https://doi.org/10.1186/s10033-018-0219-4 Chinese Journa of Mechanica Engineering ORIGINAL ARTICLE Open Access Mutivariate Rationa Response Surface Approximation of Noda Dispacements of Truss Structures Shan Chai 1, Xiang Fei Ji 2, Li Jun Li 1* and Ming Guo 3 Abstract Poynomia-basis response surface method has some shortcomings for truss structures in structura optimization, concuding the ow fitting accuracy and the great computationa effort. Based on the theory of approximation, a response surface method based on Mutivariate Rationa Function basis (MRRSM is proposed. In order to further reduce the computationa workoad of MRRSM, focusing on the aw between the cross-sectiona area and the noda dispacements of truss structure, a conjecture that the determinant of the stiffness matrix and the corresponding eements of adjoint matrix invoved in dispacement determination are poynomias with the same order as their respective matrices, each term of which is the product of cross-sectiona areas, is proposed. The conjecture is proved theoreticay for staticay determinate truss structure, and is shown corrected by a arge number of staticay indeterminate truss structures. The theoretica anaysis and a arge number of numerica exampes show that MRRSM has a high fitting accuracy and ess computationa effort. Efficiency of the structura optimization of truss structures woud be enhanced. Keywords: Mutivariate rationa function, Response surfaces method, Truss structures, Structure optimization 1 Introduction The response surface methodoogy (RSM expores the reationships between severa expanatory variabes and one or more response variabes. The method was introduced by Box and Wison in 1951. The main idea of RSM is to use a sequence of designed experiments to obtain an optima response. Box and Wison suggest using a second-degree poynomia mode to do this [1]. Because response surface methodoogy can estabish the unknown function between design variabes and response variabes, it has been widey appied to optimization. Choon-Man Jang and Ka-Ram Choi [2] carried out the optimization of the bower impeer by using the response surface method (RSM. Heng Jiang et a. [3] carried out a dynamic and static muti-objective optimization of a vertica machining center based on response *Correspondence: iijun@sdut.edu.cn 1 Schoo of Transportation and Vehice Engineering, Shandong University of Technoogy, Zibo 255049, China Fu ist of author information is avaiabe at the end of the artice surface method. Quadratic poynomias are empoyed to construct response surface (RS mode, which refects the reationship between design inputs and structura response outputs, according to the response outputs of these sampes obtained by anayzing the dynamic and static characteristics of the machining centre at these sampes with the software ANSYS. In the fied of vehice optimization, RSM aso get a widey appication [4 6]. Chun-Lin Wang et a. [7] carried out a variabe curvature bade of fire pump based on experimenta design theory and response surface approximation. Dong-Sheng Jia et a. [8] estabished the mass function of impeer of water vave-controed hydrodynamic couping with RSM and optimized the impeer key structura parameters. Response surface method is appied in the fied of robust optimization design [9, 10]. Reiabiity anaysis has an improvement based on response surface method [11, 12]. Generay speaking, in the mathematica modes of structura optimization, the objective function is the expicit function of the design variabes and the constraint function is the impicit function of the design The Author(s 2018. This artice is distributed under the terms of the Creative Commons Attribution 4.0 Internationa License (http://creativecommons.org/icenses/by/4.0/, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the origina author(s and the source, provide a ink to the Creative Commons icense, and indicate if changes were made.
Page 2 of 14 variabes. This characteristic is aso the important difference between the structure optimization probem and the mathematica program probem and it greaty increases the compexity of the structure optimization probem [13]. In order to sove this probem, the impicit function must be made expicit and the response surface methodoogy, is the most commony used method [14]. Hong-Wu et a. [15] used response surface methodoogy to estabish the function expressions of defection and structura weight,and optimized the truss structure. Roux et a. [16] carried out RS-based optimization of truss structures and concuded that RS accuracy depends on the range of the design area and the form of the expicit function. Zhang et a. [17] estabish a fitted regression mode for the dynamic transmission error(dte fuctuations to quantify the reationship between modification amounts and DTE fuctuations by using response surface method. In order to increase the accuracy and efficiency, Fan, et a. [18] derived a criterion for judging the existence of cross terms and proposed an adaptive response surface method. Houten