continuum mechanics, basic programming

This course introduces the main concepts and numerical tools required to model and solve nonlinear problems in solid mechanics using the finite element method. It aims to give students both a global understanding of the nonlinear simulation workflow and the ability to make informed modelling and numerical choices, with particular emphasis on plasticity and viscoplasticity and on applications relevant to industrial practice.

By the end of the course, students should be able to: 

- understand the theoretical foundations and overall structure of a finite element solver; 

- understand and formulate plasticity and viscoplasticity models, including yield criteria, flow rules, hardening laws and rate dependence; 

- apply numerical algorithms for the integration of nonlinear constitutive equations; 

- understand incremental–iterative nonlinear solution procedures, including Newton-type methods; 

- formulate and analyse basic contact problems; design an appropriate finite element mesh and assess its influence on the numerical results; 

- diagnose difficulties in nonlinear computations and assess, correct, adapt or improve the solution strategy to achieve robust and accurate simulations; 

- apply these concepts to industry-relevant nonlinear mechanics problems and critically interpret the resulting numerical predictions. 

The course combines lectures and hands-on computer tutorials in approximately equal proportions. 

The lectures first review the theoretical foundations of the finite element method and the main components of a finite element solver. They then introduce the specific numerical difficulties associated with nonlinear problems and the corresponding incremental and iterative solution strategies. 

Particular attention is devoted to the detailed modelling of plasticity and viscoplasticity, including constitutive formulation, numerical integration and coupling with the global finite element solver. Complementary lectures on contact mechanics and meshing provide a broader view of nonlinear computational mechanics and of the complete simulation workflow. 

The tutorials are carried out using the commercial finite element software Z-set. They allow students to apply the concepts introduced in the lectures, analyse convergence difficulties, and test different modelling and solution strategies. 

The course concludes with a two-day project carried out in small groups. With guidance from teaching assistants, students define, solve and critically analyse a nonlinear mechanics problem, potentially involving plasticity, viscoplasticity, damage or contact. This project is intended to develop both practical modelling skills and increasing autonomy in the use of nonlinear finite element methods. 

Working in small groups and with limited support from the teaching assistants, students will address a series of questions related to their project. Each group will then prepare and deliver a presentation of its work to the rest of the class.