Steel Profile

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Description: A Semi-discretized Heat Transfer Problem for Optimal Cooling of Steel Profiles

 

Figure 2

A Semi-discretized heat transfer problem for pptimal cooling of steel profiles. Several generalized state-space models arising from a semi-discretization of a controlled heat transfer process for optimal cooling of steel profiles are presented. The models oder differs due to different refinements applied to the computational mesh.

Model Equations

We consider the problem of optimal cooling of steel profiles. This problem arises in a rolling mill when different steps in the production process require different temperatures of the raw material. To achieve a high production rate, economical interests suggest to reduce the temperature as fast as possible to the required level before entering the next production phase. At the same time, the cooling process, which is realized by spraying cooling fluids on the surface, has to be controlled so that material properties, such as durability or porosity, achieve given quality standards. Large gradients in the temperature distributions of the steel profile may lead to unwanted deformations, brittleness, loss of rigidity, and other undesirable material properties. It is therefore the engineers goal to have a preferably even temperature distribution.

The scientific challenge here is to give the engineers a tool to pre-calculate different control laws yielding different temperature distributions in order to decide which cooling strategy to choose.

We can only briefly introduce the model here for details we refer to or . We assume an infinitely long steel profile so that we may restrict ourselves to a 2D model. Exploiting the symmetry of the workpiece, the computational domain $\Omega \subset \mathbb{R}^2$ is chosen as the half of a cross section of the rail profile. The heat distribution is modeled by the unsteady linear heat equation on $\Omega$:

$\begin{align} c \rho \partial_t x(t,\chi) - \lambda \Delta x(t,\chi) &= 0 \in \mathbb{R}_{>0} \times \Omega \\ x(0,\chi) &= x_0(\chi) \in \Omega \\ \lambda \partial_\nu x(t,\chi) &= g_i \in \mathbb{R}_{>0} \times \gamma_i, \partial \Omega = \bigcup_i \Gamma_i \end{align}$

where $x$ is the temperature distribution $(\in H^1([0,\infty],X)$ with $X:=H^1(\Omega)$ the state space), $c$ the specific heat capacity, $\lambda$ the heat conductivity and $\rho$ the density of the rail profile. We split the boundary into several parts $\Gamma_i$ on which we have different boundary functions $g_i$, allowing us to vary controls on different parts of the surface. By $\nu$ we denote the outer normal on the boundary.

We want to establish the control by a feedback law, i.e. we define the boundary functions $g_i$ to be functions of the state $x$ and the control $u_i$, where $(u_i)_i =: u = Fy$ for a linear operator $F$ which is chosen such that the cost functional

$J(x_0,u) := \int_0^\infty (Qy,y)_Y + (Ru,u)_U \operatorname{d}t$

with $y=Cx$ is minimized. Here, $Q$ and $R$ are linear self-adjoint operators on the output space $Y$ and the control space $U$ with $Q \geq 0$, $R > 0$ and $C \in L(X,Y)$. The variational formulation of (1) with $g_i(t,\chi) = q_i(u_i- x(\chi,t))$ leads to:

$(\partial_t x,v) = -\int_\Omega \alpha \nabla x \nabla v \operatorname{d}\chi + \sum_k \Big(q_k u_k \int_{\Gamma_k} (c \rho)^{-1} v \operatorname{d}\sigma - \int_{\Gamma_k} q_k(c\rho)^{-1} xv d\sigma\Big)$

for all $v \in C_0^\infty(\Omega)$. Here the $u_k$ are the exterior (cooling fluid) temperatures used as the controls, <ath>q_k</math> are constant heat transfer coefficients (i.e. parameters for the spraying intensity of the cooling nozzles) [2] and $\alpha := \lambda /(c\rho)$. Note that $q_0 = 0$ gives the Neumann isolation boundary condition on the artificial inner boundary on the symmetry axis. In view of (3) we can apply a standard Galerkin approach for discretizing the heat transfer model in space, resulting a first-order ordinary differential equation. This is described in the following section.

Discretized Model

For the discretization we use the ALBERTA-1.2 fem-toolbox (see for details). We applied linear Lagrange elements and used a projection method for the curved boundaries. The initial mesh (see xx--CrossReference--dft--fig1--xx) was produced by MATLABs pdetool which implements a Delaunay triangulation algorithm. The finer discretizations were produce by global mesh refinement using a bisection refinement method. The discrete LQR problem is then: minimize (2) with respect to:

$\begin{align} E \frac{\partial}{\partial x} x(t) &= A x(t) + B u(t), \\ y(t) &= C x(t), \\ x(0) &= x0, \end{align}$

with $t>0$.

Acknowledgements

This benchmark example serves as a model problem for the project A15: Efficient numerical solution of optimal control problems for instationary convection-diffusion-reaction-equations of the Sonderforschungsbereich SFB393 Parallel Numerical Simulation for Physics and Continuum Mechanics, supported by the Deutsche Forschungsgemeinschaft. It was motivated by the model described in . A very similar problem is used as model problem in the LYAPACK software package .

Origin

This benchmark is part of the Oberwolfach Benchmark Collection; No. 38881, see .

Data

This benchmark includes four different mesh resolutions. The best FEM-approximation error that one can expect (under suitable smoothness assumptions on the solution) is of order $O(h^2)$ where $h$ is the maximum edge size in the corresponding mesh. This order should be matched in a model reduction approach. The following table lists some relevant quantities for the provided models:

	# nonzero in A	# nonzero in E	max. mesh width
rail1357.zip (95kB)	8985	8997	5.5280 10-2
rail5177.zip (299kB)	35185	35241	2.7640 10-2
rail20209.zip (1011kB)	139233	139473	1.3820 10-2
rail79841.zip (3.7MB)	553921	554913	6.9100 10-3

Note that $A$ is negative definite while $E$ is positive definite, so that the resulting linear time-invariant system is stable.

The data sets are named rail_(problem dimension)_C60.(matrix name). Here C60 is a specific output matrix which is defined to minimize the temperature in the node numbered 60 (refer to the numbers given in xx--CrossReference--dft--fig1--xx) and keep temperature gradients small. The latter task is taken into account by the inclusion of temperature differences between specific points in the interior and reference points on the boundary, e.g. temperature difference between nodes 83 and 34. Again refer to xx--CrossReference--dft--fig1--xx for the nodes used. The definitions of other output matrices that we tested can be found in . The problem resides at temperatures of approximately 1000 degrees centigrade down to about 500-700 degrees depending on calculation time. The state values are scaled to 1000 being equivalent to 1000. This results in a scaling of the time line with factor 100, meaning that calculated times have to be divided by 100 to get the real time in seconds.

Dimensions

System structure:

$\begin{align} E\dot{x}(t) &= Ax(t) + Bu(t) \\ y(t) &= Cx(t) \end{align}$

System dimensions:

$E \in \mathbb{R}^{N \times N}$, $A \in \mathbb{R}^{N \times N}$, $B \in \mathbb{R}^{N \times 7}$, $C \in \mathbb{R}^{6 \times N}$

System variants:

rail1375: $N = 1375$, rail5177: $N = 5177$, rail20209: $N = 20209$, rail79841: $N = 79841$