| Week | Dates Topics | Notes | Others |
| 1 - 5 | Module 1: Structural design with finite-variable optimization | ||
| 1 |
Aug. 8, 2023:
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. 10,2023: Finite variable optimization vs. calculus of variation; and how they relate to structural optimization |
Lecture notes 2 Lecture notes 3 |
Article on strctural hierarchy by Prof. Lakes
Article on Eiffel's optimal structures |
| 2 |
Aug. 15, 2023:
No class due to Indian Independence Day.
Aug. 17, 2023: Unconstrained and constrained minimization |
Lecture notes 5 |
- |
| 3 |
Aug. 22, 2023:
Constrained minimization; duality; two-bar truss optimization
Aug. 24, 2023: Multi-bar truss optimization |
Lecture notes 6 extra (duality) |
- |
| 4 |
Aug. 29, 2023:
Size (and topology) optimization of trusses and the optimality criteria algorithm
Aug. 31, 2023: 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. 5, 2023:
Dual formulation of truss optimization for statically determinate trusses Concepts of states of self stress and Maxwell's rule Sep. 7, 2023: Dual method for truss optimizaition Simultaneous geoemtry and material optimization of trusses |
Lecture notes 9 Truss optimization using the dual method Dual truss opt code |
Geormetry+material optimization paper |
| 6 - 11 | Module 2: Structural optimization in the framework of calculus of variations | ||
| 6 |
Sep. 12, 2023:
Genesis of calculus of variations Formulating variational problems in geometry and mechanics Sep. 14, 2023: Mathematical preliminaries of calculus of variations: vectors spaces, function spaces, etc. |
Geometry and mechanics problems cast as calculus of variations problems Mathematical preliminaries to calculus of variations; functional and function spaces Banach, Sobolev, etc., spaces |
- |
| 7 |
Sep. 19, 2023:
No class due to Vinayaka Chavithi Sep. 21, 2023: Gateaux (first) variation, Frechet differential, etc. Fundamental lemma of calculus of variations Some examples to practise taking variation Fundamental lemma of calculus of variations |
Fundamental lemma of calculus of variations |
- |
| 8 |
Sep. 26, 2023:
Euler-Lagrage equations; multiple functions and multiple derivatives
Practice with Gateaux variation and EL equations
Sep. 28, 2023: Euler-Lagrange equations when there are two and three independent variables Euler-Lagrage equations with global (functional) and local (function) constraints |
Euler-Lagrange equations with global (functional) constraints Euler-Lagrange equations with local (function) constraints |
- | 9 |
Oct. 3, 2023:
Variable end conditions in Calculus of Variations; transversality conditions; broken extremals; Weirstrass-Erdmann conditions Oct. 5, 2023: Optimization of a bar (a dozen problems) Bar optimization code in Matlab |
A dozen optimization problems of a bar Solutions of two bar optimization problems |
Bar optimization code | 10 |
Oct. 10, 2023:
Beam optmization problems Oct. 12, 2023: 2D frame optimization problems; stiffness, strength, and flexibility Beam optimization code in Matlab |
Solution to beam optmization for stiffness and flexibility 2D frame optimization problem Beam optimization for strength |
Beam optimization code | 11 |
Oct. 17, 2023:
Homogenization method and its role in topology optimization; power law and material interpolation Oct. 19, 2023: 1D homogenization using asymptotic expansion method Playing with 99-line code to understand the influence of the penalty parameter and sensitivity filter YinSyn code for stiff structures and compliant mechanisms |
|
99-line code for 2D stiff-structure optimization YinSyn 2D code for topology optimization of structures and compliant mechanisms |
| 12 - 15 | Module 3: Multiphysics design problems; Sensitivity analysis for shape and topology optimization | 12 |
Oct. 24, 2023:
No class due to Dasara Oct. 26, 2023: Topology optimization of compliant mechanisms in 2D continuum Multiphysics design problems |
|
- | 13 |
Oct. 31, 2023:
Pressure load problem and electro-thermal-elastic problem Ananlytical expressions for sensitivity from the Calculus of Variations framework Nov. 2, 2023: Topology optimization for electro-elasto-statics, fluids, etc. |
|
14 |
Nov. 7, 2022:
Adjoint method of sensitivity analysis; and using the Lagrangian Sensitivity of dynamic compliance; electro-thermal-elastic analysis Nov. 9, 2022: Reversal of the sequence of design sensitivity of adjoint variables in electro-thermal-elastic problems; Coupled multi-physics problems: electrostatic-elastostatic Verification of sensitivity using finite-difference methods and its pitfalls and remedy |
Topology optimization of coupled electrostatic and elastostatic problem |
- | 15 |
Nov. 14, 2022:
Taking variation with vector and other shorthand notation COMSOL overview and demonstration; explanation of HW4 and PA2 problems Nov. 16, 2022: Topology optimization with COMSOL Sensitivity analysis: parameter, shape; 1D and 2D; Jacobian and its derivatives (material and space derivatives) |
Parameter sensititvity of a functional Shape derivative with a 1D example Derivatives of the Jacobian and its other forms Shape optimization in 2D |
- | 16 |
Nov. 21, 2022:
Homogenization; SIMP; topology optimization TopOpt, YinSyn; optimality criteria method Discussion for project-cum-programming assignment, term paper Nov. 23, 2022: Topological derivatives and their use Convex linearization leading to the method of moving asymptotes (MMA) |
Topological derivative Optimization with topological derivative Instructions for the term paper Project presentation template ConLin, MMA, and GCA papers |
- |
Lecture notes of 2022 offering of this course