Problem 1 - Concepts and Applications of Finite Element Analysis

Question

Consider the equations \(x+y=2\) and \(x+1.01 y=2.01\). Obtain the solution of these equations. Then aiter the second equation, first to (a) \(x+1.02 y=2.01\), then to (b) \(x+1.01 y=2.02\). Obtain the solution in each case, thus illustrating sensitivity of the equation set to small changes in (a) the coefficient matrix, and (b) the vector of constants.

Step-by-Step Solution

Answer

The solutions for the systems are: for the original system \(x=1, y=1\); when altering the coefficient matrix the solution is \(x=1.5, y=0.5\); and when altering the vector of constants the solution is \(x=0, y=2\). Therefore, small changes in either the coefficients of the equations or the constants can lead to significant changes in the solution of the system.

Step 1: Solve Original Equations

Start with the system of equations:\n1. \(x+y=2\)\n2. \(x+1.01y=2.01\)\nSubtract the first equation from the second to isolate y:\n\(0.01y=0.01\)\nTherefore, \(y=1\). Substitute \(y=1\) into the first equation to find that \(x=1\). So the solution to the original system is \(x=1, y=1\).

Step 2: Alter Coefficient and Solve

Now, alter the second equation to the form \(x+1.02y=2.01\), and solve it again with the first equation unchanged:\n1. \(x+y=2\)\n2. \(x+1.02y=2.01\)\nIsolate y the same way by subtracting the first equation from the second:\n\(0.02y=0.01\)\nThis gives \(y=0.5\). Substitute \(y=0.5\) into the first equation to find that \(x=1.5\). So by changing the coefficient matrix the solution changes to \(x=1.5, y=0.5\).

Step 3: Alter Constant and Solve

Next, restore the second equation to \(x+1.01y=2.01\) but change the constant to obtain a new system:\n1. \(x+y=2\)\n2. \(x+1.01y=2.02\)\nUse the same method to solve the new system:\n\(0.01y=0.02\)\nThis gives \(y=2\). Substitute \(y=2\) into the first equation to find that \(x=0\). So by changing the vector of constants the solution changes to \(x=0, y=2\).

Key Concepts

Sensitivity Analysis

Sensitivity Analysis is an important concept in mathematical modeling and engineering that measures how the output of a system is affected by changes in input parameters. It's a way to predict the impact of minor or major adjustments on the final result, helping in understanding the robustness of models and solutions.

In the context of the provided exercise, we're looking at a system of linear equations. By altering one equation slightly, we can evaluate how sensitive the solution is to changes in the system's parameters. For instance, when the coefficient in front of the variable 'y' changes from 1.01 to 1.02, or when the constant term changes from 2.01 to 2.02, we note significant changes in the solutions. This investigation reveals the stability of the solution: a small change can lead to large variations in the answer, indicating a sensitive system.

Conducting sensitivity analysis helps in several ways, such as optimizing system design, identifying critical parameters that need precise control, and estimating error bounds for predictions. It's a vital tool for engineers and scientists who need to ensure that their designs and processes can tolerate inaccuracies and variations in input values.

Coefficient Matrix

The Coefficient Matrix is a central concept in linear algebra, particularly when dealing with systems of linear equations. In our exercise, the coefficient matrix comes from the coefficients of the variables in the equations. For the original system with equations \(x+y=2\) and \(x+1.01 y=2.01\), the coefficient matrix would be formed by arranging the coefficients in a grid:\
\[\begin{bmatrix}1 & 1 \1 & 1.01\end{bmatrix}\]
A key property of the coefficient matrix is that slight changes in its values can significantly influence the solutions of the system, if the system is sensitive. This is what was explored in the given exercise by changing 1.01 to 1.02, which altered the matrix to:\
\[\begin{bmatrix}1 & 1 \1 & 1.02\end{bmatrix}\]
Such changes have a pronounced effect on the system's solution, as we saw with the solution shifting from \(x=1, y=1\) to \(x=1.5, y=0.5\). Understanding the impact of the coefficient matrix on a system of equations is crucial to comprehend the system's behavior under modifications—whether those are intentional adjustments or unintentional fluctuations due to real-world uncertainties.

System of Equations

A system of equations is a set of two or more equations involving the same set of variables. The solution to a system of equations is the set of variable values that satisfies all the equations in the system simultaneously. In our case, we're dealing with a simple two-equation system. Solving such systems can be approached in multiple ways, including graphically, using substitution, elimination, or more advanced methods like matrix operations for larger systems.

The original system of the exercise is:\
1. \(x+y=2\)\
2. \(x+1.01y=2.01\)
The solution to this system, as we performed, is a pair of values for \(x\) and \(y\), which we found to be \(x=1, y=1\). Altering parts of this system showcases how the solution can shift drastically - a valuable lesson in the interconnectedness of system elements. In practical terms, understanding how to solve a system of equations and predict the behavior of its solutions under changes is fundamental for tasks ranging from economics to engineering, where these equations can represent anything from market equilibrium to physical forces in mechanical structures.