ES 622: Finite Element Methods - Spring 2022

In a world where a large number of simulation tools are available and most people assume they know how to perform a finite element analysis, this course focuses on the fundamentals and aims to equip the students with the 'how' and 'why' of finite element simulations. It covers the fundamental aspects of the finite element method that are required to understand what goes on under the hood in available simulation software.

• To learn the fundamentals of the finite element method.
• To develop a computer program to perform finite element analysis of simple PDEs.
• To develop an understanding of what goes on under the hood of commercial finite element packages.

• know an alternative representation of physical laws (called weak form)
• be able to convert any given linear partial differential equation (PDE) to its corresponding weak form
• be able to develop finite element formulation for any physical problem governed by a PDE
• have developed a computer code for solving linear PDEs using the finite element method
• know about different types of finite elements and their suitability for different types of physical problems
• have some idea of potential problems that can arise in typical finite element solutions
• know how much error can be expected in finite element analysis

• Review of basic mathematical preliminaries.
• Strong and weak forms, Galerkin's approximation.
• 1D, 2D, and 3D finite element formulations, isoparametric elements. Error behavior.
• Finite element formulation of elliptic PDEs (elasticity equation), parabolic PDEs (heat equation), and hyperbolic PDEs (wave equation).

• An Introduction to the Finite Element Method - J.N. Reddy.
• Introductory Finite Element Method - C.S. Desai.
• Finite Element Procedures - K.-J. Bathe.
• The Finite Element Method: Linear Static and Dynamic Finite Element Analysis - T.J.R Hughes.

• Knowledge of basic linear algebra:
  • rank, column space, null space of a matrix,
    • solving system of linear algebraic equations,
    • computing eigenvalues and eigenvectors.
• Knowledge of a programming language:
  • Matlab is a good option. Here is an excellent tutorial.
  • Scilab is an open source alternative to Matlab.
• In case you feel adventurous, you can use C++ or FORTRAN as well.

• Please be considerate about everyone's time.
• In all emails pertaining to this course, please have "ES622" in the subject line.
  • (note that there is no space or hyphen or anything between ES and 622)

Cheating cases (assignments/codes/exams/project) will be awarded an F grade and will be reported to academic office. It is expected that this will never happen and everyone will uphold the honor code.

All delays beyond the defined deadlines will attract reduction in marks as per the following curve. The reduction factor, \(R\) will be multiplied to the obtained marks. Mathematically, it is given by: \(R = \exp(-D^2)\), where \(D\) is the total delay in days (will be counted hourly, i.e. fractional days are possible).

Following will be the weightage of different components of assessment

• There will be two types of assignments: analytical/hand calculation and coding.
• For coding assignments, submission of source code will be required.
• Expect one assignment per week.
• Spot quizzes will primarily be objective type. Expect one quiz per week.
• Exams will primarily be subjective / coding type. They may be in-class or take-home.

It is important to develop the habit of self-learning. A number of reading assignments and self-exercises will be given during the course. These will not be formally graded and it will be expected that students will go through them on a regular basis on their own.

Project is to be done in groups of not more than 3. Following timeline must be adhered to for all submissions. (this timeline will be updated during the first week of classes)

\(^*\) In case the instructor delays in giving feedback, every group gets 10 bonus points.

• Free Vibration Of Thin Plates
• Finite Element Modelling of Thermal Management Systems of Laptops for effectiveness analysis
• Analysis of Cold Rolling process using Finite Element Analysis in Ansys Workbench
• Static Analysis of Leaf and Coil Spring
• Design and Analysis of Rolling Process
• Thermal expansion and Stress analysis of the Radial Turbine
• Modelling Reaction in Batch Reactor