A numerical integration technique used in FEM for evaluating integrals over elements, chosen for its accuracy with polynomials of higher degrees.

Gaussian Quadrature

Gaussian Quadrature is a numerical integration method used to approximate the integral of a function, particularly when the function is difficult to integrate analytically or when high accuracy is required. It is widely used in various fields of engineering and applied mathematics, including finite element analysis (FEA), where it plays a crucial role in calculating integrals over elements. Gaussian Quadrature is known for its efficiency and accuracy, especially when dealing with polynomial functions.

Key Points about Gaussian Quadrature:

1. Definition:Gaussian Quadrature is a method of numerical integration that approximates the integral of a function as a weighted sum of function values at specific points (called Gauss points) within the integration interval. The method is designed to give the exact result for polynomials of degree 2n-1 or less, where n is the number of integration points used.The integral of a function f(x) over the interval [-1, 1] is approximated as:

   ∫-11 f(x) dx ≈ Σ wi f(xi)

   Where:

   • w_i: Weights associated with the integration points.
   • x_i: Gauss points where the function is evaluated.
2. Gauss Points and Weights:The Gauss points (x_i) and corresponding weights (w_i) are determined based on the order of the Gaussian Quadrature. These points are the roots of the Legendre polynomials, and the weights are chosen to ensure the highest possible accuracy for the given number of points.For example, the Gauss points and weights for the two-point Gaussian Quadrature are:
   • Gauss Points: x₁ = -√(1/3), x₂ = √(1/3)
   • Weights: w₁ = w₂ = 1

   For the three-point Gaussian Quadrature:

   • Gauss Points: x₁ = -√(3/5), x₂ = 0, x₃ = √(3/5)
   • Weights: w₁ = w₃ = 5/9, w₂ = 8/9
3. Advantages of Gaussian Quadrature:Gaussian Quadrature offers several advantages over other numerical integration methods:
   • High Accuracy: Gaussian Quadrature provides exact results for polynomials of degree 2n-1 or less, making it highly accurate for smooth functions, especially when compared to other methods like the Trapezoidal Rule or Simpson’s Rule.
   • Efficiency: Because it achieves high accuracy with fewer points, Gaussian Quadrature is computationally efficient, reducing the number of function evaluations required for a given accuracy level.
   • Versatility: Gaussian Quadrature can be applied to a wide range of problems, including those involving non-polynomial functions, by increasing the number of Gauss points or by transforming the integration interval.
4. Application in Finite Element Analysis (FEA):In finite element analysis, Gaussian Quadrature is used to perform numerical integration over elements to calculate quantities such as stiffness matrices, mass matrices, and load vectors. The method is particularly useful when integrating functions that are complex or when elements have curved boundaries.For example, to integrate over a quadrilateral element, the integral is transformed into the natural coordinate system (ξ, η), and Gaussian Quadrature is applied using the appropriate Gauss points and weights for each direction.
5. Numerical Integration Process:The general process of applying Gaussian Quadrature involves the following steps:
   • Transform the Integral: If the integration interval is not [-1, 1], transform it to this interval using a suitable substitution.
   • Select Gauss Points and Weights: Choose the appropriate number of Gauss points and their corresponding weights based on the desired accuracy.
   • Evaluate the Function: Calculate the function values at the selected Gauss points.
   • Sum the Weighted Function Values: Multiply each function value by its corresponding weight and sum the results to obtain the approximate value of the integral.
6. Challenges and Considerations:
   • Non-Polynomial Functions: While Gaussian Quadrature is highly accurate for polynomials, its accuracy may decrease for non-polynomial functions, especially those with singularities or discontinuities. In such cases, higher-order quadrature or specialized techniques may be required.
   • Choice of Gauss Points: The number of Gauss points needed depends on the function being integrated and the desired accuracy. Using too few points may result in significant errors, while using too many points can increase computational cost without significant gains in accuracy.
   • Integration over Complex Domains: For integration over complex geometries or higher-dimensional domains, the application of Gaussian Quadrature requires careful mapping and the use of multi-dimensional Gauss points, which can complicate the integration process.

Summary:

Gaussian Quadrature is a powerful and efficient numerical integration method that approximates the integral of a function by evaluating it at specific points (Gauss points) within the integration interval. Known for its high accuracy and efficiency, especially for polynomial functions, Gaussian Quadrature is widely used in engineering applications such as finite element analysis (FEA). By carefully selecting the number of Gauss points and applying the method to transformed intervals, Gaussian Quadrature can provide accurate results for a wide range of integration problems. However, challenges such as handling non-polynomial functions and complex domains require careful consideration and the use of appropriate techniques to ensure reliable results.