I would like to show that in 1D the quadratic finite elements lead to the following lumped mass matrices, obtained summing all line coefficients of the corresponding mass matrices onto the diagonal:
$$ A = \left( \begin{matrix} \frac{1}{6} & 0 & 0 \\ 0 & \frac{4}{6} & 0 \\ 0 & 0 & 1 \end{matrix} \right) $$
where the mass matrix is given as $M_{ij}=\int_J \phi_i(x)\phi_j(x)dx$ for $\phi_i$ the shape functions where $J$ is the respective bar/domain that we consider.
At first, I considered the simple case of the bar $I$ where the three grid points are given as $e_1=0; e_2=1/2$ and $e_3=1$. For the shape functions we need $S_i(e_j)=\delta_{ij}$. By solving a linear system of equations I get that $S_1(x)=1-3x+2x^2$ and $S_2(x)=4x-4x^2$ and $S_3(x)=-x+2x^2$. Now I just solve the integrals for the mass matrices and get that $$ M = \left( \begin{matrix} 2/15 & 1/15 & -1/30 \\ 1/15 & 8/15 & 1/15 \\ -1/30 & 1/15& 2/15 \end{matrix} \right) $$. By summing up the line coeffiecents I get the desired lumped mass matrix.
I would now like to generalize this result to arbitrary $x_1,x_2,x_3$ on the bar $I'$ where $x_3-x_1=l$. Therefore I considered the following coordinate transformation: \begin{equation} f(x)=x_1+(-3x_1+4x_2-x_3)x+(2x_1-4x_2+2x_3)x^2 \end{equation} Here each $e_i$ is mapped to $x_i$. Let $\phi_i$ denote the respective shape functions for the bar $I'$. Now we get for the mass matrix (doing integral transformation) \begin{equation} M_{{new}_{ij}}=\int_{x_3}^{x_1} \phi_i(x) \phi_j(x) dx=\int_0^1 S_i(x) S_j(x) |det(J_f)| dx. \end{equation} Simple derivation shows that $|det(J_f)|=|-3x_1+4x_2-x_3+4x_1-8x_2+4x_3|$ which means that \begin{equation} M_{new} = |-3x_1+4x_2-x_3+4x_1-8x_2+4x_3|M. \end{equation} But shouldn't they be equal to show that they have the same lumped mass matrix. Is there something wrong in my calculation?
Edit: When substituting $x_2=x_1+l/2$ and $x_3=x_1+l$ then I get that $|-3x_1+4x_2-x_3+4x_1-8x_2+4x_3|=l$. So is it correct that \begin{equation} M_{new} = lM? \end{equation} Do I then get the desired mass matrix?
