Computing Method and Hardware Circuit Implementation of Neural Network on Finite Element Analysis

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1 I.J. Intellgent Systems and Applcatons, 0, 5, 4-47 Publshed Onlne August 0 n MECS ( Computng Method and Hardware Crcut Implementaton of Neural Network on Fnte Element Analyss Lkun Cu College of scence of Inner Mongola unversty of echnology, Hohhot, Chna; Emal: lekuncu@sna.com We Wang College of Astronautcs Northwestern Polytechncal Unversty, Xan,Chna Emal: mdule@mal.nwpu.edu.cn Zhuo L College of scence of Inner Mongola unversty of echnology, Hohhot, Chna Emal: l_zhuo@63.net Abstract - he fnte element analyss n theory of elastcty s corresponded to the quadratc programmng wth equalty constrant, whch can be further transformed nto the unconstraned optmzaton. In the paper, the neural network of fnte element solvng was obtaned on the bass of Hopfeld neural network that was reformed. And the no-error solvng of fnte element neural net computaton was realzed n theory. And a desgn method to construct an artfcal neuron by usng electronc devces such as operatonal amplfer, dgtal controlled potentometer and so on was presented. A programmable hardware neural network of fnte element can be buld up by usng analog swtches to nterconnect nputs/outputs of hardware neurons. he weghts, bases and connecton n the hardware neural network of fnte element can be adjusted automatcally by mcroprocessor accordng to the results of tran to controllng system, hs programmable hardware neural network of fnte element has some more adaptablty for dfferent systems. In addton, the authors present the computer smulaton and analogue crcut experment to verfy ths method. he results are revealed that: ) he results of mproved Hopfeld neural network are relable and accuracy; ) he mproved Hopfeld neural network model has an advantage on crcut realzaton and the computng tme, whch s unrelated wth complexty of the structure, s constant. It s practcal sgnfcance for the research and calculaton. Index erms - Fnte element method; Hopfeld neural network; analogue crcut; smulaton; operatonal amplfer; dgtal controlled potentometer I. INRODUCION Wth the rapd development of computer software and hardware technology, fnte element method has been wdely appled to the structure analyss and desgn. It s very slow to deal wth large-scale and complex engneerng structure and the effcency s low because the tradtonal calculaton program for structure optmzaton s based on all seral algorthms n a sngle computer []. And the contradcton between the large computes scale and low computatonal ablty becomes more and more urgent, so the parallel computng method has mportant meanng for fnte element method. However, common methods cannot create enough computatonal effcency needed n practce []. he neural network s a large-scale nonlnear dynamcs system, and t has own characters, such as hgh parallel processng ablty, strong robustness, fault-tolerance, learnng ablty and adaptablty, and shows to superorty n settng up model for complcated system, so t s suted for many aspects n ndustry and professon. In addton, the neural network s consdered as a dynamc crcut, and t can be used for hgh-speed parallel computaton and s easy to attan the solutons wthn the tme constant [3]. So, the fnte element analyss n theory of elastcty s solved by modfed Hopfeld neural network and verfed by the computer smulaton and analogue crcut experment n the paper. he artcle also provdes a desgn method to construct an artfcal neuron n theory by usng electronc devces such as operatonal amplfer, dgtal controlled potentometer, analogy swtch and so on. A programmable hardware neuron networks can be bult up by usng analog swtches to nterconnect nputs/outputs of hardware neurons. he weghts, bases, actvaton functons and connecton n the hardware neuron networks can be adjusted automatcally by mcroprocessor accordng to the results of tran to controllng system. hs programmable hardware neuron networks has some more adaptablty for dfferent systems. II. ELASICIY FINIE ELEMENS MODEL USING IMPROVED HOPFIELD NEURAL NEWORK he fnte element analyss n theory of elastcty s corresponded to the quadratc programmng wth equalty constrant, whch can be further transformed nto the unconstraned optmzaton and the Hopfeld

