Space-Time Finite Element Method for wave equation

Solving the wave equation as a first-order system in time using space-time finite elements.

Problem statement

The strong form of the wave equation is given by

$$ \partial_{tt} u - \partial_{xx} u = f \qquad \forall x \in \Omega, t \in [0,T]. $$

As a firs-order system in time this reads

$$ \partial_t v - \partial_{xx} u = f, $$

$$ \partial_t u = v. $$

Usually the wave equation is complemented with the initial conditions for $u$ and $v$ which are given as

$$ u(x, 0) = u^0(x), $$

$$ v(x, 0) = v^0(x). $$

However, in a recent publication on space-time variatonal material modeling Junker and Wick [1] derived that instead one should have an initial and a final condition for the velocity $v$, i.e.

Running the code locally

Install FEniCS, NumPy and matplotlib. Running the normal wave equation:

jupyter notebook Wave_Normal.ipynb

Running the wave equation with initial and final condition for velocity:

jupyter notebook Wave_FinalCondition.ipynb

Running the code on Google Colab

Link for the normal wave equation:

https://githubtocolab.com/mathmerizing/SpaceTimeWave/blob/main/Wave_Normal.ipynb

Link for the wave equation with initial and final condition for velocity:

https://githubtocolab.com/mathmerizing/SpaceTimeWave/blob/main/Wave_FinalCondition.ipynb

References

[1] Junker, P., Wick, T. Space-time variational material modeling: a new paradigm demonstrated for thermo-mechanically coupled wave propagation, visco-elasticity, elasto-plasticity with hardening, and gradient-enhanced damage. Comput Mech 73, 365–402 (2024). https://doi.org/10.1007/s00466-023-02371-2