The teaching unit is mandatory.
The teaching unit is taught in French.
The teaching unit is taught in English.

Outline

This document describes the planning and structure of the course: MAterials & Structures Computation by FEM (MASC-FEM).

3 ECST / 30 Hours total in 10, 3 hour sessions.

Team

Coord. Jan Neggers (jan.neggers@centralesupelec.fr, CentraleSupelec)

Juan Pablo Marquez Costa (juan-pablo.marquez_costa@ensam.eu, ENSAM Paris)

Overview

The MASC-FEM course, is aimed at providing the theoretical and mathematical background required to understand the FEM framework, how it is constructed from a PDE, and how it can be applied to material and structural computations. The course contains two parts. The first is on linear finite element methods and ends with the full development of a vector field PDE, using the isoparametric mapping for general element types with Gauss integration. The second part is on non-linear finite element methods, especially those involving non-linear material models. Introducing explicit and implicit euler time integration methods and Runge-Kutta and Newton-Raphson methods required to solve for the consistent tangent operator required to implement such non-linear material models in the linear framework from part 1. For this second part, full 3D tensor elasto-plasticity and damage models are used to demonstrate the numerical methods required when dealing with material nonlinearity..

Structure of MASC-FEM per session

Each session is 3 hours, with a variable mix of lectures and guided exercises, approximately in a 50/50 ratio.

1. Introduction to the finite element method using the direct approach
2. PDE’s, the strong and weak formulations and Galerkin’s method
3. Shape functions, isoparametric space and Gauss quadrature in 1D for scalar fields
4. Shape functions, isoparametric space and Gauss quadrature in 3D for vector fields
5. Practical session, programming the FEM from scratch
6. Elasto-plasticity using explicit integration
7. Elasto-plasticity using implicit integration using the Newton-Raphson method
8. Elasto-plasticity using implicit integration using an analytic explicit formulation
9.  Practical session, programming non-linear integration schemes from scratch
10. Invited lecture of a renowned researcher in the field showcasing applications of the learned material

Literature used to develop the course

• Hughes, T. J. R. Finite Element Method
• Zienkiewicz O. C., Taylor R. L., The Finite Element Method: Its Basis and Fundamentals
• Dunne F. and Petrinic N., Introduction to computational plasticity
• Lemaitre J. and Desmorat R., Engineering damage mechanics