thermal optimization of the continuous casting …

THERMAL OPTIMIZATION OF THE CONTINUOUS CASTING PROCESS

USING DISTRIBUTED PARAMETER IDENTIFICATION APPROACH –

CONTROLLING THE CURVATURE OF SOLID-LIQUID INTERFACE

R. TAVAKOLI

Abstract. Thermal optimization of the vertical continuous casting process is considered in thepresent study. The goal is to find the optimal distribution of the temperature and interfacial

heat transfer coefficients corresponding to the primary and secondary cooling systems, in additionto the pulling speed, such that the solidification along the main axis of strand approaches to theunidirectional solidification mode. Unlike many thermal optimization of phase change problems inwhich the desirable (target) temperature, temperature gradient or interface position are assumed

to be a-priori known, a desirable shape feature of the freezing interface (not its explicit position)is assumed to be known in the present study. In fact, the goal is equivalent to attain a nearlyflat solid-liquid interface that is featured by its zero mean curvature. The objective functionalis defined as a function of the temperature and mean curvature of the freezing interface. The

solidus and liquidus iso-contours of the temperature field are used to implicitly determine thesolid-liquid mushy region, i.e. the freezing interface. The temperature field is computed by solvinga quasi steady-state nonlinear heat conduction equation. A smoothing method is employed toensure the differentiability of heat equation. The explicit form of first order necessary optimality

conditions is derived. A gradient descent algorithm is introduced to find local solutions of thecorresponding optimization problem. Numerical results support the feasibility and success of thepresented method.

Keywords. Adjoint sensitivity analysis; Curvature control; Effective heat capacity; Phase change

optimization; Regularization; Solidification control.

1. Introduction

Phase change is an important physical phenomenon that happens during many problems in scienceand engineering, e.g. the shape casting, ingot casting and crystal growth technologies. In these cases,the freezing conditions have a significant impact on the quality of products. Therefore, the controland optimization of solidification is an important problem that should be taken into account duringthe process design stage. The optimal design of casting components in the view of manufacturingconstraints were recently considered in the structural optimization community. For instance , themolding (casting) constraints were considered in [1–6], the control on the connectivity (integrity)and minimum thickness of components were considered in [7, 8] and considering process constraintsin mold free processes like additive manufacturing was studied in [9]. Because the process designconditions significantly vary the practical performance of casting parts, for instance , due to differentgrain sizes, heavy sections commonly exhibit lower mechanical properties in contrast to thin sections,the ultimate goal will be considering the process and product design optimization simultaneously.The process design optimization of casting components is less considered in the literature, in spiteof its importance. The optimal design of feeding system in the shape casting process was consideredin [10–14].

The continuous casting is one of the important mass production industrial processes that producesraw ingots for subsequent forming procedures. Thermal history of ingots during the continuous cast-ing process has a significant effect on the quality of final products [15, 16]. The thermal optimizationof the vertical continuous casting process based on the distributed parameter identification approach

Date: April 20, 2017.Department of Materials Science and Engineering, Sharif University of Technology, Tehran, Iran, P.O. Box 11365-

9466, tel: 982166165209, fax: 982166005717, email: [email protected]

2 R. TAVAKOLI

will be considered in the present work. Because this process involves the change of phase, it is worthto briefly mention related literature on the optimization of phase change problems.

Using the classic Stefan problem [cf. 17] is a common way to model phase change phenomena inthe case of pure materials. The control and optimization of Stefan problem based on the distributedparameter identification approach have been considered extensively in the literature [cf. 18, 19].The distributed parameter here means that control parameters are defined on a codimension-zero orcodimension-one subset of the spatial domain, i.e. there are an infinite number of control parameters(in the continuum formulation). The inverse solution of Stefan problem to attain a desired solid-liquid interface by controlling the boundary heat fluxes has been considered in [20]. In [21] theoptimal control of Stefan problem has been considered in which unknown boundary heat flux hasbeen determined such that the space-time mismatch between the solid-liquid interface and its targetdistribution (desired space-time interface position) has been minimized. The same problem hasbeen considered in [22], including the effect of melt convection in the liquid region. The inversedetermination of unknown solid-liquid interface in the steady-state Stefan problem to maintain thedesired temperature distribution and interfacial contact angle has been considered in [23]. Thedetermination of optimal boundary heat flux in the Stefan problem to control the evolution of solid-liquid interface has been considered in [24]. In [25], the optimal determination of a codimension-oneimmersed boundary in the spatial domain, that acts as the cooling element (in fact , an unknowndistributed heat sink), to maintain a desired temperature distribution in the Stefan problem hasbeen considered.

Many real-world phase change phenomena , however , include the phase transition in an alloyingsystem. Unlike the classic Stefan problem, there is not an explicit interface between the solid andliquid phases in these cases, but there is a mushy region with non-zero thickness. There are manymathematical models and numerical approaches to solve phase change problems in alloying systems[cf. 26–30]. The optimal determination of boundary heat flux to maintain the desired temperaturedistribution in a binary alloy solidification problem has been considered in [31]. To benefit fromthe formerly developed method in [20], authors in [31] have used the sharp interface assumption forthe solid-liquid boundary. There are a few efforts on the thermal optimization of solidification inalloying systems corresponding to the shape casting process. In these works, the shape of spatialdomain was identified by the shape optimization [32, 10, 11] or topology optimization [14, 13, 33, 34]methods to establish the directional solidification from the cast part to the feeding system.

The goal of present study is to optimize the heat transfer in the vertical continuous casting process[cf. 15]. Figure 1 (left) schematically shows essential features of a vertical continuous casting process.In this production method, the molten material is poured into a water cooled mold, that is calledthe primary cooling system (region), where the strand (the product of process) obtains a sufficientlytick solid shell. After exiting the mold, the strand is supported by rollers and it is cooled by watersprays, that is called the secondary cooling system (region), to complete the solidification. Whenthe strand passes the end of water sprays region, it cools down by the natural convection to theenvironment temperature. After the short initial transient stage, the continuous casting processreaches to a quasi steady-state condition in which the strand leaves the machine with a constantvelocity that is proportional to the casting rate [15, 35].

There are many works in the literature on the mathematical modeling and numerical simulationof the vertical continuous casting process, for instance , see: [37–40, 36, 41]. There were severalattempts to improve the performance of continuous casting machines by the real-time adjustment ofpulling and cooling systems parameters using feedback control approaches, see for instance [41–49].In [50, 51] a multi objective non-smooth optimization approach has been introduced to optimizethe secondary cooling system parameters in the continuous casting process. The objective functionis defined as the weighted sum of the following items: desired surface temperature distribution inthe strand, bounds on the strand surface temperature, bounds on the strand surface temperaturegradient, bound on the depth of the liquid pool and a lower bound on the temperature of thebending section. Note that the depth of the liquid pool is not directly controlled in these works.

THERMAL OPTIMIZATION OF THE CONTINUOUS CASTING PROCESS 3

solid

mushy

melt

x

y

0 lx

lm

lyξ

m = −j

v = vm

∆x

∆y

q = h(T − Ti) q = h(T − Ti)

q = 0

T = TP

Figure 1. Left: schematic of a vertical continuous casting machine [36], Right:schematic of computational domain corresponding to model problem considered inthe present study.

But, its upper bound is controlled by imposing a pointwise upper bound constraint on the tem-perature of a defined point in the strand. In [52], the difference between the temperature field anda target temperature distribution has been minimized by optimal identification of the solid-liquidinterface position. Using the genetic algorithm, in [53, 54] the parametric optimization of the sec-ondary cooling system in the continuous casting process has been considered to establish the desiredtemperature distribution in the strand.

The remainder of this paper is organized as follows. A smooth quasi steady-state mathematicalmodel to find the temperature field in the vertical continuous casting process is presented in section2. The optimization problem in the present study is formulated in section 3. A new objectivefunction is introduced in this section such that the optimization problem is defined with minimalinformation about the optimal solution. For this purpose, the objective functional is defined asthe weighted integration of the squared mean curvature of the freezing interface. To the best ofour knowledge, it is the first effort on posing control on the second order derivative of temperaturefield in heat conduction problems. The rigorous theoretical study, for instance to investigate theexistence of solution, on such a problem is a very difficult job in the field of mathematical analysis.This issue is not considered in the present work, but the success of introduced method is investigatedthrough numerical experiments. The first order necessary optimality conditions is derived in section4. The solution algorithm based on the regularized projected steepest descent approach is presentedin section 5. The spatial discretization schemes are discussed in section 6. Section 7 is devotedto numerical study on the feasibility of the presented method. Finally, section 8 summarizes thepresent work.

2. Mathematical Modeling of Continuous Casting Process

Without the loss of generality, for the purpose of convenience and to avoid the complexity offormulation, the convection of molten metal in the strand is ignored in the present study. Therefore,

4 R. TAVAKOLI

the temperature distribution in the strand can be computed by solving the heat transfer equa-tion including the phase change effect with appropriate boundary conditions. The heat transfer tothe primary cooling system (mold) and secondary cooling system are taken into account using theinterfacial boundary condition approach [cf. 55, 30]. In this way, the strand senses these envi-ronments as some convective boundaries with different temperatures and interfacial heat transfercoefficients. This approach is very common in the literature of heat transfer to model discontinuitiesand interfaces.

