanced

Hideyuki AZEGAMI Professor

Department: Graduate School of Information Science

 Class Time: 2013 Spring Friday Recommended for: Informatics students majoring in complex system science

## Course Overview

### Course Aims

This course focuses on adapting phenomena of distributed parameter systems (systems with an infinite degree of freedom) such as elastic continua and flow fields which are modeled as boundary value problems of partial differential equations. Optimization problems of domains in which boundary value problems are defined become mathematical representations of adapting phenomena. We aim to understand the mathematical theories and methods to solve numerically the optimization problems.

### Key Features

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• Although this course places an emphasis on the understanding of the shape optimization problem, I made efforts give students a better understanding of the fact that the techniques and methods introduced beforehand- the nonlinear optimization theory, space function, the finite element method, just to name a few- are actually widely used in the modern world.
• I showed the students, through several analytical samples, that the shape optimization problem, in mathematical form, could be applied to familiar problems.
• I had students figure out for themselves that 3-dimensional measurement and modeling can be done with the 3-dimensional input device (advanced computer system for research and education).
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## Syllabus

### Course Objectives

Understanding the methods for changing phenomena obtained in numerical analysis into desired outcome by using mathematics.

### Lesson Outline

Each chapter consists of two halves. The first half consists of lectures using slides. The second half sets aside some time to solve exercise problems. The problems will be reviewed after that.

### Course Requirements

Basic foreknowledge on analytics and linear algebra is recommended.

### Related Resources

Arora, J. S. : Introduction to optimum design, McGraw-Hill, 1989.

Rao, S. S.: Engineering optimization: theory and practice, 3rd ed., John Wiley & Sons, 1996.

Haftka, R. T. and Gürdal, Z.: Elements of structural optimization, 3rd ed., Kluwer Academic, 1992.

Brezis Haim, Analyse Fonctionnelle –Theorie et Applicationes, Dunod, 2002

Yosida, K. : Functional analysis, 6th ed, Springer, (1980)

Cea, J. : Numerical methods of shape optimal design, Edited by Haug, E. J. and Cea, J., Optimization of Distributed Parameter Structures, Vol. 2, Sijthoff & Noordhoff, Alphen aan den Rijn, (1981), pp. 1049-1088.

Pironneau, O.: Optimal Shape Design for Elliptic Systems, Springer-Verlag, New York, (1984).

Armijo, L.: Minimization of functions having Lipschitz-continuous first partial derivatives, Pacific Journal of Mathematics, 16, (1966), pp. 1-3.

Y. C. Juan (Translation by Y .Ohashi, S. Murakami, N. Kamiya)::Kotai no Rikigaku/Riron, Baifuu-Kan, 1970

Hughes,T.J.R : The Finite element method : Linear static and dynamic finite element analysis, Prentice-Hall, (1987)

Ciarlet, P. G. : Handbook of Numerical Analysis, Vol. 2, Finite Element Methods (Part1), edited by Ciarlet, P. G. and Lions, J. L., North-Holland, (1991).

Kardestuncer, H. editor-in-chief : Finite element handbook, McGraw-Hill, (1987)

Pironneau, O.: Optimal Shape Design for Elliptic Systems, Springer-Verlag, New York, (1984).

Strang, Gilbert and Fix, George J. : An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, New Jersey, (1973)

Zienkiewicz, O. C. and Taylor, R. L. : The Finite Element Method - Fourth Edition -, Volumes 1 and 2, McGraw-Hill, London, (1989)

Hughes, Thomas J. R. : The Finite Element Method, Prentice-Hall, Englewood Cliffs, New Jersey, (1987)

Sneddon, Ian N. : Fourier Transforms, McGrow-Hill, New York, (1951).

Cea, J., Numerical methods of shape optimal design, Optimization of Distributed Parameter Structures, edited by Haug, E. J. and Cea, J., Sijthoff & Noordhoff, Alphen aan den Rijn, volume 2, 1981, pp. 1049-1088.

Zolésio, J. P., Domain variational formulation for free boundary problems , ibid, pp. 1152-1194.

Sokolowski, J. and Zolésio, J. P., Introduction to Shape Optimization: Shape Sensitivity Analysis, Springer-Verlag, New York, 1991.

Haug E. J., Choi, K. K. and Komkov, V. : Design Sensitivity Analysis of Structural Systems, Academic Press, Orland, (1986).

Pironneau, O.: Optimal Shape Design for Elliptic Systems, Springer-Verlag, New York, (1984).

