63
ISSN 1392 - 1207. MECHANIKA. 2010. Nr.2(82)
Application of GRID computing for optimization of grillages D. Šešok*, R. Belevičius**, A. Kačeniauskas***, J. Mockus**** *Institute of Mathematics and Informatics, Akademijos str. 4, 08663 Vilnius, Lithuania *Vilnius Gediminas Technical University, Saulėtekio al. 11, 10223 Vilnius, Lithuania, E-mail:
[email protected] **Vilnius Gediminas Technical University, Saulėtekio al. 11, 10223 Vilnius, Lithuania, E-mail:
[email protected] ***Vilnius Gediminas Technical University, Saulėtekio al. 11, 10223 Vilnius, Lithuania, E-mail:
[email protected] ****Institute of Mathematics and Informatics, Akademijos str. 4, 08663 Vilnius, Lithuania:
[email protected] 1. Introduction Some engineering optimal design problems despite of significant advance in optimization methods in recent years still require an unacceptably long computation time or even can not be solved to the very end. A good example for this is one specific class of optimization problems in civil engineering – optimal design of grillage-type foundations, which are called “grillages” throughout this paper. This problem of global optimization is also attractive from the mathematical point of view, because here the ideal global solution (or a lower bound for the minimum) is known in advance, so the designer can always estimate obtained results. Grillages are the most popular and effective scheme of foundations, especially in the case of weak grounds. The grillages consist of supporting piles and connecting beams. Exhaustive technical details on the grillages can be found in [1] and [2]. The idealizations on real grillages taken in the present mathematical model are listed in [3 - 4]. The optimal grillage should meet twofold criteria: the number of piles should be minimal, and connecting beams should receive minimal possible torques. In fact, here we encounter two separate optimization problems: search for the minimal number of piles and search for the minimal volume of beams. Both problems can be integrated into one with a compromise objective function. We assume that the characteristics of piles and connecting beams are given and consider the first optimization problem. Initial data for the grillage optimization problem are the following: the geometrical scheme of connecting beams; cross-section data of all the beams (area, moments of inertia); material data of all the beams (material in one beam is treated as isotropic); positions of immovable piles (if any); maximum allowable reactive force at any pile; minimum possible distance between adjacent piles; stiffnesses of a pile (vertical, rotational); loading data. Active forces can be applied in the form of concentrated loads and moments at any point on the beam, or in the form of distributed trapezoidal loadings at any segment of the beam. The results of optimization are the number of required piles and their positions. To solve this problem it is necessary to find such a placement of the given number of piles that reactive forces do not exceed carrying capacities of the piles. If such a placement is not possible the number
of piles should be increased. Therefore we formulate a problem of placement of piles searching for appropriate pile positions under connecting beams. In an ideal grillage reactive forces at all the piles are identical. Practically this is hardly feasible, particularly when a designer introduces the so-called “immovable supports” that have to retain their positions and cannot change it during optimization process. Some technological constraints may also make the ideal scheme nonachievable, for example the distance between adjacent piles cannot be too small due to the specific capacities of a pile driver. In the present work we do not consider the immovable supports and allow for a pile to take whatever position in the grillage, thus typically the piles are not placed at the joints of grillage. This fact confines the pile placement problem scope to a low-rise buildings without significant overturning moments due to horizontal thrust, e.g. due to earthquake loading or wind loads. We assume that the characteristics of all piles are equal and therefore the objective function for minimization is the maximal vertical reactive force at a pile. An even distribution of reactive forces among all piles indicates an ideal grillage. Our experience shows that the objective function for practical grillage optimization problems possesses many local minima points. Another complicated trait of the problem is that usually the objective function is very sensitive to the positions of piles: sometimes even a small alteration of one position leads to a significant change of the value of the objective function. All this makes the placement of piles in practical grillage a difficult global optimization problem. In our previous work we tried to approach the problem as “black-box” global optimization by covering methods [5 - 6] and genetic algorithms [7] but the results were not inspiring even for small-scale problems. One possible reason is that heuristic information on the problem is not employed in the case of “black-box” optimization. More promising results have been achieved when starting from random search and heuristic random search, then progressing to metaheuristics (genetic algorithms and especially simulated annealing) [8 - 9]. However, in all solution strategies of the grillage optimization problem a huge number of required numerical experiments is the main obstacle. Naturally, distributed computations are an obvious option for significantly increasing computational capabilities. In all mentioned optimization algorithms the numerical experiments are fully independent of each other. No communication between the processes is necessary therefore such optimization problems are ideally suited for grid computing [10]. The resources for optimal design of grillage-type foundations can
