kvadratuuripisteet

Kvadratuuripisteet, or quadrature points, are special points used within numerical integration schemes to approximate integrals over geometric domains such as triangles, quadrilaterals, tetrahedra, or hexahedra. In finite element analysis, these points are placed inside each element to evaluate integrals that arise when forming element stiffness matrices, mass matrices, or load vectors. The accuracy of the finite element solution depends heavily on the choice of quadrature points and their associated weights.

Standard quadrature rules are derived from orthogonal polynomials, for example the Gauss–Legendre rule for one-dimensional intervals, which is extended to multi-dimensional domains through tensor products or specialized rules for reference elements. A simple example is the 2×2 Gauss–Legendre rule for a square reference element, which uses four interior points and achieves exact integration for polynomials up to third order. Higher-order elements or curved geometries often require more points—such as 3×3 or 4×4 grids—to maintain accuracy, especially when mapping from a reference to a physical element via isoparametric transformations.

The placement of quadrature points is typically symmetric to preserve element symmetry and to avoid bias in the numerical solution. In advanced applications, such as high-order isogeometric analysis or computational fluid dynamics, optimized or adaptive quadrature schemes are employed to reduce the number of point evaluations while still meeting error tolerances. Choosing appropriate kvadratuuripisteet is therefore essential for efficiently obtaining reliable results in numerical simulations.