Assume the spatial domain, the strand from the upper surface to a sufficiently large length (e.g.the cutting length), is denoted by Ω that is an open bounded subset of Rd (d = 2 or 3). Moreover,assume that there are no elastic or plastic strains in the liquid and solid regions. Under theseassumptions, the enthalpy conservation law in the strand has the following form [cf. section 3.2 of56]:

∂

∂t(ρe) +∇ · (ρev) = ∇ · (k∇T ) in Ω (1)

where ρ = ρ(x) denotes the density, k = k(x) denotes the heat conductivity coefficient, T = T (x)denotes the temperature, e = e(T (x)) denotes the enthalpy density (the enthalpy per unit mass)and v = v(x) denotes the velocity field. Ignoring the initial transient stage at starting point ofthe vertical continuous casting process, the temperature field in the strand reaches to the quasisteady-state conditions [cf. 15, 55, 35]. Therefore, (1) simplifies to the following form:

∇ · (ρev) = ∇ · (k∇T ) in Ω (2)

If the variation of density with the temperature and solidification is ignored, i.e. ρ = constant,the incompressibility assumption, ∇ · v = 0, simplifies (2) as follows:

ρ (v · ∇e) = ∇ · (k∇T ) in Ω (3)

Assume that the longitudinal axis of the strand is denoted by ξ (cf. Figure 1); i.e. ξ is a curvilinearaxis which is orthogonal to every cross section of the strand. Moreover, assume that m(x) denotesthe local unit vector along ξ axis directed from the upper section to the lower section of the strand.Considering the above mentioned assumptions, v(x) is parallel to m(x) and v(x) is fixed on everyξ − constant plane. Therefore:

v(x) = v m(x) (4)

where v = |v| and v is equal to the pulling (casting) speed, at the lower cross section of the strand.Assuming the variation of configurational entropy with the temperature is negligible, at the fixed(environmental) pressure, the enthalpy density can be computed as follows:

e(T ) = er +

∫ T

Tr

cp dT + Lffl(T ) (5)

where Tr denotes the reference temperature, er denotes the entropy density at the reference temper-ature, cp denotes the specific heat capacity at fixed pressure, Lf denotes the latent heat of freezingand fl denotes the local volume fraction of the liquid phase (obviously fs+fl = 1, where fs denotesthe local volume fraction of solid phase). An alloying system with a nonzero freezing interval isconsidered in the present study. In the case of isothermal solidifications, like pure elements or eu-tectic alloys, a virtual narrow interval can be considered to use the method presented in the currentwork [cf. 29].

There are several physical models to compute fl as a function of the temperature within thefreezing interval. Some instances are the level-rule, Scheil and linear models [cf. section 7.2 of28]. The later model has been extensively used in the literature. The main benefits of this modelare: simplicity, satisfactory outcomes and applicability when there exists no explicit informationabout the variation of fl as a function of the temperature, that is the case in many real-worldproblems. In this approach, it is assumed that the latent heat of freezing is distributed linearly over


the solidification interval, i.e.:

fl(T ) =

1, T > Tl

(T − Ts)/(Tl − Ts), Ts ⩽ T ⩽ Tl

0, T < Ts

(6)

where Tl and Ts denote the liquidus and solidus temperatures respectively. Without regard tothe specific choice of fl(T ) in the freezing interval, the conventional treatments of fl(T ) are non-differentiable with respect to T at T = Ts, Tl. Unfortunately there are many papers in the literaturein which this differentiability problem has been ignored without care, see [57, 38, 58, 59, 28, 60, 45, 61]as a few instances. It appears that zero values have been considered as the derivative in these non-differentiability points, whenever it was applicable. However, using non-differentiable form of flcan leads to serious problems during the solution of nonlinear system of equations correspondingto the discretized form of continuum model. To resolve this issue, either the regularized form offl should be considered or a non-smooth solver with sufficient cares should be employed. Thisissue is well discussed in [35]. It is worth noticing that, in the above mentioned works, a transientheat solver was employed and as a result of intrinsic regularization by the time stepping procedure,the negative impact of the mentioned non-differentiability issue did not seriously overshadow thenumerical results.

Equation (6) can be stated equivalently in the following semi-smooth analytic form [35]:

fl(T ) = 1−max

(Tl − T

Tl − Ts, 0

)+max

(Ts − T

Tl − Ts, 0

)(7)

where max(a, 0) returns 0 for a < 0 and a for a ⩾ 0. Now, (7) is regularized by replacing max functionwith its regularized form, that is shown by maxε here, where ε denotes the regularization parameterand max(a, 0) = limε→0 maxε(a, 0). There are many forms of regularized maximum function in theliterature. The monotonicity of maxε(a, 0) with respect to a is essential for the physical consistencyof fε

l [cf. 35]. A new form of regularized max function with C2 continuity and required propertythat was introduced in [62] is adopted in the present work:

maxε(a, 0) =

0, a ⩽ 0a3

ε2 − a4

2ε3 , 0 < a < εa− ε

2 , a ⩾ ε

(8)

Using (8), fl(T ) can be approximated by the following smoothed form:

fεl (T ) = 1−maxε

(Tl − T

Tl − Ts, 0

)+maxε

(Ts − T

Tl − Ts, 0

)(9)

Using above approach, (5) can be approximated by the following C2(R) function:

eε(T ) = er +

∫ T

Tr

cp dT + Lffεl (T ) (10)

where eε denotes the regularized enthalpy density. Without the loose of generality, assume that thespecific heat capacity is temperature invariant, i.e. cp = constant. The substitution of (10) into(3) results in:

ρcp (v · ∇T ) + ρLf (v · ∇fεl ) = ∇ · (k∇T ) in Ω (11)

Considering that ∇fεl (T ) = (dfε

l /dT )∇T , (11) can be expressed in the following equivalent form:

ρcεe (v · ∇T ) = ∇ · (k∇T ) in Ω (12)

where cεe denotes the regularized effective specific heat capacity which is computed as follows:

cεe = cp + Lfdfε

l

dT(13)

Appropriate boundary conditions are required to solve (12). Assume the boundaries of Ω aredenoted by Γ = Γt ∪ Γb ∪ Γl where Γt, Γb and Γl denote upper, lower and lateral boundaries ofthe strand respectively. In practice, the upper boundary of strand is maintained at the constant

6 R. TAVAKOLI

pouring temperature. The lateral boundaries transfer heat to the primary and secondary coolingsystems. From a physical point of view , the length of strand is infinite and there is no lower surfacefor it. However, to save the computational resource, a finite length is considered and an appropriateboundary condition is imposed on it. When the length of strand is sufficiently large the isolatedboundary condition is a reasonable choice for the lower surface of strand [cf. 38, 58–60, 45, 61, 35].Therefore, the heat equation together with its boundary conditions in the present study can bestated as follows:

ρcεe (v · ∇T ) = ∇ · (k∇T ) in Ω (14)

T (x) = TP on Γt (15)

−k∇T (x) · n(x) = 0 on Γb (16)

−k∇T (x) · n(x) = h(x)(T (x)− Ti(x)

)on Γl (17)

where TP denotes the pouring temperature, n = n(x) denotes the outward unit normal on theboundaries, h(x) denotes the interfacial heat transfer coefficient on the lateral boundaries (variesspatially) and Ti(x) denotes the environmental temperature on the lateral interfaces (varies spa-tially). In fact, Ti is equal to the mold and cooling water temperatures in the primary and secondarycooling regions respectively. It is worth mentioning that n = m on Γb. Equations (14)-(17) arecalled the direct problem henceforth.

3. Formulation of Optimization Problem

As it is formerly mentioned, h(x), Ti(x) and v are considered as the design variables in thepresent study. To avoid possible nonphysical solutions, one may impose bound constraints on thesedesign variables. For instance , these values should not be negative and they should be smaller thansome appropriate upper bounds. These constraints can be defined as follows:

hL(x) ⩽ h(x) ⩽ hU (x) on Γl (18)

TL(x) ⩽ Ti(x) ⩽ TU (x) on Γl (19)

vl ⩽ v ⩽ vU (20)

where hL, hU , (0 ⩽ hL ⩽ hU < ∞), TL, TU , (0 ⩽ TL ⩽ TU < ∞) and vL, vU , (0 < vL ⩽ vU < ∞)denote the lower and upper bounds corresponding to h, Ti and v respectively.