et a. [19] considered fifth-order poynomias to check if higher-order poynomias improved RSM accuracy. Venter et a. [20] soved pate optimization probems using higher order RS approximations. Sui and his coaborators [21 25] deveoped the centra point accurate response surface method where the response vaue at the center point of the approximate mode is equa to the true response vaue. This method was successfuy appied to optimization of membrane structures, 2D-continuum structures, pate and she structures subject to frequency constraints, and mutidiscipinary optimization. Using the property of the Kreissemerier-Steinhauser function, RS approximation was buit with function derivation, Tayor expansion near experimenta points and numerica iteration, trying to minimize the argest difference between response function vaues and true response vaues. In the finite eement dispacement method, the noda dispacement is the basic unknown quantity. The stress, dynamic response and so on can be derived from the dispacement, so this paper is devoted to discuss the response surface of truss node dispacement, its research method and resuts can be easiy extended to other types of structures. It shoud be noted that the anaytica form of poynomia-basis response surface approximation is often very different from the functions describing the actua dispacement fied. As this may resut in ow quaity approximation, it is important to assess the rea anaytica form of the dispacement fied. This study wi show that the noda dispacement fied determined by finite eement method for truss structures can be expressed as mutivariate rationa functions. In view of this, a response surface methodoogy based on mutivariate rationa functions wi be deveoped. It wi be iustrated that the determinant of the stiffness matrix and the corresponding eements of adjoint matrix invoved in dispacement determination are poynomias depending on cross-sectiona areas, with the same order as their respective matrices, each term of which is the product of crosssectiona areas. The vaidity of the proposed approach is verified for many design exampes of staticay determinate and staticay indeterminate truss structures. 2 Poynomia basis Response Surface Approximation for Noda Dispacement of Truss Structures 2.1 Poynomia basis RSM The commony used forms of poynomia-basis response surface methodoogy are as foows. Linear form ỹ = β 0 + k β i x i, separabe quadratic form ỹ = β 0 + k β i x i + k β ii xi 2, and the compete quadratic form (containing cross terms ỹ = β 0 + k β i x i + where ỹ is the response function to be constructed, and β are the unknown RS coefficients. 2.2 RSM Evauation Standards The most commony used parameters used to evauate accuracy of response surface approximation are as foows. The mutipe fitting coefficient R 2 = 1 δ γ, k k 1 β ii xi 2 + j=i+1 and the modified mutipe fitting coefficient ( m 1 δ Radj 2 = 1 m k γ, k β ij x i x j, where δ is the sum of the square of the difference between the true response vaue and the response estimated vaue; γ is the sum of the square of the difference between the response estimated vaue and the response mean vaue; m is the tota number of experimenta points; and k is the number of unknown parameters: (1 (2 (3 (4 (5
Page 3 of 14 ( m 2 γ = Y T Y y i /m. R 2 can refect the extent to which the response surface matches the data given. It ranges from 0 to 1, and the coser it is to 1, the smaer the infuence of various errors. For the approximation accuracy of check points, the most commony used parameters are as foows. The maximum reative error ζ max = max{ ζ r r = 1, 2,, S}, minimum reative error ζ min = min{ ζ r r = 1, 2,, S}, average reative error Mr=1 ζ r ζ avg =, S where S is the tota number of check points. The reative error is ζ r = δr δ r / δ r 100%, δ r is the true response vaue of the rth check ponit, δ r is the response estimated vaue of the rth check point. 2.3 Evauation of Poynomia Basis RSM Accuracy In the foowing, fitting accuracy of poynomiabasis RS method is anayzed for some simpe truss structures. The cross-sectiona area of the truss structure is seected as the design variabe. The crosssectiona area ranges from 0.01 m 2 to 0.05 m 2, and a cross sections are circuar. The eastic moduus is E = 2.1 10 11 Pa, and Poisson s ratio is 0.3. The noda dispacement is taken as the structura response. The vaue of the ith design variabe of design points P D is A PD,i {0.01, 0.02, 0.03, 0.04, 0.05} and the vaue of the ith design variabe of check points P E is A PE,i {0.015, 0.025, 0.035, 0.045}. Exampe 1: Panar 2 bar Truss Structure The first exampe is the very simpe two-bar truss structure shown in Figure 1. The given ength of bar 1 is 2, and the ength of bar 2 is 5. Let = 1 m. (1 Anaytic expression of noda dispacement (6 (7 (8 (9 Using the anaytic method of structura mechanics, the dispacement of node A aong the force F direction is as foows: B C Figure 1 Schematics of the panar 2-bar truss structure δ F = 8F + 5 5F EA 1 EA 2 = 1.0 10 7 (0.380 952 381 1A1 + 0.532 397 