ments of hypothetical soil tests and an actual construction sequence. ...... The Van der Corput sequence in base b is defined by the following formula: xn = Ïb n( ).
Received: 18 January 2016
Revised: 12 December 2016
Accepted: 18 June 2017
DOI: 10.1002/nag.2717
RESEARCH ARTICLE
Model identification and parameter estimation of elastoplastic constitutive model by data assimilation using the particle filter Akira Murakami1 | Hayato Shinmura2 | Shintaro Ohno3 | Kazunori Fujisawa1 1
Graduate School of Agriculture, Kyoto University, Kitashirakawa‐oiwakecho, Sakyo‐ku, 606‐8502 Kyoto, Japan 2
NTT Data System Technologies, Inc, Financial System Division, Nipponbashi‐Muromachi 4‐5‐ 1, Chuo‐ku, Tokyo 103‐0022, Japan 3
Kajima, Co, Kanto Branch, Shimokawabe 1201‐1, Sakura, Saitama 329‐1402, Japan
Summary Data assimilation, using the particle filter and incorporating the soil‐water coupled finite element method, is applied to identify the yield function of the elastoplastic constitutive model and corresponding parameters based on the sequential measurements of hypothetical soil tests and an actual construction sequence. In the proposed framework of the inverse analysis, the unknowns are both the particular parameter within the exponential contractancy model, nE, which parameterizes various shapes
Correspondence Akira Murakami, Graduate School of Agriculture, Kyoto University, Kitashirakawa‐oiwakecho, Sakyo‐ku, 606‐ 8502 Kyoto, Japan. Email: murakami.akira.5u@kyoto‐u.ac.jp
for the yield function of the competing constitutive models, including the original/ the modified Cam‐Clay models and in‐between models and the parameters of the corresponding constitutive model. An appropriate set, consisting of the yield function of the constitutive model and the parameters of the constitutive model, can be simultaneously identified by the particle filter to describe the most suitable soil behavior. To examine the validity of the proposed procedure, hypothetical and actual measurements for the displacements of a soil specimen were obtained for consolidated and undrained tests through a synthetic FEM computation and for consolidated and drained tests, respectively. After examining the applicability of the proposed procedure to these test results, the present paper then focuses on the actual measured data, ie, the settlement behavior including the lateral deformation of the Kobe Airport Island constructed on reclaimed land. K EY WO R D S data assimilation, inverse problem, model identification, particle filter, soil‐water coupled FEM
1 | INTRODUCTION In geotechnical practice, numerical simulation based on the soil‐water coupled FEM with a sophisticated constitutive model for the elastoplastic materials is a versatile tool prior to the construction stage to predict and evaluate the performance of the soil structure for its design. In the early stages of construction, on the other hand, the observational method is set up to reevaluate the design assumptions and to update predictions of the behavior for accuracy in future performances. Discrepancies between the numerical predictions and the corresponding field measurements, in deformation and pore pressure, are due not only to the uncertainty of the initial and the boundary conditions of the governing partial differential equation and/or the parameters of the elastoplastic constitutive model but also to the selection of an appropriate constitutive model to be adopted in the numerical predictions. For example, the computed lateral deformation, based on the original Cam‐Clay model, Contact/grant sponsor: the Japanese Society for the Promotion of Science (JSPS); contact/grant number: Scientific Research (A), # 23248040
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Copyright © 2017 John Wiley & Sons, Ltd.
wileyonlinelibrary.com/journal/nag
Int J Numer Anal Methods Geomech. 2018;42:110–131.