To define an appropriate objective functional is a key for the success of every design optimizationproblem. The macroscopic shape of solid-liquid interface is one of the most important factors inthe vertical continuous casting process. Ideal solidification conditions happen in the case of flatsolid-liquid interface [15]. In fact, the unidirectional solidification from upper to lower boundaries ofingot defines the ideal solidification conditions. By approaching to the flat solid-liquid interface, thesolidification defects, like shrinkage voids, gaseous porosities, macro segregation and the susceptibil-ity of internal hot tearing, are decreased [15]. Furthermore, the maximum production rate of healthyingot can be attained when the solidification has near unidirectional mode [15]. The depth of theliquid pool in the strand is commonly called the metallurgical length in the materials engineeringliterature (cf. Figure 1). In the case of near ideal solidification conditions, the metallurgical lengthis minimum [cf. 15]. Therefore, it is an implicit factor to measure the quality of strand. However,there is no explicit definition for this factor as a function of the temperature. The absolute value ofmean curvature of the solid-liquid interface is a good measure for the violation of the solidificationfrom its ideal conditions. However, in the case of alloying systems with large freezing interval (e.g.when Tl − Ts > 10 oC), there is not a distinct interface between the solid and liquid phases. But,there exists a mushy region between these phases. Obviously, the solidus and liquidus iso-contoursof the temperature field determine the boundaries of the mushy zone. Taking these facts intoaccount, the objective functional can be defined as follows:

J (T ) =1

2

∫Ω

ϕ(T )∣∣κ(T )∣∣2 dx (21)


where κ(T (x)

)denotes the mean curvature of the temperature field iso-contours at x. In fact,

it measures the local mean curvature of Tc − constant temperature iso-contour at x = x0, whereTc = T (x0). According to differential geometry principles κ can be computed as follows [cf. 63]:

κ(T ) = ∇ ·(

∇T

|∇T |

)(22)

Moreover, the weighting function ϕ(T ) denotes the classic Gaussian function that is centered atTm = (Tl + Ts)/2. It is defined as follows:

ϕ(T ) = exp

(−(T − Tm)2

2σ2

)(23)

where σ > 0 controls the width of Gaussian peak. In fact, ϕ(T ) in the present study, increases theemphasis of term |κ|2 near iso-contour T = Tm and decreases its emphasis gradually by approachingto Ts and Tl iso-contours. Figure 2 shows the graph of ϕ(T ) for Ts = 1360, Tl = 1460 and differentvalues of σ. According to the plot, for σ = 18, the weight function is nonzero in the mushy zoneand is near zero out of this interval, for this choice of Ts and Tl. It is important to note that thetemperature field is assumed to be sufficiently regular such that the local mean curvature of thetemperature iso-contours are well defined everywhere in Ω.

1200 1250 1300 1350 1400 1450 1500 15500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

T

φ (

T )

σ = 18

σ = 30

Figure 2. The variation of ϕ(T ) with T for Tm = 1410 and σ = 18, 30.

Considering that h and Ti are codimension-one field variables, it is worth to add regularizationterms to the objective function to avoid a sudden spatial jump in these parameters. Moreover,because distributed parameter identification problems are commonly ill-posed, such a regularizationcan stabilize the numerical optimization algorithm. For this purpose, the regularized objectivefunctional, F , is defined as follows:

F(T, h, Ti) = J (T ) +H(h, Ti) (24)

H(h, Ti) :=ζh2

∫Γl

|∇Γh|2 dx+ζi2

∫Γl

|∇ΓTi|2 dx (25)

where ζh, ζi ⩾ 0 denote the Tikhonov regularization parameters corresponding to the control pa-

rameters h and Ti respectively and ∇Γ denotes the gradient operator on the manifold Γl .

8 R. TAVAKOLI

The splitting approach introduced in [64, 65] is used for the numerical solution of the optimizationproblem in the present study. This method includes two sequential steps. In the first step, the firstpart of F , i.e. J (the most important component of F), is minimized subject to the correspondingcontrol and PDE constraints. In the second step, regularization step, the second part of F , i.e. H isminimized subject to the control constraints and the solution of former step as its initial guess. Inpractice, these two subproblems are solved incompletely at every step of the underlaying optimizationalgorithm. Further details are presented in section 5. In fact, the mentioned regularization step doesthe postprocessing (smoothing) of distributed design variables. Therefore, there are two optimizationproblems that should be solved sequentially in the present study. They can be expressed as follows:

min(v, h, Ti,T ) ∈ (Dv, Dh, DT , U)

J (T ) subject to : (14), (15), (16), (17) (26)

min(h, Ti) ∈ (Dh, DT )

H(h, Ti) subject to : ∇Γh · t = ∇ΓTi · t = 0 on ∂Γl (27)

where t denotes the outward unit normal on the boundaries of codimension-one manifold Γl (infact , the boundaries of the upper and lower cross section of the strand), Dv, Dh, DT denote theadmissible design spaces of control variables v, h and Ti respectively. They can be defined as follows:

Dv := u ∈ R | vl ⩽ u ⩽ vU

Dh := u ∈ V | hL(x) ⩽ u(x) ⩽ hU (x), ∀x ∈ Γl

DT := u ∈ V | TL(x) ⩽ u(x) ⩽ TU (x), ∀x ∈ Γl

Moreover, V = V(Γi) and U = U(Ω) denote two function spaces with sufficient regularities thatare required for the solution of (26). Because the theoretical analysis on the above mentionedoptimization problems is beyond scope of this paper, no explicit form for these function spaces arepresented here. Obviously, to study the existence of minimizers and convergence of optimizationalgorithm, their explicit definitions are essential. Therefore, in the reminder of this work, it isassumed that the optimization problems (26) and (27) are well-defined and each of them possessesat least one minimizer1.

4. First order necessary optimality conditions

In the present study, a gradient based optimization algorithm is employed to solve problems (26)and (27). Therefore, it is required to find the first order necessary optimality conditions correspond-ing to these problems. It is worth mentioning that even a black-box gradient based optimizationsolver is used for this purpose, the first variation of objective functional with respect to the designvariables is the minimal requirement of every gradient based solver. It is commonly called the sen-sitivity analysis in the engineering literature [cf. 66–71]. Although there exists extensive literatureon the design sensitivity analysis and optimization of static and transient heat conduction equations[cf. 72–81], there is no related report on the sensitivity analysis of the problem (26). To the bestof our knowledge, this work is the first effort on the design sensibility analysis of optimization prob-lems constrained to heat conduction equation and objective functional that includes second orderderivatives of the temperature field.

Consider sufficiently regular functions Θ, Θt, Θb and Θl as the Lagrange multipliers corre-sponding to equations (14), (15), (16) and (17) respectively. Moreover, consider sufficiently regularfunctions λL, λU , µL, µU , ϑL and ϑU as the Lagrange multipliers corresponding to bound con-straints on h, Ti and v respectively. Using these multipliers and augmenting the objective functionalJ with constraints (14), (15), (16), (17) and control constraints results in the following augmented

1It is easy to show that the optimization problem (27) is convex and has at least one minimizer in(L2(Γl)

)2.


Lagrangian:

L = L(T, h, Ti, v,Θ,Θt,Θb,Θl, λL, λU , µL, µU , ηL, ηU )

= J (T ) +

∫Ω

Θ(ρcεe u · ∇T −∇ · (k∇T )

)dx

+

∫Γt

Θt (T − TP ) dx+

∫Γb

Θb k∇T · n dx

+

∫Γl

Θl

(k∇T · n+ h(T − Ti)

)dx

+

∫Γl

λL (hL − h) dx+

∫Γl

λU (h− hU ) dx

+

∫Γl

µL (TL − Ti) dx+

∫Γl

µU (Ti − TU ) dx

+ ϑL (uL − v) + ϑU (v − uU )

(28)

Every local solution of the optimization problem (26) is a stationary point of L subjected to thecomplementarity conditions corresponding to the bound constraints. In fact, local solutions of (26)satisfy the first order necessary optimality conditions based on the Karush-Kuhn-Tucker (KKT)conditions [cf. 82, 18, 19]. Therefore, it is required to compute the directional derivative of L withrespect to its arguments, and to equate them to zero. The root of the first variations of L withrespect to Θ, Θt, Θb and Θl leads to (14), (15), (16) and (17), in fact , the direct PDE togetherwith its corresponding boundary conditions.

The first variation of L with respect to T results in:

LT δT =1

2

∫Ω

ϕ′(T ) |κ(T )|2 δT dx+

∫Ω

ϕ(T ) κ(T ) ∇ ·(∇(δT )

|∇T |−

∇T(∇T · ∇(δT )

)|∇T |3

)dx

+

∫Ω

ρ cεe′ Θ (v · ∇T ) δT dx+

∫Ω

ρ cεe Θ(v · ∇(δT )

)dx−

∫Ω

Θ ∇ ·(k ∇(δT )

)dx

+

∫Γt

Θt δTdx+

∫Γb

Θb k(∇(δT )

)· ndx+

∫Γl

Θl k(∇(δT )

)· ndx+

∫Γl

Θl h δTdx

:= I1 + I2 + I3 + I4 − I5 + I6 + I7 + I8 + I9

(29)

where δT denotes an arbitrary test function with sufficient regularity and,

ϕ′(T ) = −T − Tm

σ2exp

(−(T − Tm)2

2σ2

), cεe

′ = Lfdfε

l

dT

Consider the definition of the following local (pointwise) operator, P (T ), for the purpose of brevityin notations:

P (T ) := 1d −∇T

|∇T |⊗ ∇T

|∇T |(30)

where 1d ∈ Rd×d denotes the identity matrix and ⊗ denotes the tensor product of d-vectors, i.e.,a ⊗ b = [aibj ]d×d. In fact, applying this operator on the temperature field returns a second ordertensor field. Using (30) in the expression of I2 and doing integration by parts followed by applyingthe divergence theorem results in:

I2 =

∫Ω

ϕ(T ) κ(T ) ∇ ·((

P (T ))· ∇(δT )

|∇T |

)dx =

∫Γ

ϕ(T ) κ(T )

((P (T )

)· ∇(δT )

|∇T |

)· n dx

−∫Ω

∇(ϕ(T ) κ(T )

)·((

P (T ))· ∇(δT )

|∇T |

)dx := I2,1 − I2,2

(31)

Considering that ϕ(T ) is non-zero in the vicinity of the solid-liquid interface, it is a reasonable toassume that ϕ(T ) ≈ 0 on Γ. Therefore, I2,1 can be neglected in (31). Doing some algebra on I2,2

10 R. TAVAKOLI

results in:

I2,2 =

∫Γ

∇(ϕ(T ) κ(T )

)·(P (T )

|∇T |

)· n δT dx−

∫Ω

∇ ·(∇

(ϕ(T ) κ(T )