138 1A2, (2 Noda dispacement computed with poynomiabasis RSM Tria vaues of cross-sectiona areas are given in input to Eq. (10 to determine the dispacement of node A in the direction of appied force. Design points form the matrix X whie the corresponding dispacements form the response matrix Y. The RS coefficient vector β is determined for various types of poynomia-basis approximation of dispacement δ F. Linear form: ỹ = 9.240 1 10 6 7.047 6 10 5 A 1 9.849 3 10 5 A 2. Separabe quadratic form: ỹ = 1.373 1 10 5 2.310 2 10 4 A 1 3.228 6 10 4 A 2 + 0.002 7A 2 1 + 0.003 7A2 2. Compete quadratic form: 1 2 2 (10 (11 (12 ỹ = 1.373 1 10 5 2.310 2 10 4 A 1 3.228 6 10 4 A 2 + 0.002 7A 2 1 + 0.003 7A2 2 8.456 8 10 18 A 1 A 2. (13 The soving accuracy of poynomia-basis response surface method and the seection of the response surface forms are shown in Tabe 1. A F=1 kn
Page 4 of 14 Tabe 1 Accuracy of inear, separabe and compete quadratic RS modes for the 2-bar truss structure probem Evauation standards Linear form Separabe quadratic form Compete quadratic form Mutipe fitting coefficient R 2 0.7962 0.9727 0.9713 2 0.7962 0.9727 0.9713 Maximum reative error ζ max /% 37.29 15.56 15.56 Minimum reative error ζ min /% 10.13 6.56 6.56 Average reative error ζ avg /% 23.56 11.59 11.59 Using the MATLAB software, the surface comparison figure of the anaytic vaue and the response surface estimated vaue and reative error surface figure of the check points are shown in Figure 2. The accuracy of these RS modes is evauated in Tabe 1 by means of the performance indicators described in Section 2.2. Figure 2 compares the variation of noda dispacement over the design space determined from the anaytica expression (10 or by means of the three RS modes. In particuar, the MATLAB pots show the dispacement map and the reative error map. The foowing concusions can be drawn from the resuts obtained for this exampe. (1 The anaytica form of the poynomia-basis RS is competey different from the anaytic soution (10 which depends on the inverse of cross-sectiona areas. (2 The coefficients of the inear RS mode are sma. The quadratic RS modes are much more accurate than the inear RS mode athough reative errors at check points remain rather high. Adding cross-terms in the quadratic mode does not yied substantia improvements. Exampe 2: Panar 8 bar Truss Structure In order to check the vaidity of the above anaysis for more compicated structures, the staticay determinate panar 8-bar truss structure shown in Figure 3 was considered. The ength of a bars is 2 m, except for bars 6 and 8 which are 2 2 m ong. Loads and kinematic constraints for this structure are shown in Figure 3. The anaytica soution for the dispacement of node C aong the direction of appied force F is as foows: δ y = 8F + 2F + 2F + 2F + 4 2F + 2F + 4 2F EA 1 EA 2 EA 3 EA 5 EA 6 EA 7 EA 8 [ = 1.0 10 7 0.380952381 1 + 0.09523809524 A 1 ( 1 + 1 + 1 + 1 ( 1 + 0.269340119 + 1 ] A 2 A 3 A 5 A 7 A 6 A 8 (14 Poynomia-basis response surface approximations of dispacement give the foowing resuts. Linear form: ỹ = 1.315 81 10 5 1.0 10 5 (7.047 6A 1 + 1.761 9 (A 2 + A 3 + A 5 + A 7 + 4.982 8 (A 6 + A 8. (15 Separabe quadratic form: ỹ = 1.955 2 10 5 1.0 10 4 (2.310 2A 1 + 5.775 5 (A 2 + A 3 + A 5 + A 7 + 1.633 4 (A 6 + A 8 + 0.002 7 A 2 1 + 6.689 3 10 4 (A 2 2 + A2 3 + A2 5 + A2 7 + 0.001 9 (A 2 6 + A2 8. (16 For the sake of brevity, the expression of compete quadratic RS is not reported in the artice. Performance indicators for these RS modes are isted in Tabe 2. The quadratic RS modes are again much more accurate than the inear RS mode but reative errors at check points remain rather high, thus not satisfying the requirements on overa accuracy. Adding cross-terms in the quadratic mode again does not yied substantia improvements. As expected, the anaytica form of noda dispacement (14 is competey different from the expressions of RS modes. Since the two truss exampes refer to staticay determinate structures, a staticay indeterminate structure is now anayzed to draw more genera concusions. Exampe 3: Panar 6 bar Truss Structure Figure 4 shows the schematic of a staticay indeterminate 6-bar truss structure, in which is the ength and = 1 m. The dispacement to be computed is that of node B aong the direction of appied force F. The accuracy of inear, separabe quadratic and fu quadratic RS modes is isted in Tabe 3. The data isted in the tabe ead to simiar concusions as in the case of staticay determinate structures. However, introducing cross-terms in the quadratic RS mode
Page 5 of 14 Figure 2 Evauation of RS approximation accuracy for the 2-bar structure probem. a Linear RS mode. b Separabe quadratic RS mode. c Fu quadratic RS mode
