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.

Recorded video lectures of some topics area available in this playlist. Some of them will be assigned during the classes where students will watch the video lecture before coming to the class.

• 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.
  • Octave and Scilab are open source alternatives to Matlab that can be used for coding related to FEM.
• 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.

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; dates will not be announced.
• 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

• Introduction to FEM
• Different formulations of the same physical problem
• Stretching of 1D bar - force equilibrium, virtual work method, total potential energy
• Notion of strong and weak forms

• Class times rescheduled.

• Matlab session.

• Matlab session.

• Stretching of 1D bar
  • Equivalence of virtual work and force equilibrium
• Function vs. functional
• Notion of dot product
• Fundamental lemma of calculus of variations
• Properties of virtual / weight functions
• Nomenclature of boundary conditions

• Ritz method of solving weak form
• Ritz-Galerkin method
  • Sub-division of physical domain and domain integrals
  • Element-wise view vs. global view
• Shape functions / interpolation functions
  • global functions
  • piece-wise linear functions
  • Properties: Kronecker delta; partition of unity; localized nature

• FE approach of solving the 1D weak form / virtual work formulation
• Approaches to enforce essential boundary conditions
• General process for utilizing FEM
• Reference element for element-level formulations
• Coordinate transformation jacobian and global formulation
• Gauss quadrature

• Assignment 1 given here (due 5 Feb)
• Construction of 1D shape functions
• Error computations and convergence
• Computer implementation of 1D FEM

• Computing weak form of 2D Poisson equation

• Isoparametric bilinear element

• Treatment of boundary integral in 2D problems
• Consideration of vector fields in 2D problems

• Consideration of vector fields in 2D problems

• Weak form for time-dependent problems

• Weak form for time-dependent problems

• Time-marching methods

• 1D functionals and considering the energy formulations
• Minimization of functionals

• 2D functionals

• Locking behavior and issues with incompressible materials
• B-bar method

• Variational principles
• Hu-Washizu variational principle

• Shear locking in elasticity problems

• Maharvir Jayanti

• Numerical artifacts in different FE elements

• Method of manufactured solutions for code verification

• Calculation of secondary quantities like strain, stress, flux, etc.

• Consideration of 3D problems

• Nonlinear FE methods

• Nonlinear FE methods

• Nonlinear FE methods