)·(P (T )

)|∇T |

)δT dx

:= I2,2,1 − I2,2,2

(32)

Similar to former reasoning, assuming that ϕ(T ) is negligible on Γ, the boundary term I2,2,1 in (32)can be ignored in the present study. Simplifying I4 results in:

I4 =

∫Γ

ρ cεe Θ (v · n) δT dx−∫Ω

∇ ·(ρ cεe Θ v

)δT dx =

∫Γ

ρ cεe Θ (v · n) δT dx

−∫Ω

ρ cεe Θ (∇ · v) δT dx−∫Ω

ρ cεe (∇Θ · v) δT dx−∫Ω

ρ Θ(∇cεe · v) δT dx

(33)

Considering that ∇cεe = cεe′ ∇T and using incompressibility constraint (∇ · v = 0), simplifies (33) to

the following form:

I4 =

∫Γ

ρ cεe Θ (v · n) δT dx−∫Ω

ρ cεe (v · ∇Θ) δT dx−∫Ω

ρ cεe′ Θ (v · ∇T ) δT dx

:= I4,1 − I4,2 − I4,3

(34)

Applying the integration by parts and divergence theorem to I5 two times results in:

I5 =

∫Γ

k Θ(∇(δT )

)·n dx−

∫Γ

k (∇Θ ·n) δT dx+

∫Ω

∇·(k ∇Θ

)δT dx := I5,1 − I5,2 + I5,3 (35)

Collecting all terms in LT results in the following equation:

LT δT =(I1 + I2,2,2 + I3 − I4,2 − I4,3 − I5,3

)︸︷︷︸in Ω

+(I4,1 − I5,1 + I5,2 + I8 + I9

)︸︷︷︸on Γl

+(I4,1 − I5,1 + I5,2 + I7

)︸︷︷︸on Γb

+(I4,1 − I5,1 + I5,2 + I6

)︸︷︷︸on Γt

(36)

Note that the terms inside each parenthesis in (36) are defined on the same spatial domain. Equatingthe first parenthesis of (36) to zero (in an almost everywhere sense) results in the following adjointPDE [cf. 83, 84]:

−ρ cεe (v · ∇Θ) = ∇ ·(k ∇Θ

)− 1

2ϕ′(T ) |κ(T )|2 −∇ ·

(∇(ϕ(T ) κ(T )

)·(P (T )

)|∇T |

)It is assumed that the temperature field is sufficiently smooth such that the above equation is welldefined and is uniquely soluble. Now, we examine the second parenthesis of (36). Because the testfunction δT is arbitrary (in fact, the second parenthesis should be equal to zero for every choice oftrace of δT on Γl), we take δT = 0 on Γl and vary term k

(∇(δT )

)·n on Γl. As a result, Θ should be

equal to Θl on Γl. Consequently, I5,1 = I8 on Γl and the second and forth terms in this parenthesiscancel each other. Because m is orthogonal to every cross section of strand by assumption, i.e.m⊥n on Γl, v · n = 0 on Γl. Therefore, I6,1 = 0 on Γl. Then, varying the trace of δT on Γl resultsin the following boundary condition:

−k ∇Θ · n = h Θ on Γl

Consider the third parenthesis of (36) here. Taking the trace of δT equal to zero on Γb and varyingk(∇(δT )

)· n on Γb, results in Θ = Θb on Γb. Then, varying the trace of δT on Γl results in the

following boundary condition:

−k ∇Θ · n = ρ cεe v Θ on Γb

Taking the trace of δT equal to zero and varying k(∇(δT )

)·n on Γt, results in Θ = 0 on Γt. Then,

varying the trace of δT on Γt results in: Θt = −ρ cεe Θ (v · n)− k (∇Θ · n) on Γt.


Therefore, equating LT to zero leads to the following adjoint problem:

−ρ cεe (v · ∇Θ) = ∇ ·(k ∇Θ

)− 1

2ϕ′(T ) |κ(T )|2 −∇ ·

(∇(ϕ(T ) κ(T )

)·(P (T )

)|∇T |

)in Ω (37)

Θ(x) = 0 on Γt (38)

−k∇Θ(x) · n(x) = ρ cεe(x) v Θ(x) on Γb (39)

−k ∇Θ(x) · n(x) = h(x) Θ(x) on Γl (40)

Comparing (37) to (14) illustrates that the direction of advection in the adjoint PDE is in thereverse advection direction of the direct PDE. It is important to note that considering the advectiondirection is essential for the stable numerical solution of (37) and (14).

Computing the first variations of L with respect to h, Ti and v results in:

Lh δh =

∫Γi

(Θ (T − Ti) + λU − λL

)δh dx (41)

LTi δTi =

∫Γi

(−Θ h+ µU − µL

)δTi dx (42)

Lv δv =

(ϑU − ϑL +

∫Ω

ρ cεe Θ (m · ∇T ) dx+

)δv (43)

Equating (41)-(43) to zero (in an almost everywhere sense) leads to the following equations:

Θ (T − Ti) + λU − λL = 0 on Γi (44)

−Θ h+ µU − µL = 0 on Γi (45)

ϑU − ϑL +

∫Ω

ρ cεe Θ (m · ∇T ) dx = 0 (46)

The complementarity conditions [cf. 82, 18, 19] corresponding to the inequality constraints canbe expressed as follows:

λL, λU ⩾ 0, λL(hL − h ) = λU (h − hU ) = 0 on Γi (47)

µL, µU ⩾ 0, µL(TL − Ti) = µU (Ti − TU ) = 0 on Γi (48)

ϑL, ϑU ⩾ 0, ϑL(uL − ub) = ϑU (ub − uU ) = 0 on Ω (49)

Summarizing the above derivations, the first order KKT necessary optimality conditions corre-sponding to the optimization problem (26) can be stated as follows:

equations : (14), (15), (16), (17)equations : (37), (38), (39), (40)equations : (44), (45), (46)equations : (18), (19), (20)equations : (47), (48), (49)

(50)

Following [85, 64, 65] the first order necessary optimality conditions can be alternatively expressedbased on the projected gradient approach, as bellow [c.f. proposition 2.4 of 65]. In this way, thecontrol constraints and complementarity conditions are managed by projection operators. There-fore, the set of constrained stationary points of augmented Lagrangian L, that are denoted by(T ∗,Θ∗, h∗, T ∗

i , v∗)T satisfy the following optimality conditions:

12 R. TAVAKOLI

(OC1)

ρcεev∗(m · ∇T ∗) = ∇ · (k∇T ∗) in Ω

T ∗(x) = TP on Γt

−k∇T ∗(x) · n(x) = 0 on Γb

−k∇T ∗(x) · n(x) = h∗(x)(T ∗(x)− T ∗

i (x))

on Γl

−ρcεev∗(m · ∇Θ∗) = ∇ ·

(k ∇Θ∗)− 1

2 ϕ′(T ∗) |κ(T ∗)|2

−∇ ·(

∇(ϕ(T∗) κ(T∗)

)·(P (T∗)

)|∇T∗|

)in Ω

Θ∗(x) = 0 on Γt

−k∇Θ∗(x) · n(x) = ρ cεe(x) v∗ Θ∗(x) on Γb

−k ∇Θ∗(x) · n(x) = h∗(x) Θ∗(x) on Γl

h∗(x) = PDh

[h∗(x)− γ Θ∗(x)

(T ∗(x)− T ∗

i (x))]

on Γl

T ∗i (x) = PDT

[T ∗i (x) + γ Θ∗(x) h∗(x)

]on Γl

v∗ = PDv

[v∗ − γ

(∫Ωρ cεe(x) Θ

∗(x) (m(x) · ∇T ∗(x)) dx

)]

where γ ∈ R+ is an arbitrary scaling factor and PX [a] denotes the orthogonal projection operatorof trial point a (with respect to L2 norm) onto the convex set X . It is easy to show that theorthogonal projection onto bound constraints is a convex separable optimization problem with anexplicit solution. Using simple derivations, the above mentioned projection operators have thefollowing explicit forms [cf. 86, 87]:

PDh[a] = min

(hU ,max(hL, a)

), PDT

[a] = min(TU ,max(TL, a)

), PDv [a] = min

(uU ,max(uL, a)

)By straightforward derivations, the first order necessary optimality conditions corresponding to

the optimization problem (27) has the following form:

(OC2)

PDh

[h∗(x) + γζh∇2

Γh∗(x)

]= h∗(x) on ∂Γl

PDT

[T ∗i (x) + γζi∇2

ΓT∗i (x)

]= T ∗

i (x) on ∂Γl

∇Γh∗(x) · t(x) = 0 on ∂Γl

∇ΓT∗i (x) · t(x) = 0 on ∂Γl

where ∇2Γ(·) denotes the laplacian operator, called the Laplace-Beltrami operator, on Γl.

