I want to solve that equation for the dynamic analysis of a beam. Moreover I will use the generalized alpha method for the time evolution. However, i can’t figure out how I may express the variational form of this equation into fenics:
(see the comments)
Where W is the test fun, and v the solution. If i use a green identity, i get higher order terms. Thanks for the help!
(see this comments)
This is the part i have trouble
No idea how to shape it to a variational form that can be used in FEniCS
The 4^\text{th}-order term is basically the 1D counterpart to the biharmonic operator, which is implemented here using a discontinuous Galerkin method to weakly impose continuity of the first derivatives.
To implement it using a standard Bubnov–Galerkin method (i.e., directly posing the weak form you’ve written over a finite dimensional function space, without the extra dS terms of the demo linked above), you would need a function space with C^1 basis functions. This is not directly supported in FEniCS, but I wrote a library to implement arbitrary piecewise polynomial bases on top of FEniCS, which includes the Euler–Bernoulli beam problem as a demo.
Thanks for your help, could not have wished for a better answer. I was stuck trying to use the standard Bubnov/Galerkin, and came to the conclusion that was not feasible from the function space available. I will keep this post updated and use the code of your library!
