基于广义bayes理论地基参数的powell反演力学模型.pdf

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1、Zhang et al. / J Zhejiang Univ-Sci A (Appl Phys & Eng) 2017 18(7):567-578 567Powell inversion mechanical model of foundation parameters with generalized Bayesian theory*Jian ZHANG1, Chu-wei ZHOU1, Chao JIA2, Jing LIN3(1Department of Mechanics and Structural Engineering, Nanjing University of Aeronau

2、tics and Astronautics, Nanjing 210016, China) (2School of Civil Engineering, Shandong University, Jinan 250061, China) (3Institute of Structural Engineering, Tongji University, Shanghai 200092, China) E-mail: Received June 14, 2016; Revision accepted Oct. 26, 2016; Crosschecked June 26, 2017 Abstra

3、ct: The inversion mechanical model of foundation parameters based on Powell optimizing theory was studied with generalized Bayesian theory. First, the generalized Bayesian objective function for foundation parameters was deduced with maximum likelihood theory. Then, the expectation expression and th

4、e covariance expression of the foundation parameters were obtained. After selecting the Winkler foundation as representative, the governing differential equations of the typical foundation were derived. With the orthogonal series transform method, the Fourier closed form solution of a moderately-thi

5、ck plate on the Winkler foundation was achieved. After the optimal step length was determined with the quadratic parabolic interpolation method, the Powell inversion mechanical model of foundation parameters was resolved, and the corresponding inversion procedure was completed. Through particular ex

6、ample analysis, the highlight is that the Powell inversion mechanical model of foundation pa-rameters with generalized Bayesian theory is correct and the derived Powell inversion model has universal significance, which can be applied in other kinds of foundation parameters. Besides, the Powell inver

7、sion iterative processes of foundation parameters have excellent numerical stability and convergence. The Powell optimizing theory is unconcerned with the partial derivatives of systematic responses to foundation parameters, which undoubtedly has a satisfying iterative efficiency compared with the a

8、vail-able Kalman filtering or conjugate gradient inversion of the foundation parameters. The generalized Bayesian objective function can synchronously take the stochastic property of systematic parameters and systematic responses into account. Key words: Powell inversion; Mechanical model; Foundatio

9、n parameter; Bayesian objective function; Stochastic property http:/dx.doi.org/10.1631/jzus.A1600440 CLC number: TU451 1 Introduction In geotechnical engineering, the necessary me-chanical parameters used in geotechnical analysis include typical kinds of fuzzy, indeterminate, and discrete properties

10、 (Kim et al., 2015; Fathi et al., 2016). How to evaluate the selected parameters effi-ciently becomes a fundamental task. Up to now, when the plate on the foundation or on the road surface is tackled, the general soil medium models include the Winkler foundation, the Pasternac foundation, the Heteny

11、i foundation, the Reissner foundation models, the elastic half sphere and finite depth elastic com-pression layer models, etc. (Chen et al., 2015; Ong and Choo, 2016). Among these models, the Winkler analytical model has been widely used in engineering because it precisely simulates the factual work

12、ing status of the foundation and is comparatively con-venient in mathematical manipulation (Zhao, 2007; Ching et al., 2016). In engineering, some parameters such as the thickness of the foundation plate and the concrete strength can be directly achieved through the Journal of Zhejiang University-SCI

13、ENCE A (Applied Physics & Engineering) ISSN 1673-565X (Print); ISSN 1862-1775 (Online) ; E-mail: *Project supported by the Fundamental Research Funds for the Cen-tral Universities of China (No. NS2014003) ORCID: Jian ZHANG, http:/orcid.org/0000-0003-2457-9558; Chao JIA, http:/orcid.org/0000-0002-2

14、448-894X Zhejiang University and Springer-Verlag Berlin Heidelberg 2017 万方数据Zhang et al. / J Zhejiang Univ-Sci A (Appl Phys & Eng) 2017 18(7):567-578 568spot experiment while some parameters such as the foundation parameter cannot be concisely gained. The parameters are sometimes determined by expe-

15、riences which cannot take the influence of stochastic factors into account. Thus, if the parameters can be accurately inverted with systematic responses of dif-ferent measuring times and spatial spots, it helps to evaluate and forecast performance of the foundation more precisely (Xin et al., 2014;

16、Zong et al., 2014; Jia and Chi, 2015; Li et al., 2015). The parameters might be determined through some specific methods in-cluding parameter inversion methods (Sarp Arsava et al., 2016). In recent years, there has been some re-search emphasis on the inversion problems and some achievement in dealin

17、g with the inversion of the Winkler foundation parameter (Xie, 2011). Never-theless, the conjugate gradient theory has been suc-cessfully used in parameter inversion (Zhao, 2007), and the Kalman filtering theory, which has the ad-vantages of auto revision and auto optimization, has also been success