Page 6 of 14 F 5 E 4 D A 4 B F=1kN 6 7 8 3 3 6 1 C 1 2 A B Figure 3 Schematics of the panar 8-bar truss structure F=1kN 5 D 2 C Figure 4 Schematics of the panar 6-bar truss structure now yieds a more cear improvement over separabe quadratic RS mode. The above exampes demonstrate that noda dispacements of both staticay determinate and indeterminate truss structures are mutivariate rationa functions of cross-sectiona areas. This mode is totay different from the anaytica form of poynomia-basis response surfaces which cannot hence describe correcty the noda dispacement fied of truss structures. 2.4 Evauation of the Accuracy of Poynomia Basis RSM with Reciproca Variabes The dispacement of a certain node in a given direction computed using Mohr s theorem can be expressed as δ i = M N 0 j N j j EA j, (17 where N j is the interna force of jth eement under the action of true oad, N j 0 is the interna force of the jth eement under the action of unit force with the computed dispacement direction, j is the eement ength, and M is the number of eements. Let x j = 1 / A j, consequenty, Eq. (17 can be expressed as The most commony used RS approximation function of noda dispacements of truss structures can be obtained by taking the reciproca variabes x j as design variabes and substituting them into Eqs. (1, (2 and (3. In order to anayze the accuracy of the poynomiabasis response surface method with reciproca variabes, the accuracy anaysis is carried out by reusing Exampe 1 3 as the exampes. Exampe 4: Panar 2 bar Truss Structure For the 2-bar truss structure studied in Exampe 1, the computing accuracy of poynomia-basis RSM with reciproca variabes is isted in Tabe 4. Exampe 5: Panar 8 bar Truss Structure For the panar 8-bar truss structure studied in Exampe 2, accuracy of poynomia-basis RSM with reciproca variabes is isted in Tabe 5. Exampe 6: Panar 6 bar Truss Structure For the panar 6-bar truss structure studied in Exampe 3, accuracy of poynomia-basis RSM with reciproca variabes is isted in Tabe 6. The foowing concusions can be drawn from the resuts of the above these exampes. δ i = N 0 j N j j E x j. (18 Tabe 2 Accuracy of inear, separabe and compete quadratic RS modes for the 8-bar truss structure probem Evauation standards Linear form Separabe quadratic form Compete quadratic form Mutipe fitting coefficient R 2 0.8132 0.9773 0.9773 2 0.8132 0.9773 0.9773 Maximum reative error ζ max /% 37.292 15.563 15.563 Minimum reative error ζ min /% 0.039 6 2.292 10 4 2.292 10 4 Average reative error ζ avg / % 16.096 4.747 4.747
Page 7 of 14 Tabe 3 Accuracy of inear, separabe and compete quadratic RS modes for the 6-bar truss structure probem Evauation standards Linear form Separabe quadratic form Compete quadratic form Mutipe fitting coefficient R 2 0.8026 0.8991 0.9996 2 0.8026 0.8990 0.9996 Maximum reative error ζ max /% 28.68 22.66 14.43 Minimum reative error ζ min /% 2.76 3.98 5.01 Average reative error ζ avg /% 18.93 11.82 8.96 (1 For the staticay determinate structure, the evauation coefficients are very good. The a reative errors are ess than 10 10, therefore the approximation accuracy is very high. This is mainy because the poynomia-basis RSM with reciproca variabes equation have a high consistency in the form with the anaytic soution, and the response surface equation refects the true reationship between dispacement and design variabes. (2 For the staticay indeterminate structure, the approximation accuracy is very ow. This is mainy because interna forces are changing as the changes of design variabes, and these changes don t be refected in RSM equation, which ead to the difference in the form between RSM approximation with reciproca variabes and anaytic soution. In order to improve the approximation accuracy, a new method of high accuracy response surface shoud be deveoped. Tabe 4 Accuracy of inear, separabe and compete quadratic reciproca variabes RSM modes for the 2-bar truss structure probem Evauation standards Linear form Separabe quadratic form Compete quadratic form Mutipe fitting coefficient R 2 1 1 1 2 1 1 1 Maximum reative error ζ max /% 5.78 10 14 1.68 10 12 1.13 10 12 Minimum reative error ζ min /% 1.14 10 14 1.09 10 12 5.22 10 13 Average reative error ζ avg /% 3.38 10 14 1.40 10 12 8.21 10 13 Tabe 5 Accuracy of inear, separabe and compete quadratic reciproca variabes RS modes for the 8-bar truss structure probem Evauation standards Linear form Separabe quadratic form Compete quadratic form Mutipe fitting coefficient R 2 1 1 1 2 1 1 1 Maximum reative error ζ max /% 2.76 10 11 2.70 10 9 4.72 10 11 Minimum reative error ζ min /% 9.90 10 12 4.15 10 14 5.23 10 13 Average reative error ζ avg /% 1.90 10 11 6.70 10 10 2.10 10 11 Tabe 6 Accuracy of inear, separabe and compete quadratic reciproca variabes RS modes for the 6-bar truss structure probem Evauation standards Linear form Separabe quadratic form Compete quadratic form Mutipe fitting coefficient R 2 0.897 0.908 0.998 2 0.897 0.908 0.998 Maximum reative error ζ max /% 23.194 27 5.967 Minimum reative error ζ min /% 7.07 10 4 3.36 10 3 3.167 10 4 Average reative error ζ avg /% 5.096 4.065 1.024
