ISSN ( I Vol

*---:-I I

F O U N D R ' IEERING -.

u b l i h d quarterly as the organ of the Foundry Commission or the Pdish Amdemy oTSciences

Explicit and implicit approach of sensitivity analysis in numerical

modelling of solidification E. Majchrzak '' b*, G. Kalaka

a Department for Strength of Materials and CornputationaI Mechanics Silesian University of Technology, Konarskiego 18a, 44- 100 Gliwice, Poland

"nstitute of Mathematics and Computer Science Czeslochowa University of Technology, Dabrowskiego 73,42-200 Czesrochowa, Poland

* Corresponding auth0r.E-mail address: ewa.majchrzak@polsl.pl

Received 15.02.2008; accepted in revised form 22.03.2008

Abstract

The explicit and implicit approaches of sensitivity analysis using thc boundary clement mcthod arc prcscntcd. In p,~rlicul,ir. thc problcm of casting solidification is considcrcd. A perturbation of an input parameter (For cxample the thermal conductivity of casting 111:acrial) cnuscs thc changes of transient ternpxaturc Field in thc domain analyzed. The methods of scnsitivity analysis allows to dctcrminc in maihcmniic~~I way the mutual connections bctwcen parameters pcfiurbations and final results. In the papcr some significant aspccls of compi~~ntional nlporithrns associated with cxplicit and implicit approachcs of sensitivizy analysis are demonstrated.

Keywords: Application of Information Technology to the Foundry Industry: Solidification Process: Numerical Tcchniqucs: Sensitivity Analysis; Borzndary Elcmcnt Mcthod

1. Introduction

The thermal pmcesses proceeding in the casting domain arc described by the energy equalion (Fourier-KirchhoFf lype equation) and bor~ndary initial conditions resulting from thc technology considcml [ I , 2. 31. Thc transient tcmpcraturc field in thc casting domain is de~nden t on the set of thermophysical parameters of material, coefficients appearing in the boundary condirions and initial (pouring) tcrnpx-aturc. The pcrtnhations of above selected input data cause the change of the course of solidification process. To analyzc thc conncctions bawccn the parameters perturbations and results of numerical simulations the sensitivity methods are applied [2,4,5.6.7. Xj.

There are two basic approaches to scnsitivi~y analysis using b u n d a r y clement formulation: tlrc colriini~ous npproach and ~ h c . . discretized one [9]. In the continuous approach (explicit dirferen~iation method) thc anaIy~ical cxprcssions Ibr scnsizivitics are derived and then they are calculated numerically using BEM. They have rhe form of boundary integrals wilh integrands that dcpnd only on thc variables of ~ h c primary and mldilion;\l probkms. The implicit dirfcrcntiation methoti, which bclongs to ~ h c discrelized approach, bascs on the dirfcrcntialion of the rhgebraic boundary clcrnenr rnalrix cqi~ations. Thc dcrivativcs of thc boundary element system matrices can be calct~liilcd ciihcr analyticdl y or semi-aaaly~icall y. In thc papcr sumc significml aspects of formulations and computational algorithms aswciatcd wilh both methods an: dcmonstrfircd and thc ndvantsgcs and disadvmtages of both tcchniqucs arc rliscuswd.

A R C H I V E S of F O U N D R Y E N G I N E E R I N G V o l u m e 8 . S p e c l a l Issue 1 t 2 0 0 8 , 1 8 7 - 1 9 2 187

2, Governing equations A casting-mould-cnvimnmcnt system is considcred. A transient

tcmpcrature field in a casting sub-domain is dcscribcd by thc following equation

where k (T) is thc ~herrnal conductivity. c (T) is thc volumetric specific hcat, Q = Q (.r, I ) is thc sourcc function, T= T (,Y, t), .r, t denotc rcmpcraturc, spatid co-odinatcs and timc, rcspcctivcly.

As is well known. the source term Q(x. I) is proporlional to [he local solidification rate [I. 2-31. this means

wherc lq is the solid state Fraction at the neighborhood of the p int considcred from casting domain. Lv is the woli~metric latent heat.

