# Smooth PDE¶

## Governing Equations¶

The SmoothPDE can be described as a quasi static mechanic. Hence, the difference to the governing equation of the Mechanic PDE is that the SmoothPDE is not time dependent nor under the influence of external forces, therefore $\frac{\partial^2 \rho \mathbf{u}}{\partial t^2}=0$ and $\mathbf{g}=0$ at the force density $\mathbf{f}=\nabla \cdot \sigma + \mathbf{g}$. With these assumptions, we obtain $$\nabla \cdot \mathbf{\sigma} = 0 \label{eq:govSmooth},$$ where $\mathbf{\sigma}$ is the stress tensor (rank 2). It has to be noted that $\mathbf{f}$ is a body force density which is exerted from the outside. We assume linear, elastic material behaviour which may be anisotropic and express it via Hooks Law which is given by $$\mathbf{\sigma} = \mathbf{C} : \mathbf{s}, \label{eq:stresstensor}$$ where $\mathbf{C}$ denotes the stiffness tensor (rank 4) and $\mathbf{s}$ the strain tensor (rank 2). Furthermore, we assume small strains in order to use the linearized strain displacement relationship given as $$\mathbf{s} = \frac{1}{2} ( \nabla\mathbf{u} + (\nabla\mathbf{u})^{\textrm{T}}). \label{eq:straintensor}$$ Multiplying \eqref{eq:govSmooth} with a test function $\mathbf{u}^\prime$ and inserting \eqref{eq:stresstensor} and \eqref{eq:straintensor} finally leads to the weak form of the PDE given by $$\frac{1}{2} \int_{\Omega} (\nabla \mathbf{u}^\prime) : \mathbf{C} : \left( \nabla \mathbf{u} + (\nabla \mathbf{u})^{\textrm{T}} \right) \mathrm{d}\Omega - \int_{\Gamma} (\mathbf{u}^\prime \cdot \mathbf{\sigma}) \cdot \mathbf{n} \mathrm{d}\Gamma~=~0.$$

## Boundary conditions¶

The boundary conditions can be given as

$$\mathbf{u}(t) = \mathbf{u}_{\textrm e}(t) \qquad \textrm{on } \Gamma_{\textrm e},$$
$$\mathbf{\sigma}(t)\cdot\mathbf{n} = \mathbf{\sigma}_{\textrm n}(t) \qquad \textrm{on } \Gamma_{\textrm n},$$

with $\mathbf{u}_{\textrm e}$ as the Dirichlet (essential) BC-value, $\Gamma_{\textrm e}$ the essential boundary, $\mathbf{n}$ the normal vector on the Neumann (natural) boundary $\Gamma_{\textrm n}$ and $\mathbf{\sigma}_{\textrm n}$ the traction at the boundary. Please note that all surfaces that are not manually defined will automatically be free of traction. The boundary conditions can be defined in the xml-file within the <bcsAndLoads> tag.

        <bcsAndLoads>
<fix name="">
<comp dof=""/>
</fix>
<displacement name="">
<comp dof="" value=""/>
</displacement>

• fix: Fixes the nodes of the given geometry where only the direction (e.g. "x", "y", "z") specified by the tag <comp dof =""> is locked. This allows to fix the geometry in one or more directions (homogeneous Dirichlet value $\mathbf{u} = \mathbf{0}$).
• displacement: Similar to <fix>, but instead of setting the value of the displacement to zero a certain value can be specified (inhomogeneous Dirichlet value $\mathbf{u} = \mathbf{u}_{\textrm e}$).

## Material¶

In the general case the material used for the simulation is anisotropic which can be defined in the material-xml as (same as in the Mechanic PDE). Due to the anisotropy, one can influence and control the grid deformation and thus the element quality.

  <material name="material2">
<mechanical>
<density>
<linear>
<real> 1.0e+3 </real>
</linear>
</density>
<elasticity>
<linear>
<tensor dim1="6" dim2="6">
<real>
1.34615E+03 5.76923E+02 5.76923E+02 0.00000E+00 0.00000E+00 0.00000E+00
5.76923E+02 1.34615E+03 5.76923E+02 0.00000E+00 0.00000E+00 0.00000E+00
5.76923E+02 5.76923E+02 1.34615E+03 0.00000E+00 0.00000E+00 0.00000E+00
0.00000E+00 0.00000E+00 0.00000E+00 3.84615E+02 0.00000E+00 0.00000E+00
0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 3.84615E+02 0.00000E+00
0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 3.84615E+02
</real>
</tensor>
</linear>
</elasticity>
</mechanical>
</material>


For this example, all 21 individual components of the elasticity tensor can be defined. For a simpler case, e.g. an isotropic material, the definition of the elasticity reduces to

<elasticity>
<linear>
<isotropic>
<elasticityModulus>
<real> 3e7 </real>
</elasticityModulus>
<poissonNumber>
<real> 0.2 </real>
</poissonNumber>
</isotropic>
</linear>
</elasticity>


where the stiffness tensor is defined via

$$\mathbf{C} = \begin{bmatrix} 2 \mu + \lambda & \lambda & \lambda & 0 & 0 & 0\\ \lambda & 2 \mu + \lambda & \lambda & 0 & 0 & 0\\ \lambda & \lambda & 2 \mu + \lambda & 0 & 0 & 0\\ 0 & 0 & 0 & \mu & 0 & 0\\ 0 & 0 & 0 & 0 & \mu & 0\\ 0 & 0 & 0 & 0 & 0 & \mu \end{bmatrix}.$$

Where $\mu$ and $\lambda$ are the Lam\'{e} constants given by

$$\mu = \frac{E}{2(1+\nu)}$$
$$\lambda = \frac{E \nu}{(1+\nu)(1-2\nu)}$$

has been used. In these equations, $E$ denotes the elasticity modulus and $\nu$ the poisson ration.

## Analysis Types¶

As analysis type the transient analysis can be used

• Transient: $\frac{\partial }{\partial t}\neq 0$
    <analysis>
<transient>
<numSteps>10</numSteps>
<deltaT>0.1</deltaT>
</transient>
</analysis>


## Postprocessing Results¶

#### Node Results¶

• Smooth Displacement: Displacement vector for each node. (primary solution)
<nodeResult type="smoothDisplacement">

• Smooth Velocity: Velocity vector for each node. $$\mathbf{v} = \frac{\partial \mathbf{u}}{\partial t}$$
<nodeResult type="smoothVelocity">

• Smooth Acceleration: Acceleration vector for each node. $$\mathbf{a} = \frac{\partial^2 \mathbf{u}}{\partial t^2}$$
<nodeResult type="smoothAcceleration">

• Smooth ZeroStress: nodal zeroStress The SmoothZeroStress is for test purposes to define a fictive back coupling which does not have any influence.
<nodeResult type="smoothZeroStress">