Page 8 of 14 3 Determination of Noda Dispacements of Truss Structures According to the finite eement method theory, the baance equation is as K δ = F. The dispacement equation is as δ = K 1 F. (19 (20 Where the inverse matrix can be obtained by the formua as K 1 = adjk (21 K. According to the theory of determinants, the n-order determinant is a number firmed by n 2 eements of a ij. a 11 a 12 a 1n a 21 a 22 a 2n K =... a (22 n1 a n2 a nn = ( 1 τ(j 1j 2 jn a 1j1 a 2j2 a njn, j 1 j n where τ(j 1 j 2 j n is the inverted sequence number of natura numbers 1,2,,n, and j 1 j n is the sum of a permutations of natura numbers 1,2,,n. Since eements a ij of stiffness matrix incude products of inear terms corresponding to cross-sectiona areas, the determinant can be simpified as K = f (A n, (23 where f(a n is an n-order poynomia of cross-sectiona area variabes A, and n is the order of the stiffness matrix, equa to the DOF of the structure system. Eements of the adjoint matrix adjk are poynomias g ij (A n 1 which are one order ower than the determinant of the stiffness matrix. Thus, the form of the adjoint matrix is g 11 (A n 1 g 1n (A n 1 adjk =... g n1 (A n 1 g nn (A n 1 (24 Combining a of the above formuas, it can be obtained the equation to anayticay determine noda dispacements of truss structures: δ = K 1 F = adj K F. (25 K Consequenty, the dispacement of a certain node in a given direction can be expressed as δ i = 1 K 4 Conjecture about the Determinant Expression of Stiffness Matrix and Eements of Adjoint Matrix The previous derivation indicate that the determinant of the stiffness matrix of the truss structure and the eements of adjoint matrix invoved in the determination of noda dispacements are respectivey n-order and (n 1-order poynomias. Stiffness matrix determinant is an agebraic sum incuding n! terms each of which is the product of n factors beonging to different rows and coumns. The sign of each term depends on the corresponding inverted sequence number. Therefore, there are at east n! coefficients. Simiary, determinants of adjoint matrix eements are agebraic sums of (n 1! terms each of which corresponds to the product of (n 1 factors beonging to different rows and different coumns. The adjoint matrix is an n-order square matrix. Equation (26 impies that 2n! function evauations must be done. However, if matrix order changes, the number of coefficients aso wi change in a factoria way. Hence, if the equation order gets arge, too many combinations shoud be considered. Variabe inking and reationships between cross-sectiona area variabes may greaty reduce computationa compexity of combinations and mutivariate rationa function response surfaces can be buit based on these reationships. The above mentioned reationships wi be iustrated by the foowing exampes. (1 For the 2-bar truss structure considered in Exampe 1, dispacement of node A in the direction of appied oad F has the foowing anaytica expression: where, δ AF = 1 K f (A 2 = g ij (A n 1 F j = 1 f (A n g ij (A n 1 F j. g 1j (A n 1 F j = 1 f (A 2 g 12(AF, E2 50 2 5A1 A 2, (26 (27 ( g 12 (A = E A 1 2 + 4 5A2 25 (2 For the staticay indeterminate 6-bar truss structure considered in Exampe 3, the anaytica expression for the noda dispacement of node B aong the direction of appied force F is as foows: δ BF = 1 K g 1j (A n 1 F j = 1 f (A 5 g 33(A 4 F, (28
Page 9 of 14 where f (A 5 = E5 16 5 (2A 1A 2 A 3 A 5 A 6 + 2A 1 A 2 A 4 A 5 A 6 + A 1 A 3 A 4 A 5 A 6 + A 2 A 3 A 4 A 5 A 6 + 4 2A 1 A 2 A 3 A 4 A 5 + 4 2A 1 A 2 A 3 A 4 A 6, By substituting Eq. (30 into Eq. (26, it foows δ i = C 1 A 1 + C 2 A 2 + + C n A n = C j A j, (31 g 33 (A 4 = E4 16 4 (16A 1A 2 A 3 A 4 + 4 2A 1 A 2 A 3 A 6 + 4 2A 1 A 2 A 4 A 6 + 4 2A 1 A 3 A 4 A 5 + 2A 1 A 3 A 5 A 6 + 2A 1 A 4 A 5 A 6 + 4 2A 2 A 3 A 4 A 5 + 2A 2 A 3 A 5 A 6 + 2A 2 A 4 A 5 A 6. The foowing conjecture can be made from the previous reationships. The determinant of stiffness matrix and the adjoint matrix term entaied in the determination of a certain noda dispacement are poynomias of the same order as the corresponding matrices. Each term of these poynomias is the product of first power terms corresponding to cross-sectiona areas. m 1 f (A n = α i A i1 A i2 A in, (29 m 2 g(a n 1 = β i A i1 A i2 A i(n 1, where i 1 = i2 = i3 = = i(n 1 = in. The number of DOF n is the same as the order of stiffness matrix; m is the number of design variabes. A n are the possibe combinations of crosssectiona areas A i seected from the set of m design variabes; m 1 is the number of these combinations, namey, m 1 = Cm n = m!/ [(m n!n!]. Simiary, A n 1 are other possibe combinations of cross-sectiona areas A i ; m 2 is the number of these combinations, namey, m 2 = Cm n 1 = m!