FINITE ELEMENTS (a first course)
Mathworks award
Mathworks (the MATLAB's company) has rewarded the Numerical Factory project with an Academic Developement Support Grant. Congratulations to all team members!!!.
Finite Element course (official name: Mètodes Numèrics)
Introductory course to the Finite Element Method applied to engineering problems using Matlab. It corresponds to undergrad second year course at the ETSEIB (UPC-BarcelonaTech).
This material is only for educational purposes. You are free to download its contents.
Please, let me know if you find some typos or you have some idees to improve it (toni.susin at upc.edu).
At the end of this page you will find all the meshes and extra plot functions required for this course.
Index:
Introduction to FEM
Interpolation: Shape Functions
Finite Elements 1D-2D
Numerical Integration
Plane Elasticity
Extra material Files
T0 - Introduction
Theory
First presentation: Short history of Finite Element Methods. Divide and Conquer. General Modelization Idea. Fields of application. Mathematical models. Numerical Solution.Error sources.
Practices
P0.1 - Read and Write numbers using Matlab
Short introduction to managing numerical results using a txt file or a spreadsheet.
P0.2 - Circle Length Approximation
First version of the numerical approximation of the circunference length.
Second version of the numerical approximation of the circunference length.
Exercise of the numerical approximation of the circunference length.
T1 - Interpolation: Shape Functions
Theory
In this presentation the following concepts are introduced: Interpolation. Lagrange Interpolation. Splines. Shape Functions 1D-2D. Triangular Elements. Quadrilateral Elements.
Practices
P1.1 - Interpolation 1D
Interpolation 1D . Introduction of the polyfit and polyval Matlab functions. Splines and interp1.
P1.2 - Triangle Shape Functions
Shape function for triangles . Load triangular mesh files. Barycentric coordinates. Is inside function.
P1.3 - Quadrilateral Shape Functions
Shape function for quadrilaterals . Barycentric coordinates for quadrilateral elements. Is inside function.
T2 - Finite Elements 1D-2D
Theory
In this presentation Finite Element Theory. Weak formulation. Assembly problem.
In this other presentation, some application examples.
Practices
P2.3 - Truss Elements (1D linear structures)
1D linear elements living in the 2D space.
Additional files needed for the practice
P2.4 - Poisson's Equation (2D linear triangles)
Solving 2D Poisson's equation using triangle elements .
P2.5 - Thermal Equation (2D linear triangles)
Solving 2D thermal distribution using triangle elements .
P2.6 - Thermal Equation with convection (small example)
Solving 2D thermal distribution with convection boundary conditions.
P2.7 - Thermal Equation with convection (mesh version)
Solving 2D thermal distribution with convection boundary conditions.
P2.9 - Thermal equation with bilinear quadrilateral elements
Solving 2D thermal problems using quadrilateral elements.
P2.10 - Thermal equation with bilinear quadrilateral elements and convection BC
Solving 2D thermal problems using quadrilateral elements with convection boundary conditions.
P2.11a - The same problem using a triangular and quadrilateral elements and convection BC
Solving 2D thermal problems using triangular elements with convection boundary conditions.
P2.11b - The same problem using a triangular and quadrilateral elements and convection BC
Solving 2D thermal problems using Quadrilateral elements with convection boundary conditions.
P2.12 - Boundary nodes, edges and elements: Complex mesh boundaries
Find all the nodes in a general mesh boundary.
T3 - Numerical Integration
Theory
In this presentation Numerical Integration. Gauss quadrature 1D-2D. Integration on quadrilateral and triangular elements.
Practices
P3.2 - Gauss Integration on Quadrilaterals
Gauss integration for quadrilaterals . Isometric change of variables. Jacobian.
P3.3 - Gauss Integration on Triangles
Gauss integration for triangles . Shape functions. Constant solutions.
T4 - Plane Elasticity
Theory
In this presentation second order PDE 2D systems are introduced. Finite Element 2D Structural elements. Plane stress and plane strain deformation. Engineering notation.
Practices
P4.1 - One triangle element: Plane stress example
Solving 2D deformation problems using one triangular element.
P4.2 - Triangle mesh: Plane stress example
Solving 2D deformation problems using a triangular mesh. Von Misses stress.
P4.3 - Quadrilateral mesh: Plane stress example
Solving 2D deformation problems using a quadrilateral mesh. Von Misses stress.
Extra material Files
