Steel Profiles (38881)
A Semidiscretized Heat Transfer Problem for Optimal Cooling of Steel Profiles (38881)
Summary
Several generalized statespace models arising from a semidiscretization 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.
The 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 precalculate 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 [Saa03] or [BS04]. 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 Ω⊂R^{2} is chosen as the half of a cross section of the rail profile. The heat distribution is modeled by the instationary linear heat equation on Ω
cρ ∂_{t} x(t,ξ)  λΔx(t,ξ) = 0 in R_{>0} × Ω,  
x(0,ξ) = x_{0}(ξ) in Ω,  (1) 
λ ∂_{ν}x(t,ξ) = g_{i} on R_{>0}× Γ_{i}, ∂Ω = ∪_{i} Γ_{i} 
where x is the temperature distribution (∈H^{1}([0,∞],X) with X:=H^{1}(Ω) the state space), c the specific heat capacity, λ the heat conductivity and ρ the density of the rail profile. We split the boundary into several parts Γ_{i} on which we have different boundary functions g_{i}, allowing us to vary controls on different parts of the surface. By ν 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
is minimized. Here, Q and R are linear selfadjoint operators on the output space Y and the control space U with Q≥0, R>0 and C∈ L(X,Y). The variational formulation of (1) with g_{i}(t,ξ) = q_{i}(u_{i} x(ξ,t)) leads to:
(∂_{t}x,v)=∫_{Ω} α∇x ∇v dξ + ∑_{k} { q_{k} u_{k} ∫_{Γk} (cρ)^{1} v dσ ∫_{Γk} q_{k}(cρ)^{1} xv dσ}  (3) 
for all v∈C_{0}^{∞}(Ω). Here the u_{k} are the exterior (cooling fluid) temperatures used as the controls, q_{k} are constant heat transfer coefficients (i.e. parameters for the spraying intensity of the cooling nozzles)^{[2]} and α:=λ/(cρ). 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 firstorder ordinary differential equation. This idescribed in the following section.
The discretized mathematical model
For the discretization we use the ALBERTA1.2 femtoolbox (see [SS00] for details). We applied linear Lagrange elements and used a projection method for the curved boundaries. The initial mesh (see Fig. 1) 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
This benchmark includes four different mesh resolutions. The best femapproximation 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
matrix dimension  num. nonzero in A  num. nonzero in E  max. mesh width 
File 1 
8985  8997  5.5280 10^{2} 
File 2 
35185  35241  2.7640 10^{2} 
File 3 
139233  139473  1.3820 10^{2} 
File 4 
553921  554913  6.9100 10^{3} 
Note that A is negative definite while E is positive definite, so that the resulting linear timeinvariant 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 Fig 1.) 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 Figure 1. for the nodes used. the definitions of other output matrices that we tested can be found in [Saa03]. The problem resides at temperatures of approximately 1000 degrees centigrade down to about 500700 degrees depending on calculation time. The state values are scaled to 1000 being equivalent to 1.000. 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.
Acknowledgements
This benchmark example serves as a model problem for the project A15: Efficient numerical solution of optimal control problems for instationary convectiondiffusionreactionequations 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 [TU01]. A very similar problem is used as model problem in the LYAPACK software package [Pen00b].
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