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often overestimates the corresponding measurement of the foundation ground due to dilation, whereas the modified Cam‐Clay model can often provide better predictions for the embankment construction. The models should be assessed from site investigations and from the results of laboratory tests conducted prior to the construction. Our work is motivated by the need to identify both the constitutive model and its parameters to explain such ground behavior in a rational way within the framework of an inverse analysis. Inverse analyses1,2 have been applied to performance observations in geotechnical construction works to modify uncertain conditions and parameters to bridge the gap between observations and predictions and to obtain feedback regarding the reevaluation of the numerical simulations for subsequent construction sequences as a quantifiable observational method. Over the last few decades, numerical strategies used to solve inverse problems have been addressed in various fields of engineering, and many of them have also been applied to geotechnical problems, as seen in the review by Gioda and Sakurai.3 To carry out an inverse analysis, the optimization problem has to be solved to minimize the error function, ie, the difference between the experimental/observation data and the predictions obtained from a numerical computation. In addition to the nonlinear optimization, many applications have been made, for example, by Arai et al4 to identify the elastic parameters and the permeability of a soft clay deposit under embankment loading, by Calvello and Finno5 to identify the parameters of an elastoplastic model for glacial clays, and by Tang and Kung Gordon6 for braced excavations. Several alternative strategies have also been proposed, for example, in statistics, the maximum likelihood estimation7,8 for the excavation of a tunnel in stiff overconsolidated clay; the extended Bayesian method9,10 for embankment loading on soft clay; the Kalman filter11-16 and nonlinear Kalman filters, eg, the ensemble Kalman filter17 and the particle filter,18,19 for the parameter identification of an elastoplastic constitutive model for a foundation ground under embankment loading; the Markov chain Monte Carlo simulation,20,21 to identify the strength parameters for slope stability; the particle swarm optimizer, a population‐based stochastic optimization technique,22 developed to solve the nonlinear constrained optimization problem23 and applied to unsaturated soil,24 slope movement,25 and so on; the genetic algorithm for the identification of parameters for the Mohr‐Coulomb model26,27; the evolution strategy algorithm28 for underground works; and recently, by Hashash et al and Reccha et al,29,30 and references contained therein. In such existing strategies, however, some difficulties still remain in identifying both the constitutive models and their parameters simultaneously. To tackle the above‐mentioned problem, the present paper focuses on the discrimination among competing constitutive models that follow the ground behavior by identifying the parameter within the exponential contractancy (EC) model proposed by Ohno et al,31 nE, which classifies the different yield functions. The reason why this constitutive model is adopted is as follows: For example, it can be found that in many analyses of embankment on soft ground for soil‐water coupled problem, the lateral deformation of subsoil beneath the toe of embankment is often overestimated, whereas computed settlement of subsoil shows a good agreement with field measurements. Ohno et al31 assumed that the factor of such overestimation is due to the contractancy of clays and proposed the constitutive model derived from the nonlinear function capable of describing contractancy of clays such as the EC model that employs a nonlinear function to describe the contractancy of soils in a broad sense and to lead to the original model, the modified Cam‐Clay model, or models in‐between them based on the choice of parameters, nE. Herein, the particle filter (PF) is used to identify both this parameter and the material properties of the constitutive model because the PF has priority in its application to initial/boundary value problems for elastoplastic materials.18 Inverse problems in linear elasticity have been successfully solved by both analytical and numerical means. On the other hand, there still remain some difficulties in identifying the elastoplastic parameters because for elastoplastic material, the current deformation does not have one‐to‐one correspondence with the stress state at the same moment but depends on the loading path from the initial stage to the current stage. Then, the observation of the deformation, along with full knowledge of the loading history, is necessary for the parameter identification of the elastoplastic constitutive model. Related literature has tried to identify the parameters, such as Young modulus, the friction angle, the coefficient of earth pressure at rest, poroelastic parameters, and so on, based on the minimization of the objective function, by means of the gradient method, genetic algorithm, and a sensitivity analysis. A few works29,32 have considered the deformation associated with the loading history from the initial stage up to the current stage for the identification of the plastic parameters. Sequential data assimilation techniques, such as the PF, are applicable to this type of inverse problem because the time evolution of state variables, ie, displacement and pore pressure for geotechnics, under the controlled input, like the external loading, is incorporated into the system equation in a rational manner without any limitations. The PF can easily deal with nonlinear state equations and is robust when employing the Monte Carlo method in conjunction with a numerical simulation, eg, the soil‐water‐coupled finite element analysis with the elastoplastic model. Saturated soil is treated as an elastoplastic material, and its behavior relies on the parameters of an elastoplastic constitutive model, the initial stress, and the stress paths up to the current stress state, due to the different patterns of loading history, while the deformation of an elastic material is independent of such factors. From the viewpoint of an inverse analysis, it seems sensible for an optimization procedure to take the path traveled into account even if it may not be necessary to obtain a reasonably robust procedure, whereas those of an elastic material can be identified by the deformation and the stress state at the current stage.