If onc assumes 1hatL~ is thc known runction of tcmperaturc (thc scopc ofJT i s From 0 lo 1, of coursc) then

where the parameter

is called a substitute thermal capacity of mushy ronc sub-domain [ I , 21. In the casc of 'binary alloys the mushy zone sub-domain cornsponds to lhc temperarum interval [TI, TL 1, where TS. TL are lhc bordcr tcrnpcnjturcs dc~crrnining thc cnd and thc beginning of the solidification process.

In lilcrature the several hypothcscs concerning ~ h c function describing a substitute thermal capacity of the mushy zone are discussed [I, 21. In this papcr the s~~bstitute thcrmal capacity for cast steel is dcfincd as follows

On a casting surface r the Robin condition is given

X E r : - A n - V T = U ( T - T , , ) (8)

~vhcn: a is a substitute hcat t ~ n s f c t cocR?cicnt (influencc of mould). T, is a convcntionalIy xsumcd ambicnt wmFrarurc.

For thc moment t = 0 thc initial tcrnpcratilrc distribution (pouring temperature) is known, namely

3. Boundary clement method

To solvc the equation (7) the boundq elcmcnt method has been used. If the thcrmal dirrusivity is constant, this mcans a (T) = IrlC(T) = a = const, thcn for thc partial diffcrcntiat cquation (7) a fundamental solution is available [ IQ, 1 I, 3 21

where m is the problem dimension (rn = 1, 2, 3 correspnnds to ID. 2D, 3D problem, rcspcctivcly), [O, t ] is ~ h c time intcrval under consideration. 5 is the obscrvnrion point, r is thc distance bctwccn the points 6 and ,r.

In thc case of aon-constant thermal diffusivity the use of the fundamental solution ( IO) should bc accornpanicd by a timc marching technique in which a /T ) is assumcd conspit at thc beginning of each time step [ 131. So, thc fimc grid is introduced

wilh constant time step At = If- rJ-' . Starting from the initial timc I" over each timc step [tf-'. t'], the

value of a is taken as the mean avcragc, namcly

Baring on the approximation (12), the cquation (7) can be tmsformcd into the following integral equation for cach timc step [tf-I, f j ]

whcrc q . CI are the constant volumetric specific heats of liquid IT' (c,.~, r ' , ) ~ ( x . r " )d fl and solid state. n

Because the .mlidification process proceeds in a rather small interval of tempcraturc one can assumc the constant value of whcrc for E; E R: (Q = 1 and for g E T: R (5) E (0, 1 ), g (x. I ) =

thcrrnal conductivity of cast stccl and thcn thc cquation (4) takcs -L".VTb, 4'*(5*xp r< = - L ~ + ~ F * ( C * X * f! t) . a form Fundamental solution T*(s, x. [I. I ) has the following form

(7) T' = I

[4nof (r' - r ) ] " 1 2 40' (r" I )

A R C H I V E S o f F O U N D R Y E N G I N E E R I N G Volume 8 , Spec ia l Issue l E 2 0 0 8 , 187 -192

Thc function rl*(& x, lf. I) can be calculakd analytically [ I 11 internal cells and ~hm wc obtain lhc following syslcm o f algebraic equations ( i = 1,2. .... N)

r -I

where whcrc

and and cosak are dimional cosines of h e normal vcctor n. For conslam clcrncnts with rcspcct to limc 110, I I], the

equation (13) can hc writrcn in thc Form

while

whcrc

Thc system ofcquations (22) can be wtirten in thc matrix form

and After determining the 'missing' boundary valucs (Tf or q,!). the tcrnpcraturcs at intcrnal nodcs .r ' EQ for time I { arc ca!culatd using thc lorrnula(i=N+ 1 , N + 2, .... N + L)

Eunclions h (5 .r), 8 (5 s) arc dctcrmincd in analytical way [ I I ] and then

4. Sensitivity anaIysis - explicit differentiation method

Let us consider thc scnsitivity o f he solidification problem sotuiion with rcspcct to thc pnrnmcrcr p (c.g, p cornsponds to rhc thcrmal conductivity or hcat rmnsrcr coelficicnl or amhicnt tcmpcraturc).

