# How can I validate my time integration scheme for my dynamic linear elasticity FEM code with a manufactured solution or similar?

I am solving for a dynamic linear elasticity problem: $$$$\dfrac{\partial^2 u}{\partial^2 t} - \nabla \cdot \sigma = f$$$$ where $$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: $$$$M \ddot{u} + k u = f$$$$
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:

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

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

• You should have a second derivative in the first equation. It doesn't let me edit it because the edits are too small.
– NNN
Aug 23, 2023 at 4:00
• 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. Aug 23, 2023 at 5:30
• 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. Aug 28, 2023 at 16:30
• Your statement in 2) is unclear. what are you trying to explain here? Aug 28, 2023 at 16:31
• @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. Aug 29, 2023 at 17:45