Grenzelemente

Grenzelemente (boundary elements) are elements that lie on the boundary of a domain in numerical simulations. They form the portion of a mesh that coincides with the boundary surfaces and are used to apply boundary conditions, model surface phenomena, or participate in boundary integral formulations.

In conventional finite element methods (FEM), a domain is discretized into interior elements, and boundary elements help impose Dirichlet or Neumann conditions on the boundary or model thin boundary layers. In boundary element methods (BEM), the problem is reformulated so that the unknowns reside only on the boundary, and the interior is represented by integral equations. This leads to a dimensional reduction (for a 3D problem, the boundary is 2D) and is advantageous for unbounded or exterior problems, but requires the fundamental solution of the governing equation and is typically limited to linear, homogeneous media.

Boundary elements are typically line elements in 2D and surface elements in 3D. They can be employed alone in BEM or in hybrid schemes with FEM, where boundary elements connect to interior elements to enforce boundary conditions or interface continuity. The use of Grenzelemente can reduce meshing complexity for boundary phenomena, but can result in dense system matrices and challenges for nonlinear or heterogeneous materials.

Applications span structural mechanics, acoustics, electrostatics, elastodynamics, and fluid-structure interactions, particularly where accurate boundary representation or exterior domains are important.

See also: finite element method, boundary element method, Dirichlet condition, Neumann condition, Green's function.