Combinatorial synthesis approach employing graph networks

Transcription

1 ville online t VN NGINRING INFORMTIS dvnced ngineering Informtics (00) omintoril synthesis pproch employing grph networks Offer Shi, *, Noel Titus, Krthik Rmni Mechnicl ngineering School, Tel-viv University, Isrel School of Mechnicl ngineering, Purdue University, West Lfyette, IN, US Received ugust 00 strct The pper proposes methodology to ssist the designer t the initil stges of the design synthesis process y enling him/her to employ knowledge nd lgorithms existing in grph network theory. The proposed method comprises three min stges: trnsforming the synthesis prolem into grph theoretic prolem; devising the topology possessing specil engineering properties corresponding to the system requirements; finding the geometric configurtion of tht topology tht will possess the desired properties. To clrify the ide nd to demonstrte its generlity, the pproch is presented through three synthesis cse studies from different engineering domins: electricl networks, sttics nd kinemtics. s is highlighted in the pper, the pproch of employing grph theory in the synthesis process offers severl unique dvntges. mong these dvntges re: gining generl perspective on different synthesis prolems from different engineering domins y trnsforming them into the sme grph prolem; employing the sme grph lgorithms for different synthesis prolems; estlishing the existence of configurtions with specil properties solely from the topology of the system; trnsferring knowledge nd methods etween different engineering disciplines for oth the topology nd the geometry genertion steps. Ó 00 lsevier Ltd. ll rights reserved. Keywords: omintoril synthesis; Grph theory; ssur groups; Self-dul grphs; Sttics; Kinemtics; lectricl circuits. Introduction esign synthesis of engineering rtifcts hs een significnt reserch re, especilly in the fields of structures [,,,9], mechnicl systems [,,,,], MMS nd VLSI good summry of which is ville in []. Techniques rnging from serch lgorithms, stochstic nd grdient sed optimiztion, evolutionry nd genetic lgorithms, hve een used in comintion with discrete nd continuum pproches, ond grph methods, grph grmmrs, design rules nd more for the purpose of synthesis of these systems. However, s is evident from the pulictions cited ove nd others, no universl or common synthesis method hs een pplied to different engineering domins. ch hs its own distinct pproch tht utilizes the properties of the * orresponding uthor. Tel.: +9 0 ; fx: mil ddress: shi@eng.tu.c.il (O. Shi). engineering domin, through synthesis technique suited to tht domin. The current pper proposes common representtion method pplicle for different engineering systems, rendering possiility for devising stndrd synthesis process for multi-domin engineering system. The method employs grph theoreticl models to trnsform the design prolem from the engineering domin to the domin of discrete mthemtics. The engineering design prolems of vrious engineering domins then ecome prolems in grph theory, solutions of which my employ the vriety of comintoril lgorithms developed in the field. similr pproch hs lredy een employed for developing dvnced engineering nlysis techniques, s reported in [,,,] nd more, while the current pper focuses entirely on the issues of synthesis. The outline of the pper is s follows. Section descries the steps in the proposed pproch followed in the pper. It is explined how synthesis -0/$ - see front mtter Ó 00 lsevier Ltd. ll rights reserved. doi:0.0/j.ei

