Finite elements and structural design codes

The finite element method is the method used in structural analysis. This course introduces the mathematical foundations of this method. First of all, the equations of the mechanics of deformable solids are put into a matrix form that lends itself well to numerical processing. This integral form lends itself well to the introduction of approximations on small domains called finite elements. The approximations are in the form of nodal interpolations, which enable each finite element to be represented by a matrix known as the elementary stiffness matrix, which is then assembled into a global stiffness matrix for the structure, as in the displacement method seen in the first year.
The aim of the course is to construct "on paper" the elementary stiffness matrix of a finite element, bearing in mind that this depends on the initial mechanical model (beam, plate, 2D or 3D solid) and the degree of nodal interpolation. Students program the finite element method for lattices.

Matrix calculation.
Beam theory and, more generally, the technique of building structural models.
Displacement method.

Chapter I - Reminders on the displacement method

Elementary stiffness matrix, assembly, resolution.

Chapter II - Construction of the finite element bar

Weak formulation, strong formulation, link with the principle of virtual work, construction steps for the bar element

Chapter III - Construction of the finite element beam

Weak formulation, strong formulation, construction steps for the beam element

Chapter IV - Generalization of the finite element method, techniques and tools (interpolation of geometry, change of variables), application to the triangular element T3

Engineering Sciences and Techniques

"Introduction à la méthode des éléments finis", J.-C. Cuillère, DUNOD
"Comprendre les éléments finis", A. Chateauneuf, Ellipses
"Une présentation de la méthode des éléments finis", Gouri Dhatt, Gilbert Touzot,
"Analyse des structures et milieux continus - Vol. 6 Méthode des éléments finis", François Frey
"The finite element method: linear static and dynamic finite element analysis",
Thomas J. R. Hughes