Define boundary conditions in Hinged Euler–Bernoulli Beam

[LiutianZhang]

Thank you for your helpful work! I tried to be familiar with the package by a simple Euler–Bernoulli Beam. Hinged boundary conditions are used at both ends, which means $y(0,t)=0$, $y(L,t)=0$, $\frac{\partial^2y(0,t)}{\partial x^2}=0$ and $\frac{\partial^2y(L,t)}{\partial x^2}=0$. Let $x_1 = y$ and $x_2=\frac{\partial y}{\partial t}$, boundary conditions are defined by 'bc_ops', but it shows that 'Unused BCs: ['x1:*']', how can I implement such boundary conditions in this package?

$$m \frac{\partial^2 y(x,t)}{\partial t^2}+ c \frac{\partial y(x,t)}{\partial t}+ EI\frac{\partial^2}{\partial x^2}\left(\frac{\partial^2y(x,t)}{\partial x^2}\right)=0$$

from pde import PDE, FieldCollection, ScalarField, CartesianGrid
import math

m = 1
c = 0.05
EI = 1
eq = PDE(
    {"x1": f"x2",
      "x2": f"-{c/m}*x2-{EI/m}*laplace(laplace(x1))"
    },
    bc_ops=
    {
        'x1:*': {"curvature": 0},
        'x1:*': {"value": 0}
    }
)


# initialize state
xnum = 50
L = 1
grid = CartesianGrid([[0,L]],xnum)

statex1 = ScalarField.from_expression(grid, f"0.1*sin({math.pi / L}*x)")
statex2 = ScalarField.from_expression(grid, f"0")

state = FieldCollection([statex1, statex2])

# simulate the pde
t_interval = 1e-3
trackers = [
"progress",
# show progress bar during simulation
]
tmax = 1
sol = eq.solve(state, t_range=tmax, dt=1e-5, tracker = trackers, solver = 'implicit')

[david-zwicker]

Well, the current approach cannot work since you use the same key twice in the dictionary for the bc_ops argument. Are you trying to provide different BCs for the two different laplace operators in the definition of the rate equation for x2?

[LiutianZhang]

Thank you. I tried to define different BCs for x1 and x2, but it still show that 'Unused BCs: ['x1:*']', i don't know when a defined BC will be ignored in the package.

    bc_ops=
    {
        'x1:*': {"curvature": 0},
        'x2:*': {"value": 0}
    }

[david-zwicker]

The BC for 'x1:*' defines boundary conditions in the right-hand side of the evolution rate of x1. However, since there are no differential operators in this right hand side, the boundary condition cannot used. The package warns you here to prevent mistakes.

[LiutianZhang]

Thank you for your answer!