5. Solution algorithm

The two-step projected steepest descent algorithm developed in [64, 65] is used in the presentstudy to solve the optimization problem (25). As it is already mentioned in section 3, this algorithmincludes two sequential steps. Starting from an appropriate initial guess, at the first step, (26) issolved by one iteration of the projected steepest descent algorithm [65, cf.]. In the second step, theregularization step, the unconstrained gradient flow of (27) is solved by one iteration of the Eulerimplicit time integration algorithm. The current solution of the first step is used as the initial guess(value) in the second step. As it has been proven in [64, 65], the second step , in fact , performsthe optimization on the null space of bound constraints (18) and (19). Therefore, if the initialguess of the second step satisfies (18) and (19), the solution of second step will be strictly feasiblewith respect to these bound constraints. The root of this property is due to this fact that, theunconstrained gradient flow of (27) leads to the diffusion equation with the homogeneous Neumannboundary condition and the Euler implicit time integration of this equation (in both continuousand discrete forms) satisfies the maximum principles. Assuming that the superscript (·)k counts theoptimization iterations, the above mentioned solution procedure is summarized in algorithm 1:


Algorithm 1. Regularized projected steepest descent

Step 0. Initialization: Given Ω, Γt, Γb, Γl, ρ, cp, Lf , k, TP , Ts, Tl, ε, hL, hU , TL, TU , vL, vU , Tt,σ, ζT , ζG, ζκ, ζh, ζi, initial guesses h0(x), T 0

i (x), v0 and stopping criteria kmax and δ. Moreover,

given spatial discretization data and convergence threshold for direct and adjoint PDEs solvers.h0 = PDh

[h0], T 0i = PDT

[T 0i ], v

0 = PDv [v0]. Set k → 0.

Step 1. Iterations:

1.1 Given hk(x), T ki (x) and vk, solve (14)-(17) for T k(x).

1.2 Given hk(x), T ki (x), v

k and T k(x), solve (37)-(40) for Θk(x).

1.3 Given hk(x), T ki (x), v

k(x), T k(x) and Θk(x), compute h(x), Ti(x), v as follows:

h(x) = hk(x) + αk

(PDh

[hk(x)− γk Θk(x)

(T k(x)− T k

i (x))]

− hk(x)

)(51)

Ti(x) = T ki (x) + αk

(PDT

[T ki (x) + γk Θk(x) hk(x)

]− T k

i (x)

)(52)

vk+1 = vk(x) + αk

(PDv

[vk − γk

(∫Ω

ρ cεe(x) Θk(x) (m(x) · ∇T k(x)) dx

)]− vk

)(53)

1.4 Given h(x) and Ti(x) compute hk+1(x) and T k+1i (x) by solving the following equations:

hk+1(x) =

[I − αkγkζh ∇2

Γ

]−1

h(x) in Γl, ∇Γhk+1(x) · t(x) = 0 on ∂Γl (54)

T k+1i (x) =

[I − αkγkζi ∇2

Γ

]−1

Ti(x) in Γl, ∇ΓTk+1i (x) · t(x) = 0 on ∂Γl (55)

Step 2. Stopping criteria: If ∥ hk+1 − hk ∥L∞(Γl) + ∥ T k+1i − T k

i ∥L∞(Γl) + | vk+1 − vk | < δ or

k = kmax stop and return (hk+1, T k+1i , vk+1)T , else k → k + 1 and goto step 1.

In algorithm 1, I denotes the identity operator and the scaling factor γk is computed based on theBarzilai-Borwein approach [cf. 85, 65]. Moreover, αk ∈ (0, 1] in step 1.3 of algorithm 1 is computedbased on the line search algorithm [cf. 88]. As it is shown in [64, 65], step 1.4 of algorithm 1 lifts

the distributed parameters h and Ti from L2(Γl) to H1(Γl). This method has some similarities withthe Sobolev gradient method [70, 71, 89] in which the projection of directional derivative onto H1

space is used to update control parameters. Although it is more easier to establish the convergencetheory of Sobolev gradient methods, it is not easy to take into account inequality constraints in thismethod. On the other hand, the presented approach naturally holds inequality constraints whilesimultaneously keeps the H1 regularity of the solution. Of course, to prove the convergence of thisapproach is not an easy job. The convergence of the presented algorithm is not studied theoreticallyin the present study. But, it is investigated numerically through a model problem.

It is easy to show that iterations (51), (52) and (53) keep the feasibility of solutions with respectto Dh, DT and Dv respectively (in fact , hold bound constraints). It is due to the convexity of design

spaces. For instance, consider (51). Assume that hk := PDh

[hk − γk Θk (T k − T k

i )]. Because Dh

is convex, projection onto Dh has unique solution hk ∈ Dh. Substituting hk into (51) results in:

h = hk + αk(hk − hk) = (1− αk)hk + αkhk

Because hk, hk ∈ Dh and αk ∈ (0, 1], due to the convexity of Dh, h will be in Dh.

14 R. TAVAKOLI

6. Two-dimensional Model Problem and Spatial Discretization

To study the practical behavior of presented algorithm, a simple two dimensional version of theoriginal optimization problem is considered in the reminder of this paper. Assuming the strandis sufficiently thick along the z-axis, the temperature gradient in this direction can be ignored.Therefore, the problem can be considered in the xy-plane. This approximation is feasible in thecase of continuous casting of billets and slabs [cf. 15]. Furthermore, it is assumed that ξ is parallelto the y-direction, i.e. the strand is vertical without the bending, and all physical properties areconstant in the solid and liquid phases, independent of the temperature. Considering the aboveassumptions, Ω is a rectangle in the xy-plane with height ly and width lx. The bottom left corner of

Ω is considered as the origin of coordinate system. Consequently m = (0,−j) in the xy-plane and

v(x) = −v, where j denotes the unit vector along positive direction of y-axis. Furthermore, v · ∇Tand v · ∇Θ terms in the direct and adjoint PDEs are simplified to −v ∂T

∂y and −v ∂Θ∂y respectively.

Assume that Ω is divided into an nx×ny uniform rectangular grids with the grid spacings ∆x and∆y along x and y directions respectively, i.e. ∆x = lx/nx and ∆y = ly/ny. The state variables, i.e.T (x) and Θ(x) are defined at the cell centers and the control variables (except v that is not a fielddata), i.e. h(x) and Ti(x) are defined on the edge centers of lateral boundary cells. Figure 1 (right)shows the schematic of computational domain corresponding to the model problem considered inthe present study.

Assume that Ui,j denotes the value of U(x) at x =((i − 1/2)∆x, (j − 1/2)∆y

), i.e., Ui,j :=

U((i − 1/2)∆x, (j − 1/2)∆y

). Moreover, Ui+1/2,j := U

(i∆x, (j − 1/2)∆y

), Ui−1/2,j := U

((i −

1)∆x, (j − 1/2)∆y), Ui,j+1/2 := U

((i − 1/2)∆x, j∆y

)and Ui,j−1/2 := U

((i − 1/2)∆x, (j − 1)∆y

).

The cell centered finite volume method (FVM) [90, 91] is used to discretize the direct and adjointPDEs in the present study. The diffusive terms are approximated by the second order centraldifference scheme, e.g.,(

∇ · (k ∇T )

)i,j

≈qi+1/2,j − qi−1/2,j

∆x+

qi,j+1/2 − qi,j−1/2

∆y, i = 1, . . . , nx, j = 1, . . . , ny

where,

qi+1/2,j =

k(Ti+1,j − Ti,j

)/∆x, 1 ⩽ i < nx, j = 1, . . . , ny

hi+1/2,j

(Ti,j − (Ti)i,j

), i = nx, j = 1, . . . , ny

qi−1/2,j =

k(Ti,j − Ti−1,j

)/∆x, 1 < i ⩽ nx, j = 1, . . . , ny

hi−1/2,j

(Ti,j − (Ti)i,j

), i = 1, j = 1, . . . , ny

qi,j+1/2 =

k(Ti,j+1 − Ti,j

)/∆y, 1 ⩽ j < ny, i = 1, . . . , nx

k(TP − Ti,j

)/∆y, j = ny, i = 1, . . . , nx

qi,j−1/2 =

k(Ti,j − Ti,j−1

)/∆y, 1 < j ⩽ ny, i = 1, . . . , nx

0, j = 1, i = 1, . . . , nx

The advection terms are discretized by the first order upwind scheme [cf. 90, 91]. For instance, theadvection term of the direct PDE (14) is discretized as follows:(

− v ρ cεe(T )∂T

∂y

)i,j

≈ −v ρ cεe(Ti,j

) Ti,j+1 − Ti,j

∆y


Note that the pulling direction, i.e. the advection velocity, is negative here (from positive to negativedirection of y-axis). Similarly, the advective term in the adjoint PDE (37) is approximated as follows:(

v ρ cεe(T )∂Θ

∂y

)i,j

≈ v ρ cεe(Ti,j

) Θi,j −Θi,j−1

∆y

Note that the advection velocity in the adjoint equation is in the reverse direction of that of directPDE.

The remained terms in the adjoint PDE are discretized by the second order central differ-ence scheme. To avoid possible singularity during the numerical solution, |∇T |i,j is replaced by√|∇T |2i,j + 10−10 in this study. Assume that ∆l is defined as follows, ∆l := min(∆x, ∆y). Accord-

ing to [92], the maximum absolute value of the mean curvature that can be resolved by a uniformCartesian grid with spacing ∆l is equal to 1

∆l . Therefore, if the computed value of κi,j locates

outside of interval [− 1∆l ,

1∆l ], it is merely replaced with either of − 1

∆l or 1∆l depending on the sign

of κi,j .After the spatial discretization, the numerical solution of direct PDE is equivalent to the solution

of a nonlinear system of equations. The source of nonlinearity in this system is due to the existenceof cεe(Ti,j) in the discretized form of advective term. To solve this system of nonlinear equations,a hybrid iterative procedure is used in the present study. In the first step of this procedure thatproceeds up to 50 iterations, the system of equations is linearized by the Picard method, i.e. thelast available value of Ti,j is used to compute cεe(Ti,j) and then the corresponding linear system ofequations is solved by an appropriate sparse direct solver. Moreover, the temperature is updatedby the under-relaxation factor 0.75 in this step. After approaching to the solution sufficiently, inthe second step of this hybrid procedure, the system of nonlinear equations is solved by the classicNewton method. The Newton iteration is terminated whenever the infinity norm of differencebetween two consecutive temperature fields is below of threshold 10−3. See [35] for further detailsabout this solution procedure. At the first iteration of optimization, the pouring temperature isused as the initial guess of this iterative procedure. In the subsequent optimization iterations,the temperature field of former iteration is used as the initial guess. Unlike the direct PDE, thediscretized form of adjoint PDE is linear (with respect to Θ). Therefore, it can be solved at theexpense of solving a system of linear equations. All linear system of equations are solved by thesparse direct solver UMFPACK2 [93] in the present study.