64 be provided, e.g., by European Grid Infrastructure [11] based on gLite middleware [12]. The design of distributed algorithms and the software deployment in grid presents a new challenge to computational scientists [13-15]. Recent progress resulted in the dramatic increase of computational capacity, which approached the petaflop level. Hereunder we describe the optimization problem formulation, software deployment in the grid, and present numerical results of optimization of 10 practical grillages which were optimized earlier [8] using one-processor computers. 2. Problem formulation The optimization problem is formulated as follows: f * = min f ( x )
(1)
x∈D
where f(x) is a nonlinear objective function of continuous variables f : ℜ n → ℜ , n is the number of design parameters x defining positions of the piles, D ⊂ ℜ n is a feasible region of design parameters. Besides of the global minimum f * one or all global minimizers x * : f x * = f * should be found. No assumptions on unimodality are included into formulation of the problem – many local minima may exist. In this paper the maximal vertical reactive force at a pile is considered as the objective function
( )
f ( x ) = max Ri ( x ) i =1,..., N a
(2)
where N a is the number of piles, Ri (x ) is the reactive force at i-th pile. Since a supporting pile may reside only under connecting beams, there are evident restrictions on the positions of piles: during the optimization process the piles can move only along the connecting beams. Therefore, a two-dimensional beam structure of the grillage is “unfolded” to a one-dimensional construct, and the piles are allowed to range through this space freely. In such a formulation one design parameter corresponds to a position of one pile in the one-dimensional construct (n = Na). The backward transformation restores the positions of piles in the two-dimensional beam structure of the grillage. The constraints for the design parameters are as follows
0 ≤ x i ≤ L,
i = 1,..., N a
(3)
where xi is a design parameter defining the position of ith pile. L is the total length of all beams in the grillage. If the minimal possible distance δ between adjacent piles is specified, there are additional constraints xi − x j ≥ δ, i ≠ j
(4)
where xi are one-dimensional coordinates of the piles and
xi − x j denotes the distance between piles. To cope with
this constraint a penalty is included in the objective function. A finite element program is used as a “black-box” routine to the optimization program for solution of direct problem to find reactive forces in the grillage. In the direct problem the connecting beams in the grillage are idealized as the beam elements, while the piles are treated as supports, i.e. finite element mesh nodes with given elastic boundary conditions. Since time of optimization crucially depends on solution time of the direct problem, fast problem-oriented original FORTRAN programs with a special mesh pre-processor have been developed and used. The beam elements have 2 nodes with 6 degrees of freedom each (3 displacements along the coordinate axes and 3 rotations about these axes). The stiffness matrix for element can be found in many textbooks, e.g. by [16]. The main statics equation is:
[K ]a {u}a = {F }a
(5)
where a stands for the ensemble of elements (not shown in equation below), {u} are the nodal displacements, and {F} are the active forces. The reactive forces at piles are available after obtaining the nodal displacements Ri = ∑ ⎡⎣ K ij ⎤⎦ u j .
(6)
j
3. Solution strategy employing grid infrastructure As mentioned earlier, the grillage optimization is an ideal application for grid computation since for each computer in the grid one task can be assigned with no communication required between them. The grillage optimization program consists of two independent parts: the optimization algorithm (in the form of starting program), and finite element program connected to optimization program as the “black-box” routine. In the present work the simulated annealing algorithm is employed as the optimization program because our earlier results show its potential compared with other stochastic global optimization algorithms. Solution scheme in grid is shown in Fig. 1. Each processor obtains its task, which is completely independent from the others. Each process reads its data files and produces its result files in ASCII format. The produced files are not large, therefore, it is not necessary to store them in storage elements. At the end of computations the result files are automatically processed in order to sort numerical solutions, considering the computed values of objective function, and to store the best results. Accuracy required for industrial design sometimes requires running from 100 to 2000 processes. It is quite difficult to handle such amount of tasks and result files, therefore good software including graphical user interface is required for automatic job submission, monitoring and result processing. Thus, the discussed application requires infrastructure providing extendable computing resources as well as advanced services for job handling and management of computed results. Grillage optimization software has been deployed in BalticGrid infrastructure by using Gridcom [17]. It is a simple web interface for launching complex applications
65 needed to prepare the data, control the application, merge the results, etc. The third idea defines that applications are launched in HTTP-accessible directories. There is no sophisticated application–system and system–user interfaces. Any file included in an application or created by the application automatically gets its URL and can be accessed through a web browser. According to this, application authors can simply include HTML or PHP pages with links to result or intermediate files, with different ActiveX or Java file viewers for result visualization.
