Control of heating power in ATP with virtual charts

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Control of heating power in ATP with virtual charts Nicolas Bur, Pierre Joyot, Pierre Villon, Francisco Chinesta

To cite this version:

Nicolas Bur, Pierre Joyot, Pierre Villon, Francisco Chinesta. Control of heating power in ATP with virtual charts. 12e colloque national en calcul des structures, May 2015, Giens, France. �hal-01178841�

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Giens 2015

Control of heating power in ATP

with virtual charts

N. Bur

(a,?)

, P. Joyot

(a)

, P. Villon

(b)

, F. Chinesta

(c)

(a) ESTIA-Recherche, Technopôle Izarbel, 64210 Bidart, France

(b) UTC, Département GSM, Centre de Recherches de Royallieu, CS 60319, 60203 Compiègne cedex, France (c) GeM Institute, École Centrale de Nantes, 1 rue de la Noë, 44321 Nantes cedex 3, France

(?)n.bur@estia.fr

Problem

Using thermoplastic composites within an Automated Tape Placement process requires controlling drastically the temperature in order to heat the matrix enough to melt it, but not too much to avoid burning. Thus, to reach the optimal temperature within a given zone, we want defining the power of the source by

• integrating in the robot’s control loop • the temperature field

• computed for many values of different parameters

Method

Based on separated representation of the unknown, the Proper Generalised Decomposi-tion (PGD) method allows to handle param-eters as extra-coordinates, leading to virtual charts, without incurring the curse of dimen-sionality.

For example, instead of computing u on a 2D domain (x, z), at each time step (t) and for any value of a parameter (k), we write the unknown under the form

u(x, z, t, k) = X

α

Xα(x)Zα(z)T α(t)Kα(k). Then, assuming first functions are known, the approximation is enriched with a new mode XZT K within a greedy algorithm:

un+1 = n+1_X α=1 Xα(x)Zα(z)T α(t)Kα(k) = n X α=1 Xα(x)Zα(z)T α(t)Kα(k) | {z }

known from initialisation or previous computations + + X(x)Z(z)T (t)K(k) | {z } to be computed

# of plies as coordinate

• Unitary domain in the thickness, discre-tised with the Least Common Multiple be-tween the numbers of layers

• Double nodes at each possible interface

• Temperature continuity on fake interfaces (depending on # of plies) enforced with Nitsche’s method 0 1/5 1/4 1/3 2/5 1/2 3/5 2/3 3/4 4/5 1 1 pli C C C C C C C C C 2 plis C C C C RTC C C C C 3 plis C C RTC C C C RTC C C 4 plis C RTC C C RTC C C RTC C 5 plis RTC C C RTC C RTC C C RTC 1 ply 2 plies 4 plies _{5 plies}

Multi-parametric virtual charts

For a given trajectory, the robot’s velocity is known: we want to determine the best asso-ciated power to ensure the optimal temperature. We assume a static solution is enough when acceleration is small, using a transient model otherwise.

Separating space, # of plies, power and Separating space, # of plies, time, power,

speed: speed and acceleration:

u = P XZP QV u = P XZP T QCV Γ 0 µ_{− r} µ µ + r L 0 qmin q qmax x Φ t v v_min vmax v t v v +γ(t −t0) γ v q q + c₍_{t − t} 0) q c t₀

Source and velocity parametrisation Power and speed parametrisation

Running the PGD algorithm leads to virtual charts depicted below

0 0.2 0.4 0.6 0.8 1 1.2 2 000 4 000 6 000 8 000 10 000 12 000 Velocity m_·s−1 P ow er (W ) uopt = 400 K uopt = 500 K uopt = 600 K

Process window when considering 8 plies

Within the control zone, for a given

p, v and Γ are known. Then q is

com-puted with static chart. The optimum

c is solution of a minimisation problem

0 ₅ _{· 10}−2 _0.1 _0.15 _0.2 _0.25 _0.3 _0.35 _0.4 _0.45 300 320 340 360 380 400 420 time of interest optimum determined by min_c q (u _{− u}_opt)2 time (s) T emp er ature (K) uopt c = 10 000 W_·s−1 c = 20 000 W·s−1 c = 28 760 W_·s−1 c = 35 000 W·s−1

Choosing the power evolution

This research is part of the Impala project, which is a FUI 11 project, funded by Conseil régional d’Aquitaine.

Numerical results

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 280 300 320 340 360 380 400 uopt ± 5 K Time (s) T emp eratu re (K) V elo cit y× 100 + 280 m ·s − 1 3 000 4 000 5 000 6 000 7 000 8 000 P ow er (W ) Velocity Temperature Temperature – static Power Temperature

evo-lution within the

control zone, using

only static chart

(purple dash-dotted line) or both static and transient (plain green line)