Mechanical Engineering, Indian Institute of Science, Bangalore 560 012, India
Optimization hinders evolution!
ME256 Variational Methods and Structural Optimization Jan.-May, 2019
Instructor: G. K. Ananthasuresh, Room 106, ME Building, suresh at iisc.ac.in
Teaching assistant: Vageesh Singh Baghel, M2D2 Laboratory, Mechanical Engineering vageeshb at iisc.ac.in
Lectures: Tu, Th: 08:30 AM - 09:55 AM; Venue: ME MMCR

1. The first lecture will be on 3rd January, 2019, at 8:30 AM in ME Multimedia Classroom.

Course outcome: What would you have learned after taking this course?

1. Understand the difference between ordinary calculus and calculus of variations as well as functions and functionals.
2. Get a quick grasp of the terminology of function spaces, energy spaces in particular.
3. Be able to take the first variation of a functional.
4. Write down necessary conditions of functionals involving multiple functions; multiple derivatives of a function; one, two, or three independent variables on which the functions depend.
5. Understand how to write the boundary conditions, including variable end conditions and transversality conditions.
6. Appreciate energy and variational methods in mechanics as well as the interconnection between force-balance (differential equation), weak form (principle of virtual work and D' Lambert principle), and energy principles (minimum potential energy and Hamilton's principle) in mechanics.
7. Be able to think about the inverse problem of writing the minimization principle from the differential equation.
8. Gain a thorough understanding of Karush-Kuhn-Tucker (KKT) conditions for constrained minimization problems and the concept of Lagrange multipliers and their various interpretations.
9. Be able to analytically obtain the necessary conditions for optimizing a bar of variable cross-section profile for different objective functions and constraints.
10. Be able to the same for beams, plates, 2D and 3D continuous structures.
11. Understand the sensitivity analysis in structural optimization.
12. Be able to implement a numerical optimization algorithm, namely, optimality criteria method, to obtain optimized geometry of bars, beams, plates, 2D continua, 2D and 3D trusses, 2D and 3D frames and grillages.
13. Be able to consider transient and multiphysics problems in structural optimization.
14. Become familiar with Optimization Toolbox in Matlab.
15. Be able to formulate optimization problems in the framework of calculus of variations and then convert into the discretized form as a finite-variable continuous optimization.

What are variational methods? And, how are they used for shape and topology optimization of structures?

Optimization is a way to get the best out of what is available. It is thus no wonder that everybody tries to optimize almost everything. Even in Nature, it appears as if everything is optimized based on some criteria and subject to some constraints. What then are the mathematical tools that help us analyze and obtain such optima? Variational methods, or more precisely the calculus of variations, is a primary mathematical tool that helps us in this regard.

While the ordinary calculus considers functions of finite number of variables, the calculus of variations, a phrase coined by Euler, considers functions of functions themselves. That is, in optimization of functions of finite number of variables of ordinary calculus, we find minimizing values of such finite number of variables whereas in calculus of variations, we find the minimizing function on which another function (called a functional) depends. What the derivative is to ordinary calculus the variation is to calculus of variations. There are subtle similarities and profound differences between the two.

The recorded scieintific history shows that ancient Greeks had formulated some problems that now fall under calculus of variations. Galileo had considered a few problems of which one, the Brachstochrone problem, has become the most popular. Fermat put forth a variational problem for the refraction of light rays saying they follow the path of least time and not the least distance. Newton had considered the surface of revolution of a solid that experiences least resistance when traveling in a fluid along its axis. John and James Bernoullis, Leibnitz, L'Hopital, Newton, and others revived the Brachistochrone problem. Euler and Lagrange laid the firm analytical foundation for it. Lagrange, Hamilton, Jacobi and others used it in mechanics. Legendre, Jacobi and others worked further on calculus of variations.

Structures are continuous. More recent studies on compliant mechanisms show that they are also very much like structures except that they are flexible. Both the governing equations of equilibrium of structures as well as the methods of their optimum shape and topology can be cast as variational problems.

This course will be an exciting journey from Fermat's principle of least time to a practical implementation of topology optimization of structures and compliant mechanisms--a journey from the classical to the contemporary times.

And, remember that optimization hinders evolution! Nature takes many, many years to evolve to an optimum whereas engineers design optimum structures as fast as the computers can churn the numbers.