Fig. 2 Gridcom operation principles Fig. 1 Solution scheme in grid on grid. Gridcom splits input data into intervals; generates and submits as many jobs as needed. It can scatter parametric jobs into simple jobs or resubmit aborted jobs. Finally, Gridcom collects, merges and visualizes the computed results. All these functions are performed automatically, including transparent upload of large files to storage elements. Gridcom also allows group workspace access representing groupware properties. Every Gridcom instance can have many read-only users. They can read application and grid log files, study and download application results but can not launch applications on the grid. This is useful for sharing the results and for troubleshooting as administrators can look for any required log file themselves. Since Gridcom is a web-based system, it can be accessed from any computer or cell-phone, connected to the Internet. Gridcom is the main component of BalticGrid SIGs (Special Interest Groups) portal [18]. This portal provides an international, interdisciplinary forum for the exchange of ideas about the field of grid computing in Baltic region. Members of a SIG portal arrange regular meetings and discussions. The portal is used for information dissemination, for education purposes and for different grid forums. The architecture of Gridcom is illustrated in Fig. 2. It can be defined by three main ideas: full level system; application controlling code is executed on a system computer; HTTP-accessible directory interface. The first two ideas define that Gridcom application code consists of two parts - job code and application controlling code. Application controlling code is executed directly on system computer allowing freeing user computer from long operations, such as submission of thousand jobs. Controlling code is regular Linux executable code. That allows using shell scripts and gives almost absolute flexibility for application authors to implement any actions
Gridcom launches specially developed Gridcom applications. Once created, an application can be launched many times with different input data. Generally it takes some hours to adapt grid application for Gridcom. Web form for grillage optimization application is shown in Fig. 3. Thus, the form contains 13 fields to be filled. Work name – unique experiment title whereby the experiment results will be found in the future. Job Count – number of processors for one experiment. Gridcom allows up to 1000 processors for each experiment. In case more processors are needed, several experiments should be performed. For example, if we need to perform 4000 runs of an algorithm, it is better to intend performing 4 experiments with 1000 runs each or even 8 experiments with 500 runs each. Each processor creates a separate result data catalog; a large number of catalogs impedes the system functioning. Compiler g95 tag – the value of this field is entered automatically according to the “VO-balticgrid-EDEVEL-G95-0.91”, which indicates availability of compiler g95 on the node (thus, program is distributed only to the nodes having required compiler). MRS iteration number – number of iterations for obtaining the initial solution for simulated annealing (SA) algorithm. The initial solution is chosen using the Modified Random Search (MRS) algorithm suggested by authors [8]. Here, in essence, one constraint is added to the random search algorithm: the distance between two adjacent piles can not be less than the given allowable distance. Such a heuristic modification is motivated by the fact that due to the usual distribution of loading over the grillage beams, the piles also should be spread over the whole space of grillage. If the constraint is violated, the structure is considered to be nonfeasible, and the objective function is not evaluated. The best solution found is used later as a starting solution for the SA algorithm. SA1 iteration number, …, SA4 iteration number – numbers of iterations for SA algorithm with the corresponding amplitudes SA1 amplitude, …, SA4 amplitude
