Vertical Stand

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1 Description

Figure 1: CAD Geometry

The vertical stand (see xx--CrossReference--dft--fig:cad--xx) represents a structural part of a machine tool. On one of its surfaces a pair of guide rails is located. Caused by a machining process a tool slide is moving on these rails. The machining process produces a certain amount of heat which is transported through the structure into the vertical stand. This heat source is considered to be a temperature input at the guide rails. The induced temperature field, denoted by $T$ is modeled by the heat equation

$c_p\rho\frac{\partial{T}}{\partial{t}}=\nabla.(\lambda\nabla T)=0$

with the boundary conditions

$\lambda\frac{\partial T}{\partial n}=f \qquad\qquad\qquad$ on $\Gamma_{slide}$ (surface where the tool slide is moving on the guide rails),

describing the heat transfer between the tool slide and the vertical stand. The heat transfer to the ambiance is given by the fixed Robin-type boundary condition

$\lambda\frac{\partial T}{\partial n}=g_i=\kappa_i(T-T_i^{ext})$ on $\Gamma_{surf}$ (remaining boundaries).

The motion driven temperature input and the associated change in the temperature field $T$ lead to deformations $u$ within the stand structure. Since the mechanical behavior of the machine stand is assumed to be much faster than the propagation of the thermal field, it is sufficient to consider the stationary linear elasticity equations

$\begin{align} -\operatorname{div}(\sigma(u)) &= f&\text{ on }\Omega,\\ \varepsilon(u) &= {\mathbf{C}}^{-1}:\sigma(u)+\beta(T-T_{ref})I_d&\text{ on }\Omega,\\ {\mathbf{C}}^{-1}\sigma(u) &=\frac{1+\nu}{E_u}\sigma(u)-\frac{\nu}{E_u}\text{tr}(\sigma(u))I_d&\text{ on }\Omega,\\ \varepsilon(u) &= \frac{1}{2}(\nabla u+\nabla u^T)&\text{on }\Omega \end{align}$

1.1 Discretized Model

The solid model has been generated and meshed in ANSYS. For the spatial discretization the finite element method with linear Lagrange elements has been used and is implemented in FEniCS. The resulting system of ordinary differential equations (ODE), representing the thermal behavior of the stand, reads

$\begin{align} E \frac{\partial}{\partial t} T(t) &= A(t)T(t) + B(t)z(t), \\ T(0) &= T0, \end{align}$

with $t>0$ and a system dimension of $n=16\,626$ degrees of freedom and $m=6$ inputs. Note that $A(.)\in\mathbb{R}^{n\times n}$ and $B\in\mathbb{R}^{n\times m}$ are time-dependent matrix-valued functions. More precisely, here the time dependence origins from the change of the boundary condition on $\Gamma_{slide}$ due to the motion of the tool slide. The system input is, according to the boundary conditions, given by

$z_i=\begin{cases} f, i=1,\\ g_i, i=2,\dots,6 \end{cases}.$

The discretized stationary elasticity model becomes

$Mu(t)=KT(t).$

For the observation of the displacements in single points/regions of interest an output equation of the form

$y(t) = C u(t)$

is given.

Exploiting the one-sided coupling of the temperature and deformation fields, and reorganizing the elasticity equation in the form $u(t)=M^{-1}KT(t)$, the heat equation and the elasticity model can easily be combined via the output equation. Finally, the thermo-elastic control system is of the form

$\begin{align} E \frac{\partial}{\partial t} T(t) &= A(t)T(t) + B(t)z(t), \\ y(t)& = \tilde{C}T(t),\\ T(0) &= T_0, \end{align}$

where the modified output matrix $\tilde{C}=CM^{-1}KT(t)$ includes the entire elasticity information. Geometrical dimensions:

Stand: Width ($x$ direction): $519mm$, Height ($y$ direction): $2\,010mm$, Depth ($z$ direction): $480mm$

Guide rails: $y\in [519, 2\,004] mm$

Slide: Height $500mm$

The heat load $q$ induced by the slide and the external temperature $x_{ext}$ serve as the input $u$ of the corresponding state-space system.

