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