IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 8, AUGUST 2010 3265

Grid-Enabled Tabu Search for Electromagnetic Optimization ProblemsSara Carcangiu, Alessandra Fanni, Anna Mereu, and Augusto Montisci

Electrical and Electronic Engineering Department, University of Cagliari, Cagliari 09123, Italy

The use of Grid Computing to solve electromagnetic optimization problems by means of the Tabu Search strategy is proposed inthis paper. In order to significantly reduce the prohibitive computational cost of the numerical analyses required by the majority ofiterative algorithms, two different grid-enabled Tabu Search strategies have been ported in the grid. Both strategies belong to the DomainDecomposition family: the decomposition of the search space and the decomposition of the neighborhood. The performances of thedifferent parallel implementations have been evaluated on some electromagnetic benchmarks.

Index Terms—Design of electromagnetic devices, finite element methods, grid computing, optimization methods.

I. INTRODUCTION

I N THE DESIGN of electromagnetic structures, it is oftennecessary to analyze the electromagnetic field distribution

using numerical techniques such as the Finite Element Method(FEM). In order to optimize the design, it is usual to apply iter-ative techniques to search the potentially optimal configurationin the solutions domain. Moreover, when the number of designparameters to be optimized is considerable, the number of elec-tromagnetic problems to be solved could be of the order of thou-sands. Because numerical electromagnetic solutions are oftencomputationally intensive, the use of numerical solutions duringthe iterative optimization process could be unfeasible. One wayto overcome this problem is to use approximating techniques,such as neural networks [1].

The main drawback of using approximating models is repre-sented by the approximation errors, which can alter the value ofthe solution corresponding to the same design parameters. An-other way to avoid the prohibitive computational time of itera-tive optimization is to use the new Grid Computing technology.Grid computing is a family of technologies for dynamically andopportunistically provisioning computing power from a pool ofresources. The Grid is a type of parallel and distributed systemthat enables the sharing, selection, and aggregation of geograph-ically distributed “autonomous” resources. These resources aredynamically assigned at runtime depending on their availability,capability, performance, cost and user’s quality-of-service re-quirements [2].

In this paper, a Tabu Search (TS) is proposed as search algo-rithm, and its parallel implementation on a computational gridis presented. TS is a family of meta heuristic procedures, whichperform the search for the optimal solution exploring the vari-able space and storing the features that correspond to bad pre-vious moves. Such features are labeled as tabu and they areavoided during the search for the optimum [3]. In literature, dif-ferent approaches have been adopted to implement a paralleliza-tion of the TS. In [4], a hierarchical classification of the par-

Manuscript received December 23, 2009; revised February 25, 2010;accepted March 04, 2010. Current version published July 21, 2010. Corre-sponding author: S. Carcangiu (e-mail: s.carcangiu@diee.unica.it).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMAG.2010.2045487

allel TS strategies is presented. Two main types of paralleliza-tion can be performed: the first one is the so-called Multiple TStask category, in which multiple TS algorithms are run in par-allel, which may differ for some parameters such as the initialsolution and the tabu list size; the second class is the so-calledDomain Decomposition.

In this work, we will compare the performances of two par-allel TS strategies both belonging to the Domain Decompositionfamily: the decomposition of the search space and the decom-position of the neighborhood. The decomposition of the searchspace implies that the domain space is decomposed in a numberof smaller domains. Each sub-domain has to be solved by sep-arate TS. The decomposition of the neighborhood is performedby assigning to each task a different portion of the neighborhoodto be evaluated. In this work the performances of the two typesof Domain Decomposition strategies will be compared.

II. GRID-ENABLED TABU SEARCH ALGORITHMS

The optimization problem under investigation consists infinding a set of design parameters that allows the device toproduce the desired electric or magnetic field in prefixed points,under feasibility constraints.