Cea, J. : Numerical methods of shape optimal design, Edited by Haug, E. J. and Cea, J., Optimization of Distributed Parameter Structures, Vol. 2, Sijthoff & Noordhoff, Alphen aan den Rijn, (1981), pp. 1049-1088.

Zolésio, J. P. : The material derivative (or speed) method for shape optimization, ibid., pp. 1089-1151.

Azegami, H., Shimoda, M., Katamine, E. and Wu, Z. Q. : A domain optimization technique for elliptic boundary value problems, Computer Aided Optimization Design of Structures IV, Structural Optimization, edited by Hernandez, S. and El-Sayed, S. and Brebbia, C. A., Computational Mechanics Publications, Southampton (1995), pp. 51-58.

Azegami, H., Kaizu, S., Shimoda, M. and Katamine, E. : Irregularity of shape optimization problems and an improvement technique, Computer Aided Optimization Design of Structures V, edited by Hernandez, S. and Brebbia, C. A., Computational Mechanics Publications, Southampton (1997), pp. 309-326.

Azegami, H. : Solution to boundary shape identification problems in elliptic boundary value problems using shape derivatives, Inverse Problems in Engineering Mechanics II, edited by Tanaka, M. and Dulikravich, G. S., Elsevier, Tokyo, (2000), pp. 277-284.

Azegami, H., Yokoyama, S. and Katamine, E. : Solution to shape optimization problems of continua on thermal elastic deformation, Inverse Problems in Engineering Mechanics III, edited by Tanaka, M. and Dulikravich, G. S., Elsevier, Tokyo, (2002), pp. 61-66

### Course Schedule

Session Contents
1 Guidance
2 Chapter 1: the Basics of Nonlinear Optimization Theory
- Lagrange Multiplier Method
- Kuhn -Tucker Condition
- The Duality Theorem
3
4 Chapter 2: the Basics of Optimum Design
- the State Equation
- Gâteaux derivative, Fréchet derivative
- Gradient
- the Ajoint Variable method
5 Chapter 3: Nonlinear Programming Method
- the Gradient Method
- Newton Method
- the Sequential Quadratic Programming Method
6 Chapter 4: Variational Principle and the Basics of Functional Analysis
- Variational Principle
- Linear space, Banach space, Hilbert Space
- the Dual space
- Riesz representation theorem
7
8 Chapter 5: The Boundary Value Problem of the Partial Differential Equation
- Poisson Problem
- the Abstract Variational Problem
- Lax-Milgram Theory
- the Minimum Variational Problem
- the Linear Elastic Problem
- Stokes Problem
- the Saddle Point Variational Problem
- the Dynamic Linear Elastic Problem
- Navier-Stokes Problem
9
10 Chapter 6: the Finite Element Method
- Galerkin Method
- the Finite Element Method
- the Isoparametric Element
- Gaussian Quadrature Rule
- the Error Estimation
11
12 Chapter 7: Shape Optimization Problem
- the Abstract Optimization Problem
- the Abstract Gradient Method
- the Optimum Shape Variation Problem
- the Shape Derivative
- the H1 Gradient Method
- the Linear Elastic Problem
- Stokes Problem
- the Dynamic Linear Elastic Problem
- Navier-Stokes Problem
13
14 Chapter 8: Topology Optimization Problem
- the Optimum Density Variation Problem
- the Linear Density Problem
- the H1 Gradient Method
15 Practice with a 3-dimensional input device

### Grading

Grading will be based on the final examination.

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## Class Materials

### Lecture Handouts

Note: All files are in Japanese.

Session #1
Guidance (PDF, 7602KB)

Session #2-#3
Chapter 1: the Basics of Nonlinear Optimization Theory (PDF, 413KB)

Session #4
Chapter 2: the Basics of Optimum Design (PDF, 123KB)

Session #5
Chapter 3:Nonlinear Programming Method (PDF, 151KB)

Session #6-#7
Chapter 4: Variational Principle and the Basics of Functional Analysis (PDF, 212KB)

Session #8-#9
Chapter 5: The Boundary Value Problem of the Partial Differential Equation (PDF, 274KB)

Session #10-#11
Chapter 6: the Finite Element Method (PDF, 327KB)

Session #12-#13
Chapter 7: Shape Optimization Problem (PDF, 3011KB)

Session #14
Chapter 8: Topology Optimization Problem (PDF, 147KB)

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Page last updated February 11, 2009

The class contents were most recently updated on the date indicated. Please be aware that there may be some changes between the most recent year and the current page.