2 4 Computng Method and Hardware Crcut Implementaton of Neural Network on Fnte Element Analyss neural network s a good method n the feld of optmzaton calculaton, so that problem can be solved by ths method. Besde ths, the prevous work on neural network has ndcated that the potental energy functonal equals to the objectve functon of the fnte element method and the maxmum or mnmum pont, whch s the stable equlbrum pont of the network system, s the soluton. A. Quadratc Optmal Method for Elastcty heory Problems It s well known that system optmzaton s the nature of computaton. In ths secton, a detal explanaton s proposed for ths pont n the context of elastcty [4]. ) Equlbrum Equaton σ + f = ( =,,3, V V) () j, j 0 Where σ j, j s stress and f s volume force. ) Geometrcal Equaton ε j = ( u, j + uj, ) () (, j=,,3, V V) Where ε j s stran and u, js dsplacement. 3) Physcs Equaton σ = D ε (, j, k, l =,,3, V V) (3) j jkl kl D Where jkl s elastc constant 4) Boundary Condton = ( =,,3, S S σ ) (4) u = u ( =,,3, S S u ) (5) Where s area force, S σ s force boundary and S u s dsplacement boundary. Accordng to the prncple of the mnmum potental energy, elastomer under external force s balanceable and the total potental energy s gven by: = ( D εε fu ) dv uds p V jkl j kl S σ (6) Equaton (6) depcts the equlbrum equaton and force boundary of elastc body and ts mnmum value s true soluton n all possble dsplacement feld,whch s meet the condtons () and (5). hs means that the fnte element analyss n theory of elastcty can be transformed nto the optmzaton problems under an equalty constrant condton. he mathematcal model s gven by: mn p = ( Djklεε j kl fu ) dv V uds Sσ εj = ( u, j + uj, ) (V V) (7) st. u = u (S Su) he authors dvded contnuum nto m elements (ncludng m force boundary element, n nodes and n dsplacement boundary nodes) by fnte element wth respect to mathematcal model transformaton [5].he new mathematcal model s gven by: mn ( δ )= δ Kδ f δ (8) st. Aδ = δ ( S Su ) Where K s global stffness and ts value s K value m B DBdV Ve e= =, f s node load array and ts m m ( ) ( ) V e Sσ e e= e= f = N f dv + N ds, s surface force, δ s node dsplacement array, A s constraned matrx, B s stran matrx, D s elastcty matrx and N s shape functon matrx. he above optmzaton problem s equvalent to the followng lnear equaton [6]: K f δ = (9) δ ( =,, L, s). Where δ x s equal to For statc problem, compulsory boundary condtons are drectly substtuted nto (9) and the new equaton s gven by: K f (0) δ = Where K s postve defnte matrx. Fnally, (8) s transformed nto the followng unconstraned optmzaton problem and ts soluton s confrmed when K s the postve defnte matrx accordng to the optmzaton theory. mn ( δ )= δ K δ f δ () B. Improved Hopfeld Neural Network he prevous work has ndcated that the energy functonal of neural network equals to the objectve functon of the fnte element method, so ts mathematcal form, Lyapunov form [7-9], and dynamc equaton are gven by (), (3) and (4): E=V( δ )= δ K δ f δ () = δ W δ b L δ (3) E [ (,,,)] s

