1. Introduction#

In Code_Aster, we call a « finite element », a triplet (phenomenon, modeling, mesh type). There are three main phenomena: MECANIQUE, THERMIQUE, and ACOUSTIQUE.

There are numerous models; for example, for the phenomenon MECANIQUE: 3D, C_PLAN,, D_PLAN, AXIS, DKT, POU_D_E,…

For a given modeling (for example 3D) of a phenomenon (for example MECANIQUE), there are in general several finite elements: one element per type of supported mesh: HEXA8, HEXA20, PENTA6,…

In the end, there are therefore a large number of finite elements (more than 500 in July 2004).

On the other hand, the types of mesh are in small number: POI1, SEG2,,, SEG3,,, SEG4, TRIA3,, TRIA6, TRIA7, QUAD4, QUAD8, HEXA8 HEXA20 TETRA4 TETRA10.

In general, each finite element, to carry out its elementary calculations, uses the concepts of interpolation function (or shape function) and integration diagram. In general too, these form functions and integration schemes are defined on a so-called « reference » element whose geometry is defined in a coordinate system often called: \((\xi ,\eta ,\zeta )\). The transition from the reference element to the real element is done thanks to a geometric transformation that uses the same interpolation functions. The element is then said to be « iso-parametric ». These concepts are very well explained in [bib1] or in [bib2] (which is available for free online).

The high number of finite elements in the code, combined with the limited number of mesh types, leads to the fact that there are several finite elements based on the same type of mesh; for example the 8-node quadrilateral called QUAD8 supports more than 60 different finite elements.

In theory, each finite element can choose its interpolation functions and integration schemes as it sees fit. But in practice, almost all finite elements based on the same type of mesh use the same reference element, the same form functions and the same integration schemes. The purpose of this document is to describe these various reference elements.

For each reference element (referred to later in document ELREFE), we will indicate:

  • the support mesh, the number of nodes, their local numbering and their coordinates,

  • algebraic expressions for form functions and their first (and sometimes second) derivatives

  • the families of integration points that will be called. For each family, the number of points, their coordinates and their integration « weights » will be given. The sum of these weights should give the « volume » of the reference element. For example, the « volume » of the reference quadrangle (\(-1\le \xi \le +1\), \(1<\eta <+1\)) is 4.