18、fully applied in the inversion of the foundation parameter (Zhang et al., 2008). However, it is unfortunately a deficiency that in both the Kal-man filtering theory and the conjugate gradient the-ory, the partial derivatives of the systematic responses to the Winkler foundation parameters must be re

19、-peatedly computed in iterative processes, which will consequentially lead to lower computational effi-ciency and error accumulation. As an improvement, the Powell optimizing method is not concerned with the perplexing partial derivatives. In addition, it has been proved that the Pasternac foundatio

20、n model as well as some other foundation models are undoubt-edly more precise than the Winkler foundation model (Bouderba et al., 2013; Bousahla et al., 2014; Mezi-ane et al., 2014; Zidi et al., 2014). The main point of this study is how to derive the Powell optimizing inversion mechanical model of

21、foundation parameters with generalized Bayesian theory, and in order to make comparisons with other research results (Zhao, 2007; Zhang et al., 2008) in a convenient manner, the concise Winkler foundation model is chosen the same as Zhao (2007) and Zhang et al. (2008). Certainly, the following Powel

22、l optimizing inversion model based on generalized Bayesian theory has universal signif-icance for different kinds of foundation parameters and only the corresponding foundation model should be considered. Thus, in this paper, the generalized Bayesian objective function for foundation parameters is d

23、e-duced with maximum likelihood theory. Then, the expectation expression and the covariance expression of the foundation parameters are obtained. With the Fourier closed form solution for the foundation, the Powell inversion mechanical model of foundation parameters is resolved and some typical exam

24、ples are analyzed in detail. 2 GeneraIized Bayesian objective function of foundation parameters During the process of Powell optimizing inver-sion of the soil medium model, the foundation pa-rameters can be treated as random variables, which are noted as the random vector X=x1x2 xmT(m is the dimensi

25、on of the vector X) to carry the parameter inversion into execution. In inversion research studies such as evaluation of models performance of fracture toughness of polymeric particle nanocomposites, de-tection of material interfaces in piezoelectric struc-tures and detection of flaws in piezoelectr

26、ic structures using extended finite element method (FEM) (Nan-thakumar et al., 2013; 2016; Hamdia et al., 2016), Bayesian theory is widely applied because one of its superior properties lies in taking the stochastic prop-erty of systematic parameters and systematic re-sponses into account efficientl

27、y. However, a much more efficient Bayesian objective function is derived below. From Bayesian theory, it can be noted: *(|)()(| ) ,()ffff=WX XXWW(1) where f(X) is the priori information distribution, f(W*|X) is the conditional distribution of the system-atic response, f(W*) is the systematic respons

28、e dis-tribution, and f(X|W*) is the posterior information distribution. It is presumed that the foundation pa-rameters X conform to a Gaussian normal distribu-tion, and then the priori information distribution f(X) is expressed: 1/2/2T100()(2)1exp ( ) ( ) ,2mf-= - -I XXXCXX CXX(2) 万方数据Zhang et al. /

29、 J Zhejiang Univ-Sci A (Appl Phys & Eng) 2017 18(7):567-578 569where X0and CXare respectively the expectation vector and covariance matrix of foundation parame-ters X. In engineering practice, the systematic responses at the testing nodes must be measured several times and the measured systematic re

30、sponse data Wi*from each measure are all extracted from the total data W*. If the ordinary Bayesian objective function is set up to inverse parameters, there is much repeated and worthless work. Thus, the generalized Bayesian ob-jective function of the foundation parameters is de-duced. The joint de

31、nsity function of Wi*is 1()n*ii=f |WX, where n is the number of times of the measured systematic response data. From maximum likelihood theory, it can be obtained: 1/2/21T11(|)(2)1exp ( ) ( ) ,2*iin*min*ii iiif*-=-=- - -I WWWX CWWCWW(3) where Wi=Wi(X) is the systematic response vector of the computa

32、tional results. Substituting Eqs. (2) and (3) into Eq. (1), the generalized Bayesian objective function J can be derived as T11T100()()()().in*ii iiiJC*-=-=- - -WXWWCWWXX XX(4) The generalized Bayesian objective function J in Eq. (4) is utilized in the Powell optimizing inversion. In order to attain

33、 the variance inversion result of the foundation parameters X, from Eq. (4) the partial differentiations of the generalized objective function J to the foundation parameters X are expressed: T11012()2().*in*iiiiJC-= =-l） XWWWW CXXXX(5) When Wi(X) is submitted with a Taylor formula expansion at the e