2 O. Shi et l. / dvnced ngineering Informtics (00) prolem is trnsformed into terms of grphs nd then solved through known grph-theoretic theorems nd lgorithms. This process is then followed y the topology genertion of the physicl system tht possesses unique properties relevnt to the originl synthesis prolem. Section introduces the notion of using common representtion schem for different engineering domins through exmples in structures, kinemtics nd electricl circuits. This is followed up in Section with n explntion of how systems in different engineering domins cn e represented in the common mthemticl lnguge of grph theory nd shows further elucidtion of the exmples introduced in Section. Section descries the process of synthesizing solution to given engineering prolem strting with root grph structure. This section highlights one of the unique properties of this pproch in which the sme lgorithm cn e pplied for solving prolems from different engineering domins. Section employs comintoril methods to chrcterize the configurtions of engineering systems possessing the specil engineering properties tht correspond to the system requirements. n exmple in structures is used to demonstrte how the comintoril pproch goes eyond topology, to incorporte lso geometricl tretment of the prolems.. Generl description of the proposed comintoril synthesis method The method introduced in this pper consists of three min stges (Fig. ): trnsforming the synthesis prolem into prolem in grph theory; devising the proper topology which possesses the required engineering property; the configurtion synthesis step, which involves finding the geometry tht exhiits the desired engineering property. notle feture of this pproch is tht during the synthesis process, the engineering prolems re trnsformed Theorems in Grph Theory Grph representtions Known prolems in Grph Theory lgorithms in Grph Theory Prolem trnsformtion step efining the synthesis prolem in terms of grph network theory. Topologicl synthesis step - evising the topologies with the specil properties. Geometric synthesis step Finding the geometry of the network possessing the desired engineering properties. Fig.. Min spects of the proposed synthesis method nd the mthemticl steps employed throughout the proposed synthesis process. into known prolems in grph theory, the solutions of which cn e otined from existing grph theory lgorithms. Fig. presents the flow digrm for the suggested synthesis pproch in top down mnner, while outlining the mthemticl foundtion ech level requires. The right side of the figure lists the three min stges, while the left side of the figure lists the mthemticl sujects employed through the process. The dshed rrows connecting the left nd the right sides of the figure indicte which mthemticl sujects re employed t ech step of the proposed synthesis procedure. The mthemticl sujects from grph theory tht re employed in the synthesis process re s follows:. Theorems in grph theory system of theorems, xioms nd rules developed in grph theory underlying the mthemticl ehvior of grphs [0].. Grph representtions grphs ugmented y dditionl mthemticl vriles nd properties. The vriles nd properties ssocited with such representtions re employed in mpping the vriles nd properties underlying the physicl ehvior of engineering systems. s is explined in [], different sets of the ugmented vriles nd their properties yield different types of grph representtions. The sic ide of replcing the tretment of n engineering system with the tretment of network grph is thoroughly elorted in [,,,].. Known prolems in grph theory s ny other mthemticl field, grph theory is chrcterized y the set of prolems treted through its mthemticl pprtus [0]. From the point of view of the current pper, the prolem in grph theory, once converted from the engineering domin, is seen s prolem fully formulted in the terminology of comintoril mthemtics. The known grph theory prolems, or t lest their specil cses, re tretle through known grph-theoreticl tools, s descried elow.. lgorithms in grph theory known methods nd lgorithms developed to solve prolems in grph theory [0,]. The stges of the proposed synthesis methodology tht re supported y the ove mthemticl foundtion re descried s follows:. Prolem trnsformtion step This step involves the formultion of the trnsformed engineering synthesis prolem to purely mthemticl prolem in grph theory. The trnsformtion is performed through the trnsformtion rules used when constructing the grph representtion for the engineering system [,]. The type of grph representtion chosen depends oth on the engineering domin for which the originl synthesis prolem hs een formulted nd on the nture of the specific design requirements nd constrints.

