optimisation et simulation numérique du chauffage par ... - EPFL

Abstract
In t,his work, we are interested iri tlie optimal coritrol of tlie inrluctiori heatiiig pro-
cess. This problerri has beeri posed by our iridustrial partrier Alcari withiri the frarrie-
work of ari iritlustrial process of manufacture of aluniiriiurri parts, called tliixoforrnirig.
Duririg the rriaiiufacture, one Iieats by iriciuctiori an aliirrii~iiiirri alloy billet uritil a par-
ticular serni-solid state, called thixotropic state, before injcctirig it, uritler pressure, iri a
mould. Tlie quality of the part obtairied clepends directly on ttie qiiality of the heatirig.
The billet rriust, have t,he rriost. liorriogerieous possihle t.erriperat,itre corresporidirig to
apprmirriately 50 % of solid fraction to guararitee the coriforrriity of the niouldirig.
Thus, it is very sigriificaiit to be able to optirriize the coritrol pararrieters of the iri-
ductiori heatirig iri order to achieve this goal. Otlier pararrieters can also be takeri irito
accouiit to guararitee ttie qualit,y of tlie heat,irig. The coiitrol variables are the frequency
aiid tlie terisiori (or power) siipplirtl to tlie iritlrictor. More precisely, we seek a power
wliose arriplitude varies in tirrie.
We use riurnerical analysis techniques to solve the optirrial coritrol problern. The
whole proceedirig plienonieria, during the heating, are described by non-liriear partial
derivative equatioris. We work withiii a two-dirrierisiorial frarriework. We use t,wo riiirrierical
rnodels allowirig t,o obtairi the rriaguetic field h arid the entlialpy u iti talie piece.
From physical corisitlerations, we huild ari objective functiori J ctiaracteristic of the
control wliicli orle warits to carry out. By a perturbatioris rriethod, we tlieri seek to calculate
the gradierit of tlie cost fuiictiori followirig the coritrol paranieters. This tecliriiqiie
bririgs 11s to tlie calciilatiori of two associated acljoirit problerris, orle for electrorriagrietisrri,
the other for thermies. These problerns heirig solvetl, it is pomible to evaliiate the
gradierit of the ot>.ject,ive fuiictiori. Tlie value of coiitrol variables, the gratlierit and the
furictiori J are theri trarisrriitted to a descerit algoritlirri based or1 a quasi-Newton wliich
gives us the riew set of paranieters. Theri, we reit,erate iiritil satisfyirig a stop criteriori
deperidirig on tlie required precisioii. Firially, we obtairi the optirrial pararrieters. For
solvirig these prohlenis, we use a corribiriatiori of a firiite diff'ererice nietliotl for the the discretixation arid of a finite elemerits niethoti for the space discretiïiation.
The so developetl rriethod was irriple~rieiited iri a already existiiig two-diiiierisioilal
simulation code for iriductioii heatirig. We preserit riiirrierical tests, optirriixatiori results
and a cornparisori with experirrierital results. We show like this the effectiveriess of
our method for t,tie optimal cont,rol of inductioii Iieating. We firiisli our study t>y the