integrointipisteet

Integrointipisteet, or integration points, are specific locations within a geometric element where numerical integration is performed to approximate the behavior of a continuous system. These points are crucial in finite element analysis (FEA) and other numerical methods for solving partial differential equations. Instead of integrating over an entire element analytically, which can be complex or impossible for arbitrary shapes, the integral is replaced by a weighted sum of function values evaluated at these discrete points.

The choice of integration points and their weights is based on numerical integration formulas, such as Gaussian quadrature. The number of integration points, known as the order of integration, influences the accuracy of the approximation. Higher-order integration generally leads to more accurate results but also increases computational cost. For simple elements like quadrilaterals or triangles, specific, well-defined integration points are used.

The concept of reduced integration arises when fewer integration points are used than what is typically required for exact integration of certain polynomial terms. This can sometimes improve performance by reducing the number of calculations, but it also carries the risk of introducing spurious zero-energy modes, also known as hourglassing, which can lead to unphysical deformations. Conversely, full integration uses the standard number of points required for exact integration. The correct selection of integration points and their associated quadrature rules is therefore a critical aspect of achieving accurate and stable numerical solutions in engineering simulations.