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I am solving for a dynamic linear elasticity problem: \begin{equation} \dfrac{\partial^2 u}{\partial^2 t} - \nabla \cdot \sigma = f \end{equation} where $ u \in R^2 $ with sufficient BCs and initial values both on $ u $ and it's time derevative.

After dicritization with FEM I get the standard: \begin{equation} M \ddot{u} + k u = f \end{equation}
and I can then proceed to use a time some time integration scheme. I opted for a standard explicit Euler. How can I validate my solver?

  1. Using a manufactured solution

I am getting the expected convergence rates for a steady state problem but it's not so straightforward for a dynamic simulation because both space and time dicritization errors are present. Even when I refine my mesh to isolate the time discritization errors, but I am not getting small L2 errors. Is there a manufactured solution for such a case in the literature I can validate my code against?

  1. Plotting energies

I thought about creating energy plots but they don't seem to have a specific period. I used the boundary conditions defined by the initial solution and the right hand side:

\begin{equation} u = \begin{bmatrix} \sin(2\pi x) \cos(2\pi y) \cos(20\pi t) ;\\ \sin(2\pi y) \cos(2\pi x) \cos(20\pi t) \end{bmatrix} \quad f = \begin{bmatrix} 1200\pi^2\sin(2\pi x)\cos(20\pi t)\cos(2\pi y) ; \\ 1200\pi^2\sin(2\pi y)\cos(20\pi t)\cos(2\pi x) \end{bmatrix} \end{equation}

on a square domain where I have chosen my lame parameters to be $ \lambda= 0 $ and $ \mu =100 $

enter image description here

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    $\begingroup$ You should have a second derivative in the first equation. It doesn't let me edit it because the edits are too small. $\endgroup$
    – NNN
    Aug 23, 2023 at 4:00
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    $\begingroup$ You can use a manufactured solution where your spatial field is homogeneous, i.e., the whole field osciallates. That way your spatial discretization error will not bleed into your measurement. $\endgroup$
    – MPIchael
    Aug 23, 2023 at 5:30
  • $\begingroup$ 1) For the dynamic problem, use a fixed number of mesh elements and compute the error norms with different time steps at a specific time-instant, say t = 0.3. The trend in the resulting error norms should provide information about temporal discretization error. $\endgroup$ Aug 28, 2023 at 16:30
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    $\begingroup$ Your statement in 2) is unclear. what are you trying to explain here? $\endgroup$ Aug 28, 2023 at 16:31
  • $\begingroup$ @SthavishthaBhopalam This is a dynamic linear elasticity problem so the plots for energy can give me some insight. I guess. If I start have an initial condition pertaining to a fundamental frequency I'd expect the solution to oscillate at that same frequency. Right? Also, thanks everyone for the comments. $\endgroup$ Aug 29, 2023 at 17:45

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