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Course Co-ordinated by :
IISc Bangalore
Course Available from :
05-August-2016
NPTEL
Mechanical Engineering
NOC:Variational Methods in Mechanics (Video)
Lec1-Part I-Classification of optimization problems and the place of Calculus of Variations in it
Modules / Lectures
Overview of Optimization Calculus of Variations
Lec1-Part I-Classification of optimization problems and the place of Calculus of Variations in it
Lec2-Part II-Classification of optimization problems and the place of Calculus of Variations in it
Lec3-Part I - Genesis of Calculus of Variations
Lec4-Part II - Genesis of Calculus of Variations
Lec5-Part I - Formulation of Calculus of Variations problems in geometry and mechanics and design
Lec6-Part II - Formulation of Calculus of Variations problems in geometry and mechanics and design
Summary of finite-variable optimization
Lec7-Part I - Unconstrained minimization in one and many variables
Lec8-Part II - Unconstrained minimization in one and many variables
Lec9-Part I - Constrained minimization KKT conditions
Lec10-Part II - Constrained minimization KKT conditions
Lec11-Part I - Sufficient conditions for constrained minimization
Lec12-Part II - Sufficient conditions for constrained minimization
Mathematical preliminaries for calculus of variations
Lec13-Part I-Mathematical preliminaries function, functional, metrics and metric space, norm and vector spaces
Lec14-Part II-Mathematical preliminaries function, functional, metrics and metric space, norm and vector spaces
Lec15-Function spaces and Gateaux variation
Lec16-First variation of a functional Freche?t differential and variational derivative
Lec17-Part I-Fundamental lemma of calculus of variations and Euler Lagrange equations
Lec18-Part II-Fundamental lemma of calculus of variations and Euler Lagrange equations
Euler-Lagrange equations with and without constraints
Lec19-Extension of Euler-Lagrange equations to multiple derivatives
Lec20-Extension of Euler-Lagrange equations to multiple functions in a functional
Lec 21-Part I-Global Constraints in calculus of variations
Lec22-Part II-Global Constraints in calculus of variations
Lec23-Part I-Local (finite subsidiary) constrains in calculus of variations
Lec 24-Part II-Local (finite subsidiary) constrains in calculus of variations
Size optimization of a bar & Calculus of variations in 2D and 3D
Lec25-Part I-Size optimization of a bar for maximum stiffness for given volume
Lec26-Part II-Size optimization of a bar for maximum stiffness for given volume
Lec27-Part III-Size optimization of a bar for maximum stiffness for given volume
Lec28-Part I-Calculus of variations in functionals involving two and three independent variables
Lec29-Part II-Calculus of variations in functionals involving two and three independent variables
Advanced concepts in calculus of variations & General framework for optimal structural design
Lec30-Part I-General variation of a functional, transversality conditions. Broken extremals, Wierstrass-Erdmann corner conditions
Lec31-Part II-General variation of a functional, transversality conditions. Broken extremals, Wierstrass-Erdmann corner conditions
Lec32-Variational (energy) methods in statics; principles of minimum potential energy and virtual work
Lec33-Part I-General framework of optimal structural designs
Lec34-Part II-General framework of optimal structural designs
Lec35-Optimal structural design of bars and beams using the optimality criteria method
First integrals, invariants, and Noether’s theorem & Minimum characterization of eigenvalue problems
Lec36-Invariants of Euler-Lagrange equations and canonical forms
Lec37-Noether’s theorem
Lec38-Minimum characterization of Sturm-Liouville problems
Lec39-Rayleigh quotient for natural frequencies and mode shapes of elastic systems
Lec40-Stability analysis and buckling using calculus of variations
Optimal structural design & “Inverse” of Euler-Lagrange equations
Lec41-Strongest (most stable) column
Lec42-Dynamic compliance optimization
Lec43-Electro-thermal-elastic structural optimization
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
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