| Week | Dates Topics | Notes | Others |
| 1 - 6 | Module 1: Structural design with finite-variable optimization | ||
| 1 |
Aug. 6, 2026:
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 Finite variable optimization vs. calculus of variation; and how they relate to structural optimization |
Lecture notes 1 Lecture notes 2 Lecture notes 2A Lecture notes 3 |
Article on strctural hierarchy by Prof. Lakes
Article on Eiffel's optimal structures |
| 2 |
Aug. 11, 2026:
Unconstrained and constrained minimization
Aug. 13, 2026: KKT conditions; fmincon to solve constrained minimization problems Truss FEA; size and shape optimization in Matlab. |
Lecture notes 5 |
Matlab files for three-bar truss optimization
Quiz 1 Quiz 2 |
| 3 |
Aug. 18, 2026:
Constrained minimization; duality; two-bar truss optimization
Aug. 20, 2026: Multi-bar truss optimization |
Lecture notes 6 extra (duality) |
Quiz 3 Quiz 4 |
| 4 |
Aug. 25, 2026:
Size (and topology) optimization of trusses and the optimality criteria algorithm
Aug. 27, 2026: 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 Quiz 5 Quiz 6 |
| 5 - 11 | Module 2: Structural optimization in the framework of calculus of variations | ||
| 5 |
Sep. 1, 2026:
Genesis of calculus of variations
Sep. 3, 2026: Formulating variational problems in geometry and mechanics |
Geometry and mechanics problems cast as calculus of variations problems |
Quiz 7 |
| 6 |
Sep. 8, 2026:
Mathematical preliminaries of calculus of variations: vectors spaces, function spaces, etc.
Sep. 10, 2026: Gateaux (first) variation, Frechet differential, etc. Fundamental lemma of calculus of variations Some examples to practise taking variation |
Banach, Sobolev, etc., spaces First variation of a functional Fundamental lemma of calculus of variations |
Quiz 8: https://connections.swellgarfo.com/game/-P0vA_0qCK_yBboQL7sb
Quiz 9 Read Ted Chiang's "Story of Your Life". Try to watch Arrival, the movie, if you can find the time. |
| 7 |
Sep. 15, 2026:
Variational derivative and derivation of Euler-Lagrange equations in the manner apparently done by Euler without using the concept of variation Minimum potential energy principle, Principle of virtual work, and force-balance: equivalence established through calculus of variations Sep. 17, 2026: Static equilibrium of a beam and interpreting the boundary conditions. EL equations for the case of multiple derivatives and multiple functions 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 |
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Quiz 10
Quiz 11 |
8 |
Sep. 22, 2026:
Generalization of Euler-Lagrnage equations to multiple functions and multiple derivatives; subtleties in Gateaux variation
Sep. 24, 2026: Discussion of the mini-project: various aspects of truss design |
|
Quiz 12 | 9 |
Sep. 29, 2026:
Euler-Lagrage equations with global (functional) and local (function) constraints Two examples: Compliant design problem of a beam and contact problem with a beam Oct. 1, 2026: Euler-Lagrange equations and boundary conditions for 2D and 3D domains. |
Euler-Lagrange euqations for local (or function) constraints Euler-Lagrange equations in two and three independent variables |
- | 10 |
Oct. 6, 2026:
Review for the midterm
Oct. 8, 2026: Midterm examination |
|
Midterm questions of variational methods course ME 256 from 2005 to 2018 Midterm questions of ME 260 from 2021 to 2025 Refer also to early homework problems of ME 256 as practice for the midterm |
11 |
Oct. 13, 2026:
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 Oct. 15, 2026: Bar optimization Principle of optimality for light-stiff structures Optimization of a bar (a dozen problems) Bar optimization code in Matlab and the optimality criteria algorithm |
A dozen optimization problems of a bar Solutions of two bar optimization problems |
Quiz 13: Grading ChatGPT's solution to the midterm on a printed copy with everything closed except one's mind, to find the mistakes, inconsistencies, and misinterpretations. Bar optimization code |
| 12 |
Oct. 20, 2026:
Beam optimization problem for stiffness, strength, and flexibility Beam optimization code in Matlab 2D frame optimization Oct. 22, 2026: Homogenization method and its role in topology optimization; 1D homogenization using asymptotic expansion method Power law and material interpolation, and SIMP 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 |
Lecture notes 8b |
Many beam optimization problems Solution to beam optmization for stiffness and flexibility 2D frame optimization problem Beam optimization for strength Homogenization of a 1D problem Beam optimization code 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. 27, 2026:
Material-property interpolation for topology optimization for electro-elasto-statics, fluids, etc. Pressure load problem and electro-thermal-elastic problem Oct. 29, 2026: Introduction to topology optimization using COMSOL and nTop |
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- | 14 |
Nov. 3, 2026:
Ananlytical expressions for sensitivity in the Calculus of Variations framework Sensitivity of dynamic compliance; electro-thermal-elastic analysis Reversal of the sequence of design sensitivity of adjoint variables in electro-thermal-elastic problems Nov. 5, 2026: Verification of sensitivity using finite-difference methods and its pitfalls and a remedy |
Topology optimization of coupled electrostatic and elastostatic problem |
- | 15 |
Nov. 10, 2026:
Sensitivity analysis: parameter, shape; 1D and 2D; Jacobian and its derivatives (material and space derivatives)
Nov. 12, 2026: 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. 17, 2026:
Numerical optimization techniques for topology optimization Convex linearization leading to the method of moving asymptotes (MMA) Nov. 19, 2026: Topological derivatives and their use |
ConLin, MMA, and GCA papers Topological derivative Optimization with topological derivative Instructions for the term paper Project presentation template |
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Schedule of lectures in previous years
Lecture notes of 2025 offering of this course
Lecture notes of 2024 offering of this course
Lecture notes of 2023 offering of this course
Lecture notes of 2022 offering of this course