2. The resolution method#

To solve the various problems mentioned, the main spatial discretization method currently implemented in code_aster is the finite element method.

2.1. A parameterized implementation of the finite element method#

A particular effort has been made to parameterize the implementation of the finite element method. The calculation options required for each analysis method (static, quasistatic, dynamic) and for each phenomenon (mechanical, thermal, acoustic) are treated globally for the entire structure, regardless of the models selected for a particular study.

Some of the possibilities offered by this architecture include:

  • The independence between the discretization topology (MAILLAGE) and the interpolation properties of the finite elements assigned to these cells (MODELE), hence the diversity of models that can be used on the same mesh;

  • The diversity of behavioral relationships and properties of materials that can be used in the same model;

  • The treatment of boundary conditions and loads by specific on-board finite elements, to allow their unambiguous location, especially for continuous media;

  • A systematic procedure for dealing with the dependence of material properties and boundary conditions on various parameters (temperature, time, space variable…) using FONCTION or FORMULE defined in Python.

  • Data structures allowing the use of all models with the various resolution algorithms.

2.2. An extensive finite element library#

The finite element library is configured to allow the assignment, to the various recognized cells, of discretized formulations of the available phenomena.

2.2.1. Continuous environments#

A portion of a three-dimensional or two-dimensional structure, treated as a volume, is called a continuous medium.

3D models are the simplest forms of continuous media because they do not require additional hypotheses. In 2D models, an equation is removed, but hypotheses must be added: for example plane deformations or plane stresses in mechanics, axisymmetry in thermal and mechanical engineering.

2.2.2. The structural components#

Structural elements are built by integrating hypotheses on three-dimensional kinematic behavior (representing more or less well the phenomena of bending, twisting, shearing, warping, etc.). They can be classified into three categories:

  1. Medium-sheet elements (plates, shells): each type of element is based on hypotheses of variation of unknowns in thickness, which makes it possible to calculate the value at any point from that taken on the middle sheet (and possibly the lower and upper faces in thermal mode);

  2. Medium-fiber elements (bars, beams, pipes, cables): the hypotheses relate, for each cross section, the value of the unknowns at all points to that taken on the average fiber;

  3. Discrete elements (masses, springs, shock absorbers, etc.): they make it possible to introduce characteristics expressed in any coordinate system onto point cells or segments.

These components require the use of the AFFE_CARA_ELEM command to define their characteristics.

2.2.3. The modeling connections#

The layout selected for the finite element method makes it possible to treat structures modelled with various types of mechanical elements (continuous media or structural elements). The connection of finite elements based on different degrees of freedom, into the same node, can be done by writing linear relationships adapted to the nature of the connection. A particular methodology has been developed to transmit force torsors as correctly as possible (in the sense of least squares). It is thus possible to represent in a satisfactory manner the connection between a 3D medium and beams, plates, shells or pipes, as well as the shell-beam, shell-pipe or beam-pipe connections.

2.2.4. Discontinuous media#

There are also elements that take into account discontinuities (e.g. crack)

  1. By the level-set method (elements XFEM) with contact and friction modeling;

  2. By methods CZM (Cohesive Zone Models) with modeling of reclosing contact and opening damage;

  3. Elements with internal discontinuity;

2.2.5. Heterogeneous models#

Homogenization techniques make it possible to represent at a lower cost a network of tubes immersed in an incompressible fluid, multilayer composite shells, or multi-fiber beams.