TS is a typically discrete algorithm, but it is possible to de-velop suitable strategies in order to apply it to solve optimizationproblems dealing with real-valued variables. The first step of theproposed methodology consists of subdividing the range of eachvariable into a finite number of sub-ranges uniformly distributed(whose size is a design parameter). A symbol of a discrete al-phabet is associated to each sub-range, so that every solutionin the continuous can be identified with an ordered sequence ofsymbols. According to this choice, the search strategy movesfrom one discrete solution to another by simply modifying thevalue of one variable at a time. At each iteration, the neighbor-hood associated to the current configuration is built performingall the possible moves that can be played starting from thatconfiguration. This simple paradigm can lead to some ineffi-ciencies and its success would be doubtful if only a straight-forward random extraction is performed in the subintervals togenerate a real-valued feasible solution. Local optimization al-gorithms, such as the mono dimensional Golden Search, maybe performed in each subinterval to find out the optimum valuecorresponding to a new current solution. The simple iterativescheme of the TS is enhanced introducing several rule-of-thumbcriteria such as aspiration (an element may be removed from

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3266 IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 8, AUGUST 2010

the Tabu List under certain conditions), intensification (deeplyexploration of a region looking promising), and diversification(leave a region that does not look promising).

A. Parallel Strategy

The idea for a straightforward parallelization scheme is to de-compose the search space into a set of disjoint subspaces, eachof them explored by an instance of TS on a different CPU. Whenall the processes terminate, the obtained results are written in asecondary memory. Then, a new process reads the secondarymemory and gathers the results to obtain the optimal solution.The resulting parallel algorithm is simple, because inter-processcommunication is not necessary, and only a final synchroniza-tion point is needed.

B. Master-Slave Strategy

The previously described parallelization scheme does notfully exploit the peculiarity of the TS meta-heuristics to makeuse of the past history of the search to enhance the solutionsdomain. An alternative use of the Grid-Computing paradigm ismade in this paper by decomposing the neighborhood. In fact,the most important requirement in using a TS algorithm con-sists in defining the set of admissible moves, i.e., in defining theneighborhood set of a given configuration. Our TS implementsCartesian moves, which consist of changing the value of onedesign parameter at a time. The exploration of a single variablecan be carried out by independent jobs, which means that,for each configuration in the neighborhood, a FEM analysishas to be performed in order to evaluate the fitness of thatsolution. It is important to notice that in the Grid the executionof cyclic jobs is not allowed. This leads us to design the coreof our algorithm to run locally in the user interface and to runthe parallel exploration of the neighborhood in the Grid. Thishas been possible by splitting the program in a master-slavearchitecture.

At the beginning, the master program sets up a catalogue thatis needed to obtain the Tabu List. In fact in the Tabu Searchalgorithm there is a list (Tabu List) that keeps trace of the pre-viously explored configurations. This is due to the idea to avoidcycling around an already explored configuration. To this end,in the master program a catalogue is initialized, to which alsothe slaves can have access. Each slave writes its own best so-lution into the catalogue. When all of them finish writing, themaster program reads and compares the data selecting the op-timal move. The creation of the database is done by using sev-eral Application Programming Interfaces (API) available forAMGA (Arda Metadata Grid Application) Metadata Catalogue[5]. After the creation of the “tabulist” catalogue, a while cyclestarts. It is governed by two parameters: the total number of callsto the objective function and the number of iterations of the TabuSearch that we want to perform. It is also possible to stop the al-gorithm when the objective function reaches a certain threshold.

Throughout the APIs, the master program launches the exe-cution of the slaves. During the first iteration the master sendsto the slaves the jobs by starting from an initial configuration.The jobs are parametric jobs where all slaves have to performthe same task, that is, given the current configuration, the neigh-borhood of one variable is explored and different FEM analyses

Fig. 1. Die press with electromagnet.

are run. When one slave finds a configuration that is a good can-didate to be stored in the catalogue, it checks the tabu list: if thatconfiguration is not already stored in the list, the slave computesthe hash code corresponding to the configuration and stores allthe information in the catalogue. When all the slaves terminatetheir jobs, the master program analyzes the entries written by theslaves in the catalogue and performs the optimal current move.

