muotofunktioiden

Muotofunktioiden, often translated as "shape functions" or "form functions" in English, are mathematical tools used in the field of finite element analysis (FEA). They are fundamental to discretizing complex continuous domains into smaller, simpler elements. Within each element, the displacement, temperature, or other field variables are approximated by a set of these functions. These functions define how the variable varies within the element based on values at specific points called nodes.

Shape functions are typically defined in terms of local coordinates, making them independent of the element's geometry and orientation in the global system. For a one-dimensional element, linear shape functions would be simple linear polynomials, and quadratic shape functions would be quadratic polynomials. In two or three dimensions, these functions become more complex, often using Lagrange polynomials or Hermite polynomials. The key property of shape functions is that they evaluate to unity at their corresponding node and zero at all other nodes within the same element.

The weighted average of the governing differential equation over each element is used in FEA, and the shape functions play a crucial role in this process. They are used to interpolate the unknown field variable within the element and to derive the element stiffness matrix and load vectors. The choice of shape functions influences the accuracy and convergence of the finite element solution. Higher-order shape functions generally lead to more accurate results but also increase computational cost due to a larger number of degrees of freedom.