7. Numerical Results

The feasibility of the presented method is studied by numerical experiments in this section.The two-dimensional model problem that is mentioned in section (6) is used for this purpose. Inpractice, the lateral boundaries of strand are composed of the primary (mold) and secondary coolingregions. Considering that they are modeled by the interfacial heat transfer boundary conditionsin this work, the interval [0, ly] is decomposed into two regions, [0, ly − lm] (lower part, in fact ,the secondary cooling system) and [ly − lm, ly] (upper part, in fact the primary cooling system).The spatial dimensions and physical parameters corresponding to the model problem in this sectionare adopted from the realistic continuous casting of steel alloys. The dimensions of spatial domain,spatial discretization parameters, physical properties and boundary conditions of this model problemare presented in Table 1. The parameter σ is taken equal to 18 here as it appropriately concentratesthe weight of objective function within the freezing interval (see Figure 2). Moreover, ε = 1 isselected based on the numerical experiment (balancing between the smoothness of model and theconvergence of temperature field to ε → 0 limit).

The initial values of distributed design parameters and their corresponding upper and lowerbounds are listed in Table 2. It is well known that increasing the pulling speed, v, increases themetallurgical length [cf. 15, 61], equivalently the value of objective function in the present study.Therefore, it is expected that this parameter assumes its lower bound as a result of the optimization

2http://faculty.cse.tamu.edu/davis/suitesparse.html

16 R. TAVAKOLI

procedure in this work. On the other hand, decreasing the pulling speed decreases the productionrate and leads to coarsening of microstructure that are not desirable issues. Considering these facts,two scenarios are considered in this section. In the first one, v is assumed to vary with v0 = 0.025, vL= 0.01 and vU = 0.1 (units are in SI scale). In the second scenario, v is assumed to be equal to theconstant value 0.025 (i.e. vL = vU = 0.025), which is a common parameter in the steel continuouscasting process.

To evaluate the presented algorithm, 10 test cases with different values of the regularizationparameters ζh and ζi are considered in this section. In the first five test cases (#1-#5), the pullingvelocity is allowed to vary and in the next five test cases (#6-#10), it is fixed to the constantvalue v0. The regularization parameters ζh and ζi in test cases #1-#10 are shown in Table 3. Theconvergence thresholds (δ, kmax) = (10−6, 200) and (10−6, 400) are considered for test cases #1-#5and #6-#10 respectively.

Table 1. Spatial data, physical properties and boundary conditions correspondingto the model problem used in the present study (units are based on the SI scale).

ρ cp k Lf Ts Tl TP ε σ lx ly lm nx ny δ kmax

7500 700 32 260000 1360 1460 1480 1 18 0.2 30 0.8 40 200 10−6 200

Table 2. Initial values of distributed design parameters and their bounds corre-sponding to the model problem in this work (units are based on the SI scale).

region h0 hL hU T 0i TL TU

primary cooling system 1000 100 2000 400 100 600secondary cooling system 300 100 1000 80 25 100

Table 3. The regularization parameters ζh and ζi corresponding to test cases #1-#10.

case ζh, ζi case ζh, ζi case ζh, ζi case ζh, ζi case ζh, ζi1, 6 0.0 2, 7 0.0005 3, 8 0.001 4, 9 0.005 5, 10 0.01

Figure 3 shows the variation of objective functional with iteration in test cases #1-#10. Further-more, Table 4 shows the values of objective functional and depth of the liquid pool (metallurgicallength) at k = 0 and k = kmax in these test cases. According to Figure 3 the objective functionalsare reduced monotonically as iterations proceed. This observation numerically confirms the successand convergence of the presented model and its numerical treatment. Considering that the ob-jective function is defined as the weighted integral of absolute value of freezing interface curvature,its reduction implicitly implies approaching to the ideal solidification conditions, i.e. flated freezinginterface that is our actual goal. According to Table 4, the final values of objective functional areslightly decreased by the smoothing of distributed control fields, however, it is increased by increas-ing the regularization parameters of control variables. Therefore, the are optimal values for theseparameters that can be computed by numerical experiments. The variation of metallurgical length,that gives physical insight on the quality of solution, shows the similar trend like the objectivefunctional. It supports the feasibility of this specific choice of objective functional. Considering thevariation of pulling velocity with iteration in test cases #1-#5 illustrates that this parameter as-sumes its lower bound after a few iterations. As it is formerly mentioned, it is a well-known outcome


and was expected a-priori (in fact , the metallurgical length is proportional to the pulling speed).The optimal values of distributed control parameters, h and Ti, are plotted in Figures 4 and 5 fortest cases #1-#5 and #6-#10 respectively (plots are corresponding to the left hand side boundary ofΩ as the right hand side boundary has the similar distribution due to the y-axis symmetry of Ω andproblem boundary conditions). According to the plots, the smoothness of control parameter fieldsincreases by increasing the regularization parameters. Furthermore these fields are feasible withrespect to the corresponding bound constraints. Considering Figures 4 and 5, the interfacial heattransfer coefficient and temperature in the primary cooling system should be sufficiently large andsmall at optimal conditions respectively. It implies that, increasing the cooling power of the strandis recommended based on our results. On the other hand, the interfacial heat transfer coefficient isgradually decreased by distancing from the strand in the secondary cooling region. Furthermore, ttsrate of variation is decayed by distance such that it reaches to a plateau-like condition. Moreover,the interfacial temperature of the secondary cooling region assumes its lower bound value (i.e. 25)in the vicinity of the strand. It gradually increases by distancing from the strand, and reaches to aplateau-like condition too. Putting all together, the optimal conditions are obtained by increasingthe cooling power of primary cooling system as much as possible, and optimizing the heat transferconsitions corresponding to the secondary cooling system. In fact, the thermal optimization of thesecondary cooling system is the most important part of the optimization problem considered in thepresent study.

To get insight on the evolution of solution with iteration, the contour plot of the temperaturefield together with that of the liquid volume fraction field during the optimization cycles for testcases #1 and #6 are plotted in Figures 6 and 7 respectively ( other cases reveal the same trend).Note that the shape and width of freezing region can be directly inferred by liquidus and solidus iso-contour of the temperature field. Equivalently, they can be deduced by looking at the region locatedbetween the zero and one iso-contours of the liquid volume fraction field. According to the plots themode of freezing tends gradually to the directional solidification as iteration proceeds. Althoughthe freezing interface is not flat and has a wedge-like shape in practice, its height is decreased as aresult of optimization procedure. Comparing the contour plots of the temperature field and/or liquidvolume fraction fields at the initial and final stages of optimization in Figures 6 and 7 obviouslyillustrates approaching of the algorithm to our pre-defined goal. Furthermore, the depth of theliquid pool decreases by iteration. This observation supports the feasibility of the presented approachfrom a practical point of view. It is worth mentioning that the perfect unidirectional solidificationcan not be essentially attainable, because it can not be essentially included in the feasible designspace.

Table 4. The values of objective functional (F) and depth of the liquid pool (D)at k=0 and k = kmax corresponding to test cases #1-#10. Moreover, the optimalvalues of pulling speed (v) is shown at k = kmax for test cases #1-#5.

case : #1 #2 #3 #4 #5 #6 #7 #8 #9 #10F0 : 21758 22746 23733 31632 41505 21758 22746 23733 31632 41505

Fkmax : 6862 6454 6509 6962 7251 9266 9585 9622 9782 9910D0 : >30 >30 >30 >30 >30 >30 >30 >30 >30 >30

Dkmax : 9.90 9.60 9.75 9.90 10.20 16.10 17.10 17.10 17.10 17.25v : 0.01 0.01 0.01 0.01 0.01 - - - - -

18 R. TAVAKOLI

20 40 60 80 100 120 140 160 180 2006000

7000

8000

9000

10000

11000

12000

13000

Iteration

Obje

ctive F

unctional

#1

#2

#3

#4

#5

50 100 150 200 250 300 350 4000.9

1

1.1

1.2

1.3

1.4

1.5

1.6x 10

4

Iteration

Obje

ctive F

unctional

#6#7#8#9

#10

Figure 3. The variation of objective functional with iteration corresponding totest cases #1-#5 (up) and #6-#10 (bottom). For the better illustration, iterations10-200 and 20-400 are shown on the plots for cases #1-#5 and #6-#10 respectively.


300 400 500 600 700 800 900 1000 11000

5

10

15

20

25

30

h (W/m2 K)

y (

m)

↑#1

↑#2↑

#3↑#4

←#5

50 100 150 200 250 300 350 4000

5

10

15

20

25

30

Ti (Degree of Centigrade)

y (

m)

↑#1

↑#2

↑#3

← #4

←#5

Figure 4. The optimal distribution of interfacial heat transfer coefficient (up) andtemperature (bottom) for test cases #1-#5 (in fact, the distributed control param-eters at k = 200). The results are corresponding to the left hand side boundary.