III. APPLICATIONS

The computing infrastructure we used is based on severalcomputation centers located in the main research institutionsand academies in Sardinia, Italy, and it consists of more than100 nodes 1U System x3455 (200 CPU AMD Opteron 2218,400 CPU core). The fiber optics connections allow to dynam-ically aggregate the distributed resources and to reach a pickaggregated computation power of some TeraFlops.

In order to compare the performance of the two grid-enabledTS codes, they have been used to find the optimal configurationof two electromagnetic devices test beds: the T.E.A.M Problem25 “Optimization of Die Press Model” [6], and the T.E.A.MProblem 22, “Optimal design of a superconducting magneticenergy storage (SMES) device” [7]. The bidimensional modelsand the computation of the objective functions of the two bench-marks are performed throughout calls to the ELFIN [8] finiteelement code. In the following, we will show the improvementin terms of computational time that has been reached by imple-menting the two algorithms in the computational Grid.

A. TEAM Workshop Problem 25

The die press with electromagnet for the orientation ofmagnetic powder is used for producing anisotropic permanentmagnet [6]. The shape of the die mold is controlled by a circleof radius R1 for the inner die and an ellipse represented byL2, L3 and L4 for the outer die (see Fig. 1). The problemconstraints are the ranges of the design parameters, which areshown in Table I. The four parameters have to be chosen so thatthe magnetic flux density be radial and equal to 0.35 T in thecavity where the magnetic powder is inserted.

The objective function to minimize is

(1)

CARCANGIU et al.: GRID-ENABLED TABU SEARCH FOR ELECTROMAGNETIC OPTIMIZATION PROBLEMS 3267

TABLE IRANGE OF THE DESIGN VARIABLES

Fig. 2. Objective function trend for different jobs obtained with Parallel TSstrategy.

(2)

where and are the components of the flux density inpoints along a circular arc of 45 having radius of

11.75 mm, whereas is the angle from the axis, and identi-fies the required values. Each design parameter can assume 100equidistant values within the range. In this way, the search spacedimension to be explored is 100 . In order to limit the computa-tion time, the size of the neighborhood can be changed. To avoidany loss of generality, we adopted a strategy in which dynam-ically varies in a given range following a ReactiveTS scheme [3]. In the present implementation these bounds are:

and .The optimization procedure terminates when a stable value

of the objective function is found, or the maximum number ofFEM analysis is reached. Said number has been fixed equal to10 000, which corresponds to 40 iterations.

As said before, the parallel strategy consists in subdivide thesearch space into disjoint sub-domains. In order to evaluate howthe search space decomposition affects the performance of thisstrategy, different subdivisions have been compared in terms ofnumber of iterations needed to the convergence of the optimiza-tion procedure. Fig. 2 shows the diagrams of objective functioncorresponding to some examples of subdivisions. As it can beseen, the number of different jobs (equal to the number of CPUs)and the way the search space is divided affect the convergencespeed, whereas the optimal value is practically the same in allthe cases.

The same stop criterion and number of jobs have been fixedfor the Master-Slave Strategy. Table II compares the perfor-mance of the proposed TS algorithms with those reported in lit-erature. As can be noted, the obtained values show a good agree-ment with those obtained by other authors. The performancesof the two Grid-enabled TS are reported in Table III in terms of

TABLE IITEAM PROBLEM 25: COMPARISON WITH RESULTS FROM OTHER AUTHORS

TABLE IIITEAM PROBLEM 25: PERFORMANCE OF THE

MASTER-SLAVE TS VERSUS THE PARALLEL TS

speedup and efficiency parameters usually used to evaluate theperformance of parallel algorithms [9]. As the Grid we used iscomposed by a number of identical CPUs, we can extend theconcept of speedup and efficiency, used in parallel systems, inthe grid environment. The speedup is the ratio of the executiontime of the serial algorithm when executed on one processor tothat when executed on processors , whereasthe efficiency is the ratio of the actual speedup versus the the-oretical maximum speedup equal to . Hereand are the total time required to run the TS algorithm ona single CPU, and on CPUs respectively. The maximum effi-ciency is obtained when the speedup is the closest to the theo-retical speedup [12]. The results highlight the superiority of theMaster-Slave approach with respect to the Parallel strategy.

