| Week | Dates Topics | Notes | Others |
| 1 - 6 | Module 1: Structural design with finite-variable optimization | ||
| 1 |
Aug. 6, 2024:
Overview of the course (Size, shape and topology optimization) Template of a structural optimization problem: What we need to learn and how to formulate problems Identifying size, shape, and topology optimization problems Aug. 8,2024: Finite variable optimization vs. calculus of variation; and how they relate to structural optimization |
Lecture notes 2 Lecture notes 2A Lecture notes 3 |
Article on strctural hierarchy by Prof. Lakes
Article on Eiffel's optimal structures |
| 2 |
Aug. 13, 2024:
Unconstrained and constrained minimization
Aug. 15, 2024: No class due Independence Day |
Lecture notes 5 |
- |
| 3 |
Aug. 20, 2024:
Constrained minimization; duality; two-bar truss optimization
Aug. 22, 2024: Multi-bar truss optimization |
Lecture notes 6 extra (duality) |
- |
| 4 |
Aug. 27, 2024:
Size (and topology) optimization of trusses and the optimality criteria algorithm
Aug. 29, 2024: Dual formulation of truss optimization for statically determinate trusses Concepts of states of self stress and Maxwell's rule Simultaneous geoemtry and material optimization of trusses |
Lecture notes 7b Lecture notes 8a |
FEA theory notes (trusses and beams) Matlab truss analysis code Matlab truss optimization code |
| 5 |
Sep. 3, 2024:
Dual formulation of truss optimization for statically determinate trusses Concepts of states of self stress and Maxwell's rule Sep. 5, 2024: Dual method for truss optimizaition |
|
Dual truss opt code |
| 6 |
Sep. 10, 2024:
Simultaneous geoemtry and material optimization of trusses
Sep. 12, 2024: Discusison of Homework 1, problem 1 Overview of the programming assignment 1 Summary of Module 1. |
|
Geometry+material optimization paper
Geometry+material statically-determinate truss optimization paper |
| 7 - 11 | Module 2: Structural optimization in the framework of calculus of variations | ||
| 7 |
Sep. 17, 2024:
Genesis of calculus of variations Formulating variational problems in geometry and mechanics Mathematical preliminaries of calculus of variations: vectors spaces, function spaces, etc. Sep. 19, 2024: Gateaux (first) variation, Frechet differential, etc. Fundamental lemma of calculus of variations Some examples to practise taking variation Fundamental lemma of calculus of variations |
Geometry and mechanics problems cast as calculus of variations problems Mathematical preliminaries to calculus of variations; functional and function spaces Banach, Sobolev, etc., spaces |
Read Ted Chiang's "Story of Your Life". Try to watch Arrival, the movie, if you can find the time. |
| 8 |
Sep. 24, 2024:
Some discussion on definitions of functional spaces The concept of variation and its mathematical interpretation Distinguishing Gateaux variation, Frechet differential, and Frechet derivative Fundamental lemma of calculus of variations. Sep. 26, 2024: Variational derivative and derivation of Euler-Lagrange equations in the manner apparently done by Euler without using the concept of variation Static equilibrium of a beam and interpreting the boundary conditions. |
Fundamental lemma of calculus of variations Euler-Lagrange equations in multiple functions and multiple derivatives but in a single domain variable |
- | 9 |
Oct. 1, 2024:
Examples of a bar and a beam static equilibrium equations Three ways of writing static equilibrium: minimum potential energy, principle of virtual work, and force balance Three ways for dynamic equilibrium: Hamilton's principle, D'Lambert's principle, and Newton's second law. Dynamic equilibrium of a bar to motivate functionals of two independent variables. Oct. 3, 2024: Euler-Lagrange equations when there are two and three independent variables Euler-Lagrage equations with global (functional) and local (function) constraints Two examples: Compliant design problem of a beam and contact problem with a beam |
Euler-Lagrange euqations for global (or functional) constraints Euler-Lagrange euqations for local (or function) constraints |
- | 10 |
Oct. 8, 2024:
Optimization of a bar (a dozen problems) Bar optimization code in Matlab and the optimality criteria algorithm Oct. 10, 2024: Beam optimization problem for stiffness, strength, and flexibility Beam optimization code in Matlab 2D frame optimization |
Solutions of two bar optimization problems Many beam optimization problems Solution to beam optmization for stiffness and flexibility 2D frame optimization problem Beam optimization for strength |
Bar optimization code Beam optimization code |
11 |
Oct. 15, 2024:
Midterm examination: open-everything including the internet. Oct. 17, 2024: Variable end conditions in Calculus of Variations; transversality conditions; broken extremals; Weirstrass-Erdmann conditions Examples: Fermat's refraction problem and the generalization of the tractrix problem |
|
- | 12 |
Oct. 22, 2024:
Homogenization method and its role in topology optimization; 1D homogenization using asymptotic expansion method Power law and material interpolation, and SIMP Taking variation with vector and other shorthand notation Playing with 99-line code to understand the influence of the penalty parameter and sensitivity filter Practice with YinSyn code for stiff structures and compliant mechanisms Oct. 24, 2024: Review of module 2 and an overview of Programming Assignment 2 |
|
Homogenization of a 1D problem 99-line code for 2D stiff-structure optimization YinSyn 2D code for topology optimization of structures and compliant mechanisms |
| 13 - 16 | Module 3: Multiphysics design problems; Sensitivity analysis for shape and topology optimization | 13 |
Oct. 29, 2024:
Pressure load problem and electro-thermal-elastic problem Ananlytical expressions for sensitivity in the Calculus of Variations framework Oct. 31, 2024: Material-property interpolation for topology optimization for electro-elasto-statics, fluids, etc. |
|
- | 14 |
Nov. 5, 2024:
Sensitivity of dynamic compliance; electro-thermal-elastic analysis Reversal of the sequence of design sensitivity of adjoint variables in electro-thermal-elastic problems Nov. 7, 2024: Verification of sensitivity using finite-difference methods and its pitfalls and a remedy COMSOL overview; and demonstration of topology optimization using COMSOL |
Topology optimization of coupled electrostatic and elastostatic problem |
- | 15 |
Nov. 12, 2024:
Sensitivity analysis: parameter, shape; 1D and 2D; Jacobian and its derivatives (material and space derivatives)
Nov. 14, 2024: Shape sesitivity and shape optimization |
Shape derivative with a 1D example Derivatives of the Jacobian and its other forms Shape optimization in 2D |
- | 16 |
Nov. 19, 2024:
Numerical optimization techniques for topology optimization Convex linearization leading to the method of moving asymptotes (MMA) Nov. 21, 2024: Topological derivatives and their use Discussion for project, term paper; course overview |
ConLin, MMA, and GCA papers Topological derivative Optimization with topological derivative Instructions for the term paper Project presentation template |
- |
Schedule of lectures in previous years
Lecture notes of 2023 offering of this course
Lecture notes of 2022 offering of this course