3 O. Shi et l. / dvnced ngineering Informtics (00) Once the prolem formultion in the terminology of grph theoretic form hs een otined, the prolem is correlted with known grph theoreticl prolems. If such correltion is not strightforwrd, one my employ known theorems of grph theory s intermedite steps in the prolem formultion until known prolem in grph theory is otined.. Topologicl synthesis step t this stge, grph theoreticl lgorithms re employed to devise the solution to the prolem formulted in the previous stge. s will e illustrted y the synthesis prolems in the current pper, the grph theoretic equivlent of n engineering prolem will in most cses e grph topology stisfying set of mthemticl requirements. Some of the prolems will involve developing topology from scrtch, while others require no more thn djusting given topology.. Geometric synthesis step The finl step ridges the gp etween the found topology of system nd its design description. In some cses when the engineering domin is one dimensionl or sclr, such s in electric networks, this step is irrelevnt nd cn e omitted. In cses of multidimensionl systems such s structures, their geometricl properties re mpped s the vector vriles nd constnt properties ssocited with the elements of the corresponding grph representtion. s is shown lter in the pper, determining or djusting these vector properties nd vriles cn e chieved y employing lgorithms known in grph theory. s cn e seen, the initil steps of the synthesis methodology shown here re sed on grph theory entities, while the lter steps involve the engineering elements. The lst step dels exclusively with geometry nd configurtion issues. The proposed pproch cn e used oth s stnd lone (s is demonstrted in the cse studies shown in the pper), ut for higher effectiveness cn lso e used in comintion with known engineering design techniques. For instnce, t the stge of topology genertion one cn e ided y stochstic methods such s genetic lgorithms nd simulted nneling [,,]. lterntively, introducing new connections nd joints in systemtic mnner during topology genertion is chllenge, primrily due to the dynmic nture of the prolem, s pointed out y [9]. The grph theoretic pproch llows for dynmic representtion of the topologicl synthesis prolem. dditionlly, since specific engineering properties re utilized in generting new topologies, the topology synthesis prolem is systemtic procedure. Hence, in the pproch presented in the pper, oth the topologicl synthesis nd geometricl synthesis steps re menle to linking with evolutionry optimiztion methods.. The synthesis cse studies used in the pper In this section three synthesis prolems re introduced while their solutions re shown in the susequent sections, using the pproch descried in Section. To highlight the generlity of the pproch, the prolems were chosen from three different engineering domins: electricl networks, structurl mechnics nd kinemtics. In ddition, the solutions of these three prolems will demonstrte tht new reltions etween different engineering domins cn e reveled, so tht the sme lgorithms cn e pplied to solve prolems in different engineering domins. It should e pointed out tht, of the presented exmples, only the two prolems from sttics nd kinemtics involve geometricl considertions, while the electricl prolem is onedimensionl... Synthesis of multi-functionl electricl networks The first cse study is the synthesis of n equivlent electricl circuit tht exhiits the property of interchngeility etween the voltges nd currents of the originl circuit with the currents nd voltges, respectively, of the equivlent circuit. In these specific electricl circuits, when the voltge nd current sources re switched nd the resistnces ecome conductivities with the sme vlues, specil property is reveled, nmely the voltge ehvior of the elements in the originl circuit ecomes identicl to the current ehvior of the elements in the second circuit. Fig. depicts two circuits where the first one (Fig. ) possesses the ove property while the seemingly similr circuit in Fig. does not. The synthesis procedure to construct such types of electricl circuits is given in Section.. R R I R R R R R V R R R R I R R R V R R Fig.. xmple of n electricl circuit tht possesses the interchngeility property etween voltges nd currents () nd one tht does not ().

4 O. Shi et l. / dvnced ngineering Informtics (00).. Synthesis of trusses with specil geometricl properties In the previous prolem we referred to the synthesis of networks tht possess specil topologicl property. In the second prolem, we extend this methodology nd introduce wy to construct topology tht possesses specil geometricl properties. In other words, this pproch is directed towrds resoning s to whether given topology fter reliztion my hve geometric configurtion with specil engineering property oth in sttics nd kinemtics. Specificlly, in the domin of structures, for exmple, y using the proposed methodology it will e shown how to conclude from the topology of structure whether there exists geometric configurtion in which n internl force in one of its rs exerts forces in ll the other rs nd in which ll the inner joints hve infinitesiml motions. In the pper, this property will e referred to s the self-stress moility property. Using this property it will e possile to conclude tht the truss in Fig. possesses this property, while the truss in Fig., since only joint of the truss cn move, does not possess the self-stress moility property. In the following sections it will e shown how topologies of trusses possessing the self-stress moility property cn e constructed. Fig. depicts topology of truss, for which there exist numerous configurtions possessing the self-stress moility property. On the other hnd, for the topology of Fig. there is no configurtion with