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The rest of the paper is arranged as follows. Section 2 describes the details of the EC model. The computational procedure for the PF is outlined in Section 3. Section 4 presents a numerical example of the identification of a constitutive model for synthetic and actual soil tests to demonstrate the performance of the proposed strategy. Analyses for the behavior of the foundation ground of the Kobe Airport Island are carried out to examine the applicability to actual construction project. Conclusions will follow in the last section.
2 | T H E EX P O N E N T I A L C O N T R AC TA NC Y M O D E L We will outline the EC/logarithmic contractancy (LC) elastoplastic model for clays.31 Ohta and Hata33 derived the elastoplastic constitutive model for clays based on 2 experimental results concerning the volume change of saturated clays: (i) It is independent of effective stress path,34 and (ii) it consists of consolidation and contractancy.35 The contractancy of soils is the negative dilatancy usually modeled by linear or bilinear relation between the volumetric strain εv and stress ratio η (=q/p′) under the constant effective mean principal stress p′, and such widely adopted assumption has physically clear meaning. Meanwhile, the experimental behavior of drained shear tests under constant p′ for isotropic consolidated clays in Figure 1 insists that the contractancy of soils is not necessarily limited to linear or bilinear but exponential or logarithmic interpolation. It is then noteworthy to examine the applicability of such nonlinear functions to the description of the contractancy. Ohno et al31 developed the following 2 alternative nonlinear functions to be capable of describing contractancy of soils:
or
η ¼ kη−η0 k; η ¼
MD η nE ðexponentialÞ εv ¼ nE M
(1)
nL 2MD M þ ηnL ln ðlogarithmicÞ εv ¼ M nL nL
(2)
s s0 1 1 ; η0 ¼ ′ ; s¼σ′ −p′ 1; s0 ¼ σ′0 −p′0 1; p′ ¼ σ′ 1; p′0 ¼ σ′0 1 ′ p 3 3 p0
(3)
where s0 is the initial deviatoric stress tensor, σ′0 is the effective initial stress tensor, p0 is the preconsolidation pressure, M is the critical state parameter, D is the coefficient of dilatancy after Shibata,34 η* is the stress parameter by Sekiguchi and Ohta,36 1 is the third‐order identity tensor and denotes a magnitude, and nE and nL are the fitting parameters. D can be related to other parameters in the following expression: D¼
λ−κ λΛ ¼ M ð1 þ e0 Þ M ð1 þ e0 Þ
(4)
where κ, λ, and Λ = 1 − κ/λ denote the swelling index, the compression index, and the irreversible ratio, respectively.
FIGURE 1 Model description for drained shear experiments
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Figure 1 demonstrates the applicability of these functions by using a set of parameters in Table 1 to the measurement of drained shear experiments under constant p′. Ohno et al31 extended these nonlinear functions to the yield functions for general state as follows: 0 p′ MD η nE p −εv ¼ 0 ðexponentialÞ (5) f σ ; εv ¼ MD ln ′ þ nE M p0 or
0
f σ ; εv
nL p′ 2MD M þ ηnL −εpv ¼ 0 ðlogarithmicÞ ln ¼ MD ln ′ þ M nL nL p0
(6)
where σ′ is the effective stress tensor, p′0 is the effective initial stress tensor at which anisotropic consolidation completed, and εVp is the hardening parameter (plastic volumetric strain). Equation 5 is called the EC model and is equivalent to the yield function of Sekiguchi‐Ohta's model36 in case of nE = 1.0. Equation 6 is, on the other hand, called the LC model and is equivalent to the yield function of the modified Cam‐Clay model in case of nL = 2.0, where nE and nL ≧ 1.0. Figure 2 shows various shapes of the yield surface of the EC/LC model on the p′‐q space where isotropically consolidated soil is assumed (η0 = 0.0, M = 1.0); η0 is the initial stress ratio at which anisotropic consolidation completed. It is seen in Figure 2 that the description of contractancy is closely related to the shape of the yield surface.
