Test of numerical minimization package for theshape optimization of a paper making machineheader
Jan Stebel
July 16, 2007

1

Description of the model

We are interested in ﬁnding an optimal shape of the back wall of a paper
making machine header (see Figure 1). The header is the ﬁrst component ina paper making machine headbox. Its function is to deliver the ﬂuid (water)
and wood ﬁbres equally in the cross direction of a paper making machinein order to produce good quality paper. The cost function to be minimized
reads:
J(α, v(α), p(α)) :=|v2 (α) − vopt |2 ,
Γoutwhere α is a function describing the shape of the back wall, (v(α), p(α)) isthe velocity and the pressure of the mixture and vopt is a given target velocity
on the outlet Γout .
The header domain Ω(α) is of the form2Ω(α) = x = (x1 , x2 ) ∈ R ; 0 < x1 < L, 0 < x2 < α(x1 ) .

H1ΓDΓαα(x1 )
Γout

L1

L2

ΓD
L3

Figure 1: Geometry of the header Ω(α).
1H2We say that Ω(α) is an admissible domain iﬀ α ∈ Uad , where
Uad = α ∈ C 0,1 ([0, L]); αmin ≤ α ≤ αmax ,
α|[0,L1 ] = H1 , α|[L1 +L2 ,L] = H2 , |α | ≤ γ a.e. in [0, L] , (1)and L := L1 + L2 + L3 .
The motion of the mixture is modelled using the generalized Navier-Stokessystem− div T(p, D(v)) + ρ div(v ⊗ v) = 0in Ω(α).
(2)
div v = 0Here v := v(α) means the velocity, p := p(α) the pressure, ρ is the densityof the ﬂuid and the stress tensor T is deﬁned by the following formulae:T(p, D(v)) = −pI + 2µ(|D(v)|)D(v),
2|D(v)|, µ0 > 0,µ(|D(v)|) := µ0 + µt (|D(v)|) = µ0 + ρlm,αwhere µ0 is a constant laminar viscosity and µt (|D(v)|) stands for a turbulent
viscosity. The function lm,α represents a mixing length in the algebraic modelof turbulence and it has the following form:2
4 12dα (x)2dα (x)lm,α (x) = α(x1 ) 0.14 − 0.08 1 −− 0.06 1 −,2α(x1 )α(x1 )
where dα (x) = min {x2 , α(x1 ) − x2 } , x ∈ Ω(α). The following boundary
conditions are assumed:vvv · τ = v1
T22 := Tν · ν====0vD0−σ|v2 |v2

on Γf ∪ Γα ,on ΓD ,on Γout ,on Γout ,(3)where ν, τ stands for the unit normal, tangential vector to Γout , respectively
and σ > 0 is a given suction coeﬃcient. The condition (3)4 originates in thehomogenization of a complex geometry.

2

Approximation and test results

The state problem (2) is discretized using the ﬁnite element method on atriangular mesh. In order to compute the gradient of J with respect toα, the adjoint equation technique is applied to the discrete state problem.
2Domain shapes
1optimalinitial

0.8
0.6
0.4
0.2
01

2

34

56

78

9

Figure 2: Initial and optimal shape.
All necessary partial derivatives are provided with help of the automatic
diﬀerentiation. The function α is approximated by a Bézier function αM oforder M. Admissibility of αM is imposed by simple lower and upper bounds
only.
Finally, our problem can be formulated as a nonlinear, bounds-constrained
programming problem for which the cost function value and gradient areavailable. We used the NAG C library, routine e04wdc [1] to solve thisproblem, using the default parameter values.
In the test computation we used the following dimensionless parameters:
L1 = 1.0, L2 = 8.0, L3 = 0.5, H1 = 1.0, H2 = 0.1, αmin = H2 , αmax = H1 ,2µ0 = 10−3 , ρ = 103 , σ = 103 , the inlet velocity v D|{0}×(0,H1 ) = (4(1 − ( 2x−H12x2881) ), 0), v D|{L}×(0,H2 ) = (1 − ( H2 − 1) , 0). The order of the Bézier function
αM was set M = 20, and the target velocity vopt := −0.433.
The optimization started from the traditional linearly tapering shape.
The initial and optimal shape (found by e04wdc) is depicted in Figure 2 andthe initial and optimal velocity proﬁle v2|Γout is depicted in Figure 3.
The NAG routine found the solution after 73 major iterations, yielding
the optimality error smaller than 10−6 .

3Outlet velocity proﬁles1
initialoptimal
target

0.8
0.6
0.4
0.2
01234

56789Figure 3: Initial and optimal velocity proﬁle.

3Conclusion

The NAG optimization routine shows itself to fully meet the demands of theproblem.

References[1] P. E. Gill, W. Murray, and M. A. Saunders. SNOPT: An SQP Algorithm
for Large-scale Constrained Optimization. SIAM J. Optim., 12:979-1006,
2002.

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