this property for ny choice of the lengths of the rs or their inclintions... Synthesis of kinemtic linkges with specil locking position property The third synthesis exmple exploits nother unique feture of the comintoril synthesis pproch which enles utiliztion of solutions to engineering prolems in one engineering domin to synthesize solutions in nother engineering domin. This is done y performing trnsformtions on the grph network representing the system in the originl engineering domin. This ide is illustrted y trnsforming the results otined in the previous synthesis prolem in sttics, into different engineering domin, nmely kinemtics. Fig.. The topology of sttic truss for which there exist configurtions with the self-stress moility property () nd the topology of the truss for which there is no such configurtion (). In the current kinemtic synthesis prolem it is desired to construct topologies of linkges for which there exists t lest one configurtion in locking position which possesses two properties: First property: The configurtion will e such tht the force exerted y the driving link is resisted y forces in ll the other links of the system. Second property: ny of the links in the linkge should e le to drive the linkge out of the locked position. These two properties re importnt in vriety of pplictions, including the synthesis of deployle structures []. Fig. depicts two kinemtic systems; the left one () does not possess the ove properties while the right one () does. The topologicl resoning nd the lgorithm to construct such kinemtic systems pper in Section.. In the next section the method of trnsforming the synthesis prolem into prolem in network grph theory will e explined.. Trnsforming the synthesis prolems into grph theory prolems This initil step trnsforms the synthesis prolem from n engineering domin into grph theory prolem, V/0 V/0 V/0 Fig.. The configurtion of truss () tht hs the self-stress moility property while the configurtion in () does not.

5 O. Shi et l. / dvnced ngineering Informtics (00) Fig.. Kinemtic linkge tht does not hve the specil locking property () nd one tht does (). enling the use of knowledge nd lgorithms tht exist in grph nd network theories to solve the engineering synthesis prolems. In mny cses, different synthesis prolems ecome the sme grph prolem, enling the sme lgorithm to e employed to solve different prolems tht re currently considered to e completely unrelted. lthough the synthesis prolems in this pper elong to three different engineering domins, the lgorithms for solving them will e shown to e similr. This opens up new wy for trnsforming knowledge etween engineering domins nd otining generl perspective of the synthesis prolem... The multi-functionl electricl synthesis prolem in grph terminology Throughout the pper, prticulrly in this section, we use one of the sic topics in grph theory grph dulity. For more detils on grph dulity the reders re referred to introductory textooks on grph theory, such s [0]. For every plnr grph G =(V, ) (i.e., grph tht cn e drwn in the plne in such mnner tht its edges do not intersect), there exists nother grph, G *, termed dul grph tht is constructed s follows: for every edge in the originl grph there is corresponding edge in the dul grph nd for every fce (circuit without inner edges) in the originl grph there exists vertex in the dul grph. ny two vertices re djcent in the dul grph if nd only if the corresponding fces in the originl grph re djcent. For exmple, in Fig., the fces defined y links {, } nd {,, } re djcent in the originl grph (Fig. ), thus the corresponding vertices nd, respectively, re djcent in the dul grph (Fig. ) on the sme edge. The dulity etween the grphs pplies lso to the elements of the grph. For exmple, the edges constituting cutset in the originl grph ( set of edges whose removl disconnects the grph) correspond to edges tht constitute circuit in the dul grph. For exmple, the edges {,,} in the originl grph, Fig., form cutset while the corresponding edges in the dul grph { 0, 0, 0 } form circuit (Fig. ). When deling with the engineering domin of electricl circuits, the cutset equtions govern the ehvior of the electric currents (Kirchhoff s urrent Lw) nd the circuit equtions govern the ehvior of the voltges (Kirchhoff s Voltge Lw). y mens of the dulity reltion, the current equtions in the originl grph re identicl to the voltge equtions in the dul grph []. Since we re looking for circuits in this synthesis cse for which there exists correspondence etween the currents nd voltges, we need to construct them so tht their corresponding grphs re dul to themselves. Following this topologicl rule it cn e verified tht the known diode ridge rectifier circuit (Fig. ) is multi-functionl, i.e., it cn serve s rectifier of oth voltges nd currents while nother circuit, the full wve rectifier (Fig. ), does not possess this property s it rectifies only voltges... The synthesis of trusses with self-stress moility property in grph theory terms The cse study presented in this nd the following sections possess cler distinction from the multi-functionl electricl networks descried in the previous section. s the electricl circuits lck geometricl informtion, determining the system topology is sufficient for devising the design description. On the other hnd, when deling with multidimensionl systems, such s structures nd kinemticl systems, the geometricl informtion is essentil nd n dditionl step the geometric synthesis step is eing used. In the previous cse study the topologicl synthesis ws trnsformed into serch for specific clss of grphs ' ' ' ' ' ' ' ' ' ' Fig.. The correltion etween the originl grph nd its dul grph. () The originl grph nd the highlighted cutset (in lue). () The dul grph nd the highlighted dul circuit (in red).

