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)
| 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 |