The motion of the tool slide and the associated variation of the affected input boundary are modeled by two different system representations. The following specific model representations have been developed and investigated in , .

1.2 Switched linear system

File:Slide stand scheme.pdf

Figure 2: Schematic segmentation

For the model description as a switched linear system, the guide rails of the machine stand are modeled as 15 equally distributed horizontal segments with a height of $99mm$ (see a schematic depiction in xx--CrossReference--dft--fig:segm--xx). Any of these segments is assumed to be completely covered by the tool slide if its midpoint (in y-direction) lies within the height of the slide. On the other hand, each segment whose midpoint is not covered is treated as not in contact and therefore the slide always covers exactly 5 segments at each time. This in fact defines 11 distinct, discrete boundary condition configurations for the stand model that are prescribed by the geometrical dimensions of the segmentation and the tool slide. These distinguishable setups define the subsystems

$\begin{align} E_{th}\dot{T}&=A_{th}^{\alpha}T+B_{th}^{\alpha}z,\\ y&=\bar{C}T, \end{align}$

of the switched linear system , where $\alpha$ is a piecewise constant function of time, which takes its value from the index set $\mathcal{J}=\{1,\dots,11\}$. To be more precise, the switching signal $\alpha$ implicitly maps the slide position to the number of the currently active subsystem.

2 Acknowledgement & Origin

The base model was developed , in the Collaborative Research Centre Transregio 96 Thermo-Energetic Design of Machine Tools funded by the Deutsche Forschungsgemeinschaft .

3 Data

3.2 Parametric System Data

The data file Data_VertStand.tar.gz contains a MAT_File matrices.mat which consists of the matrices

$E,A\in\mathbb{R}^{n\times n},B_{slide}\in\mathbb{R}^{n\times 1},B_{surf}\in\mathbb{R}^{n\times 5}, n=16\,626$

in sparse format and a file with the coordinates of the mesh nodes called coord.txt.

Here $B_{slide}$ consists of all nodes located on the guide rails.

In order to get a parameter dependent matrix $B_{slide}(\mu)$ one has to pick the "active" nodes (nodes hit by tool carriage) at vertical position $\mu$. The "active" nodes are in the interval of $[\mu-\frac{d}{2},\mu+\frac{d}{2}]$, where $d$ is the heigth of the slide.

The file coord.txt provided in Data_VertStand.tar.gz includes a column with indices followed by three additional columns containing the spatial coordinates $x,y,z$ of the corresponding nodes.

The matrix $B_{surf}$ describes the locations where the external temperatures act on. The first column is responsible for the input of the temperature at the clamped bottom slice of the structure. Column 2 describes the ... part of the stand. Columns 3 to 5 describe different thresholds with respect to the height of ambient air temperature. The third column includes the nodes of the lower third $(y\in[0,670)mm)$ of the stand. In column 4 all nodes of the middle third $(y\in[670,1\,340)mm)$ of the geometry are contained and the fifth column of $B_{surf}$ includes the missing upper $(y\in[1\,340,2\,010]mm)$ part.

4 Citation

To cite this benchmark, use the following references:

• For the benchmark itself and its data:

The MORwiki Community. Vertical Stand. MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/Vertical_Stand

   @MISC{morwiki_vertstand,
    author = {The {MORwiki} Community},
    title = {Vertical Stand},
    howpublished = {{MORwiki} -- Model Order Reduction Wiki},
    url = {http://modelreduction.org/index.php/Vertical_Stand},
    year = {2014}
   }

• For the background on the benchmark:

   @Article{morLanSB14,
     author =       {Lang, Norman and Saak, Jens and Benner, Peter},
     title =        {Model Order Reduction for Systems with Moving Loads},
     journal =      {at-Automatisierungstechnik},
     year =         2014,
     volume =       62,
     number =       7,
     pages =        {512--522},
     month =        {June},
     publisher =    {deGruyter},
     doi =          {10.1515/auto-2014-1095}
   } 

6 Contact

Jens Saak