6 O. Shi et l. / dvnced ngineering Informtics (00) ' ' ' ' ' ' ' ' ' ' ' Fig.. xmple of diode ridge rectifier circuit tht is self-dul () nd the full wve rectifier circuit tht is not (). clled self-dul grphs. In the current cse study the serch for the proper topology is performed in su-clss of known clss of grphs. In generl, computtionl resources needed to find the proper topology cn e reduced when grphs elonging to specific clsses re utilized in synthesis, since properties ssocited with those clsses of grphs cn e tken dvntge of. Since this topologicl synthesis dels with trusses with specific property, we need to serch for clss of grphs corresponding to trusses which re rigid, i.e., ny generic reliztion of the grphs the trusses will e stle. This clss of grphs is clled Lmn grphs [] nd is defined y the following criteri:. e(g) = * v(g), where e(g) nd v(g) stnds for the numer of edges nd vertices in grph G, respectively.. ll the sugrphs, G 0 G stisfy the reltion: e(g 0 ) * v(g 0 ). F Fig.. xmple of grph stisfying oth Lmn grph criteri () nd the one tht stisfies only criterion (). F Fig. 9. The hierrchic order showing the reltion etween the desired grphs, the grphs corresponding to the trusses with the self-stress moility property, nd other possile grph fmilies. For exmple, the grph in Fig. is not Lmn grph since the sugrph defined y the set of vertices {,,,} does not stisfy condition. The serch for finding proper truss possessing the stility nd self-stress moility properties, ecomes serch for grph mong desired set of grphs. This set is defined to e the intersection of Lmn grphs, those grphs whose reliztion is lwys stle, corresponding to the stility property, with the set of self-dul grphs, corresponding to the property of the self-stress moility. This desired set of grphs is depicted in Fig. 9 s the intersection of the two grph sets nd is designted with gry color. Recently, it hs een noticed tht specil schemtic structures developed for kinemtic systems, termed ssur Groups [] re ctully specil clss of Lmn grphs []. These grphs re defined in grph terminology s miniml rigid in reltion to vertices, i.e., deleting ny set of vertices results in grph tht is not rigid. It hs een mthemticlly estlished [0] on the sis of numer of conjectures [] tht ll the ssur grphs nd only ssur grphs possess configurtions with the self-stress moility property. In other words, if given configurtion hs this property it is mthemticlly proven tht its topology is n ssur grph. For exmple, the truss in Fig. 0 is n ssur grph while tht in Fig. 0 is not, since fter deleting the vertices {,,F,G} the remining truss is rigid. Thus, from the ove theorem it follows tht only for the truss ppering in Fig. 0 it is possile to find configurtion hving tht property... The synthesis of linkges with specil locking position property in grph theory terms Section. outlined the discovery of new su-clss of grphs the desired grphs, depicted in Fig. 9 within known clss of grphs (Lmn grphs) tht defines ll the engineering systems possessing specific engineering property. In this section it ws lso shown tht the trnsformtion of engineering synthesis prolem into prolem in grph theory mkes possile use of known clss of grphs tht is widely used in different engineering domin. In similr mnner, the current section shows tht n engineering prolem cn e trnsformed not only into mthemticl representtion, ut lso further into nother engineering domin. It is first shown (Section..) how to trnsform every linkge in kinemtics into determinte truss in sttics thus mking it possile to serch for the desired topology of the linkge in the domin of sttics, s is demonstrted in Section...

