MEAM 427/527: FINITE ELEMENTS AND APPLICATIONS
Instructor: Haim H. Bau
Prerequisites: MATH 240 and 241 and PHYS 151
Textbook: Lecture notes posted on Blackboard
Description: The objective of this course is to equip students with the background needed to carry out
finite elements-based simulations of various engineering and science problems with finite elements
packages. The first part of the course will outline the theory of finite elements. The objective here is to
acquaint the students with the theory rather than to equip them with the programming skills needed to write
multi-dimensional finite elements codes.
The second part of the course will address the solution of the classical equations of mathematical
physics such as the Laplace, Poisson, Helmholtz, wave, and heat equations. The general properties of the
solutions will be described, and potential pitfalls will be addressed. This part of the course will also
address issues such as code verification and convergence. The students will gain hands-on experience
working with Femlab. This part of the course will also address relevant topics of numerical analysis such as
the solution of initial value problems and their precision, the solution of algebraic equations, and the
calculation of eigenvalues and eigenvectors.
The third part of the course will consist of case studies taken from various areas of engineering and the
sciences. Presentation of each case study will start with a description of the pertinent physical and
engineering background and how finite elements are being used to solve the problem. For example, the
module on biological interaction will describe the operation of an optical bio-sensor (the BIACore). Then
the corresponding convection-diffusion equations and the equations that describe the reaction kinetics will
be introduced. Subsequently, the problem will be solved with Femlab. The module will also address the
verification issue (how do we know that the solution is right?) and the analysis and post-processing of the
computational data. In parallel with the case studies, the students will work on a pre-approved project
selected by the students.
The grading will be based on homeworks, a quiz, and a project.
COURSE OUTLINE
1.
An introduction to modeling and simulation and various modeling tools
2.
Interpolation functions and the finite element space, convergence
3.
The method of weighted residuals and the weak form
4.
Solution of linear systems of algebraic equations (direct, iterative, and semi-direct methods)
5.
Case study: heat transfer in an extended surface (fin)
6.
Time-dependent problems and their solution
7.
Case study: the thermosyphon, model reduction, and the Lorenz equations
8.
Introduction to nonlinear problems: predictor-corrector methods, bifurcations, and linear stability
9.
Analysis of beams under tension, eigenvalue problems, and the wave equation
10.
Beam bending – fourth order differential equations with various boundary conditions
11.
The classical equations of mathematical physics: Laplace, Poisson, and Helmholtz. Boundary
conditions and compatibility. Weak formulation and various variational forms. Superposition.
12.
Case study: Advection-diffusion equations with heterogeneous reactions modeling protein-protein
interactions in a biosensor (BIACore).
13.
Case study: Problems in electrostatics