Manufactured Solution Q4 dynamic - GCMLab/GCMLab-FEM GitHub Wiki
Manufactured solution
This test calculates the response for a plane stress manufacture solution in which the displacements in the $x$ and $y$ directions are
$$ u(x, y, t) = - \cfrac{1}{1000} \sin{\left( \cfrac{1}{2} \pi x \right) } \sin{\left(\cfrac{1}{2}\pi y\right)} \sin{\left( 2\pi t\right)} $$
$$ v(x, y, t) = \cfrac{1}{1000} \cos{\left( \cfrac{1}{2} \pi x \right) } \cos{\left(\cfrac{1}{2}\pi y\right)} \cos{\left( 2\pi t\right)} $$
By substituting $u(x, y, t)$ and $v(x, y, t)$ into the governing equations
$$ \nabla \cdot \boldsymbol{\sigma} + \textbf{b} = \rho \ddot{\textbf{u}}, $$
where
$$ \boldsymbol{\sigma} = \textbf{D} \boldsymbol{\varepsilon}, $$
so that the body forces of the system are obtained as:
$$ b_x(x, y, t) = \frac{\pi^2}{8000(\nu^2 - 1)} \sin\left(\frac{1}{2} \pi x\right) \sin\left(\frac{1}{2} \pi y\right) \left[\left(32\rho(\nu^2 - 1) + E(3 - \nu ) \right) \sin(2\pi t) + E\left(\nu + 1 \right) \cos(2\pi t)\right] $$
$$ b_y(x, y, t) = \frac{\pi^2}{8000(\nu^2 - 1)} \cos\left(\frac{1}{2} \pi x\right) \cos\left(\frac{1}{2} \pi y\right) \left[\left(32\rho( 1 -\nu^2 ) + E( \nu - 3 ) \right) \cos(2\pi t) - E\left( \nu + 1 \right) \sin(2\pi t)\right] $$
Geometry and Mesh
The domain is a $1 \times 1 ~\text{m}^2$ square. The element type is Q4. Three uniform meshes where used:
Mesh 1: 2x2
Mesh 2: 10x10
Mesh 3: 20x20
Material Properties
| Parameter | Symbol | Value |
|---|---|---|
| Young's Modulus | $E$ | $200~GPa$ |
| Poisson's Ratio | $\nu$ | $0.3$ |
| Density | $\rho$ | $2400$ $kg/m^3$ |
Plane stress conditions applied.
Simulation Properties
| Parameter | Value |
|---|---|
| Start Time ($t_{0}$) | $0~s$ |
| End Time ($t_{end}$) | $3.123~s$ |
| Number of time steps | 100 |
| $\alpha $ | $-\cfrac{1}{3}$ |
Boundary and Initial Conditions
The boundary conditions are fixed at their manufactured solution values on all boundaries (top, bottom, left, and right edges).
The initial conditions are prescribed as zero at all but the external nodes where the boundary conditions are set according to the manufacture solution.
Results for Mesh 3
Magnitude of displacements
/Verification/images/mesh_3_displacement.gif
Displacements on $x$
/Verification/images/mesh_3_displacement_X.gif
Displacements on $y$
/Verification/images/mesh_3_displacement_Y.gif
Warped system (factor = 100)
/Verification/images/mesh_3_distorted.gif
Displacements across the domain along the line $y = 0.51~m$
/Verification/images/mesh_3_line.gif
