Dreieckselement

DreiecksElement is a two-dimensional triangular element used in mesh generation and finite element analysis to discretize a geometric domain. It is defined by three non-collinear vertices in the plane and serves as the simplest polygonal element for representing 2D regions. In practice, a DreiecksElement may include additional nodes along its edges to support higher-order interpolation.

The element supports interpolation of field variables (such as displacement, temperature, or hydraulic potential) inside its interior through shape (basis) functions. For a linear DreiecksElement, the field value is a linear combination of the three vertex values, with barycentric coordinates acting as the shape functions. In higher-order variants, additional mid-edge or interior nodes are used, and corresponding higher-order shape functions are employed to capture more complex variation within the element.

Geometrically, the element is conveniently described by its vertices (x1,y1), (x2,y2), and (x3,y3). Important derived quantities include the area, given by 0.5 times the determinant formed by edge vectors, and the centroid, typically the average of the vertices. When used in finite element analysis, an affine mapping from a reference triangle to the physical triangle is common, enabling standardized integration and assembly procedures.

Variants include linear (three-node) and quadratic (six-node) DreiecksElement, with quadratic elements incorporating mid-edge nodes. DreiecksElement are widely employed in 2D simulations of structural mechanics, heat transfer, fluid flow, and computer graphics, where robust meshing and accurate interpolation are essential. See also finite element method, barycentric coordinates, and mesh generation.