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.

Conoscenza dei contenuti dei corsi di Matematica (studio di funzioni e calcolo di derivate e integrali, calcolo matriciale e problemi agli autovalori/autovettori), Fisica (concetti basilari di cinematica e statica) e Fondamenti di meccanica strutturale (caratteristiche meccaniche e di resistenza dei materiali, stato di tensione e di deformazione in campo lineare elastico)

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. Modal analysis, eigenvalues and eigenvectors. Direct integration of dynamic equations (explicit and implicit methods) 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

Notes are available to students in folder format. Reference textbooks: - Zienkiewicz O.C., Taylor R.L., Zhu J.Z. The finite element method: its basis and fundamentals, Elsevier Butterworth-Heinemann, 2005. - Zienkiewicz O.C., Taylor R.L., Zhu J.Z. The finite element method: for solid and structural mechanics, Elsevier Butterworth-Heinemann, 2005. - Reddy, J.N. An introduction to the finite element method, McGraw-Hill, 2006.

Exam: Written test; Compulsory oral exam;

The final examination of the course is characterized by: - 1 written work (20 points): 1 hour of duration without notes or books - 1 oral examination (10 points or more): 15-20 minutes of duration The oral exam can be carried out only by students having a minimum score of 15/30 (score of the written work). The oral exam consists in 2-3 questions on the contents of the homework. The final score of the exam is given by the sum of the score of the written and the oral exam. A score larger than 30/30 will be registered as 30/30 with laudem.

In addition to the message sent by the online system, students with disabilities or Specific Learning Disorders (SLD) are invited to directly inform the professor in charge of the course about the special arrangements for the exam that have been agreed with the Special Needs Unit. The professor has to be informed at least one week before the beginning of the examination session in order to provide students with the most suitable arrangements for each specific type of exam.