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Ant Stigmergy on the Grid - Computer Systems @ JSI - IJS

————————————————————————————————————————————————- bestSoluti**on** = {p 1,...,p i,...,p D} change = True while change do change = False for i =1to D do if Evaluate({p 1,...,p i + p step i ,...,p D}) < Evaluate(bestSoluti**on**) **the**n change = True while Evaluate({p 1,...,p i + p step i ,...,p D}) < Evaluate(bestSoluti**on**) do bestSoluti**on** = {p 1,...,p i + p step i ,...,p D} p i = p i + p step i end while else if Evaluate({p 1,...,p i − p step i ,...,p D}) < Evaluate(bestSoluti**on**) **the**n change = True while Evaluate({p 1,...,p i − p step i ,...,p D}) < Evaluate(bestSoluti**on**) do bestSoluti**on** = {p 1,...,p i − p step i ,...,p D} p i = p i − p step i end while end if end for end while ————————————————————————————————————————————————- Figure 2. Functi**on** LocalSearch() mutual c**on**sistency between sub-col**on**y matrices is ensured. After **the** clients are stopped with ReceiveSoluti**on**s&StopAllClients() functi**on**, **the**y return **the**ir current best soluti**on** and am**on**g **the**m **the** server chooses **the** best **on**e. On this best soluti**on** **the** server performs **the** local search. The LocalSearch() pseudo code is given in Figure 2. 2.4 **Grid** implementati**on** The DMASA was designed to run **on** any kind of **Grid** that uses **the** TCP/IP protocol for communicati**on**. One can notice that **the** DMASA c**on**sists of a parallel part (runs **on** clients) and a sequential part (local search – runs **on** **the** server). C**on**sequently, **the** search process can **on**ly be speeded up in **the** parallel part while **the** sequential part remains c**on**stant. But we can go fur**the**r than that. The so-called software pipeline can be used. The pipeline c**on**sists of two stages: **the** first stage is **the** parallelized part of **the** algorithm and **the** sec**on**d stage is local search. So while **the** server is running local search, **the** clients can already search for new soluti**on**s. Therefore, new soluti**on**s can be acquired as fast as **the** local search is performed. Of course, with multiple servers, **the** time needed to acquire new soluti**on**s would be fur**the**r decreased. 3 C**on**tinuous steel casting 3.1 Ma**the**matical model Figure 3 represents a schematic picture of a c**on**tinuous casting machine. In **the** c**on**tinuous casting process **the** molten steel is poured into a bottomless mold which is cooled with internal water flow. The cooling in **the** mold extracts heat from **the** molten steel and initiates **the** formati**on** of **the** solid shell. The shell formati**on** is essential for **the** support of **the** slab after mold exit. After **the** mold **the** slab enters into **the** sec**on**dary cooling area in which it is cooled by water (mist) sprays. The sec**on**dary cooling regi**on** is divided into cooling z**on**es where **the** amount of **the** cooling water can be c**on**trolled separately. Figure 3. C**on**tinuous casting machine The simulati**on** model calculates **the** temperature