66 (for description of solution algorithm via SA refer to the 4th chapter). Output frequency – output frequency of intermediate results. The code is automatically compiled by the provided shell script and executed. The obtained result files are transferred to the Gridcom server and stored in different catalogs. Then, the result files are automatically processed by the special shell script incorporated in Gridcom as the post-processing stage. This shell script sorts results considering the computed values of objective function and saves only the user defined number of files that store the best values. A text file including the sorted values of objective function is prepared in order to provide for users the ability to investigate all aspects of global optimization process. Fig. 4 Job execution statistics on small PC clusters of Lithuanian Grid (LitGrid) [19] collected from less than hundred processors. 4. Algorithm and numerical results
Fig. 3 Web form for grillage optimization application Screenshot displayed in Fig. 4 illustrates job execution statistic. As seen from an example (Fig. 4), only two jobs were aborted and one job failed while one thousand jobs were accomplished successfully. Only 45% of jobs were computed on large computer clusters having more than 1000 processors. The big part of computations was carried
In this chapter numerical results of optimization of 10 practical grillages on the grid, which were earlier optimized with 7 different deterministic and stochastic global optimization algorithms using one-processor computers [8], are presented. Earlier the most promising results were achieved with SA, therefore this algorithm is chosen also for computations on the grid. All optimized grillages are of small-to-medium scale, requiring from 17 to 55 piles. Data for these problems are obtained from several Dutch design bureaus (courtesy of Consultancy W.F.O.B.V., Paauw B.V. Aannemingsbedrijf V. Dijk, Bouwtectuur West Friesland, Stabo Bouw B.V., Aannemingsbedrijf A. Tuin Den Helder and others) which use the analysis and design package MatrixFrame. MatrixFrame has some capabilities for optimization of pile placement schemes employing local search methods [20]. One trait is common to all these problems: the current optimization routine of MatrixFrame was not capable to yield even a rational scheme of pile placement. Algorithm of numerical experiments. The SA is a stochastic global optimization algorithm, i.e., generally the results of optimization can not be repeated. Premature convergence of algorithm to the local solution can be avoided starting from a good initial solution and choosing appropriate parameters of the algorithm. Therefore our optimization algorithm allows obtaining a starting solution on the basis of some random search, and allows exploring a wide range of SA parameters. Thus, one optimization experiment is performed in 5 stages. Firstly, the initial solution for SA is sought for which the Monte - Carlo method is employed with one experiential heuristics that the distance between adjacent piles must exceed the value half of grillage length divided by the number of piles. For obtaining of initial solution MRS iteration number (Fig. 3) of random tests are executed. The best obtained result is used as the initial solution. Secondly, the initial solution is improved using SA. In this stage the SA1 iteration number iterations are performed. In each iteration the new pile position is chosen
67 from interval [old_position_value-SA1 amplitude, old_position_value+SA1 amplitude]. The 3rd, 4th and 5th stages allow exploring other sets of SA parameters. These stages repeat the operations of the 2nd stage with different parameters SA2 iteration number, SA2 amplitude, …, SA4 iteration number, SA4 amplitude. Two numerical experiments with different strategies were performed. First, each grillage optimization problem is solved using a large number of independent numerical tests but with comparatively short lifespan of each population of solutions (iteration numbers in Fig. 3). Second, each grillage is optimized using lesser number of independent tests but with longer lifespan of population. Experiment No 1. 10 earlier mentioned grillages [8] were reoptimized. As earlier, 5000 SA iterations were employed for the optimization of each grillage. However, now the grid infrastructure allows performing much more computations. Thus, in order to exploit all capabilities of grid that was used here, 1000 independent runs of algorithm were undertaken instead of former 28 runs.
