502861 - INTRODUCTORY COMPUTATIONAL MECHANICS

courses

ID:

502861

Duration (hours):

73

CFU:

SSD:

BIOINGEGNERIA INDUSTRIALE

SCIENZA DELLE COSTRUZIONI

Year:

2025

Date/time interval

Secondo Semestre (02/03/2026 - 12/06/2026)

Course Objectives

The aims of the course are basically twofold: 1) to introduce the theoretical foundations of computational mechanics; 2) to teach how to develop a finite element code in Matlab/Python environment.

Course Prerequisites

Using Matlab/python, and in particular: creating functions, handling matrices, inverting linear systems, plotting functions
you can refer to available Matlab tutorials.
Euler Bernoulli and Timoshenko beam theory, development of a corresponding finite element, with direct construction of the corresponding elementary stiffness matrices and elementary load vector
Knowledge of FEM codes in Matlab (e.g., codes developed during previous courses)
Analytical solution of simple problems (cantilever beam, supported beam, etc.)
Three-dimensional deformable body theory (kinematics, equilibrium, constitutive bonding)
Particularisation of three-dimensional deformable body theory to the two-dimensional case, in particular for plane state of tension and plane state of deformation
Reissner-Mindlin thick plate theory.

Teaching Methods

Both blackboard and slide-supported lectures. Practical lectures with laptops/computers for coding.

Assessment Methods

Computer-based exam with exercises to prove coding abilities in the context of finite elements + oral examination on theoretical topics of numerical analysis starting from technical reports prepared by the students during the course.

Texts

T.J.R. Hughes, “The Finite Element Method: Linear Static and Dynamic Finite Element Analysis". Dover, 2000.

Zienkiewicz, O. C., & Taylor, R. L. (2000). "The finite element method: solid mechanics"
Butterworth-heinemann.

Euler Bernoulli beam
Weak form and FEM
Introduction to symbolic Matlab
Nodal load vector for distributed load (constant, linear)
Timoshenko beam, shear locking, methods for solving it
Displacement formulation with shear locking
Linked displacement formulation
Sub-integrated formulation
Mixed formulation
Enhanced displacement formulation
Constraints: master node, rigid plane
FEM formulation for 3D/2D elasticity
Strong vs. weak form for 3D problems
Displacement FEM approximation, element viewpoint, local and global numbering, assembly
Recalls of 2D problems: stress plane state, strain plane state
Element TRI-3
Area coordinates, FEM formulation at displacements
Comparison with analytical solutions for simple problems
Solving complex problems: mesh generation (slab with circular/elliptical hole, bending beam, etc), comparison of TRI-3 code numerical solutions with commercial calculation codes
Element QUAD-4
Isoparametric map, bilinear approximations, displacement formulation, numerical integration
Comparison with analytical solutions for simple problems
Solution of complex problems: mesh generation (slab with circular/elliptical hole, bending beam, etc.)
Comparison of TRI-3 numerical solutions with commercial calculation codes
Element TRI-6
Area coordinates, displacement FEM formulation
Numerical problems FEM formulation at displacements and possible solution
Volumetric Locking
Sub-integrated formulation
Mixed u-p formulation
Enhanced formulation
Pian-Sumihara element
Plate models
QUAD-4 formulation for thick plate problem
Evidence of shear locking problem
Discussion of possible solutions
2D coupled thermal and thermo-mechanical problems