| 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:
Euler-Lagrage equations with global (functional) and local (function) constraints Two examples: Compliant design problem of a beam and contact problem with a beam Sep. 24, 2026: Euler-Lagrange equations when there are two and three independent variables |
Euler-Lagrange euqations for local (or function) constraints Euler-Lagrange equations in two and three independent variables |
Quiz 12 | 9 |
Sep. 29, 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 Oct. 1, 2026: 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 |
10 |
Oct. 6, 2026:
Elastic continuum for stiff-light optimization Review for the midterm Oct. 8, 2026: Midterm examination |
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- | 11 |
Oct. 13, 2026:
Dynamic and multi-physics problems in calculus of variations framework
Oct. 15, 2026: Electro-thermal-elastic structural optimization 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 |
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| 12 |
Oct. 20, 2026:
Simultaneous geoemtry and material optimization of trusses 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. 22, 2026: Overview of the programming assignment 2 Dual formulation of truss optimization for statically determinate trusses Concepts of states of self stress and Maxwell's rule Dual method for truss optimizaition |
Lecture notes 8b |
Homogenization of a 1D problem 99-line code for 2D stiff-structure optimization YinSyn 2D code for topology optimization of structures and compliant mechanisms Geometry+material optimization paper Geometry+material statically-determinate truss optimization paper Dual truss opt code |
| 13 - 16 | Module 3: Multiphysics design problems; Sensitivity analysis for shape and topology optimization | 13 |
Oct. 28, 2025:
Pressure load problem and electro-thermal-elastic problem Ananlytical expressions for sensitivity in the Calculus of Variations framework Oct. 29, 2025: Material-property interpolation for topology optimization for electro-elasto-statics, fluids, etc. |
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- | 14 |
Nov. 4, 2025:
Sensitivity of dynamic compliance; electro-thermal-elastic analysis Reversal of the sequence of design sensitivity of adjoint variables in electro-thermal-elastic problems Nov. 6, 2025: 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. 11, 2025:
Sensitivity analysis: parameter, shape; 1D and 2D; Jacobian and its derivatives (material and space derivatives)
Nov. 13, 2025: 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. 18, 2025:
Numerical optimization techniques for topology optimization Convex linearization leading to the method of moving asymptotes (MMA) Nov. 19, 2025: 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 |
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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