20 R. TAVAKOLI

200 400 600 800 1000 1200 1400 1600 18000

5

10

15

20

25

30

h (W/m2 K)

y (

m)

↑#6

↓

#7↑#8

← #9

← #10

20 30 40 50 60 70 80 90 1000

5

10

15

20

25

30

Ti (Degree of Centigrade)

y (

m)

↓

#6

↓#7

↑#8

→#9

↑#10

Figure 5. The optimal distribution of interfacial heat transfer coefficient (up) andtemperature (bottom) for test cases #6-#10 (in fact, the distributed control pa-rameters at k = 400). The results are corresponding to the left hand side boundary.


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

900

950

1000

1050

1100

1150

1200

1250

1300

1350

1400

800

900

1000

1100

1200

1300

1400

600

700

800

900

1000

1100

1200

1300

1400

500

600

700

800

900

1000

1100

1200

1300

1400

500

600

700

800

900

1000

1100

1200

1300

1400

Figure 6. The contour plot of liquid volume fraction (top row) and temperature(bottom row) fields at optimization iterations 0, 5, 10, 50 and 200 (left to rightrespectively) corresponding to test case #1 in the present study.

22 R. TAVAKOLI

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

900

950

1000

1050

1100

1150

1200

1250

1300

1350

1400

900

950

1000

1050

1100

1150

1200

1250

1300

1350

1400

850

900

950

1000

1050

1100

1150

1200

1250

1300

1350

1400

800

900

1000

1100

1200

1300

1400

700

800

900

1000

1100

1200

1300

1400

Figure 7. The contour plot of liquid volume fraction (top row) and temperature(bottom row) fields at optimization iterations 0, 5, 10, 50 and 400 (left to rightrespectively) corresponding to test case #6 in the present study.


8. Summary

The optimization of heat transfer in the vertical continuous casting process is considered in thepresent study. The temperature field is computed by solving a quasi steady-state nonlinear heattransfer equation. The effect of phase change is modeled by the smoothed effective heat capacitymethod. The interfacial heat transfer coefficients and temperatures together with the casting rateare considered as the design parameters. The objective function is defined such that no priorinformation about the optimal temperature distribution is required. It is defined as the weightedintegral of squared mean curvature of freezing interface. The minimization of this function propelsthe solidification to the ideal condition, that is the mono-directional freezing. The first order adjointsensitivity analysis is performed and the optimization problem is solved by the regularized projectedsteepest descent algorithm. The feasibility of presented method was demonstrated by numericalexperiments using a two-dimensional model problem.

Acknowledgement

We would like to thanks two anonymous reviewers for their constructive comments that improvethe presentation of our work.

References

[1] L Harzheim and G Graf. A review of optimization of cast parts using topology optimization ii-topologyoptimization with manufacturing constraints. Struct Multidisc Optim, 31(5):388–399, 2006.

[2] Q Xia, T Shi, M Wang, and S Liu. A level set based method for the optimization of cast part. StructMultidisc Optim, 41(5):735–747, 2010.

[3] Q Xia, T Shi, M Wang, and S Liu. Simultaneous optimization of cast part and parting direction usinglevel set method. Struct Multidisc Optim, 44(6):751–759, 2011.

[4] A Gersborg and C Andreasen. An explicit parameterization for casting constraints in gradient driventopology optimization. Struct Multidisc Optim, 44(6):875–881, 2011.

[5] O Schmitt, J Friederich, S Riehl, and P Steinmann. On the formulation and implementation of geometricand manufacturing constraints in node–based shape optimization. Struct Multidisc Optim, pages 1–12,2015.

[6] S Liu, Q Li, W Chen, R Hu, and L Tong. H-dgtpa heaviside-function based directional growth topologyparameterization for design optimization of stiffener layout and height of thin-walled structures. StructMultidisc Optim, 52(5):903–913, 2015.

[7] G Allaire, F Jouve, and G Michailidis. Thickness control in structural optimization via a level setmethod. Struct Multidisc Optim, 53(6):1349–1382, 2016.

[8] Q Li, W Chen, S Liu, and L Tong. Structural topology optimization considering connectivity constraint.Struct Multidisc Optim, pages 1–14, 2016.

[9] J Liu and Y-S Ma. 3d level-set topology optimization: a machining feature-based approach. StructMultidisc Optim, 52(3):563–582, 2015.

[10] T Morthland, P Byrne, D Tortorelli, and J Dantzig. Optimal riser design for metal castings. MetalMater Trans B, 26(4):871–885, 1995.

[11] S Ebrahimi, D Tortorelli, and J Dantzig. Sensitivity analysis and nonlinear programming applied toinvestment casting design. Appl Math Model, 21(2):113–123, 1997.

[12] R Lewis, M Manzari, R Ransing, and D Gethin. Casting shape optimisation via process modelling.Mater Design, 21(4):381–386, 2000.

[13] R. Tavakoli and P. Davami. Automatic optimal feeder design in steel casting process. Comput MethAppl Mech Eng, 197(9):921–932, 2008.

[14] R. Tavakoli and P. Davami. Optimal riser design in sand casting process by topology optimization withsimp method i: Poisson approximation of nonlinear heat transfer equation. Struct Multidisc Optim,36(2):193–202, 2008.

[15] J. Hejazi. Ingot casting (in persian). Iranian Foundrymen Society, 1982.[16] H Fredriksson and U Akerlind. Materials processing during casting. John Wiley & Sons Inc, 2006.[17] J. Crank. Free and moving boundary problems. Clarendon Press, 1984.[18] F. Troltzsch. Optimal control of partial differential equations: theory, methods, and applications. Amer-

ican Mathematical Society, 2010.

24 R. TAVAKOLI

[19] A. Borzi and V. Schulz. Computational optimization of systems governed by partial differential equations.SIAM, 2012.

[20] N. Zabaras, Y. Ruan, and O. Richmond. Design of two-dimensional stefan processes with desiredfreezing front motions. Numer Heat Trans B, 21(3):307–325, 1992.

[21] M. Hinze and S. Ziegenbalg. Optimal control of the free boundary in a two-phase stefan problem. JComput Phys, 223(2):657–684, 2007.

[22] M. Hinze, O. Patzold, and S. Ziegenbalg. Solidification of a gaas meltoptimal control of the phaseinterface. J Cryst Growth, 311(8):2501–2507, 2009.

[23] O. Volkov and B. Protas. An inverse model for a free-boundary problem with a contact line: Steadycase. J Comput Phys, 228(13):4893–4910, 2009.

[24] M. Bernauer and R. Herzog. Optimal control of the classical two-phase stefan problem in level setformulation. SIAM J Sci Comput, 33(1):342–363, 2011.

[25] X. Peng, K. Niakhai, and B. Protas. A method for geometry optimization in a simple model of two-dimensional heat transfer. SIAM J Sci Comput, 35(5):B1105–B1131, 2013.

[26] H. Hu and S. Argyropoulos. Mathematical modelling of solidification and melting: a review. ModelSimul Mater Sci Eng, 4(4):371, 1996.

[27] V. Voller. An overview of numerical methods for solving phase change problems. Adv Numer HeatTrans, 1(4):341–380, 1996.

[28] C.P. Hong. Computer Modelling of Heat and Fluid Flow in Materials Processing. CRC Press, 2004.[29] R. Tavakoli and P. Davami. Unconditionally stable fully explicit finite difference solution of solidification

problems. Metal Mater Trans B, 38(1):121–142, 2007.[30] R. Tavakoli and P. Davami. A fast method for numerical simulation of casting solidification. Commun

Numer Meth Eng, 24(12):1723–1740, 2008.[31] G. Yang and N. Zabaras. The adjoint method for an inverse design problem in the directional solidifi-

cation of binary alloys. J Comput Phys, 140(2):432–452, 1998.[32] D. Tortorelli, J. Tomasko, T. Morthland, and J. Dantzig. Optimal design of nonlinear parabolic systems.

part ii: Variable spatial domain with applications to casting optimization. Comput Meth Appl MechEng, 113(1):157–172, 1994.

[33] R. Tavakoli and P. Davami. Optimal riser design in sand casting process with evolutionary topologyoptimization. Struct Multidisc Optim, 38(2):205–214, 2009.

[34] R. Tavakoli and P. Davami. Feeder growth: a new method for automatic optimal feeder design ingravity casting processes. Struct Multidisc Optim, 39(5):519–530, 2009.

[35] R Tavakoli. Smooth modeling of solidification based on the latent heat evolution approach. Int J AdvManuf Tech, inpress,, 88(9):3041–3052, 2017.

[36] B.G. Thomas. Modeling of continuous casting. The Making, Shaping and Treating of Steel, 11th ed.,Casting Volume, The AISE Steel Foundation, Pitsburg, 2003.

[37] E. Laitinen and P. Neittaanmaki. On numerical simulation of the continuous casting process. J EngMath, 22(4):335–354, 1988.

[38] S. Louhenkilpi, E. Laitinen, and R. Nieminen. Real-time simulation of heat transfer in continuouscasting. Metal Trans B, 24(4):685–693, 1993.

[39] J. Hill and Y-H. Wu. On a nonlinear stefan problem arising in the continuous casting of steel. ActaMech, 107(1-4):183–198, 1994.