34、xpectation point X and only the zero- and first-order items are reserved, it can be derived as () () ()( ),iii= -WX WX SX X X (6) where the sensitivity matrix ()ii=X XWSXX. Sub-stituting Eq. (6) into Eq. (5) gives *T1 *1102( )2( ),iniiiiiWiJ-=-=-XSC W SX SX WXCXX(7) where ()ii=WWX. When the generali

35、zed Bayesian objective function J reaches the minimum value, Eq. (7) equals zero. Then, *T1 1=1T1 * 10=1().iiniiiniiiii*-=l）- XWXWSC S C XSC W W SX C X(8) Assuming T1 11iniii*-= XWHSCSC and M=121T1 T1 T112,nn* *- - WW WHSCSC SC , from Eq. (8) the inversion value X of the foundation parameters X can

36、be written as *0() ( ),=- -XIMSXMWMWSX (9) where I is a unit matrix and W*=W1*, W2*, , Wn*T, and Wi*is the vector of the measured systematic re-sponse data of the ith time. T12, ,n=WWW W where iW is the systematic response vector of com-putational data of the ith time at the expectation point X. S=S

37、1, S2, , Sn, where Siis the sensitivity matrix of the measured systematic responses of the ith time. Assuming the priori information X0of the foundation parameters X is unrelated to the measured systematic response data Wi*, from Eq. (9) the variance of X can be written as TT,*=- - XX WC I MS C I MS

38、 MC M (10) 万方数据Zhang et al. / J Zhejiang Univ-Sci A (Appl Phys & Eng) 2017 18(7):567-578 570where 12diag( , , , )n*=WWWWCCCC and i*WC is the covariance matrix of the measured systematic re-sponse data of the ith time. Using the non-singularity property of CXand *WC , Eq. (10) can be transformed into

39、 the summation form of 11T11.iniii*-=I XX WCC SCS (11) 3 Fourier cIosed form soIution for the foun-dation modeI In Eq. (4), the systematic response vector Wiof the computational results must be grasped ahead of inversion iterative analysis. The Pasternac foundation model and some other foundation mo

40、dels are con-sidered to be more precise than the Winkler founda-tion model of moderately-thick plate supposition (Belabed et al., 2014; Hebali et al., 2014; Bourada et al., 2015; Hamidi et al., 2015; Yahia et al., 2015; Bennoun et al., 2016). Although in order to make comparisons with the achievemen

41、ts conveniently (Zhao, 2007; Zhang et al., 2008), the Winkler foun-dation model is deliberately chosen, it will not affect the research emphasis of how to derive the Powell optimizing inversion mechanical model of foundation parameters with generalized Bayesian theory. The main suppositions is that

42、the normal line of the middle surface before deformation remains a straight line after deformation, but it may be not ver-tical to the middle surface, which means the trans-versal shearing deformation effect is admitted and the stress vertical to the middle surface can be ignored (Zhang et al., 2008

43、; Xie, 2011). The displacement field of the moderately-thick plate in Fig. 1 can be expressed: (, , ) (, ),(, , ) (, ),(, , ) (, ),xyuxyz z xyvx yz z x ywx y z wx yr =I=I=L(12) where x, y are the rectangular coordinates of the plate plane and z is the thickness coordinate. u, v, and w are the displa

44、cements along the x, y, and z directions, respectively. xand yare the rotational displace-ments of the normal lines of x-z plane and y-z plane, respectively. The equations between generalized stress and strain of the moderately-thick plate are derived as fff,= D (13) sss,= D (14) where f=MxMyMxyTand

45、 fyxx y=I Tyxyx are defined as generalized bending stresses and generalized bending strains, respectively. s=QxQyTand Ts xywwx y = I are de-fined as generalized shearing stresses and generalized shearing strains, respectively. Dfis defined as the bending elastic matrix which is connected to the elas

46、tic modulus, Poissons ratio, and the thickness of the moderately-thick plate. Dsis defined as the shearing elastic matrix which is connected to the shearing elastic modulus, the thickness, and the shearing correction coefficient of the moderately- thick plate. When the forces including surface tensi

47、on are neglected, the governing differential equations for the moderately-thick plate on the Winkler foundation can be written as 0,0,0,yxxyxxxy yyqkwxyxyxyr - =IIII-=III -=ILQQMMQMMQ(15) where k is the Winkler foundation parameter, and q is the load density in the z direction. The boundary conditio

48、ns of the plate are considered as simply supported. Directly solving the governing differential Eq. (15) is very difficult and therefore the Fourier transform method is used. Substituting Eqs. (13) and (14) into Eq. (15), the important governing differen-tial Eq. (15) can be turned into the differential equa-tions with the variables named x, y, and w. Then, the 万方数据Zhang et al. / J Zhejiang Univ-Sci A (Appl Phys & Eng) 2017 18(7):567-5