The aim of the module is to provide the basis for the comprehension of the methods and procedures the finite element method is based on. The basic knowledge acquired during the course will give the student the possibility to explore the possible applications of the method to solve engineering problems in several different fields, in particular, the structural field.

Students attending the course will learn about the wide capabilities of the numerical methods in the virtual simulation of the behaviour of the material world. On successfully completing this course unit, students will be able to solve structural problems of medium complexity by using commercial codes widely present in industries, to say, they will be able to:
• prepare the numerical models of simple structures
• set up the boundary conditions for static and dynamic analyses
• critically analyse the results shown and verify their reliability.

Students attending the course must have the basic knowledge in order to:
• evaluate the static equilibrium of frames with prescribed boundary conditions,
• evaluate the stress, strain and displacement fields of loaded beams within frameworks,
• use analytical and graphical methods to analyze stress and strain fields.

1 –Structural analysis by matrices
Basic concepts: nodes, generalised displacement and force vectors.
Stiffness, displacement and transfer formulations.
Reference system change and structural stiffness matrix assembly.
Boundary conditions imposition and linear equation system solution.
Stiffness matrix evaluation for rods and beams (equivalent nodal forces for distributed loads).
2 – Finite elements in the linear static field
General formulation: principle of virtual work and minimum of total potential energy (Ritz method) for structural problems, residual methods (Galerkin method) for thermo-fluid dynamic problems.
Equivalent nodal forces for distributed loads.
Mono, bi- and tri-dimensional finite elements
Parametric formulation, numerical integration with the Gauss method.
Quality of approximated solution, error estimation, strategies for improving solution.
Meshing methods, automatic meshing, adaptivity, convergence analysis.
3 – Finite elements in the linear dynamic field
Mass matrix: congruent formulation and concentrated formulation.
4 – Finite elements in the thermal field
General description of the thermal problem (boundary conditions).
Elements used in FEM analysis for thermal fields.
Applications concerning vehicle design.

During the practical exercises at the Information Technology laboratories, which represent a fundamental component of the learning process, students will carry out exercises on the topics dealt with during the lessons. It is essential, in order to draw maximum advantage from such exercises, that the students apply the concepts and notions learned during the lessons autonomously. IT laboratory work will concerns the following subjects:
• use of a basic software for FEM pre and post processing (mesh generation, data pre-processing, problem solution, post-processing),
• application of finite elements as introduced during the course in order to analyse simple structures under the linear elastic hypothesis and static loading conditions,
• modal analysis of simple components

The exam of the Numerical Modelling and Simulation course is characterized by:
- a written exam (maximum score 28/30),
- 2 laboratory reports concerning the 2 parts of the course (overall score in the range 0:+2 for each report),
- an oral exam (on student’s request).
A score larger than 30/30 will be registered as 30/30 with laudem.
The deadline of the laboratory reports upload on the course web site is mandatory, laboratory reports uploaded after the deadline will not be taken into consideration. If any laboratory report will be missing the maximum score for the written exam will be automatically reduced to 24/30.
The oral exam has to be attended the same date of the correction of the written exam. The oral exam can be carried out only by students having a minimum score of 18/30 (score of the written exam). The oral exam consists in 2÷4 questions on the contents of the complete course and is evaluated with a maximum score of 28/30. The final score of the exam is given by the mean score of the written and oral exam + the score for the laboratory reports.
The score of the written exam can be refused whereas the score of the written + oral exam can not be refused.
The absence (no show) of the student at the registration date is considered as an implicit acceptance of the published score (written exam + laboratory reports).