3 Computng Method and Hardware Crcut Implementaton of Neural Network on Fnte Element Analyss 43 dδ ε = Kδ + f dt (4) δ (0) = p Where W s equal to K and f s [ b (,, L,)].In addton, stablty of ths network s dscussed by: s de dδ dδ dδ = ε[ ] = ε ( ) 0 (5) dt dt dt dt = Accordng above theory, the dagram of mproved Hopfeld neural network s proposed n fg.: R, / R,..., / R r )of potentometer resstance s used to represent the weght of nput sgnal, the product(u j /R j ) of the nput sgnal and ts weght s nput current flow of operatonal amplfers, then, he mathematcal model s gven by: u u ur u o = R f ( + + L + ) (8) R R R o acheve automatc adjustment of neuron nput sgnal weghts, the potentometer R R,..., Rr n fg.3 can be dgtal potentometer AD8403. he prncple chart of the nternal structure of AD8403 s shown by fg. 4. r Fg. Dagram of mproved Hopfeld neural network III. NEURAL NEWORK HARDWARE IMPLEMENAION A. Neurons Hardware Implementaton Fg. 3 Neuronal nput weghted summaton crcut Neurons are basc processng unts of artfcal neural network. It s usually a non-lnear element has many nputs but only sngle output. A typcal neural model s shown n fg.. Its nput/output relatonshp s gven by (6): r a = f ( ω j p j + b) (6) Its matrx form s j= A = f ( W P + b) (7) Fg. Sngle neural model B. crcut of the sgnal weght sum For neural model n fg., ts nput sgnal weghted summaton parts are realzed by the hardware crcut showed n fg.3. Operatonal amplfer OP s connected nto adder form. he nput sgnal u, u,, u r are supposed as the standard output voltage sgnal (- 5V - + 5V) of varous sgnal sensors, hence, the recprocal ( / Fg. 4 Internal structure prncple dagram of dgtal potentometer AD8403 It conssts of four ndependent potentometers. he resstances (normnal value) between the fxed ends of

4 44 Computng Method and Hardware Crcut Implementaton of Neural Network on Fnte Element Analyss potentometers generally have four specfcatons: KΩ, 0KΩ,50K Ω and 00K Ω. Each potentometer has two fxed ends (A and B) and one sldng contact (W), the resstance between W end and B end s decded by the datas s placed n ts seral nput regsters, and there are 56 branch decson ponts. he bggest characterstc of dgtal potentometer s ts programmable ablty, so t also s called numercal control varable resstor. Under the acton of the clock CLK, 0 bts seral datas are placed n through seral data nput termnals SDI. he 0 bts datas formats are D7 D6 D5, A A0 D4 D3 D D D0. If sldng arm W and A s connected, the varable resstor RWB computaton formula s: R ( Dx) = ( Dx) / 56 R + R WB If sldng arm W and B s connected, the varable resstor RWA computaton formula s: R ( Dx) = (56 Dx) / 56 R + R WA AB W AB W fg.6. It s suppled power by.8-5.5v sngle power. he largest conducton resstance s less than 0.4 Ω, conducton resstance flatness s 0. Ω; Swtch hge current capacty of s 400mA; Double gude conducton; Response tme s 35ns; Low power consumpton; L/CMOS compatble nput; Analog swtch ADG80 s often open (IN = 0 dsconnect, when IN = conducton), ADG80 s often closed (IN = dsconnect, when IN = 0 conducton). C. crcut of Devaton comparson Consderng the role of devaton b, the comparson of ub and u0 can be realzed by the devaton compared crcut shown n fg.5 (potentometer Rb shown n fgure s also dgtal potentometer AD8403). In fg.5 crcut, nput resstance value s equal wth the feedback resstance valve (both are R), the devaton u b s postve voltage, and u 0 s nverse wth the nput sgnal through former treatment of the OP operatonal amplfer. herefore, D-valve of u 0 and u b s a sgnal nputted amplfer OP n practce. he comparson between them was realzed. In addton, the operatonal amplfer OP whch uses sngle polarty power (+ 5V) has output only when the sgnal u 0 s bgger than devaton u b. Fg. 6 Schemes of analog swtch pns In addton to the above analog swtches, there are ADG706, ADG707 etc. IV. NEURAL NEWORK HARDWARE IMPLEMEN ON FINIE ELEMEN Accordng above theores, the Hopfeld model crcut can be changed nto the form n fg.7, notably the neural network s a lnear system. Fg. 5 Crcut of Devaton comparson he mathematcal model s gven by: u = u + u ) (8) x ( 0 b After two reversed-phase operatons of operatonal amplfer OP and OP, u x s changed nto same phase wth u, u,... u r. D. connecton crcut between nputs and outputs he analog swtch ADG80/80[0] s chosen to control crcut for the automatc connecton realzaton of dfferent nput and output sgnals between neurons. he dagram of ADG80/80 analog swtch pns s shown as (b) Fg.7 he network structure of fnte element neurons and three nodes network