earlier. Generally, with grid the solution is improved till about 10% per similar calculation time on each node of the grid. Experiment No 2. Now the second strategy is chosen for the solution of the same 10 grillages: only 100 independent tests were performed for each task, however, with 20 times more iterations, 100000 in total. Since all numerical tests on the nodes of grid must be accomplished in real time (usually, in a few hours), and some computers in the grid may be incapable, it was not possible to explore longer lifespan of each test. The SA parameters (note, only the iteration numbers differ from those in Table 1) are presented in the Table 3. Table 3 SA parameters for the 2nd experiment Job Count Initial temperature Annealing rate MRS iteration number SA1 iteration number SA1 amplitude SA2 iteration number SA2 amplitude SA3 iteration number SA3 amplitude SA4 iteration number SA4 amplitude
Table 1 SA parameters for the 1st experiment Job Count Initial temperature Annealing rate MRS iteration number SA1 iteration number SA1 amplitude SA2 iteration number SA2 amplitude SA3 iteration number SA3 amplitude SA4 iteration number SA4 amplitude
1000 5.0 2.0 200 1400 0.6 1400 0.2 1000 0.05 1000 0.01
Table 4 compares results of the 2nd experiment with other our results. Table 4 The best results of the 2nd experiment
Table 2 presents the best obtained results together with our former results, ideal solutions, and size of the problem – number of the piles. Table 2 The best results of the 1st experiment No of grillage 1 2 3 4 5 6 7 8 9 10
Number of piles 25 18 31 31 30 37 23 34 17 55
Ideal solution 307.47 104.12 101.85 101.24 97.51 97.53 287.35 236.28 244.71 349.05
Solution (28 tests) 339.30 106.36 107.25 106.80 102.00 117.26 298.11 357.67 253.00 463.34
100 5.0 2.0 500 24500 0.6 25000 0.2 25000 0.05 25000 0.01
Solution (1000 tests) 328.57 105.61 108.30 105.98 101.07 109.71 292.05 325.24 249.11 449.93
Thus, the 35 times larger number of numerical tests allows improving solution for almost all grillages (the best results in the Table 2 are highlighted). Theoretically, with the same algorithm and with a larger number of independent tests we should obtain at least not worse results for each task. The peculiar results for the 2nd grillage may be explained by the fact that the algorithm employed in numerical experiment No 1 differs slightly from the one used
No of grillage
Ideal solution
Solution (28 tests)
1 2 3 4 5 6 7 8 9 10
307.47 104.12 101.85 101.24 97.51 97.53 287.35 236.28 244.71 349.05
339.30 106.36 107.25 106.80 102.00 117.26 298.11 357.67 253.00 463.34
Solution (1000 tests) 328.57 105.61 108.30 105.98 101.07 109.71 292.05 325.24 249.11 449.93
Solution (100 tests) 322.32 104.90 104.21 103.11 99.04 104.00 289.08 288.94 245.76 433.42
The obtained results clearly show the advantage of the present strategy: the better results were achieved for all tasks. Since the 5% discrepancy to the ideal solution is usually sufficient for engineering practice, we may assert that for 7 grillages from 10 the desired solution was found. The modifications of algorithm or solution strategy still are necessary for successful solution of awkward grillages No 6th, 8th, and 10th. All the “black-box” program and data files for all grillages are freely available from the site: http://soften.ktu.lt/~mockus/1-global.php?id=1 -> Civil Engineering Example -> GRILL. The links Data1, Data2, … , Data10 allow access to the initial data files Grillage_1.dat, Grillage_2.dat, … , Grillage_10.dat. The last number in the file name indicates
68 the number of grillage and coincides with the column “No of grillage” in the Tables 2 and 4. Files GRILLAGE.dll, GRILLAGE.exp and GRILLAGE.lib are the Fortran library for MS Windows operating system. The library contains two functions: ANALYSIS( Length, SupNum, AllowReaction ) calculates some characteristics of the grillage: Length – REAL*8, the total length of grillage beams (output), SupNum – INTEGER, necessary number of supports (output), AllowReaction – REAL*8, allowable reaction force (output). OBJECTIVE_FUNCTION(X_KOORD,SupNum,Reaction) calculates maximum reaction force depending on the given coordinates of supports: X_KOORD – REAL*8, array of coordinates of supports in 1D space (to be supplied), SupNum – INTEGER, necessary number of supports (to be supplied), Reaction – REAL*8, maximum reaction force (output). In order to optimize the desired grillage, one should rename the file Grillage_N.dat to an aa_in.dat, and should place it into the catalog of optimization program. Also, the library GRILLAGE.lib should be included into the project. The link Example gives access to the file Example.f90, which shows how to evaluate 10 values of objective function depending on the randomly generated supports’ coordinates. Objective function calculation software for Linux operating system is accessible by the link “Archive, Fortran library for Linux”. Program will generate two auxiliary files aa_mes.dat and aa_out.dat. File aa_mes.dat is intended for the analysis of possible errors (if any). File aa_out.dat is intended for temporary data output. 