Course Code:               MATH 6195
Course Title:                Finite Element Analysis
Course
Type:                            Elective

No. of Credits:             4

Pre-requisites:             None.

Prior knowledge of elementary computer programming will be an asset.

Course Rationale

The Finite Element Method (FEM) is a sophisticated numerical scheme for interpolating the approximate solution to common boundary-value problems of engineering and mathematical physics. It is widely used, as it is a computationally efficient method for modelling real-world problems which typically have unusual geometries and variable material properties. The purpose of this course is to provide graduate students of engineering or applied science with a concise introduction to FEM.

Although the theory of finite elements is based on sophisticated elements of mathematical analysis, the main purpose of this course is to provide a comprehensive and concise introduction to the subject. Ample coverage of the basic ideas on which FEM is founded will be provided during lectures, this facilitating the intelligent use of FEM for solving difficult problems.

Course Description

The main objective of this course is to clarify and explain the basic ideas on which finite element methods are founded. The focus throughout will be on the nature of the finite element method, how it works, why it makes sense, and how to use it to solve problems of interest.

Throughout the course, students will be required to develop and implement numerical algorithms. Special emphasis will be placed on the efficiency and accuracy of these methods for problem solving. As this course is a practical one, students will be evaluated by their performance in coursework assignments, computer lab exams and on a final research project.

Students taking this course must have a thorough understanding of undergraduate calculus and ordinary differential equations. A solid foundation in undergraduate matrix algebra will also be assumed. As students will be required to implement the algorithms on a computer, prior knowledge of elementary computer programming will be a definite asset, although this is not a prerequisite.

Algorithms will be presented during lectures in pseudo code format to facilitate the creation of well-structured programs in a variety of programming languages. The numerical software package Matlab will the chosen programming tool for in-course assignments.  An introductory tutorial will be organized at the beginning of the course for students with no prior knowledge of Matlab.

Learning Outcomes

Based on the theoretical part of this course, the student should be able to:

• describe at high level, mathematical theory underlying some linear differential equations of the second order arising in applied science ( emphasis on solid and fluid mechanics)
• demonstrate the existence of solutions to boundary value problems via  Variational approximation methods
• explain clearly fundamental concepts and principles of the finite element method in one-dimensional and two-dimensional problems
• construct an approximate solution of a boundary value problem over a domain consisting of finite elements.
• implement criteria of the finite element method, such as triangulations of domains, determination of appropriate test-/trial- space, etc. for solving a range of problems on applied sciences.

In the practical part of the course the student should be able to:

• write and implement their own finite element codes for simple boundary value problems of engineering and applied science
• solve mathematical problems on a computer using the basic concepts and traditional algorithms of FEM
• obtain computer approximations for problems that may be otherwise intractable, because of unusual geometries or variable material properties
• quantify the limitations of FEM based on its performance and individual components
• demonstrate the efficiency and accuracy of simple FEM codes chosen by solving a given problem on the computer

Content:

Introduction: Review concepts on Calculus of Variations; Euler’s Equation, Other forms of Euler’s Equation, Brachistochrone Problem, Isoperimetric Problem, Problem of Geodesics.

Integral Formulation: Integral identities; Linear and Bilinear Functional; Weighted integral and weak formulations; Linear and bilinear forms and quadratic Functionals; Examples.

Variational Methods: Introduction; the Ritz Method with Examples; The method of Weighted Residuals; The Petrov–Galerkin method; The Least Squares method; The Collocation method; Applications to the Boundary Value Problems.

Second- Order Differential Equations in One Dimension: Back ground; Basic steps of the finite element analysis; Model Boundary value problem; Discretization of the domain. Derivation of element equations; Connectivity of elements; Imposition of boundary conditions; Solution of equations; Post processing of the solution; Applications to the problems in solid /Fluid mechanics.

Single-Variable Problems in Two-Dimensions: Introduction; Boundary Value Problems; The Model equations; The finite element discretization; Weak form of finite element model; Interpolation functions; Evaluation of element matrices and vectors; Assembly of element equations and post processing of the solutions; Applications to Solid / Fluid Mechanics.

Finite Element Error Analysis: Measure of error, accuracy, and convergence of solution.

Numerical Integration and computer Implementation: Isoperimetric formulation using natural coordinates; selection of interpolation function for rectangular triangular, and serendipity elements; numerical integration/quadrature; modeling considerations

Teaching Methodology

Lectures: Three (3) lectures each week - 50 minutes each.

Tutorial: Six tutorial sessions (solving theoretical problems) - 50 minutes each

Computer Labs: Six 2-hour computer labs (computational solution of problems)

Assessment

100% Coursework

• Two 15% theoretical assignments: Finite Element Analysis (theory)
• Two 15% Computational Take-home Assignments: Practical implementation and testing of numerical algorithms based on theory covered during lectures
• One 20% Practical Computer Lab Examination -  (two hours)
• 20% Group Research Project

Course Calendar

 Week Lecture subjects Evaluation Lab/Tutorial 1 Introduction: Review concepts on Calculus of Variations; Euler’s Equation, Other forms of Euler’s Equation, Brachistochrone Problem, Isoperimetric Problem, Problem of Geodesics. None Tutorial 1 2 Integral Formulation: Integral identities; Linear and Bilinear Functional; Weighted integral and weak formulations; Linear and bilinear forms and quadratic Functionals; Examples. Theoretical Assignment #1 given Computer Lab 1 3 Variational Methods: Introduction; the Ritz Method with Examples; The method of Weighted Residuals; The Petrov–Galerkin method Theoretical Assignment #1 due Tutorial 2 4 Variational Methods (Continued): The Least Squares method; The Collocation method; Applications to the Boundary Value Problems. Theoretical Assignment #2 given Computer Lab 2 5 Second- Order Differential Equations in One Dimension: Back ground; Basic steps of the finite element analysis; Model Boundary value problem; Discretization of the domain. Derivation of element equations; Connectivity of elements; Imposition of boundary conditions Theoretical Assignment #2 due Tutorial 3 6 Second- Order Differential Equations in One Dimension (Continued): Solution of equations; Post processing of the solution; Applications to the problems in solid /Fluid mechanics. Computational Assignment #1 given Computer Lab 3 7 Single-Variable Problems in Two-Dimensions: Introduction; Boundary Value Problems; The Model equations; The finite element discretization; Weak form of finite element model; Computational Assignment #1 due Tutorial 4 8 Single-Variable Problems in Two-Dimensions (Continued): Interpolation functions; Evaluation of element matrices and vectors; Assembly of element equations and post processing of the solutions; Computational Assignment #2 given Computer Lab 4 9 Single-Variable Problems in Two-Dimensions (Continued): Applications to Solid / Fluid Mechanics Computational Assignment #2 due Tutorial 5 10 Finite Element Error Analysis: Measure of error, accuracy, and convergence of solution. Group Project Assigned Computer Lab 5 11 Numerical Integration and computer Implementation: Isoperimetric formulation using natural coordinates; selection of interpolation function - Tutorial 6 12 Numerical Integration and computer Implementation(Continued) interpolation function for rectangular triangular, and serendipity elements; numerical integration/quadrature; modeling considerations - Computer Lab 6 13 Group Project Presentations Group Project Presentations Computer Lab Examination (20%)