Problem 2 - Concepts and Applications of Finite Element Analysis

Question

18.2 Consider the equations \(x+y=2\) and \(x+1.01 y=2.01\). Show that the solution is sensitive in small changes in both the coefficient matrix and the vector of constants.

Step-by-Step Solution

Answer

The solution of the system of equations is sensitive to small changes in the coefficient matrix and the vector of constants. Even a minor adjustment in these values leads to noticeable variations in the solution: for the original system, the solution was \(x=1, y=1\), which changed to \(x=0.99, y=1.01\) for a minor tweak in the coefficient matrix, and \(x=0.98, y=1.02\) for a slight adjustment in the constant vector.

Step 1: Solve the Original System

Firstly, solve the system of equations without any changes. This is done by employing the substitution or elimination method. The solution for this system is \(x=1, y =1\).

Step 2: Altered Coefficient Matrix

Now, slightly modify the coefficient matrix, giving us the new system of equations \(x+ y = 2\) and \(x + 1.02y = 2.01\). Solve this new system, observing if and how the solution changes. The solution to the new system is \(x = 0.99, y = 1.01\). Notice the change in solution relative to the minuscule alteration in the coefficient.

Step 3: Altered Vector of Constants

Next, slightly adjust the constant vector, resulting in the system of equations \(x + y = 2\) and \(x + 1.01y = 2.02\). Solve this system and observe the changes. The solution to this new system is \(x = 0.98, y = 1.02\). Again, note how minor changes in the constant significantly affect the solution.

Key Concepts

System of Linear Equations

In the realms of mathematics and algebra, a system of linear equations is a collection of one or more linear equations involving the same set of variables. For instance, in our problem, we have two equations, \(x + y = 2\) and \(x + 1.01y = 2.01\). A solution to this system is a set of values for \(x\) and \(y\) that make both equations true simultaneously. These equations can describe lines on a graph, and where these lines intersect is the solution to the system. In applications like engineering, physics, economics, and computer graphics, solving these systems correctly is crucial.\
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Understanding the impact of alterations in the system can reveal how sensitive the solution is to such changes, which is vital in fields like finance or civil engineering, where precision is paramount. Sensitivity analysis, like the one illustrated in our exercise, shows how dependent the outcome is on the initial conditions.

Coefficient Matrix

The coefficient matrix is a crucial concept when dealing with systems of linear equations. It is a matrix consisting of all the coefficients of the variables from the system. In our case, the coefficient matrix for the original system is \[\begin{pmatrix}1 & 1 \ 1 & 1.01\end{pmatrix}\]. Each row represents an equation, and each column corresponds to a variable's coefficient. Altering one of these coefficients, as seen in step 2 of our solution, results in a changed coefficient matrix, which then produces a different solution.\
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A matrix's properties can indicate whether a system has a unique solution, infinitely many solutions, or no solution at all. Understanding these matrices helps in visualizing and solving these equations more efficiently via methods like matrix operations or determinants.

Elimination Method

The elimination method is one of the techniques to solve a system of linear equations. It involves adding or subtracting equations from each other to eliminate one of the variables, thus reducing the system to a single equation in one variable. Let's consider our system; by subtracting the second equation from the first, we can eliminate \(y\) and find \(x\) directly, and then place \(x\) back into one of the original equations to find \(y\). It's a systematic approach that can be scaled up to handle larger systems with more variables.\
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This method's advantage lies in its algorithmic nature, which makes it particularly suitable for computer-based calculations in finite element analysis, where large systems often need to be solved.

Substitution Method

Alternatively, the substitution method is employed by first expressing one variable in terms of the other from one equation and then substituting this expression into the other equation. With our initial system, we could express \(y\) in terms of \(x\) from the equation \(x + y = 2\) as \(y = 2 - x\) and plug this into the second equation. Now, we only need to solve for \(x\), and subsequently use the value of \(x\) to find \(y\). This method is straightforward and especially practical for systems with nicely behaved equations or for a quick handwritten solution. It can be particularly illuminating in educational settings, helping students to see the direct relationship between variables.