/ [(m n + 1! (n 1!]; α and β are unknown coefficients to be determined. The number of terms incuded in matrix stiffness determinant decreases from n! to m! / [(m n!n!]; whie the number of terms in the corresponding adjoint matrix decreases from (n-1! to m! / [(m n + 1! (n 1!], thus reducing the computationa cost of the fitting process. Let s now consider a staticay determinate structure with m bars and n DOFs. Since there are no redundant bars, it hods m = n. Hence, in Eq. (29, it hods m 1 = Cm n = Cn n = 1, m 2 = Cm n 1 equation simpifies to = Cn 1 n = n and the f (A n = b 0 A 1 A 2 A n, g ij (A n 1 = a 1 A 1 A 2 A n 1 + a 2 A 1 A 3 A n + + a i A 1 A i A n + +a n A 2 A 3 A n. (30 where C j = a n j+1 /b 0. For staticay determinate structures, if the externa force P is known, interna forces N j deveoped in each bar are constant. If the generaized dispacement δ i in the direction i is computed using Mohr s theorem, a unit generaized force N i 0 is appied in the direction i and the corresponding interna forces N j in each bar aso are constant. Eement engths j and cross-sectiona areas A j are constant vaues as we. Therefore, noda dispacements of a truss structure can be expressed as δ i = N j N 0 j j EA j = C j A j, / E. (32 where the constant C is equa to C j = N j Nj 0 j The consistency of Eqs. (31 and (32 proves that it may be reasonabe to approximate noda dispacements of staticay determinate truss structures with mutivariate rationa response surfaces. The foowing exampes wi prove the vaidity of this concusion aso for staticay indeterminate trusses. Exampe 7: Panar 10 bar Truss Structure For the panar 10-bar truss structure shown in Figure 5, the genera form of the determinant of the stiffness matrix is for this structure is as foows: f (A n = E8 8 (8A 1A 10 A 2 A 3 A 4 A 5 A 6 A 8 + 8A 1 A 10 A 2 A 3 A 4 A 5 A 7 A 9 + A 1 A 10 A 2 A 3 A 4 A 6 A 7 A 9 + + βa a A b A c A d A e A f A g A h +. (33 The corresponding adjoint matrix eement invoved in the determination of the dispacement of node C in the direction of appied oad F is as foows: g 67 (A n 1 = E7 7 (8A 1A 10 A 2 A 3 A 4 A 5 A 7 + 8A 1 A 10 A 2 A 3 A 4 A 5 A 9 + 8A 1 A 10 A 2 A 3 A 4 A 8 A 7 + + αa a A b A c A d A e A f A g +. (34
Page 10 of 14 A F 6 5 1 7 8 E B where {a, b, c, d, e, f, g, h} {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, α and β are the unknown coefficients of the products of design variabe A i, and a b c d e f g h. Exampe 8: Spatia 18 bar Truss Structure For the spatia 18-bar truss structure shown in Figure 6, determinant of stiffness matrix is as foows: f (A n = E12 12 (2175A 1A 2 A 17 A 18 A 5 A 9 A 11 A 12 A 15 A 16 A 13 A 14 + 76 2A 1 A 17 A 18 A 4 A 5 A 9 A 10 A 12 A 15 A 16 A 11 A 14 + 128 2A 1 A 3 A 6 A 8 A 2 A 9 A 10 A 4 A 11 A 12 A 15 A 16 + + αa a A b A c A d A e A f A g A h A i A j A k A + +. (35 The corresponding adjoint matrix eement invoved in the determination of the dispacement of node A in the direction of appied oad F is as foows: g 12 (A n 1 = E11 11 (4096 2A 1 A 3 A 6 A 2 A 7 A 17 A 5 A 10 A 11 A 12 A 15 + 2048A 1 A 3 A 6 A 2 A 7 A 5 A 4 A 11 A 16 A 13 A 14 + 768A 1 A 3 A 6 A 17 A 18 A 5 A 4 A 9 A 10 A 13 A 16 + 6144 2A 1 A 3 A 2 A 7 A 17 A 18 A 5 A 9 A 10 A 11 A 16 + + αa a A b A c A d A e A f A g A h A i A j A k +, (36 where {a, b, c, d, e, f, g, h, i, j, k,, m, n, o, p, q, r} {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18}, α and β are the unknown coefficients of the products of design variabe A i, and a b c d e f g h i j k m n o p q r. Other exampes not documented in this artice for the sake of brevity confirm the vaidity of the conjecture made on the mutua reationship between the stiffness matrix of the truss structure and adjoint matrix eements. 9 4 2 10 Figure 5 Schematics of the panar 10-bar truss structure 3 C D F=1kN 5 Mutivariate Rationa Response Surface Approximation of Truss Noda Dispacements Response surface approximation of truss noda dispacements can be buit as foows: where α i and β ij are the RS fitting coefficients to be determined with β ij = β ij F j ; m 1 and m 2, respectivey, are the number of terms incuded in poynomias f(a n and g ij (A n 1. Suitabe design points are defined incuding the crosssectiona areas of the M bars forming the truss. For the generic design C, it hods A k = A C k with k = 1, 2,, M. After the dispacement δ C i corresponding to design points is determined from finite eement anaysis, Eq. (37 is rewritten as foows: δ C i or δ C i n δ i = 1 f (A n g ij(a n 1 F j n ( m 2 = β ija i1 A i2 A in 1 F j m1 α ia i1 A i2 A in m2 F j =0 = β ij A i1a i2 A in 1 m1 α, ia i1 A i2 A in ( m1 α i A C i1 AC i2 AC in = α i A C i1 AC i2 AC in ( m1 11 1 9 4 6 12 5 3 8 10 15 17 Figure 6 Schematics of the Spatia 18-bar truss structure m 2 β ij AC i1 AC i2 AC i(n 1, F j =0 m 2 β ij AC i1 AC i2 AC i(n 1 = 0. F j =0 7 16 A 13 F 14 2 18 (37 (38 (39
