机械设计方面的外文参考文献

A Fuzzy Neural Network for the Analysis of Experimental Structural Mechanics Problems Ewa Pabisek, Magdalena Jakubek and Zenon Waszczyszyn Institute of Computer Methods in Civil Engineering, Cracow University of Technology, Warszawska 24, 31-155 Krak6w, Poland, e-mail: .pl Abstract. A simplified neuro-fuzzy network is formulated. The membership functions of the network weights are computed on the base of learning the single patterns. The network is applied to the interval analysis of two problems from experimental structural mechanics. 1. Introduction Results of tests on material models can be noisy, incomplete and inconsistent. Another aspect is a limited number of tests because of various difficulties and costs of arrangement of experiments on laboratory specimens or full scale structures. That is why fuzzy variables and a possibility approach seem to match better the nature of experimental results than using crisp variables 5 . This concerns also fuzzy neural networks. There are three possibilities to formulating fuzzy networks. The first one corresponds to the neural network with crisp parameters (called for short NN weights) and performing computations on interval variables 8. Much ad- vanced are NNs with crisp inputs and outputs but their processing is performed on fuzzyfied variables with fuzzy reasoning rules, cf. fuzzy inference systems 4. The third class is associated with full fuzzification of transmitted signals, NN weights and neurons of a fuzzy NN 2. A more numerically efficient ap- proach depends on joining simple membership functions of signals and NN pa- rameters with interval arithmetics 7. In the paper a simplified approach, proposed in 6 is discussed. The main idea lies in the formulation of NN weights on the basis of weight values, com- puted for single patterns taken each after the other from the training pattern set. Crisp type neurons are used for the transformation of interval values for fixed a - cuts, transmitted through NN. The above mentioned approach is developed in the paper. The correspond- ing neuro-fuzzy network approach is applied to the estimation of the stress in- tensity factor related to fracture toughness of dense concrete. 2. A fuzzy neural network The idea of a fuzzy NN, proposed in 6, is shown Fig. I where a schematic al- L. Rutkowski et al. (eds.), Neural Networks and Soft Computing Springer-Verlag Berlin Heidelberg 2003 773 gorithm is shown. It corresponds to a standard multi-layered, forward neural network and the training set of patterns: = (x,t)(p) I p= 1, . ,L Set of training patterns .c = (x, t)(p) I p = 1, . ,L Stage I Initial training ofNN Initial values ofNN weights w? Ii = 1, . ,NW Stage II Detail training of NN Set of NN weights w!p) li=1, . ,NW;p=l, . ,L Stage III Computation of membership functions for NN weights Fig. 1. Schematic algorithm of a fuzzy NN formulation FuzyNN with weights membership functions Pi = p(wp (1) Let as assume that the network was designed using a corresponding cross- validation procedure and a subset selected from (1). The formulated network is then trained on set (1) at Stage I of the algorithm shown in Fig. 1. A set ofNN weights (both synaJ?tic weights and biases) is collected as a set of initial value weights Wo= wed i= 1, . ,NW, where: NW-numberofNN weights. The weights WOi are assumed as initial weights to learn weights correspond- ing to each pattern of the training set (l). At Stage II the network is trained L times for a sequence of single patterns p = 1, . ,L. After this training a set of weights is completed, i.e. W= Wi= Wi (P) Ii = 1, . ,NW; p = 1, . , L . The membership functions for the NN weights J1 i = J1 (W;) are computed at Stage III. In 6 the triangular membership functions were assumed. In Fig.2a a triangular shape of MB function (t) is shown for the weight w (index i is om- mited). The distances 3 aL and 3 aR are measured from the mean value w, where: aL, aR-standard deviations of patterns p that are smaller or greater than 774 w, respectively. The interval values of a-cut wL , wRa are depicted in Fig. 2a as WL,WR. a) p(w) 1.0 (t) b) Jt(W) 1.0 a = 0.,5 Wmin W;s ill wfs W Wmax Fig. 2. Shapes of membership functions for NN weights: a) triangular MF (t), b) nonlin- ear MF (n) The other method of formulation of a nonlinear MF (function (n) in Fig. 2b) lies in on computation of the discrete cumulative functions for the ranges WOOn :; w(P) w, w:; w(P) :; Wmax Numbers of patterns w L , w are computed from inequalities (l-2KIL):; IX 2600 kglm used in special structures, is brittle. Esti- mation of the concrete fracture