3 | DATA ASSIMILATION FOR S OIL ‐WATER COUPLED PROBLEMS 3.1 | System equation The data assimilation technique37-40 has been considered in conjunction with the soil‐water coupled FEM to identify the constitutive model and the corresponding elastoplastic material properties of soil deposits. The following set of system equations is assumed: xk ¼ f k ðxk−1 Þ þ vk
(7a)
TABLE 1 Parameters used for model description Names of Clays
M
nE
nL
Amagasaki clay
1.32
1.0
1.2
Fukuoka clay
1.32
1.6
2.0
London clay
0.65
3.0
3.0
Weald clay
0.86
2.0
3.0
FIGURE 2
Different shapes of yield surfaces of exponential contractancy (EC)/logarithmic contractancy (LC) model
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FIGURE 3
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Computational procedure for particle filter (PF) with sequential importance sampling (SIS)
yk ¼ hk ðxk Þ þ εk
(7b)
Equation 7a is the state equation or the system model, while Equation 7b is the observation equation. Vector xk, called the state vector, includes the state of a system that is constituted by the displacement and the pore pressure for soil‐water coupled problems at a discrete time T = tk, while vector yk, called the observation vector, indicates the measured quantity. A set of unknown parameters, to be identified based on the measurements, is additionally incorporated into the state vector. Vectors vk and εkdenote system noise and observation noise, respectively, whose probabilistic density function (PDF) follows the normal distribution with an average value of 0, namely, vk eN ð0; Qk Þ εk eN 0; Rk
(8)
where Qk and Rk are predetermined covariance matrices. Operator fk represents the evolution of the states of displacement and pore pressure from time tk − 1 to time tk, according to the simulation model, ie, the FEM stiffness equation for soil‐water coupled problems (see Equation 17). Nonlinear function hk describes the measured quantity. In many cases, it is written in matrix form as yk ¼ H k xk þ εk
(9)
where Hk is the observation matrix, composed of 0 or 1 component, if parts of the state variables are directly measured for the geotechnical construction sequence; see section 3.3.2.
3.2 | Ensemble approximation Data assimilation strategies based on the PF employ an ensemble approximation technique in which a PDF of the stochastic variables is approximated with its realizations and weights. Each realization is called a “particle,” and each set is called an “ensemble.” For example, the filtered distribution at time T = tk − 1, p(xk − 1|y1 : k − 1), where y1 : k − 1 denotes n o n o ð1Þ ð 2Þ ðNÞ ð 1Þ ð 2Þ ðNÞ {y1, y2, ⋯ , yk − 1}, is approximated with ensemble xk−1jk−1 ; xk−1jk−1 ; ⋯; xk−1jk−1 and weights wk−1 ; wk−1 ; ⋯; wk−1 by the following equation: N ðiÞ ðiÞ pðxk−1 jy1:k−1 Þ≈ ∑ wk−1 δ xk−1 −xk−1jk−1 i¼1
(10)
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where p(xk − 1|y1:k − 1) is the PDF of x at k − 1th time; tk − 1, based on a set of observation y1:k − 1 from 1st to k − 1th time, ie, t1 to ðiÞ
ðiÞ
tk − 1, xk∣k−1 is an ith state estimation at kth time based on observation y1:k − 1; wk−1 is weight for ith particle at kth time; δ is Dirac's delta function; and N is the number of particles in the ensemble. In some kinds of PF, eg, sequential importance resampling, each ðiÞ
weight wk−1 is set to N1 in all time steps; as a result, the weight set is omitted.