Page 11 of 14 The above expression is a homogeneous equation. Since the ratio between numerator and denominator in Eq. (37 stays the same if both terms are mutipied by the same quantity, RS mode coefficients can be normaized with respect to any fitting coefficient. In order to keep generaity, et us set α 1 = 1 and rewrite Eq. (39 as foows: m 2 β ij AC i1 AC i2 AC in 1 F j =0 ( m1 δ C i α i A C i1 AC i2 AC in = δi C AC i1. i=2 (40 Seecting M design points (M m 1 + Km 2, where K is the number of oads F j that are not equa to zero, a system of M inear equations in the unknown coefficients α i and β ij is obtained: m 2 F j =0 m 2 β ij AC1 i1 AC1 i2 AC1 i(n 1 δc1 i F j =0 m 2 F j =0 β ij ACM i1 β ij AC2 i1 AC2 i2 AC2 i(n 1 δc2 i A CM i2 A CM i(n 1 δcm i ( m1 ( m1 (41 The determination of these coefficients by soving this system can be done in two ways. (1 If M > m 1 + Km 2, the number of design points is arger than the number of unknown coefficients. Soving accuracy is maximized by using the east square method. (2 If M = m 1 + Km 2, the number of design points is equa to the number of unknown coefficients and cassica inear agebraic sovers can be utiized. According to best uniform approximation theory, quaity of RS approximation improves as the fitting mode resembes the rea structura response. Since the mutivariate rationa RS mode deveoped in this study reproduces the actua anaytica mode behind determination of noda dispacements of truss structures, the present mode can give resuts very cose to the rea structura response of a truss structure. i=2 ( m1 i=2 i=2 α i A CM i1 α i A C1 i1 AC1 i2 AC1 in α i A C2 i1 AC2 i2 AC2 in A CM i2 A CM in = δ C1 i A C1 i1, = δ C2 i A C2 i1,. = δ CM i A CM i1. 6 Verification of Mutivariate Rationa RS Mode for Truss Dispacement Approximation The accuracy of the mutivariate rationa response surface approximation mode deveoped in this study is verified in some design exampes. The cross-sectiona area of truss eements is seected as the design variabe which can range between 0.01 and 0.05 m 2 ; cross sections of a eements are supposed to be circuar. The Young s moduus is 2.1 10 11 Pa whie Poisson s ratio is 0.3. Node dispacement is taken as the structura response to be approximated. Vaue of design variabes seected for design points P D is A PD,i {0.01, 0.02, 0.03, 0.04, 0.05} whie vaue of design variabes seected for check points P E is A PE,i {0.015, 0.025, 0.035, 0.045}. Exampe 9: Panar 2 bar Truss Structure For the truss studied in Exampe 1, accuracy of mutivariate rationa RS mode is evauated in Tabe 7. Figure 7(a compares the dispacement maps corresponding to the true structura response and the approximate mode. The reative errors on noda dispacements evauated at check points are shown in Figure 7(b. Exampe 10: Panar 6 bar Truss Structure The structura ayout, kinematic constraints and oads are the same as for Exampe 3. Tabe 8 ists the vaues of performance indicators for the mutivariate rationa RS mode. Exampe 11: Panar 12 bar Truss Structure The panar truss shown in Figure 8 is a staticay indeterminate structure constrained by fixed-hinged supports around the edge. There are ony two degrees of freedom for this design exampe. Performance indicators for the mutivariate rationa RS mode, evauated with respect to the soution provided by the commercia finite eement software ANSYS, are isted in Tabe 9. Exampe 12: Monoithic Tower For the monoithic tower schematized in Figure 9, the atera oad P 2 simuates the action of the wind whie the vertica oad P 1 simuates the weight of the tower body and other concentrated oads. Two oading conditions are considered: (i P 1 = 0, P 2 = 1 kn; (ii P 1 = 1 kn, P 2 = 1 kn. The dispacement to be computed is the atera dispacement of node A and B respectivey. Performance indicators for the mutivariate rationa RS mode, evauated with respect to the soution provided by the commercia finite eement software ANSYS, are isted in Tabe 10.
Page 12 of 14 Tabe 7 Accuracy of mutivariate rationa RS mode for the 2-bar truss structure probem Evauation standards Node A Mutipe fitting coefficient R 2 1 2 1 Maximum reative error ζ max /% 6.58 10 14 Minimum reative error ζ min /% 1.15 10 14 Average reative error ζ avg /% 2.05 10 14 Tabe 8 Accuracy of mutivariate rationa RS mode for the 6-bar truss structure probem Evauation standards Node B Mutipe fitting coefficient R 2 1 2 1 Maximum reative error ζ max /% 3.63 10 4 Minimum reative error ζ min /% 1.51 10 7 Average reative error ζ avg /% 7.46 10 5 A P Figure 8 Schematics of the panar 12-bar truss structure Exampe 13: Secondary Mirror Supporting Structure The secondary mirror supporting structure schematized in Figure 10 is a highy ightweight trianguar structure in which six supporting bars form a cosed bar system in an end to end way between the primary and secondary mirror pates. The weight of the objects on the top of the supporting structure is equivaent to the concentrated oads on the three top nodes (see Figure 10. The dispacement to be computed is the vertica dispacement of node A. Performance indicators for the mutivariate rationa RS mode, evauated with respect to the soution provided by the commercia finite eement software ANSYS, are isted in Tabe 11. Figure 7 Evauation of mutivariate rationa RS approximation accuracy for the 2-bar structure probem. a Comparison of dispacement map for exact soution and RSM approximation. b Dispacement error map at check points It can be seen that reative error between response resuts obtained by mutivariate rationa RS approximation and ANSYS is very sma for a design exampes. Evauation coefficients are aways very high, thus satisfying engineering requirements. The same concusion hods true for both staticay determinate and indeterminate truss structures.
