# MEAM 599-06 : 1999 Winter

## The Homogenization Design Method for Structures, Materials, and Mechanics

### - An Advanced Computational Method in Mechanical Engineering -

Description, Outline, Lecture Notes

### General Information

M & W, 4:30-6:00pm @ Dow 1014

No homework & No Examination

Research Oriented Advanced Course for Computational Methods and Design

### Couse Description

This course is designed to teach advanced computational methods for design optimization of structures, material microstructures, and compliant mechanisms, by using the homogenization method and the finite element method. Starting from reviewing recent development of structural optimization and various finite element formulations, we shall develop an appropriate stress/strain assumed finite elements which are appropriate for design optimization of structures and material microstructures. Then we shall develop the homogenization theory to deal with material heterogeneity using the asymptotic expansion method based on the Wu-Washizu variational principle. These are then applied to design optimization.

This course is specially designed for the students who are interested in advanced computational methods for design and mechanics of composite materials.

### Outline of the Course

1. Review of Finite Element Method

1. Brief History of the Finite Element Method
2. Concept of Clough's Finite Element
3. Argyris' Contribution to Plate/Shell Elements
4. MATHEMATICA Applications in the Finite Element Method
5. Strain/Stress Assumed Finite Elements
6. Equation Solvers for the Finite Element Method

2. Adaptive Finite Element Methods

1. Adaptive Methods : r-, h-, and p-Adaptation
2. Peano's Theory of Interpolation
3. Error Analysis : Cea's Lemma
4. Interpolation Error Analysis : An Application of MATHEMATICA
5. A Posteriori Error Estimate for Adaptation

3. Review of Structural Optimization

1. Introduction : Size, Shape, and Topology Optimization
2. Fully Stressed Design and Optimization Algorithms
3. Shape Optimization and Its Difficulty
4. Characteristics Function for Topology Optimization
5. Approximation of the Characteristic Function
6. Homogenization Theory for 1-D Problems
7. Introduction of the Homogenization Design Method

4. The Homogenization Design Method

1. Multiple Loads and Multiple Constraints Design Problems
2. Eigenvalue Related Topology Design : Vibration & Buckling
3. Material Microstructural Design
4. Compliant Mechanism Design
5. Piezo-ceramic Microstructural Design
6. Magnetic Field Design
7. Issues on Filtering and Penalty Schemes

5. Homogenization Theories

1. Asymptotic Expansion Methods
2. Homogenization in Thermo-elasticity
3. Extension to Nonlinear Problems
4. Homogenization in Mixed Formulations
5. Further Research Issues

### Lecture Notes and Handouts : 1999 Winter

1. History of the Finite Element Method in 1960s
2. A Note on Clough's Finite Element (Stress/Strain Assumed Element) : Posted on January 18, 1999 : This is a summary note of Clough's original paper on the finite element method published in 1960.
3. A Note on the Standard Displacement Method : Introduction to FEM : Posted on January 21, 1999 : Revised on January 25, 1999. This note should be read before the one for Clough's finite element. I hope that you would quickly review standard finite element method for linearly elastic solid structures. I also put MATLAB programs so that you can modify them to do quick exercises to find out the nature of the 8 node hexahedral finite element in which each displacement field is approximated by a tri-linear polynomial of the parametric coordinates.
4. A Note on the Homogenization Design Method : Posted onFeb 12, 1999

### Term Projects for the Students Who need Grade

The following problems are suggested to be solved by April 21, if you wish to have better grade than A. Problems are not set up very rigidly, and then you should make up the problem you would work based on the five suggested problems. If you have any questions and matters to be discussed with me, please contact with me, while I am in town.