Problem 14 - Concepts and Applications of Finite Element Analysis

Question

Derive the locations and weights of an order 2 Gauss rule by requiring that it integrate exactly the polynomial \(\phi=a_{1}+a_{2} \xi+a_{3} \xi^{2}+a_{4} \xi^{3}\) in the range \(-1 \leqslant \xi \leqslant 1\). Assume that weights and points are symmetric with respect to the axis \(\xi=0\).

Step-by-Step Solution

Answer

The weights w of the Gaussian quadrature are both 1 and the locations are \( \sqrt{\frac{1}{3}} \) and \(-\sqrt{\frac{1}{3}}\).

Step 1: Formulate Gaussian Quadrature equations

Assuming that the weights are \(w\) and \(-w\) for points \(x_{i}\) and \(-x_{i}\) respectively due to symmetry, the Gaussian quadrature rule for this case can be written as: \[w f(x_{i}) + (-w) f(-x_{i}) = \int_{-1}^{1} f(x) dx\]where \(f(x)\) can be any of the four polynomials \(1\), \(\xi\), \(\xi^{2}\) and \(\xi^{3}\). This will lead to 4 equations.

Step 2: Derive equations for monomials

Applying the step 1 for all monomials, we get the following 4 equations. For \(f(x)=1\), \[2wx_{i}=\int_{-1}^{1} dx =2\]For \(f(x)=\xi\), \[2wx^2_{i}=\int_{-1}^{1} \xi d\xi=0\] For \(f(x)=\xi^{2}\), \[2w{x_{i}}^{3}=\int_{-1}^{1} \xi^{2}d\xi = \frac{2}{3}\]For \(f(x)=\xi^{3}\), \[2w{x_{i}}^{4}=\int_{-1}^{1} \xi^{3}d\xi=0\]

Step 3: Solve for weights and locations

The equations derived in step 2 can now be solved which gives \[w=1\]and \[x_{i} = \sqrt{\frac{1}{3}} = -x_{i}\] Thus, the weights w are 1 and the locations are \(\sqrt{\frac{1}{3}}\) and \(-\sqrt{\frac{1}{3}}\).

Key Concepts

Numerical Integration

Numerical integration is a key technique used in computational mathematics to approximate the definite integrals of functions when an exact symbolic antiderivative is difficult or impossible to obtain. Unlike analytical integration, where we find a closed-form expression for the integral, numerical integration provides a numerical result based on evaluating the function at certain points and then combining these values following a specific formula.

One popular method for numerical integration is Gaussian Quadrature, which focuses on obtaining the highest possible accuracy for polynomials of a given degree. This method uses a set of points called nodes or abscissas and associated weights to create a weighted sum that approximates the integral. Gauging by the given exercise, we see that Gaussian Quadrature not only simplifies the integral calculation but also ensures that the approximation is as accurate as possible for a polynomial of a degree less than or equal to 2n-1, where n is the number of nodes.

Polynomial Approximation

Polynomial approximation is another foundational concept essential in numerical methods and computer simulations. In many real-world scenarios, the functions we're dealing with might be too complex, lacking a simple analytical form, or may even be only known at certain discrete points. Approximating these functions with polynomials can make them much easier to work with since polynomials are well-understood and can be integrated, differentiated, and evaluated with relative ease.

In the context of Gaussian Quadrature, as shown in the provided exercise, polynomials are used to represent the function we want to integrate. The key idea is that if a quadrature rule can integrate polynomials of a certain degree exactly, it will likely produce a good approximation for more complex functions as well. When deriving locations and weights for Gaussian nodes, the goal is to exactly integrate polynomials up to a certain degree, which directly influences how accurate the numerical integration will be.

Finite Element Analysis

Finite Element Analysis (FEA) is a sophisticated computational technique widely used in engineering and physics to predict how objects will react to external forces, such as heat, vibration, or other physical effects. Numerical integration is a cornerstone of FEA, as it allows for the approximate solution of integrals over complex geometries that represent real-world objects.

In FEA, the object of interest is divided into small pieces or 'elements', usually with polynomial shape functions defining the spatial variation of physical quantities within each element. Then, through the process of assembling all the elements' equations and solving the global system, numerical integration is used to calculate the relevant matrix entries and load vectors. This is where Gaussian Quadrature becomes extremely useful, as it provides a way to accurately integrate the polynomial approximations of the physical fields within each element, which is critical for the accuracy and efficiency of FEA.