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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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September 25, 2023

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September 25, 2023

Duration:

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FREE

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Unlimited Duration

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Description

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)

Course Curriculum

    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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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