Lecturer: Giancarlo Sangalli  

Course name: Theory and application of Finite Elements
Course code: 503004
Degree course: Bioingegneria
Disciplinary field of science: MAT/08
University credits: CFU 6
Course website: http://www-dimat.unipv.it/~sangalli/TAEF.html

Specific course objectives

The course gives the basic knowledge of the Finite Element Method and its theoretical foundation. Moreover, it discuss the implementation in MATLAB of a code for solving elliptic model problems in one and two dimensions

Course programme

Review of functional analysis

• Sobolev Spaces and their properties
• Variational formulation of elliptic problems (Poisson and elasticity)

Ritz-Galerkin method

• Mesh in one and more dimensions
• Examples of finite elements
• Isoparametric elements
• Bramble-Hilbert lemma
• Approximation properties
• A priori estimates for second-order elliptic problems

Mixed and non-conforming methods

• Strang lemma
• Saddle-point problems
• Mixed finite elements

Application of the Finite Element Method in solid mechanics

• Review of linear elasticity
• Membranes
• Beams
• Incompressible materials (Stokes Equation)
• Hints on Kirchoff and Reissner-Mindlin plates

MATLAB implemenation

• Poisson problem in one dimension
• Poisson problem in two dimensions
• Beam problem
• Stokes problem in two dimensions

Course entry requirements

Differential and integral calculus in multi-dimension. Complex numbers. Matrix and vector calculus. MATLAB programming

Course structure and teaching

Lectures (hours/year in lecture theatre): 30
Practical class (hours/year in lecture theatre): 30
Practicals / Workshops (hours/year in lecture theatre): 0

Suggested reading materials

Dietrich Braess. Finite Elements. Theory, Fast Solvers, and Applications in Solid Mechanics. Cambridge University Press.

Alfio Quarteroni. Modellistica numerica per problemi differenziali. Springer Verlag.

Thomas J.R. HUGHES. The finite element method: Linear static and dynamic finite element analysis. Dover Publications Inc..

Testing and exams

The exam consists in an oral discussion about all the topics of the course. Students are admitted to the exam after the submission of a written report on the MATLAB practice in the computer lab