5. Conclusions Computer hardware that is common to a typical civil engineering design bureaus and a reasonable computation time for engineering practice does not allow solution of the problem of pile placement to the very end with any global optimization algorithm applied. However, first, the fact that ideal solution usually is not required for the engineering purposes, and second, the increasing accessibility and popularity of the distributed computing in contemporary engineering practice makes it possible solving global optimization problems possessing to 30 - 40 design parameters. It can be expected, that allowing longer lifespan of calculations on the nodes of grid may increase the size of solvable optimization problems. Large computing resources available on European grid infrastructure can be simply and successfully used for solution of actual problems of industrial design. Acknowledgements This work was supported by Research Council of
Lithuania (contract No MOS-10/2010 “Use of stochastic global optimization methods in engineering”). Authors also wish to thank the Lithuanian State Science and Studies Foundation within the projects on S20/2009 “Use of stochastic global optimization methods in engineering” and B-03/2007 "Global optimization of complex systems using high performance computing and GRID technologies" and the joint venture Matrix Software for inspiration, financial support and permission to use the software package MatrixFrame®. The computational resources provided by the BalticGrid-II infrastructure (FP7- INFRA-2007-1.2.3: eScience Grid infrastructures contract No 223807) are gratefully acknowledged. This work is also supported by project GridTechno (Lithuanian State Science and Studies Foundation, B-07043). References 1. Bowles, J.E. Foundation Analysis and Design, 5th Ed. -McGraw-Hill International. 1995.-1024p. 2. Reese, L.C., Isenhower, W.M., Wang, S.-T. Analysis and Design of Shallow and Deep Foundations. -Wiley. 2005.-608p. 3. Belevičius, R., Valentinavičius, S. Optimization of grillage-type foundations. -J. of Civil Engineering and Management. -Vilnius: Technika, 2000, Nr.6(6), p.255261. 4. Belevičius, R., Valentinavičius, S., Michnevič, E. Multilevel optimization of grillages. -J. of Civil Engineering and Management. -Vilnius: Technika, 2002, Nr.8(2), p.98-103. 5. Baravykaitė, M., Belevičius, R., Čiegis, R. One application of the parallelization tool of Master-Slave algorithms. -Informatica. -Vilnius: Institute of Mathematics and Informatics, 2002, Nr.13(4), p.393-404. 6. Čiegis, R., Baravykaitė, M., Belevičius, R. Parallel global optimisation of foundation schemes in civil engineering. -Lecture notes in computer science, Applied Parallel Computing. State of the Art in Scientific Computing: 7th International Conference, PARA 2004, Lyngby, Denmark, June 20-23, 2004. Revised Selected Papers. Editors: Jack Dongarra, Kaj Madsen, Jerzy Wasniewski. -Berlin/Heidelberg: Springer, 2006, v.3732, p.305-312. 7. Belevičius, R., Šešok, D. Global optimization of grillages using genetic algorithms. -Mechanika. -Kaunas: Technologija, 2008, Nr.6(74), p.38-44. 8. Belevičius, R., Ivanikovas, S., Šešok, D., Valentinavičius, S., Žilinskas, J. Optimal placement of piles in real grillages: experimental comparison of optimization algorithms. Engineering optimization. 2010 (under review). 9. Šešok, D., Mockus, J., Belevičius, R., Kačeniauskas, A. Global optimization of grillages using simulated annealing and high performance computing. -J. of Civil Engineering and Management. 2010 (in press). 10. Foster, I., Kesselman, C. Grid: Blueprint for a New computing infrastructure (1st ed.). -San Francisco: Morgan Kaufmann publishers. 1998.-675p. 11. EGEE: http://www.eu-egee.org/ 12. gLite: http://glite.web.cern.ch/glite/ 13. Gentzsch, W. Grid and cloud portals for design, simulation and collaboration. -Parallel, distributed and grid
69 Computing for engineering (Eds. Topping, B. H. V., Ivanyj, P.). -Saxe-Coburg Publications. 2009, p.83-116. 14. Jankovski, V., Atkočiūnas, J. MATLAB implementation in direct probability design of optimal stell trusses. -Mechanika. -Kaunas: Technologija, 2008, Nr.6(74), p.30-37. 15. Stoian, V., Nitulescu, M., Pana C. Hexapod leg control algorithm in fault conditions. -Mechanika. -Kaunas: Technologija, 2008, Nr.3(71), p.57-61. 16. Zienkiewicz, O.C., Taylor, R.L. The Finite Element Method for Solid and Structural Mechanics, 6th Ed. -London: Elsevier Butterworth-Heinemann. 2005. -736p. 17. Juozapavičius, A., Piatov, D. Gridcom: simple grid user interface for complex applications. 2009. http://sig.balticgrid.org/gridcom/docs/articles/gridcom0.10.pdf 18. SIG: http://sig.balticgrid.org/ 19. LitGRID: http://www.litgrid.lt/ 20. Belevičius, R., Valentinavičius, S., Weener, R.J. Optimisation of grillage-type foundations. -Proc. of 2nd Worldwide ECCE Symposium “Information and Communication Technology in the Practice of Building and Civil Engineering”, 6-8 June, 2001, Finland. -Helsinki, p.251-256.