toughness is made on the basis of laboratory tests on especially prepared specimens. In Fig. 3 the so-called Model II is shown as well the equilibrium path during the specimen loading. On the basis of experimental results the stress intensity factor KIIC is computed : 5.IIPQ r=-:-r, 3/2 Kllc = -V 7C a LMN/m 2BW where all variables and parameters are shown in Fig. 3. (3) In Table 1 experimental results are given for selected tests, performed at the Institute of Building Materials and Constructions of Cracow University of Tech 775 a) p b) Fig.3. a) Model II of concrete specimens, b) Force-displacement curve nology for data: a = 60 mm, B = 90 mm and W= 150 mm (Tests I performed in 1998, test II in 1991, cf. references in 9).9 concretes ofstrengthlc were used and from each of them 6 specimens were made. The first five concretes of Nos 1-5 (Tests I) were used for the network training, the four next concretes of Nos 6-9 (Tests II) were explored for the fuzzy NN testing. Table 1. Stress intensity factor Kuc for Tests I and Tests II Tests I II No. of concrete 1 12 13 14 15 6 17 18 J9 Ic Mpa 25.8 130.1 135.1 136.6 140.2 27.2 I 30.2 135.5 141.0 2.36 2.70 2.84 3.00 3.32 2.38 2.71 2.94 3.29 2.48 2.82 2.92 3.35 3.72 2.19 3.20 2.89 3.20 2.62 3.00 2.97 3.30 3.25 3.25 3.12 3.12 3.86 Kuc MN/mw 2.52 2.65 3.10 3.12 3.82 2.66 3.37 2.85 4.35 2.48 2.90 3.05 3.05 3.42 2.55 2.05 3.04 3.29 2.63 2.69 3.00 3.46 3.35 2.34 2.79 2.96 3.20 meanKuc 2.52 12.79 12.82 13.21 13.48 2.56 I 2.87 I 2.97 I 3.53 The network was designed as a very simple network BPNN: 1-3-1, with one input x = Ic and one output y = Kuc. The sigmoidal neuron NN was trained by Levenberg-Marquardt learning method 1. After a rapid training process the error of Stage I was RMSE 0.8.10-2 and for Stage II RMSE (P) 1.10-6 for each pattern p. In Fig. 4 the MF functions (t) and (n) are shown for selected concretes. The concretes of Nos 1, 5, 6 and 9 are considered as corresponding to minimal and maximal values of strentgh Ic . Concretes 1 and 5 were used for training of neuro-fuzzy NN, concretes 6 and 9 were used for testing of the trained net- work. It is visible that the crisp values corresponding to the cut a = 1 are close to mean values of stress intensity factors computed on the base of measure- ments (they are marked at the top of figures with * among other 6 specimens marked with 0). The coincidence of mean values * and 0 is very good for the training patterns. In case of training patterns the maximal errors between * and o are 12.5 % for mean values of specimens made of concretes Nos 6-9, used as testing patterns. 776 a) Concrete No 1 b) Concrete No 5 1 I 0.8 0.8 0.6 0.6 es CS 0.4 0.4 0.2 0.2 0 0 1.8 3.1 3.9 c) Concrete No 6 d) Concrete No 9 1 1 Ie = 41.0MPa 0.8 0.8 0.6 0.6 es 0.4 es 0.2 0 2 3.2 4 4.4 K/Ie Fig. 4. Membership functions for stress intensity factor Kllc for concretes of selected strength/c In the next Fig. 5 there are shown intervals for all the concretes. The inter- vals correspond to the cuts a = 0.5, 1.0 for the nonlinear MS functions (n), cf. Fig.2b. It is visible that the experimental dispersion of six points, corresponding to concretes of the same strength, is different but all the six measurements are covered by the intervals a = 0.5 for concretes Nos 1-4 and 8 . 4.5 4 . il . 71 i T I ! S 3.5 6, . T 3 . I 9: f + 2.5 ! $ 224 .G 26 28 30 32 34 36 38 40 42 Ie Mpa Fig.5. Estimating intervals for cut a= 0.5 for concretes of Tests I and II 777 4. Final remarks and some conclusions 1. A simplified neuro-fuzzy network can be easily formulated using the algo- rithm schematically presented in Fig. 1. The neural weight membership functions are formulated on the base of weights computed for single pat- terns taken from the training set. 2. The network can be applied for both crisp and interval inputs mapping into interval outputs. 3. The network has been examined on an example of identification of the stress intensity factor associated with fracture toughness of dense concretes. 4. The intervals of parametric estimation of characteristics of of semi-rigid steel connections were successfully computed in 3 using the above dis- cussed neuro-fuzzy network. Acknowledgement Financial support by the Polish Committee for Scientific Research, Grant No 8 T07E 020 22 Applications of artificial neural networks in the analysis of steel structures is gratefully acknowledged. References 1. Denmuth H, Beale M (1998) Neural network toolbox for use with MATLAB, Users Guide, Version 3, The Math Works 2. Hayashi Y, Buckley JJ, Czogala E (1993), Int. J. Intell. Systems, 7:527-537 3. Jakubek M, Pa