Elementmethoden

Elementmethoden refer to numerical techniques for solving partial differential equations by discretizing a domain into small elements and formulating a variational problem. They transform continuous equations into algebraic systems, enabling the handling of complex geometries and heterogeneous materials.

Core workflow involves creating a mesh of elements; selecting function spaces and shape functions; deriving element equations via the weak form (for example, Galerkin); assembling the global stiffness and mass matrices and force vectors; applying boundary conditions; solving the resulting linear or nonlinear system; and post-processing results.

Element types include 1D, 2D, and 3D elements; common families are linear and higher-order; isoparametric elements use the same shape functions for geometry and field variables; isoparametric mapping enables curved geometries; time integration is used for transient problems.

Variants and methods include static analysis, dynamic analysis, and eigenproblem formulations; mixed and hybrid formulations; adaptive mesh refinement guided by error estimators; numerical stability and convergence depend on mesh quality and polynomial order.

Applications encompass structural mechanics, heat conduction, fluid dynamics, electromagnetism, and acoustics; used in engineering design, simulation, and optimization; implemented in both commercial and open-source software.

History and impact: the method originated in the 1950s–1960s, with early work by Clough and colleagues, and rapidly expanded across engineering and physics; today it is a standard tool in science, engineering, and industry.