5 Computng Method and Hardware Crcut Implementaton of Neural Network on Fnte Element Analyss 45 he crcut shown by fg.7 can be descrbed n followng equaton. m dv() t u c = vj dt R R j= j u = o R f (9) By comparson wth expresson (5), we fnd they are equvalent and meet the followng relatons: c = ε (0) j = () = kj R u R j = f () v = a (3) kj R Because can be postve, negatve or zero, j also can be postve, negatve or nfnte. An nvert and v R on-off swtch can be set between j and j n order to f solve ths problem. And also can be dsposed n ths method. he crcut s shown n fg.8(a) after dsposal. In dfferent fnte element soluton, the matrx K, F s dfferent. In order to realze generalty of the neural network n solvng the fnte element problem, resstance R must be varable resstor and u be set for a constant. A neural network solvng system can only solve the fnte element solvng problems whose matrx s n n. In order to solve the problems whose matrx s under n n, nputs and outputs between neurons be controlled by swtches. And so, neural network crcut shown n fg.8 (a) must be changed nto the form shown n fg.8(b). Fg.8 Correspondng crcut dagram Fnte element neurons and three nodes network Accordng to the hardware neural model realzaton method has been dscussed n the prevous text, the varable resstors and swtches n the crcut shown n fg.8 must be respectvely dgtal potentometer AD8403 and analog swtch ADG80 or ADG8. he hardware neural network system structure chart s llustrated by fg.9. Fg.9 he structure schematc drawng of hardware neural network system he System conssts of a hardware neurons array (contanng a number of neurons hardware crcuts), several analog swtches, a non-volatle storage (such as Flash memory), a mcrocontroller and relevant nterface crcuts. All dgtal potentometers and analog swtches have ther separate addresses n the system board.he weghts, thresholds, nput/output connectons are controlled by MCU based on dgtal potentometers, analog swtches for programmng settng (adjustment). V. ANALOGUE CIRCUI EXPERIMEN As shown s fg., consder a sheet wth thckness t=0.mm, elastc modulus E and Possons rato μ=0.3 subject to force F=0KN. Fnd the dsplacement of each pont assumng the gravtaton s zero.

6 46 Computng Method and Hardware Crcut Implementaton of Neural Network on Fnte Element Analyss the authors use the followng three dfferent ntal nput values, whch are mutually orthogonal vector. ( ) ( ) ( ) δ (0) = δ δ 3 (0) = (0) = Fg. 0 Structure schematc dagram In order to fnd the dsplacement of each pont, the boundary condtons are ntroduced to the calculaton doman. he global stffness matrx s modfed, and the followng expresson s results: v v u3 0.0 = v u u Frst, ths paper solves ths queston by usng mproved Hopfeld neural network based on the energy functon of the neural network equals to the objectve functon of the fnte element method and the mnmum pont, whch s the stable equlbrum pont of the network system, s the soluton. In addton, the fg.0 gves the crcut. And then analyss s carred out for the accuracy and precson of computng method by comparng the results of calculaton n the same ntal condton separately. In order to make the calculated result s unversal, able I lsts the results of network crcut and theoretcal values and the unt of all the crcut outputs n the followng tables are N.mm.E -. It can be observed that crcut smulaton solutons under three ntal nput values are same and the dfference s small between the smulaton and theoretcal value. hs s because the nfluence of nput offsets lke voltage and electrcty. ABLE I HE RESULS OF NEWORK CIRCUI AND HEORY Outpu V t V U 3 V 3 U 5 U 6 Case Case Case Case a a heoretcal value Partcularly, when the neural network nput s zero, the system appears the zero drft phenomenon and the values are gven by table II. In order to rase the accuracy, there needs modfy the results of mproved Hopfeld neural network to attan the true value. ABLE II HE ZERO DRIF VALUES OF NEWORK CIRCUI Output V V U 3 V 3 U 5 U Value Fg. Neural network crcut