7 O. Shi et l. / dvnced ngineering Informtics (00) G F G F Fig. 0. xmple of frmework () whose grph topology is ssur nd () whose grph topology is not ssur.... Trnsforming kinemtic linkges into sttic trusses The trnsformtion is performed y removing the driving links nd replcing them with pinned joints resulting in determinte truss. For exmple, the four r linkge (Fig. ) is trnsformed into determinte truss y removing the driving link,, nd replcing the inner joint y fixed support.... Topology synthesis of linkges with specil locking position property The synthesis prolem here is to find the topology of linkges tht possess specil geometries in which the linkges re in locked positions while there exist forces in ll the links nd the inner joints re moile. The previous section introduced methodology for the deterministic trnsformtion etween linkges nd determinte trusses, which mkes it possile to use the knowledge nd synthesis methods developed in plne sttic trusses for synthesis prolems in plne kinemtic mechnisms. In this specific synthesis cse, sed on the results reported in [,0] the trusses tht re ssur grphs, termed ssur trusses, hve specil geometry with the self-stress moility property. dding driving link to this specific configurtion yields linkge in which there exist forces in ll the links nd in which ll the joints re infinitesimlly moile, i.e., linkge in the needed locking position. Fig. depicts the min ide underlying this synthesis cse where the linkge (Fig. ) is first trnsformed into determinte truss ( more compound cse thn tht Fig.. xmple of trnsforming linkge () into determinte truss (). () The four r linkge. () The dyd. 9 depicted in Fig. ), which is then identified s n ssur type determinte truss. It hs een mthemticlly proven tht for the ltter type of trusses there exists specil configurtion which stisfies the desired property self-stress moility property.. The topologicl synthesis process 9 Fig.. xmple of finding the specil locking position in the linkge () y finding the self-stress moility property in the ssur-truss (). In this section the process of deriving the topologies of the systems tht possess the desired properties will e descried. It is importnt to notice tht lthough the three synthesis prolems re tken from three different engineering domins nd re trnsformed into different grph prolems, the methods to produce the grphs re similr... The procedure for otining the multi-functionl electricl circuits The topologicl procedure is initited from the minimum self-dul grph consisting of four vertices such tht ech of its vertices is djcent to the other three, s shown in Fig.. This grph is clled the complete grph with four vertices nd is denoted y K. The procedure strts y choosing, ritrrily, one of the vertices to e the ground vertex, (indicted in the

8 O. Shi et l. / dvnced ngineering Informtics (00) c d e R R R R R 9 I R R R0 R R R R V R R V R Fig.. xmple of producing multi-functionl electricl network from the complete grph - K () The grph K. (,c,d) fter expnding edges,, nd, respectively. (e) The electricl network corresponding to grph (d). figure s gry vertex), nd ll the edges djcent to this vertex re termed ground edges. t ech step, one of the ground edges is replced y the complete grph-k, while the ground edge is deleted, s shown in Fig.. This opertion is clled expnsion []. For the ske of clrity, only the grounded edges re mrked in Fig. since they re the only edges tht cn e expnded. It cn e proved tht ny grph otined through series of expnsions is self-dul, i.e., the grph is isomorphic to its dul grph. The strightforwrd line of proof of this property is y induction, showing tht pplying the expnsion oth on the originl nd the dul grphs results in identicl chnges. Since networks, whose corresponding grphs re self-dul, possess the interchngeility property of currents nd voltges, in ccordnce to the result devised in Section., the procedure of expnsion cn e employed for constructing multi-functionl electricl circuits. Fig. shows the process of devising the topology of multi-functionl electricl network with 9 elements nd junctions... reting the topologies of the trusses with the self-stress moility property s ws introduced in Section., only those trusses whose topology is of the ssur grph type possess the selfstress moility property. ccordingly, the grphs of this type re of vlue for engineers seeking to design trusses cple of sustining full self-stresses in ll its elements. The current section, therefore, focuses on the process of generting the ssur grphs. The extension process is done y splitting n edge, dding vertex to the edge, nd connecting the new vertex to one of the inner vertices or to the ground. This process cn continue till the needed topology is otined. Fig. illustrtes the process of constructing ssurtrusses from the known trid structure (Fig. )... The procedure for creting the linkges with the specil locking positions ue to the existence of the trnsformtion etween plne linkges nd trusses, s shown in Section.., ny c 9 0 Fig.. xmple of extending ssur trusses The trusses in () nd (c) re otined fter splitting edges nd in figures () nd (), respectively.

