# Fenics: Result of Steady state dynamic linear elastic doesn't match with actual values

I solved the steady state dynamic linear elastic model in a solid. My equation is a function of frequency and the strong form is: $$\operatorname{div}(\operatorname{stress}(\vec x, w)) + w^2 \rho u( \vec x,w)=0$$ where $\rho$ is the density.

The BC is $$\sigma(\vec x,w) \cdot n(x)=T(x,w), \qquad u(x,w)=U_0$$

The physical problem is a plate with dimension of $1 \times 1 \times 0.1$ with a harmonic load on the all the top and also the bottom of the plate is fixed. I solved this as a test with the fixed frequency $(w=200)$.

The weak form of the this equation is: $$\int \sigma(x,w)\cdot \operatorname{grad}(v)\, dv - \rho w^2 \int v(x)\cdot u(x,w) \, dv - \int v\cdot T(x,w)\, ds(2) = 0$$

where $v$ is test function, $ds(2)$ is the top of the plate.

I compared the results with the model with converged mesh from Abaqus, I checked almost everything in the code the order of the displacement is mostly correct but the signs of the displacements are wrong and also the error is large between the solution of the Abaqus and this code.

I checked the variation form and I can't find any error in it. So the only thing that I suspected to be wrong is the Neumann boundary condition but I can't find any bug in it.

I would be very thankful if someone can help me to figure out what is wrong in my code that I get different answers from the code. Or I am not sure if this is a bug and my code is right.

• Your code matches PDE+BCs given when assuming U0=0 on bottom, T=f on top and T=0 elsewhere. Did you set f[1] = Magnitude < 0 intentionally? – Jan Blechta May 12 '13 at 18:57
• I just notice this comment about my post.yes, by setting Magnitude<0 I meant to make the T.ns which make a negative number since the direction of the traction is (-). I am not sure why I am receiving different answers from Abaqus since the code is not complicated. can it be because of the bug in library? – Bahram May 22 '13 at 17:30
• I would say that direction of displacement seems correct. – Jan Blechta May 24 '13 at 10:44
• when I changed the sign of the f[1] to a positive number then I saw that the direction of the load change in plot. but when the magnitude is negative the displacement plot make sense since the direction of the displacement in toward down. – Bahram May 24 '13 at 19:51
• So where's the problem? You said that direction of displacement is different from one obtained by Abaqus but now we agreed that this is correct. – Jan Blechta May 24 '13 at 20:06