Thc cxplicit diifcrcntlat ion mcthd, which hclongs to rhc conzinuous approach, hascs on thc ditfercntiat ion or governing equations with respect to thc paramctcr p [ 14, 15. 161. So. rhc diffcrcntia~ion o f cquntion (7) lcads to thc Tollnwing cquation

l ~ e x p ( - & ) ] p 3D problcm

and

I&erfc[&} 3D problem

i? ( 5 , ~ ) = I

wbcrc U= aTBp is rhe sensitivity function. Taking into account thc formula (7) thc cquarion (30) can hc

wriaen in the form

( 2X ID problcm (2 1 )

1 2D problem

In ordcr to solve equation (141, the boundary r is divided into N linear boundw elements Ti, the interior R is divided into L linear

A R C H I V E S of FOUNDRY E N G I N E E R I N G Volume 8 , S p e c i a l I s s u e 112008. 187-1 9 2 189

J U 3) C ( T ) 57 ~ C ( T ) ~ T (31) So. tbc system of cquarions (26) has bccn dilrcrcnrimctl with .r E R : c(T)- = >.VzU +--- - --

3r 1 31 at) Jr respcct to the paramcterp and thcn

a~ aqf JH 3Tf aP 311 all

aT"l (40) The houndary condition (8) is also dirrcrcntiatcd with rcsvct to -qf + G- = - T f + 'II- + -T"" + P- p and thcn JP a l ~ JP a p

Diffcrcntiation of initial condition (9) gives

The cqualion (27) concerning inlcrnal nodm is also diffcrcntiatcd with rcspcct lop, nnrncl y

(34) 6. ID problem

In this way one obtains thc additional boundmy initial probtcm Thc casling of thickness 2D is andyzcd (ID problem). Taking conncctd with the scnsitivit y r~lnction U dcscribcd by cquations inlo acmun1 the sYmmctrY. the fol[owi% boundary initia[ ~ m b l c m (31). (33). (34). This pmblcrn has hccn solvcd hy mcans of thc boundary element method. So. one obtains thc following systcm of algehnic equarions (c.f. equation (22)) I a+ a2r o c .t c n : c(r)- = 2,-

X ar

ZG, w,! =f H , ~ U; + ke, U:-' + $z,, Q: (35) . r = o : g = h - = O ar I - l ,:I I . 1 I I ax

and In this case the boundary is rcduccd lo rhc rwo pints. namly I 7;' - T,"' p i n t 1 (.r= 0) and p i n t 2 (.Y= D). Thc domain 10. DJ is dividd into

(37) L linear internal cclls of thickncss h. 'Ihc ccntml points of thcsc cclls are nrimkml as 3, 4, ..., L+2. Thc system of cqunrions (26)

Tnc syslcm orcyua~ions (35) can k wrirlen in the rnarix form resulting from thc REM applica~lon has thc Following form

GW' = H U 1 + PU"' + ZQ1 (38) [ [ I = [ +

Aflcr dctcrmining the 'missing' boundary values ( ~ J o r yf). tbc valucs of function U! at inkmat nodes .s ~ E R for time I a r ~ hJ-' 1 calculated using the formula ( i=N + 1. N+ 2. .... N+ L)

P,Z ... PZL U! = i~,, W: - t ~ , ~ W: + i$, u/-' + t5, p: (39)

1-1 1.1 1-1 1.1

whcre (c-f. equations (23). (24))

5. Sensitivity analysis - implicit G , , = ~ [ E ; " O ) , G , , = ~ ( c ' , D )

differentiation method and

Thc implicit diffcrcntiation mcthod. which bclongs ro rhe h ( ~ ' , o ) , i # j Hi1 = -112. i = j , , , 2 = { h ( ~ 1 9 4 9 -112. I = i * j j discrcli7cd appmilch. hmcs on thc dilfcrenriation of rhc algebraic (46)

houndary clcrncnt matrix cquations 191.

