I was reading the following article about the direct stiffness method. When it comes to solving the system of equations:
The site states:
[...]There are several different methods available for evaluating a matrix equation including but not limited to Cholesky decomposition and the brute force evaluation of systems of equations[...]
After doing some research, I figured out that the most common way is the Cholesky Decomposition. The problem that I have is that we are not solving for $u$ in the equation $f=Ku$ but for $f$ and $k$. In the Image, it’s clear that for if we do not know the value of $f_i$, we know the value of $u_I$ instead.
I am not sure how to solve this for both, the unknown values in $u$ and $f$ at the same time. I am very happy for any advice
I am not sure, if I get this right, but I general you cannot solve an equation system $Ax=b$ if you don't know both $x$ and $b$. However, in your case, you can solve the system since you got $f_{x_{2}}=0, f_{x_{3}}=f_{x}, f_{y_{3}}=f_{y}$ and $u_{x_{1}}=u_{y_{1}}=u_{y_{2}}=0$. For example you know, that $k_{33}u_{x_{2}}+k_{35}u_{x_{3}}+k_{36}u_{y_{3}}=f_{x_{2}}=0$.