field of **the** steel slab as a functi**on** of casting parameters. We c**on**sider steady state casting c**on**diti**on**s, i.e. **the** parameters are c**on**stants in time. We denote **the** 3D geometry of **the** slab by V =Ω× [0,L Z ], where Ω=[0,L X ] × [0,L Y ] is a 2D cross-secti**on** of **the** slab and L Z is **the** length of **the** strand. Moreover we denote by L M **the** length of **the** mould. We divide **the** boundary Γ = ∂V into four parts: Γ 0 =Ω×{0}, Γ N = {(x, y) ∈ ∂Ω :x =0∨ y =0}×[L M ,L Z], Γ S = {(x, y) ∈ ∂Ω :x ≠0∧ y ≠0}×[0,L Z) ∪ Ω ×{L Z}, Γ M = {(x, y) ∈ ∂Ω :x =0∨ y =0}×[0,L M ]. The ma**the**matical model for **the** temperature field T = T (x, y, z, t) of **the** slab can be written as: (7) ∂H(T ) + v ∂H(T ) − ∆K(T )=0 ∂t ∂z inV×(0,t f ], T = T 0 **on** Γ 0 × (0,t f ], ∂K(T ) ∂n + h(T − T w)+σɛ(T 4 − T 4 ext) =0 **on**Γ N × (0,t f ], ∂K(T ) ∂n =0 **on**Γ S × (0,t f ], ∂K(T ) = Q ∂n **on** Γ M × (0,t f ], T (x, y, z, 0) = T 0 in V. (8) Here n is **the** unit vector of outward normal **on** ∂V, h is **the** heat transfer coefficient, v is **the** casting speed, and T w ,T ext are known temperatures. The σ is **the** Stefan-Boltzmann c**on**stant and ɛ is **the** emissivity. The cooling efficiency Q in **the** mould is known c**on**stant and t f is **the** simulati**on** time. H(T )andK(T ) are **the** temperature dependent enthalpy and Kirchoff functi**on**s, see [15]. The Eq. 8 is discretized by **the** finite-element method (FEM) and **the** corresp**on**ding n**on**linear set of equati**on**s is solved by relaxati**on** iterative methods [8]. More detailed descripti**on** of discretizati**on** and c**on**structi**on** of FEM-matrices is presented in [4]. We note that in our method it is sufficient to c**on**struct **on**ly 2D- and 1D-matrices. Therefore, it is obvious that our model is computati**on**ally much more efficient than in **the** case of using **the** ordinary 3D-brick elements. 3.2 The coolant flow optimizati**on** The sec**on**dary cooling area of **the** c**on**sidered casting device is divided into nine z**on**es. In each z**on**e, cooling water is dispersed to **the** slab at **the** center and corner positi**on**s. Target temperatures are specified for **the** slab center and corner in every z**on**e. Water flows should be tuned in such a way that **the** resulting slab surface temperatures match **the** target temperatures. Formally, a cost functi**on** is introduced to measure **the** differences between **the** actual and target temperatures. It is defined as c(T ) = 1 N ∑ Z 2 ( l i(Ti center − Ti center∗ ) 2 + i=1 N ∑ Z + l i(Ti corner − Ti corner∗ ) 2 ) (9) i=1 where N z denotes **the** number of z**on**es, l i **the** length of **the** i-th z**on**e, Ti center and Ti corner **the** slab center and corner temperatures, while Ti center∗ and Ti corner∗ **the** respective target temperatures in z**on**e i. The optimizati**on** task is to minimize **the** cost functi**on** over possible cooling patterns (water flow settings). Water flows cannot be set arbitrarily, but according to **the** technological c**on**straints. For each water flow, minimum and maximum values are prescribed. Table 1. The target temperatures and water flow intervals T ∗ p min i p max i Pos. Z**on**e no. [ ◦ C] Parameter [m 3 /h] [m 3 /h] 1 1050 p 1 7.1 26.1 2 1040 p 2 22.8 57.5 c 3 980 p 3 13.3 39.9 e 4 970 p 4 1.5 7.9 n 5 960 p 5 2.7 10.0 t 6 950 p 6 0.8 6.5 e 7 940 p 7 0.7 5.9 r 8 930 p 8 1.0 5.8 9 920 p 9 1.2 6.2 1 880 p 10 7.1 26.1 2 870 p 11 22.8 57.5 c 3 810 p 12 13.3 39.9 o 4 800 p 13 1.2 3.5 r 5 790 p 14 2.4 4.4 n 6 780 p 15 2.4 2.9 e 7 770 p 16 0.7 5.9 r 8 760 p 17 1.0 5.8 9 750 p 18 1.2 6.2 Table 1 shows an example of **the** prescribed target temperatures T ∗ and water flow intervals for c**on**tinuous casting of a selected steel grade analyzed in this study. The slab cross-secti**on** (L X ×L Y ) in this case was 1.70 m × 0.21 m and **the** casting speed v =0.23 m/s.

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