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Course11210820144 lectures

Variational Methods in Mechanics

Prof. Ananthasuresh and Prof. Safvan Palathingal · IISc Bangalore

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Lectures

Overview of Optimization Calculus of Variations

  1. Lec1-Part I-Classification of optimization problems and the place of Calculus of Variations in it Video
  2. Lec2-Part II-Classification of optimization problems and the place of Calculus of Variations in it Video
  3. Lec3-Part I - Genesis of Calculus of Variations Video
  4. Lec4-Part II - Genesis of Calculus of Variations Video
  5. Lec5-Part I - Formulation of Calculus of Variations problems in geometry and mechanics and design Video
  6. Lec6-Part II - Formulation of Calculus of Variations problems in geometry and mechanics and design Video

Summary of finite-variable optimization

  1. Lec7-Part I - Unconstrained minimization in one and many variables Video
  2. Lec8-Part II - Unconstrained minimization in one and many variables Video
  3. Lec9-Part I - Constrained minimization KKT conditions Video
  4. Lec10-Part II - Constrained minimization KKT conditions Video
  5. Lec11-Part I - Sufficient conditions for constrained minimization Video
  6. Lec12-Part II - Sufficient conditions for constrained minimization Video

Mathematical preliminaries for calculus of variations

  1. Lec13-Part I-Mathematical preliminaries function, functional, metrics and metric space, norm and vector spaces Video
  2. Lec14-Part II-Mathematical preliminaries function, functional, metrics and metric space, norm and vector spaces Video
  3. Lec15-Function spaces and Gateaux variation Video
  4. Lec16-First variation of a functional Freche?t differential and variational derivative Video
  5. Lec17-Part I-Fundamental lemma of calculus of variations and Euler Lagrange equations Video
  6. Lec18-Part II-Fundamental lemma of calculus of variations and Euler Lagrange equations Video

Euler-Lagrange equations with and without constraints

  1. Lec19-Extension of Euler-Lagrange equations to multiple derivatives Video
  2. Lec20-Extension of Euler-Lagrange equations to multiple functions in a functional Video
  3. Lec 21-Part I-Global Constraints in calculus of variations Video
  4. Lec22-Part II-Global Constraints in calculus of variations Video
  5. Lec23-Part I-Local (finite subsidiary) constrains in calculus of variations Video
  6. Lec 24-Part II-Local (finite subsidiary) constrains in calculus of variations Video

Size optimization of a bar & Calculus of variations in 2D and 3D

  1. Lec25-Part I-Size optimization of a bar for maximum stiffness for given volume Video
  2. Lec26-Part II-Size optimization of a bar for maximum stiffness for given volume Video
  3. Lec27-Part III-Size optimization of a bar for maximum stiffness for given volume Video
  4. Lec28-Part I-Calculus of variations in functionals involving two and three independent variables Video
  5. Lec29-Part II-Calculus of variations in functionals involving two and three independent variables Video

Advanced concepts in calculus of variations & General framework for optimal structural design

  1. Lec30-Part I-General variation of a functional, transversality conditions. Broken extremals, Wierstrass-Erdmann corner conditions Video
  2. Lec31-Part II-General variation of a functional, transversality conditions. Broken extremals, Wierstrass-Erdmann corner conditions Video
  3. Lec32-Variational (energy) methods in statics; principles of minimum potential energy and virtual work Video
  4. Lec33-Part I-General framework of optimal structural designs Video
  5. Lec34-Part II-General framework of optimal structural designs Video
  6. Lec35-Optimal structural design of bars and beams using the optimality criteria method Video

First integrals, invariants, and Noether’s theorem & Minimum characterization of eigenvalue problems

  1. Lec36-Invariants of Euler-Lagrange equations and canonical forms Video
  2. Lec37-Noether’s theorem Video
  3. Lec38-Minimum characterization of Sturm-Liouville problems Video
  4. Lec39-Rayleigh quotient for natural frequencies and mode shapes of elastic systems Video
  5. Lec40-Stability analysis and buckling using calculus of variations Video

Optimal structural design & “Inverse” of Euler-Lagrange equations

  1. Lec41-Strongest (most stable) column Video
  2. Lec42-Dynamic compliance optimization Video
  3. Lec43-Electro-thermal-elastic structural optimization Video
  4. Lec44-Formulating the extremization problem starting from the differential equation, self-adjointness of the differential operator, and methods to deal with conservative and dissipative system Video

Syllabus and materials

At NPTEL, as the course publishes them.

Credits and licence

Institution
IISc Bangalore
Department
Mechanical Engineering
Taught by
Prof. Ananthasuresh, Prof. Safvan Palathingal
Level
Undergraduate
Language
English
Subjects
Engineering
Source
NPTEL
Licence
CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
Attribution
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