7 Computng Method and Hardware Crcut Implementaton of Neural Network on Fnte Element Analyss 47 It s found that ths neural network s a lnear dynamc system whch nput and output accord wth the superposton prncple, so the true values are network ABLE III HE FIXED VALUES AND RELAIVE ERRORS output mnus zero drft values. he fxed values and relatve errors are lsted n followng table. Output V V U 3 V 3 U 5 U 6 Fxed value heoretcal value Relatve error 0.4% 0.49% 0.4% 0.59% 0.68% 0.9% he analog crcut experment results and relatve errors lsted n table IV. From the followng table and table III, we can come to the concluson that: ) he maxmum relatve error between neural network calculaton outputs and the conventonal algorthm results s 4%, whch meets the engneerng requrement; ) he mproved Hopfeld neural network model has an advantage on crcut realzaton and the computng tme, whch s unrelated wth complexty of the structure, s constant. ABLE IV HE ANALOG CIRCUI EXPERIMEN Results AND RELAIVE ERRORS Output V V U3 V3 U5 U6 Actual output heoretcal value relatve error % 0.4% 3.9% 0.6%.9%.7% IV. CONCLUSIONS In the paper, the fnte element analyss n theory of elastcty s solved by modfed Hopfeld neural network based on the energy functon of the neural network equals to the objectve functon of the fnte element method and the mnmum pont, whch s the stable equlbrum pont of the network system, s the soluton. In addton the authors present the computer smulaton and analogue crcut experment to verfy ths method. he results are shown that: ) he results of mproved Hopfeld neural network are relable and accuracy; ) he mproved Hopfeld neural network model has an advantage on crcut realzaton and the computng tme, whch s unrelated wth complexty of the structure, s constant. ACKNOWLEDGMEN he author s grateful to the scentfc study project of the Inner Mongola autonomous regon (NJZY08058) for ts support. REFERENCES [] N. G. Xe,.. Yu, W. D. Shao, Network parallel algorthm n optmzaton computaton of engneerng structure," Journal of East Chna Unversty of Metallargy, vol.8, no., pp. 7-0, 00 (n Chnese). [] W. L. Zhang. "Studyng status and development of computng mechancs", Journal of Anhu nsttute of archtecture, vol.9, no., pp. 7-, 00 (n Chnese). [3] X. Chen, D. H. Sun, H. Z.Huang, "Fnte element system for structural analyss and neural network", Hostng and conveyng machnery, no. 6, pp. 6-8, 999 (n Chnese). [4] X. C. Wang," Fnte element method, Bejng: snghua Unversty Press,003 (n Chnese). [5] D. H. Sun, Q. Hu, H. Xu," Real-tme Neurocomputng Model Used n Elastc Mechancs", Chna mechancal engneerng,vol.8, no., pp , 997 (n Chnese). [6] H. B. L, Neurocomputng based structural fnte element analyss, Dalan: Dalan unversty of technology, 003 (n Chnese). [7] L. K. Cu, W. Wang, Z. L, "Applcaton of Hopfeld neural network n fnte element solvng", Computer Engneerng and Applcatons,vol.46, no. 6, pp , 00 (n Chnese). [8] H. Z. Huang, H. B. L, "Research on neurocomputng method on fnte element analyss", Journal of Mechancal Strength,vol.5, no. 3, pp , 003 (n Chnese). [9] E. Z. Lang, E. W. Lang, Crcut desgn and smulaton applcaton base on Protel 99 SE,Bejng: snghua Unversty Press,000 (n Chnese). [0] Datasheet: <0.4 Ω CMOS.8V to 5.5V, SPS Swtches ADG80/ADG80. Analog Devces