9 O. Shi et l. / dvnced ngineering Informtics (00) 9 c Fig.. xmples of linkges which possess the specil locking positions derived from the ssur trusses ppering in Fig.. truss with topology possessing this self-stress nd moility property is trnsformed (y dding driving links) to linkge with the specil locking positions. xmples of constructing the topology of the linkges y the trnsformtion from sttics pper in Fig... Geometric synthesis step Fce FF F R Fce In this section, the comintoril process of finding the configurtion for the given linkge topology will e descried. Since there is trnsformtionl mpping etween linkges nd trusses (Section..), the exmple is pplied only to linkge synthesis prolem. lectricl networks re one dimensionl systems nd hence for them this step is not pertinent. t this stge, the network topology for the linkge hs een constructed (Section.). Since it is n ssur grph, it is gurnteed tht t lest one configurtion with the desired property exists. In this section the process of finding such configurtion will e shown. ue to the dulity etween trusses nd linkges the force vectors in the rs re equl to the dul reltive liner velocities in the corresponding links []. The proof underlying this reltion is the grph theory dulity tht ws employed in Section.. This dulity reltion etween different grph networks lso revels new concepts nd entities. For exmple, the solute liner velocity of the joints of the mechnism corresponds to new vrile of force in trusses, termed fce force (FF) [], cting in the corresponding fce ( nonisected contour formed y rods). One of the unique properties of this vrile is tht the force in the r is defined y sutrction etween its two djcent fce forces s depicted in Fig.. This type of vrile resemles the known vrile in electricity mesh current which is known to e sclr, while fce force is multidimensionl vrile. More properties nd detils on fce force cn e found in [,]. To illustrte nd clrify the ide of the dulity etween trusses nd linkges Fig. c nd d depicts truss nd its FF Fig.. The force in the rod is defined y its two djcent fce forces. dul linkge (superimposed) such tht the rs nd the corresponding links re perpendiculr. The ide underlying the type of fce force vrile is illustrted in Fig. y showing the correltion etween the voltge of junction in the originl system - the electricl circuit (Fig. ) nd the mesh current in the dul electricl circuit (Fig. ), while Figs. c nd d highlight the correltion etween the liner velocity of joint nd the fce force cting in the corresponding fce in the dul truss, respectively. This type of physicl vrile hs specil properties since it exhiits the qulities of oth force nd potentil (it is the counterprt of the liner velocity). It cn hve mny prcticl pplictions, one of which is in the geometric synthesis stge introduced in this pper, s follows:. Identify the fces in the linkge.. ssign n ritrry fce force vector to ech fce.. ompute the internl link forces from the fce force vectors in step. For ech link the internl force vector is equl to the vector difference etween the two fce forces ssocited with the fces seprted y this link. Fig. provides vector digrm yielding the internl forces in this mnner. In the digrm, the fce force vectors found in step, re designted y the colored

10 0 O. Shi et l. / dvnced ngineering Informtics (00) I R V Y +- V Y +- I I Y R II Y I I I Y I II Y Y R I V + - R Y I c d P ' V / 0 ' V / 0 = P Fig.. The dulity reltion etween potentil of joint nd its corresponding vrile in the dul system. (,) Voltges of electricl circuit junctions nd their corresponding mesh currents in the dul electricl circuit. (c,d) The liner velocity of joint in the linkge nd its corresponding fce force in the dul determinte truss. Fig.. xmple of finding specil configurtion using the fce-force method. () The topology of the linkge. () The fce-force digrm. (c) The corresponding configurtion of the linkge. rrows strting t the origin, while the internl forces in the links re descried y the