Page 13 of 14 Tabe 9 Accuracy of mutivariate rationa RS mode for the 12-bar truss structure probem Evauation standards Node A Mutipe fitting coefficient R 2 1 2 1 Maximum reative error ζ max /% 2.03 10 3 Minimum reative error ζ min /% 1.54 10 9 Average reative error ζ avg /% 1.67 10 4 P 2 P 1 A P 1 7 Concusions (1 The poynomia-basis response surface approximation is different from the rea function between cross-sectiona area and noda dispacement, reative error ranging from 4.747% to 23.56% which does not meet the engineering requirement. (2 Poynomia-basis RSM with reciproca variabes has a great improvement on the staticay determinate structure, with reative errors ess than 10 10. However, for the staticay indeterminate structure, the approximation accuracy is sti ow. (3 Reationship between the noda dispacement and the cross-sectiona area is derived, foowing the conjecture about the determinant expression of stiffness matrix and eements of adjoint matrix. (4 On the basis of previous conjecture, mutivariate rationa response surface approximation of truss noda dispacements is estabished. The fitting accuracy of both staticay determinate structure and staticay indeterminate structure is greaty improved, with reative errors ess than 10 3. P 2 B P 2 Figure 10 Schematics of the secondary mirror supporting structure Figure 9 Schematics of the monoithic tower Tabe 11 Accuracy of mutivariate rationa RS mode for the secondary mirror supporting structure probem Evauation standards Node A Mutipe fitting coefficient R 2 1 2 1 Maximum reative error ζ max /% 1.24 10 4 Minimum reative error ζ min /% 7.37 10 6 Average reative error ζ avg /% 4.48 10 5 Tabe 10 Accuracy of mutivariate rationa RS mode for the monoithic tower probem Evauation standards Condition 1 Condition 2 Node A Node B Node A Node B Mutipe fitting coefficient R 2 1 1 1 1 2 1 1 1 1 Maximum reative error ζ max /% 3.67 10 4 3.63 10 2 4.83 10 3 7.08 10 3 Minimum reative error ζ min /% 2.10 10 3 7.30 10 8 7.65 10 8 2.76 10 8 Average reative error ζ avg /% 5.58 10 4 2.45 10 3 6.72 10 4 5.18 10 4
Page 14 of 14 Authors contributions SC, LJL, MG carried out the mutivariate rationa response surface approximation of noda dispacements of truss structures study. XFJ paticipated in the sequence aignment and drafted the manuscript. And a authors read and approved the fina mancuscript. Author detais 1 Schoo of Transportation and Vehice Engineering, Shandong University of Technoogy, Zibo 255049, China. 2 Key Laboratory of Eectronic Equipment Structure Design, Xidian University, Xi an 710071, China. 3 Shandong Lingong Tire Co., Ltd., Zhaoyuan 265400, China. Authors Information Shan Chai, born in 1955, is a professor at Schoo of Transportation and Vehice Engineering, Shandong University of Technoogy, China. He received his Ph.D. degree from Daian University of Technoogy, China, in 1996. His main research interests incude structura optimization, computer aid engineering, vehice and mechanics dynamics. Xiang-Fei Ji, born in 1991, is currenty a Ph.D. candidate at Key Laboratory of Eectronic Equipment Structure Design, Xidian University, China. His research interests incude optimization, noise anaysis and contro. Li-Jun Li, born in 1977, is currenty an associate professor at Schoo of Transportation and Vehice Engineering, Shandong University of Technoogy, China. She received her Ph.D. degree from Shanghai Jiao Tong University, China, in 2005. Her research interests incude computer aid engineering, vehice and mechanics dynamics, noise anaysis and contro. Ming Guo, born in 1989, is currenty an engineer at Shandong Lingong Tire Co., Ltd., China. He received his master degree Schoo of Transportation and Vehice Engineering, Shandong University of Technoogy, China, in 2006. His main research interests incude structura optimization, computer aid engineering. Acknowedgements Supported by Nationa Natura Science Foundation of China (Grant No. 5150261, and Shandong Provincia Natura Science Foundation of China (Grant No. ZR2015AM013. Competing interests A the authors decare that they have no competing interests. Ethics approva and consent to participate Not appicabe. Pubisher s Note Springer Nature remains neutra with regard to jurisdictiona caims in pubished maps and institutiona affiiations. Received: 6 November 2016 Accepted: 16 January 2018 References 1. André I Khuri, Siui Mukhopadhyay. Response surface methodoogy. Wiey Interdiscipinary Reviews Computationa Statistics, 2010, 2(2: 128-149. 2. Choon-Man Jang, Ka-Ram Choi. Optima design of spitters attached to turbo bower impeer by RSM. Journa of Therma Science, 2012, 21(3: 215-222. 3. Heng Jiang, Yi-Sheng Guan, Zhi-Cheng Qiu, et a. Dynamic and static muti-objective optimization of a vertica machining center based on response surface method. 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