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Variational Methods in Mechanics and Design

Variational Methods in Mechanics and Design. Instructor: Prof. G. K. Ananthasuresh, Department of Mechanical Engineering, IIT Bangalore. This course introduces calculus of variations for a comprehensive understanding of the subject and enables the student understand mechanics from this viewpoint. It also provides basic understanding of functional analysis for rigorous appreciation of engineering optimization. After taking this course, the student will be able to formulate many problems in mechanics using energy methods. The course also reinforces the understanding of mechanics and gives hands-on experience for using variational methods. Matlab programs are part of the course. (from nptel.ac.in)

Lecture 33 - General Framework of Optimal Structural Designs


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Overview of Optimization Calculus of Vibrations
Lecture 01 - Classification of Optimization Problems and the Place of Calculus of Variations in it
Lecture 02 - Classification of Optimization Problems and the Place of Calculus of Variations in it (cont.)
Lecture 03 - Genesis of Calculus of Variations
Lecture 04 - Genesis of Calculus of Variations (cont.)
Lecture 05 - Formulation of Calculus of Variations Problems in Geometry and Mechanics
Lecture 06 - Formulation of Calculus of Variations Problems in Geometry and Mechanics (cont.)
Summary of Finite Variable Optimization
Lecture 07 - Unconstrained Minimization in One and Many Variables
Lecture 08 - Unconstrained Minimization in One and Many Variables (cont.)
Lecture 09 - Constrained Minimization KKT Conditions
Lecture 10 - Constrained Minimization KKT Conditions (cont.)
Lecture 11 - Sufficient Conditions for Constrained Minimization
Lecture 12 - Sufficient Conditions for Constrained Minimization (cont.)
Mathematical Preliminaries for Calculus of Variations
Lecture 13 - Function and Functional, Metrics and Metric Space, Norm and Vector Spaces
Lecture 14 - Function and Functional, Metrics and Metric Space, Norm and Vector Spaces (cont.)
Lecture 15 - Function Spaces and Gateaux Variation
Lecture 16 - First Variation of a Functional Frechet Differential and Variational Derivative
Lecture 17 - Fundamental Lemma of Calculus of Variations and Euler-Lagrange Equation
Lecture 18 - Fundamental Lemma of Calculus of Variations and Euler-Lagrange Equation (cont.)
Euler-Lagrange Equation with and without Constraints
Lecture 19 - Extension of Euler-Lagrange Equation to Multiple Derivatives
Lecture 20 - Extension of Euler-Lagrange Equation to Multiple Functions in a Functional
Lecture 21 - Global Constraints in Calculus of Variations
Lecture 22 - Global Constraints in Calculus of Variations (cont.)
Lecture 23 - Local (Finite Subsidiary) Constraints in Calculus of Variations
Lecture 24 - Local (Finite Subsidiary) Constraints in Calculus of Variations (cont.)
Size Optimization of a Bar for Maximum Stiffness for Given Volume
Lecture 25 - Size Optimization of a Bar for Maximum Stiffness for Given Volume I
Lecture 26 - Size Optimization of a Bar for Maximum Stiffness for Given Volume II
Lecture 27 - Size Optimization of a Bar for Maximum Stiffness for Given Volume III
Lecture 28 - Calculus of Variations in Functionals involving Two and Three Independent Variables
Lecture 29 - Calculus of Variations in Functionals involving Two and Three Independent Variables (cont.)
Advanced Concepts and General Framework for Optimal Structural Design
Lecture 30 - General Variation of a Functional, Transversality Conditions; Broken Examples, Weierstrass-Erdmann Corner Conditions
Lecture 31 - General Variation of a Functional, Transversality Conditions; Broken Examples, Weierstrass-Erdmann Corner Conditions (cont.)
Lecture 32 - Variational (Energy) Methods in Statics; Principles of Minimum Potential Energy and Virtual Work
Lecture 33 - General Framework of Optimal Structural Designs
Lecture 34 - General Framework of Optimal Structural Designs (cont.)
Lecture 35 - Optimal Structural Design of Bars and Beams using the Optimality Criteria Method
First Integrals, Invariants, and Noether's Theorem and Minimum Characterization of Eigenvalue Problems
Lecture 36 - Invariants of Euler-Lagrange Equation and Canonical Forms
Lecture 37 - Noether's Theorem
Lecture 38 - Minimum Characterization of Sturm-Liouville Problems
Lecture 39 - Rayleigh Quotient for Natural Frequencies and Mode Shapes of Elastic Systems
Lecture 40 - Stability Analysis and Buckling using Calculus of Variations
Optimal Structural Design and Inverse of Euler-Lagrange Equation
Lecture 41 - Strongest (Most Stable) Column
Lecture 42 - Dynamic Compliance Optimization
Lecture 43 - Electro-thermal-elastic Structural Optimization
Lecture 44 - Formulating the Extremization Problem