Integrointipiste

Integrointipiste, also known as the integration point or collocation point, is a concept primarily used in numerical methods, particularly in the finite element method (FEM). It refers to specific locations within an element where certain calculations, such as integration of stiffness matrices or force vectors, are performed. Instead of integrating over the entire element domain, which can be computationally expensive, numerical integration techniques like Gaussian quadrature approximate the integral by evaluating the integrand at a set of predetermined points. These points are the integrointipisteet.

The number and placement of integrointipisteet are crucial for the accuracy and stability of the numerical solution. For a given element and the desired order of accuracy, a specific number of integration points is required. For instance, a 1D element might use two or three integration points, while a 2D element might use four or nine. The selection of these points, along with their associated weights, is based on the theory of Gaussian quadrature, which aims to achieve the highest possible accuracy for a given number of points.

The choice of integrointipisteet can influence the behavior of the numerical solution. For example, using too few integration points can lead to "locking" phenomena in certain types of elements, where the element becomes overly stiff and produces inaccurate results, especially under bending loads. Conversely, using more points than necessary increases computational cost without a proportional gain in accuracy. Therefore, careful consideration of the element formulation and the expected load cases is essential when selecting the number and location of integrointipisteet.