lck lines connecting the two corresponding fce forces.. In order for the internl link forces otined in step to correspond to the rel physicl ehvior of the linkge, the ngles of the links should e ligned with the ngles of their corresponding internl force vriles. Thus, redrwing the linkge in ccordnce with these computed ngles yields geometricl configurtion of the linkge for which there exists fesile set of internl forces. In other words, the linkge configurtion otined through this process is proved to e linkge in locked (immoile) position, since it is cple of sustining the externl force pplied y the driving links. Fig. c shows linkge whose topology is identicl to tht of the linkge in Fig., ut the ngles of its links re set ccording to the ngles of the vectors otined from the vector digrm of Fig.. It cn e verified through the theorems of mchine theory [] tht the linkge configurtion of Fig. c is mechnism in locked position.. onclusions This pper introduced the potentil of employing grph network theory t the initil stge of the synthesis process. The ide ws introduced through solving different synthesis prolems from different engineering domins: electronics, kinemtics nd sttics.

11 O. Shi et l. / dvnced ngineering Informtics (00) The pper highlighted severl dvntges of using grph networks in synthesis, which re: Gining generl perspective on different synthesis prolems the first step in the proposed method (Fig. ) isto trnsform the synthesis prolem to prolem in terms of grph theory. For exmple, the prolem of synthesis of multi-functionl electricl circuits tht cn switch the ehvior etween currents nd voltges, ws trnsformed into prolem of constructing grphs whose topologies re self-dul. These grphs re well known nd re widely used oth in theory nd prctice. mploying the sme grph lgorithms for different synthesis prolems the fct tht different synthesis prolems re trnsformed into the sme grph theory prolem opens the possiility of pplying the sme lgorithm for the solution of mny nlogous prolems. Furthermore, it is possile tht different synthesis prolems re trnsformed into different grph prolems ut the sme lgorithm is used for the ltter. This cse is reported in the pper where the synthesis of interchngele electricl circuits ws trnsformed into the synthesis of self-dul grphs (Section.) nd the synthesis of trusses with the self-stress moility property ws trnsformed into ssur grphs (Section.). lthough the ltter two grph prolems re different, the lgorithm for their construction is similr, s ppers in Section.. orrowing knowledge nd methods from different engineering disciplines different grph types correspond to different engineering domins. When there re reltions etween the corresponding grphs, there is reltion etween the engineering domins. For exmple, due to the reltions etween the grphs of sttic systems nd those of kinemtic systems, truss possesses the self-stress moility property if nd only if (IFF) the corresponding linkge hs specil locking position, s ppers in Section.. The pper lso employed physicl vrile, referred to s Fce Force (Section ), which, lthough new to sttics, is multi-dimensionl counterprt of the one-dimensionl mesh current widely known in electronics. References []. ntonsson, Microsystem design synthesis, in:. ntonsson, J. gn (ds.), Forml engineering design synthesis, mridge University Press, New York, NY, US, 00, pp. 9. [] L.V. ssur, Investigtion of plnr rod mechnisms from the point of view of their structure nd clssifiction, Vol. 0, St. Petersurg Polytechnicl Institute, Izvestij, 9. [] N. lnin, T.. ickrt, lectricl Network Theory, John Wiley & Sons, NY, 99. [] M.P. endsoe, N. Kikuchi, Generting optiml topologies in structurl design using homogeniztion method, omputtionl Methods in pplied Mechnics nd ngineering (9) 9. [].R. erg, T. Jordn, proof of onnlelly s conjecture on -connected circuit of the rigidity mtroid, Journl of omintoril Theory () (00) 9. [] M.I. mpell, J. gn, K. 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