D. Šešok, R. Belevičius, A. Kačeniauskas, J. Mockus APPLICATION OF GRID COMPUTING FOR OPTIMIZATION OF GRILLAGES Summary The paper addresses one engineering design problem, the optimization of pile placement schemes under grillage-type foundations. Aim of the optimization is to find pile placement scheme where the number of piles is minimal and reactive forces at all piles do not exceed the given allowable reaction. This global optimization problem is solved here using the simulated annealing algorithm which proved to be one of the most promising algorithms for grillage optimization. Optimization of real-life grillages using PC requires usually unacceptably long for engineering practice computation time, therefore present paper investigates the possibilities of employing grid computing for that. Numerical examples presented clearly show potential of grid computing in engineering design problems. Д. Шешок, Р. Белявичюс, А. Каченяускас, Й. Моцкус ПРИМЕНЕНИЕ GRID ВЫЧИСЛЕНИЙ ДЛЯ ОПТИМИЗАЦИИ РОСТВЕРКОВ
D. Šešok, R. Belevičius, A. Kačeniauskas, J. Mockus
Резюме
ROSTVERKINIŲ PAMATŲ OPTIMIZAVIMAS GRID SKAIČIAVIMAIS
В статье обсуждается одна из задач инженерного проектирования – оптимизация распределения свай в ростверках. Целью оптимизации является получение таких схем распределения свай, при которых общее количество свай в ростверке было бы минимальным, а реакции опор, возникающие в сваях, не превышали бы заранее заданной допустимой реакции. Для решения данной задачи глобальной оптимизации используется метод имитации отжига, который по результатам предшествующих исследований оказался одним из наиболее эффективных алгоритмов для оптимизации ростверков. Оптимизация реальных ростверков, используя обычный персональный компьютер, требует слишком больших, неприемлемых для инженерной практики временных затрат, поэтому в статье для оптимизации применяется технология вычислений GRID. Представленные численные примеры явно демонстрируют большие возможности применения GRID технологии для решения задач инженерного проектирования.
Reziumė Straipsnyje nagrinėjamas vienas inžinerinio projektavimo uždavinys – polių išdėstymo po rostverkiniais pamatais schemų optimizavimas. Optimizavimo tikslas – gauti tokias polių išdėstymo schemas, kur polių skaičius būtų minimalus, o atraminės reakcijos visuose poliuose neviršytų nustatytos leistinosios reakcijos. Šis globaliosios optimizacijos uždavinys sprendžiamas naudojantis vienu efektyviausių sijynams optimizuoti atkaitinimo modeliavimo algoritmu. Realių sijynų optimizavimas įprastais asmeniniais kompiuteriais inžinerinei praktikai yra nepriimtinas dėl pernelyg gaišlių skaičiavimų, todėl šiame straipsnyje tam bandoma taikyti GRID skaičiavimo technologijas. Pateikti skaitiniai pavyzdžiai akivaizdžiai rodo dideles GRID technologijų galimybes sprendžiant inžinerinio projektavimo uždavinius.
Received December 14, 2009 Accepted March 15, 2010
