Solving the Finite Element Method Matrix with Rao - Engineering

[Chacabucogod]

I was reading the finite element method in engineering by Rao and in the first example he ends up with a matrix that is singular.

The matrix is the following:

$$\begin{pmatrix} 2 &-2 & 0\\ -2 & 3&-1\\ 0&-1& 1 \end{pmatrix}$$

Which is a symmetric matrix as far as I can remember. Now you have 3 Nodes that will be the vector that will multiply this matrix the first node you already know is zero
so:

$$\begin{pmatrix} 0\\ f1\\ f2 \end{pmatrix}$$

The solution vector is

$$\begin{pmatrix} P1\\ 0\\ 1 \end{pmatrix}$$

Now he says that you can eliminate row 1 and column 1, but if you multiply the matrix by the first element of the vector only column one disappears. I tried solving this with the transpose method, but it seems there's no way to solve this matrix. Since there are only 2 unknowns can I completely forget ably the 3*3 matrix and do a 2*2 or do I have to take into account the third equation (the one that comes out of row 1)

Thank you

 

[Stephen Tashi]

The solution vector is

$$\begin{pmatrix} P1\\ 0\\ 1 \end{pmatrix}$$

You haven't stated a mathematical problem, so it isn't clear what you mean by "the solution".

I suggest you describe the problem completely or at least give a link to the passage in the book if Google books has the text.

 

[the_wolfman]

I not familiar with this particular book and I'm not sure where the book ends and where your interpretation begins.
But something is wrong.

That being said, in FE methods we often have an algebraic system of the form:

$A\vec x = \vec b$

For a simple system with 3 degrees of freedom we can write
$A=\begin{pmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31}&a_{32}& a_{33} \end{pmatrix}$,

$\vec x=\begin{pmatrix} x_1\\ x_2\\ x_3 \end{pmatrix}$

and
$\vec b=\begin{pmatrix} b_1\\ b_2\\ b_3 \end{pmatrix}$

Now if we want to insert a Dirichlet boundary condition we have to modify the matrix A and the vector b, but the vector x is unchanged. (This is our unknown that we are solving for.) For instance if we want the displacement at $x_1=2$ then the matrix becomes


$A=\begin{pmatrix} 1 & 0 & 0\\ a_{21} & a_{22} & a_{23}\\ a_{31}&a_{32}& a_{33} \end{pmatrix}$

and the RHS vector becomes
$\vec b=\begin{pmatrix} 2\\ b_2\\ b_3 \end{pmatrix}$.

Note that if you multiple the first row of A with x you get the equation $x_1=2$.

Since we know $x_1$ we can use this information to simplify the equation. In this case we'd get a simplier 2x2 system with


$A=\begin{pmatrix} a_{22} & a_{23}\\ a_{32}& a_{33} \end{pmatrix}$,

$\vec x=\begin{pmatrix} x_2\\ x_3 \end{pmatrix}$

and
$\vec b=\begin{pmatrix} b_2-2a_{21}\\ b_3-2a_{31} \end{pmatrix}$.

 

[Chacabucogod]

Thank you. I guess That is what I was looking for

 

[blue_raver22]

for bringing up this issue with the finite element method matrix. I can provide some insights and suggestions to help solve this problem.

Firstly, it is important to note that a singular matrix means that it is not invertible, which can be problematic in solving equations. In this case, the matrix is indeed symmetric, but it is also diagonally dominant, meaning that the magnitude of the diagonal elements is greater than the sum of the absolute values of the other elements in the same row. This can cause numerical instability and lead to a singular matrix.

To solve this issue, there are a few options that I would suggest. One option is to use a different method, such as the Gaussian elimination method or the LU decomposition method, to solve the equations instead of the finite element method. These methods may provide a more stable and accurate solution.

Another option is to modify the matrix by adding a small value (known as a "pivot") to the diagonal elements. This can help to make the matrix non-singular and improve the accuracy of the solution. However, care must be taken when choosing the value of the pivot, as it should be small enough to not significantly affect the solution but large enough to prevent the matrix from becoming singular.

In terms of eliminating the first row and column, it is important to note that this will only eliminate the first equation, not the entire first row and column. This means that the third equation (coming from the third row) will still need to be considered in the solution.

In summary, there are a few options for solving this singular matrix issue, including using a different method or modifying the matrix. It is important to carefully consider the numerical stability and accuracy of the solution when dealing with singular matrices in the finite element method.