Problem 1 - Concepts and Applications of Finite Element Analysis

Question

Derive 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 interval \(-1 \leq \xi \leq 1\). Assume that sampling points and weights are symmetric with respect to the middle of the interval.

Step-by-Step Solution

Answer

The locations for an order-2 Gauss rule that exactly integrates the given polynomial are \(x_{1} = -\frac{\sqrt{3}}{3}\) and \(x_{2} = \frac{\sqrt{3}}{3}\). The corresponding weights are \(w_{1} = w_{2} = 1\).

Step 1: Express the Integral

First, express the condition that Gauss rule should exactly integrate the polynomial \(\phi=a_{1}+a_{2} \xi+a_{3} \xi^{2}+a_{4} \xi^{3}\). This results in the integral representation: \(\int_{-1}^{1} (a_{1}+a_{2} \xi+a_{3} \xi^{2}+a_{4} \xi^{3}) d\xi = w_{1}f(x_{1})+w_{2}f(x_{2})\), where \(w_{1}\) and \(w_{2}\) are the weights and \(x_{1}\) and \(x_{2}\) are the locations of the order 2 Gauss rule, and \(f(x)\) is the integrand.

Step 2: Determine the Locations

Given that the locations are symmetric, the locations \(x_{1}\) and \(x_{2}\) can be expressed as \(-\alpha\) and \(\alpha\) for some \(\alpha\). With these substitutions, the system of equations formed from Step 1 can be solved to yield \(x_{1} = -\frac{\sqrt{3}}{3}\) and \(x_{2} = \frac{\sqrt{3}}{3}\).

Step 3: Determine the Weights

By using the symmetry of the sampling points and the nature of the order-2 Gauss rule that gives equal weights to the sampling points, the weights \(w_{1}\) and \(w_{2}\) can be expressed as \(w\). From the system of equations from Step 1, we can solve for the weights to get \(w_{1} = w_{2} = 1\).

Key Concepts

Numerical Integration

Numerical integration is a technique used to approximate the definite integral of a function when an analytical solution is difficult or impossible to find. In many practical situations, functions may not have simple antiderivatives or may be complex, making it hard to integrate directly. This is where numerical methods, like the Gauss Quadrature, come into play.
To perform numerical integration, the area under the curve of a function is approximated by a sum of function values at specific points, multiplied by appropriate weights. These points and weights are specially chosen to provide accurate approximations.
There are various methods of numerical integration, each with different levels of accuracy and efficiency. The choice of method depends on several factors such as the function's behavior, the desired precision, and available computational resources.

Polynomial Integration

Polynomial integration is a specific type of numerical integration applied to polynomial functions. Polynomials are suitable for this because they can be expressed simply and are easy to manipulate. They are also commonly used to approximate more complex functions over a particular interval.
When integrating polynomials, the goal is to find the area under the polynomial curve within a specific range. Using methods like Gauss Quadrature, we can achieve exact results for polynomials up to a certain degree, by carefully selecting sampling points and weights.
Notably, polynomial functions have certain beneficial properties, such as continuity and differentiability, which make them particularly manageable for integration and ideal candidates for analytical and numerical approaches.

Symmetric Sampling Points

Symmetric sampling points are a significant concept in numerical integration methods, especially in Gauss Quadrature. These points are equally spaced around the center of the integration interval, which enhances the accuracy of the approximation by balancing the error on either side.
In the context of Gauss Quadrature, the choice of symmetric sampling points leads to simplifications in calculations. For instance, with an order 2 Gauss rule as in the provided exercise, the sampling points are at locations \(x_1 = -\frac{\sqrt{3}}{3}\) and \(x_2 = \frac{\sqrt{3}}{3}\). This specific symmetry ensures that the integral of a polynomial is precisely estimated up to its polynomial degree limits.
Symmetric points also help in reducing computational costs and improving numerical stability, which are critical factors in computational methods.

Integration Weights

Integration weights are crucial in numerical integration techniques, as they determine the influence of each sampling point on the final integral approximation. They essentially represent how much each sampled value contributes to the total area approximation under the curve.
In Gauss Quadrature, these weights are carefully chosen to maximize accuracy and efficiency. For an order 2 Gauss rule, which focuses on a quadratic polynomial integration, the weights inherently reflect the symmetry of the sampling points. For instance, with the symmetric sampling points used in the exercise, each point is weighted equally, with weights \(w_1 = w_2 = 1\).
Integration weights ensure that approximations are not biased and can accurately represent the integral's true value, especially in polynomial functions, by accounting for varying degrees and enhancing the precision of the final result.