193 A R C H I V E S 0 5 F O U N D R Y E N G I N E E R I N G V o l u m e 8 , S p e c i a l I ssue 1F2008, 187-192

tet us assume that the parameter p comspnds to the thermal conductivity of casting rnatcrial. Using the explicit approach of scnsitivity analysis one obtains rhc following additional problcm a q , ac,, q[]+ (JC(T)ldh= O - c.f. cquation (6)) -- -

a h an 3). 31. "' a. ?!i0 (53)

7. Example of computations

In numerical computations thc following data havc bccn introduced: 2D = 0.02 rn, 1- = 35 [Wl(mK)], c~ = 5.74 [ M J I ~ ~ ~ K ) ] , cs= 5.175 [ ~ ~ l r n ' K]. Lv=1957.5 [ ~ ~ l r n ~ 1. pouring tempcrnturc '1;) = 1570 " C, liquidus ternpmturc TL = 1505 " C, soIidys tternpcrature Ts= I47OUC, heat transfer cocficient a = 250 [W/(m- K)j. ambient tcmpcrature T, = 600 " C. The domains has bccn dividcd into 20 internal cells, time step At= 0.5 s.

In Figure 1 the ternpcralurc distribution in [he domain considered [or times 3 0.20. ..., 100 s is shown.

The pmhlcrn /47) is muplcd with thc primary onc Icquation (43)) hccauw in ordcr to solvc i~ [he dcrivativc aT/dr and hcat flux q should kw known.

In thc casc of implicit d i f fml iat ion method application the dcrivativcs aG, la), atfij /a), and 3Pil should hc calculatcd. Bccausc

and

Fig. I. Tcmpcr:~iurc dis~rihution and

Finally. the derivative aP;,/ah is calculated

?he system of equations connected with thc scnsitivity function has thc following form

Fig. 2. Courscs of scnsitivity runction U (explicit approach - solid Iinc, implicit approach - d o n 4 Iinc)

A R C H I V E S of F O U N D R Y E N G I N E E R I N G V o l u m e 8. S p e c i a l I ssue l J 2 0 0 8 . 187-192 191

Figurc 2 illustrates thc courscs of lunction U = 3T/3?+ at thc points .v = O ('axis of symmcrry), .r = 0.005 m md .r = 0.01 m (boundary) rrhtaincd Sry cxplicit and implicit diflcrcnt imion rncthod. It is visihlc that thcsc courscs ;YC similar hut not idcntical.

8, Conclusions

Scnsitivily annlysis is thc vcry cfkctivc loot in numerical modelling of solidification prohlcm. Ir allows to rcbuilt rhc basic solution on thc so!ulion conccrninp thc othm disturbed vduc of parameter. In thc p a r r thc cxplicit and implicit approaches using the houndary clemcnt mcthod havc h c n prcscntcd. Frum the mathcma~ical p in1 of vicw ~ h c cxplicia approach is sirnplcr, but thc implicit approach pivcs mow cxact rcsults.

Acknowledgements

This u ~ r k um hndcd by Grant No N N507 3592 33.

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Jawna i nicjawna rnctoda analizy wra?liwoSci w nurncrycznym modclowaniv pracesu krzepniecia

W art!kulc pmdstnwiono jawne i niejannc podcjScic nnnliv w~.liwoSci z ws!orjer\anicrn rnctody etcmcntb~v br.tgorrych. 1V stc/cgi~lnoki rnqatr?.tvano proces krzcpni~cia. Zaburzcnie paramclru wcc,iS~io\~~cgo (np. wspilcn.nnika pr+tc%%d/enia cicpta) powodi!jc rmiany p ~ l n tcrnpcntur_r. iv ro7wnPanym obs~uzc. Mctody analizy 1vm2liwo6ci poz\toaInj;( w matcmal!.cmy spsrib pl-/cdsraw\ ii: rvzajcmnc A.alc)noici rniqd;l!. znh~trzcniami panrnetrbw a kohcowymi wynikami. W pracp pokmino najjlwn?nicjszc alipckry algoqqmdlv obliczenio~vych rwiqmn!.ch z j a w q i niujmvnq rnetodq annlizy wvra2liwoSci.

A R C H I V E S ol F O U N D R Y E'NGI'NEERING Vo lume B . S p e c l a l I s s u e 1 1 2 0 0 8 , 1 8 7 - 1 9 2