The Master-Slave strategy has speedup limits and appears tosaturate. This is due mainly to communication overhead, andto some uncertain factors as instability of computing nodes, dy-namic change of Grid environment, total load on the Grid, and soon. The overhead can be neglected if the parallelized jobs havea high computational load. As the number of slaves increases,the speedup is more and more lower than the theoretical one.This phenomenon depends on the communication time , thenumber of variables , the size of the sub-ranges, the compu-tational cost of a single FEM calculation , and the fractionof calculus performed by the Master [12]. More specifically, thenumber for which the saturation begins depends on the ratiobetween and the total computation time , andon the ratio .

B. TEAM Workshop Problem 22

The Master-Slave strategy has been also used to optimize su-perconducting magnetic energy storage (SMES) device [7] inorder to store a significant amount of energy in magnetic fieldswith a fairly simple and economical coil arrangement which canbe rather easily scaled up. There are 8 design variables all re-lated to two coils (see Fig. 3): the radius of the coils 1 and 2, Rand R ; their height h and h ; their thickness d and d ; and

3268 IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 8, AUGUST 2010

Fig. 3. Configuration of the SMES device.

TABLE IVTEAM PROBLEM 22: PERFORMANCE OF THE MASTER-SLAVE TS

the value of the current densities J and J , respectively. In theimplemented TS algorithm each design parameter can assume100 equidistant values within its range.

The given current density and the maximum magnetic fluxdensity value on the coil must not violate, at any point of thecoils, the superconducting quench condition [7].

There are two objectives: maintaining a prescribed level forthe stored energy on the device; and minimizing the strayed fieldevaluated along the lines and in Fig. 3. These two conflictinggoals give indeed rise to a multi-objective problem, which isreduced to a single-objective one in the benchmark definition.The objective function to minimize is [7]

(3)

where the reference stored energy and stray field areMJ and mT. is defined as

(4)

where is evaluated along 22 equidistant points along thelines and in Fig. 3.

Coupling the Tabu Search algorithm with the ELFIN codefor energy and field calculations, the serial algorithm requiredabout 56 h and about 130 iterations. The stop criterion fixedthe maximum number of analysis FEM equal to 100 000. Theobtained optimal solution leads to an energy of 180 MJ and astray field of 8.9 T.

Table IV reports the performance of the Master-Slave strategyin terms of speedup and efficiency parameters, showing the

benefits of using the grid computing technologies. Note thatspeedup saturates and efficiency decreases when the number ofprocessors is much greater than the number of design parame-ters to optimize.

IV. CONCLUSION

In this paper, two grid-enabled implementations of the TSalgorithm, the Parallel and the Master-Slave strategies, havebeen tested and compared. The experiments showed that theporting of the algorithm on a computational Grid reduces thecomputation time with respect to the implementation on a singleprocessor, even considering the communication overhead, dueto the communication delay between the computing elements.The Master-Slave strategy exhibits better performance with re-spect to the Parallel strategy, but the former one is subject tothe saturation phenomenon, whereas the latter one seems tobe immune. As a consequence, the Master-Slave strategy ap-pears to be suitable when the computational resources are lim-ited, whereas the Parallel strategy allows one to exploit a greatnumber of processors.

ACKNOWLEDGMENT

This work makes use of results produced by the CybersarProject managed by the Consorzio COSMOLAB, a projectco-funded by the Italian Ministry of University and Researchwithin the PON 2000–2006. More information is available athtttp://www.cybersar.it.

The authors would like to thank Prof. Salvatore Alfonzettiand Dr. Emanuele Dilettoso for the availability of ELFIN code,and for the useful discussions.

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