Gauss Quadrature Rules Gauss quadrature rules are a set of principles and algorithms used in numerical analysis to approximate the values of functions in a...
Gauss Quadrature Rules
Gauss quadrature rules are a set of principles and algorithms used in numerical analysis to approximate the values of functions in a given domain. In the context of the finite element method, which employs isoparametric elements to approximate geometric objects, these rules play a crucial role in determining the element values and, consequently, the solution of the governing differential equations.
Principles of Gauss Quadrature:
Gauss quadrature rules are based on the observation that the value of a function in a given domain can be approximated by the average of function values at its discrete mesh nodes. The basic principle is to assign a set of equally spaced points (mesh nodes) within the domain and then evaluate the function value at these points.
Algorithm for Gauss Quadrature:
Determine the mesh size (h): Divide the domain into a set of equal-sized subintervals (elements) by choosing a value for h.
Select mesh nodes: Choose the appropriate mesh nodes within each element based on the desired order of accuracy.
Evaluate function values: Calculate the function value at each mesh node.
Calculate element values: Compute the element values as the average of function values at the mesh nodes within each element.
Repeat for all elements: Repeat the process for each element, resulting in a global approximation of the function value in the entire domain.
Examples:
Gauss-C quadrature is a commonly used method for approximating continuous functions. The values of the function at the mesh nodes are averaged to obtain the element values.
Gauss-Lobel-Simpson (GLS) quadrature is a higher-order method that provides higher accuracy than Gauss-C. It involves assigning different orders of accuracy to different mesh nodes.
Significance in Finite Element Method:
Gauss quadrature rules provide a flexible and effective approach for approximating the values of functions within the context of isoparametric elements. By determining the element values using these rules, the finite element method can approximate the solution of differential equations with high accuracy, leading to better solutions to engineering and scientific problems