Integrationspunkt

Integrationspunkt is a term used in numerical analysis and computational mechanics to denote a point within the domain of integration where the integrand is evaluated in order to approximate an integral by quadrature.

In finite element analysis, integration points are located inside each element according to the chosen quadrature rule, typically Gauss quadrature. The approximate integral of a function f over an element E is expressed as the sum over all points: sum_k w_k f(x_k), where x_k are the integration points mapped from a reference element to E, and w_k are the corresponding weights adjusted by the determinant of the element’s mapping (Jacobian). The number and placement of points depend on the polynomial order of the shape functions; higher-order elements require more points to achieve accuracy.

Common schemes include 1-, 2-, and 3-point Gauss rules in one dimension, and 2x2 or 3x3 rules in two dimensions (for quadrilaterals) or triangles, as well as 3x3x3 in three dimensions. Gauss points provide exact integration for polynomials up to degree 2n−1, where n is the number of points per dimension. In practice, the integration points are internal evaluation locations rather than degrees of freedom; the accuracy of the method depends on the chosen quadrature rule.

See also Gauss quadrature, numerical integration, and finite element method. The concept of integration points is fundamental for assembling matrices such as stiffness and mass matrices and for evaluating internal forces in computational mechanics.