[40] Y. Meng and B. Thomas. Heat-transfer and solidification model of continuous slab casting: Con1d.Metal Mater Trans B, 34(5):685–705, 2003.

[41] R. Hardin, K. Liu, C. Beckermann, and A. Kapoor. A transient simulation and dynamic spray coolingcontrol model for continuous steel casting. Metal Mater Trans B, 34(3):297–306, 2003.

[42] K. Zheng, B. Petrus, B. Thomas, and J. Bentsman. Design and implementation of a real-time spraycooling control system for continuous casting of thin steel slabs. In AISTech 2007, Steelmaking Confer-ence Proceedings, 2007.

[43] L. Guo, Y. Tian, M. Yao, and H. Shen. Temperature distribution and dynamic control of secondarycooling in slab continuous casting. Int J Min Metal Mater, 16(6):626–631, 2009.

[44] B. Petrus, J. Bentsman, and B. Thomas. Feedback control of the two-phase stefan problem, withan application to the continuous casting of steel. In Decision and Control (CDC), 2010 49th IEEEConference, pages 1731–1736, 2010.

[45] Z. Han, D. Chen, K. Feng, and M. Long. Development and application of dynamic soft-reduction controlmodel to slab continuous casting process. ISIJ Int, 50(11):1637–1643, 2010.


[46] B. Petrus, K. Zheng, X. Zhou, B. Thomas, and J. Bentsman. Real-time, model-based spray-coolingcontrol system for steel continuous casting. Metal Mater Trans B, 42(1):87–103, 2011.

[47] K. Jabri, E. Godoy, D. Dumur, A. Mouchette, and B. Bele. Cancellation of bulging effect on mouldlevel in continuous casting: Experimental validation. J Process Contr, 21(2):271–278, 2011.

[48] K. Jabri, D. Dumur, E. Godoy, A. Mouchette, and B. Bele. Particle swarm optimization based tuning ofa modified smith predictor for mould level control in continuous casting. J Process Contr, 21(2):263–270,2011.

[49] B. Petrus, J. Bentsman, and B. Thomas. Enthalpy-based feedback control algorithms for the stefanproblem. In Decision and Control (CDC), 2012 IEEE 51st Annual Conference, pages 7037–7042, 2012.

[50] K. Miettinen, M. Makela, and T. Mannikko. Optimal control of continuous casting by nondifferentiablemultiobjective optimization. Comput Optim Appl, 11(2):177–194, 1998.

[51] A. Lotov, G. Kamenev, V. Berezkin, and K. Miettinen. Optimal control of cooling process in continuouscasting of steel using a visualization-based multi-criteria approach. Appl Math Model, 29(7):653–672,2005.

[52] I. Nowak, A. Nowak, and L. Wrobel. Inverse analysis of continuous casting processes. Int J NumerMeth Heat Fluid Flow, 13(5):547–564, 2003.

[53] K. Miettinen. Using interactive multiobjective optimization in continuous casting of steel. Mater ManufProcesses, 22(5):585–593, 2007.

[54] D. S lota. Identification of the cooling condition in 2-d and 3-d continuous casting processes. NumerHeat Trans B, 55(2):155–176, 2009.

[55] R. Lewis, K. Morgan, H. Thomas, and K. Seetharamu. The finite element method in heat transferanalysis. John Wiley & Sons, 1996.

[56] Z.U.A. Warsi. Fluid dynamics: theoretical and computational approaches, 2nd edition. CRC Press,1999.

[57] C. Huang, M. Ozisik, and B. Sawaf. Conjugate gradient method for determining unknown contactconductance during metal casting. Int J Heat Mass Trans, 35(7):1779–1786, 1992.

[58] E. Majchrzak. Numerical simulation of continuous casting solidification by boundary element method.Eng Anal Bound Elem, 11(2):95–99, 1993.

[59] C Santos, J Spim, and A Garcia. Mathematical modeling and optimization strategies (genetic algorithmand knowledge base) applied to the continuous casting of steel. Eng Appl Artif Intel, 16(5):511–527,2003.

[60] M. Alizadeh, A. Jahromi, and O. Abouali. A new semi-analytical model for prediction of the strandsurface temperature in the continuous casting of steel in the mold region. ISIJ Int, 48(2):161–169, 2008.

[61] M. Sadat, A. Gheysari, and S. Sadat. The effects of casting speed on steel continuous casting process.Heat Mass Trans, 47(12):1601–1609, 2011.

[62] M. Hintermuller and I. Kopacka. A smooth penalty approach and a nonlinear multigrid algorithm forelliptic mpecs. Comput Optim Appl, 50(1):111–145, 2011.

[63] Y. Giga. Surface Evolution Equations: A Level Set Approach. Birkhauser, 2006.[64] R. Tavakoli. Multimaterial topology optimization by volume constrained allen–cahn system and regu-

larized projected steepest descent method. Comput Meth Appl Mech Eng, 276:534–565, 2014.[65] R. Tavakoli. Computationally efficient approach for the minimization of volume constrained vector-

valued ginzburg-landau energy functional. J Comput Phys, 295:355–378, 2015.[66] D. Tortorelli and R. Haber. First-order design sensitivities for transient conduction problems by an

adjoint method. Int J Numer Meth Eng, 28(4):733–752, 1989.[67] M. Giles and N. Pierce. An introduction to the adjoint approach to design. Flow Turbul Combust,

65(3-4):393–415, 2000.[68] K. Dowding and B. Blackwell. Sensitivity analysis for nonlinear heat conduction. J Heat Trans,

123(1):1–10, 2001.[69] S. Li and L. Petzold. Adjoint sensitivity analysis for time-dependent partial differential equations with

adaptive mesh refinement. J Comput Phys, 198(1):310–325, 2004.[70] B. Protas, T. Bewley, and G. Hagen. A computational framework for the regularization of adjoint

analysis in multiscale pde systems. J Comput Phys, 195(1):49–89, 2004.[71] B. Protas. Adjoint-based optimization of pde systems with alternative gradients. J Comput Phys,

227(13):6490–6510, 2008.[72] R Haftka. Techniques for thermal sensitivity analysis. Int J Numer Meth Eng, 17(1):71–80, 1981.[73] R Haftka and D Malkus. Calculation of sensitivity derivatives in thermal problems by finite differences.

Int J Numer Meth Eng, 17(12):1811–1821, 1981.

26 R. TAVAKOLI

[74] K. Dems and B. Rousselet. Sensitivity analysis for transient heat conduction in a solid bodypart i:External boundary modification. Struct Optim, 17(1):36–45, 1999.

[75] S Turteltaub. Optimal material properties for transient problems. Struct Multidisc Optim, 22(2):157–166, 2001.

[76] Y Gu, B Chen, H Zhang, and R Grandhi. A sensitivity analysis method for linear and nonlineartransient heat conduction with precise time integration. Struct Multidisc Optim, 24(1):23–37, 2002.

[77] A Gersborg-Hansen, M Bendsøe, and O Sigmund. Topology optimization of heat conduction problemsusing the finite volume method. Struct Multidisc Optim, 31(4):251–259, 2006.

[78] A Seppala, M Assad, and T Kapanen. Optimal structure for heat and cold protection under transientheat conduction. Struct Multidisc Optim, 36(4):355–363, 2008.

[79] E Dede, T Nomura, and J Lee. Thermal-composite design optimization for heat flux shielding, focusing,and reversal. Struct Multidisc Optim, 49(1):59–68, 2014.

[80] D Makhija and K Maute. Level set topology optimization of scalar transport problems. Struct MultidiscOptim, 51(2):267–285, 2015.

[81] P Coffin and K Maute. A level-set method for steady-state and transient natural convection problems.Struct Multidisc Optim, 53(5):1047–1067, 2016.

[82] K. Ito and K. Kunisch. Lagrange multiplier approach to variational problems and applications, vol-ume 15. SIAM, 2008.

[83] G. Allaire, F. Jouve, and A. Toader. Structural optimization using sensitivity analysis and a level-setmethod. J comput phys, 194(1):363–393, 2004.

[84] R Tavakoli. Optimal design of multiphase composites under elastodynamic loading. Comput Meth ApplMech Eng, 300:265–293, 2016.

[85] E. Birgin, J.M. Martınez, and M. Raydan. Nonmonotone spectral projected gradient methods on convexsets. SIAM J Optim, 10(4):1196–1211, 2000.

[86] S. Boyd and L. Vandenberghe. Convex optimization. Cambridge university press, 2004.[87] R. Tavakoli. On the coupled continuous knapsack problems: projection onto the volume constrained

gibbs n-simplex. Optim Letters (inpress), pages 1–22, 2015.[88] J. Nocedal and S. Wright. Numerical optimization. Springer, 2006.[89] J. Neuberger. Sobolev gradients and differential equations. Lecture Notes in Mathematics, 1670, 2010.[90] S. Patankar. Numerical heat transfer and fluid flow: Computational Methods in Mechanics and Thermal

Science. Hemisphere Publishing Corp., 1980.[91] J. Ferziger and M. Peric. Computational methods for fluid dynamics, volume 3. Springer Berlin, 1996.[92] S. Osher and R.P. Fedkiw. Level set methods and dynamic implicit surfaces. Springer Verlag, 2003.[93] T. Davis. Algorithm 832: Umfpack v4. 3—an unsymmetric-pattern multifrontal method. ACM T Math

Software, 30(2):196–199, 2004.

thermal optimization of the continuous casting …

Documents