3.3 | Prediction and filtering steps for particle filter 3.3.1 | Prediction steps for particle filter We obtain the ensemble approximation for predicted distribution p(xk|y1 : k − 1) at time T = tk from the filtered ensemble and weights at time T = tk − 1 by the following calculation: p ðxk jy1:k−1 Þ ¼ ∫ p ðxk−1 jy1:k−1 Þ p ðxk jxk−1 Þ dxk−1 N ðiÞ ðiÞ ≈∑ ∫ wk−1 δ xk−1 −xk−1jk−1 pðxk jxk−1 Þ dxk−1 i¼1
N ðiÞ ðiÞ ðiÞ ¼ ∑ wk−1 δ xk − f k xk−1jk−1 þ vk
(11)
i¼1
N ði Þ ðiÞ ¼ ∑ wk−1 δ xk −xkjk−1 ; i¼1
n oN ðiÞ is an independent identical distribution sample set for Equation 7a. This calculation means that each particle for where εk i¼1
ðiÞ
the prediction ensemble,xkjk−1 , is generated via the state equation (Equation 7a), ie, the stiffness equation for the soil‐water coupled FEM appearing in the previous section for geotechnical applications (see Equation 17), ðiÞ ðiÞ ðiÞ xkjk−1 ¼ f k xk−1jk−1 þ vk
(12)
and the weights are unchanged in this step.
3.3.2 | Filtering steps for particle filter We obtain an approximation of the filtered distribution, p(xk|y1 : k), from the ensemble of the predicted distribution, p(xk| y1 : k − 1), and observation yk by using Bayes theorem, namely, p ðxk jy1:k Þ ¼
p ðxk jy1:k−1 Þp ðyk jxk Þ ∫ p ðxk jy1:k−1 Þp ðyk jxk Þdxk
N 1 ðiÞ ðiÞ ðiÞ ∑ p y x δ x −x w k k k−1 k k−1 k k−1 j j ðjÞ ðjÞ ∑ p yk xkjk−1 wk−1 i¼1 j N ðiÞ ðiÞ ¼ ∑ si wk−1 δ xk −xkjk−1 ≈
(13)
i¼1
N ðiÞ ðiÞ ¼ ∑ wk δ xk −xkjk−1 i¼1
where si is defined as
ðiÞ p yk xkjk−1 si ¼ ðjÞ ðjÞ ∑ p yk xkjk−1 wk−1 j
and is calculated in the following manner in the case of linear observations:
(14)
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2 T 3 ði Þ ðiÞ −1 y −H x R y −H x k k k k k kjk−1 kjk−1 7 1 ðiÞ 6 p yk xkjk−1 ¼ pffiffiffiffiffiffiffiffiffim exp4− 5: 2 ð2π Þ jRk j ðiÞ
ET AL.
(15)
ðiÞ
Each weight wk is the product of si and the previous time weight, wk−1 , namely, ðiÞ
ðiÞ
wk ¼ si wk−1
(16)
The sampling importance resampling algorithm, one of the PF procedures, is performed according to the following steps: (1) Initialization
n o ðiÞ Draw a sample of size N from the prior density at the initial time and define the sample set as x0∣0 .
(2) Preliminaries n o ði Þ Assume that xk−1∣k−1 is a population of N particles, approximately distributed as in an independent sample from p (xk − 1|y1 : k − 1). (3) Prediction ðiÞ
Sample N values, vk , from the distribution of vk. Use them to generate a new population of particles,
n
o ðiÞ xk∣k−1 , via
Equation 13. (4) Filtering ðiÞ
Assign a weight si to each xkjk−1 . This weight is calculated by Equation 14. (5) Resampling
n o ðiÞ Resample N times with a replacement from the set of particles xkjk−1 , obtained in the filtering stage, with the probability n o ðiÞ proportional to si. The set of determined particles, xkjk , results in an ensemble approximation of p(xk|y1 : k). n o ðiÞ In this procedure, weight set wk is not required because the resampling flattens the weight difference among the particles and resets it to N1 .
n o ðiÞ ðiÞ On the other hand, filtering via SIS (sequential importance sampling)41 preserves the weight set wk and calculates wk by ðiÞ
Equation 16 instead of by resampling. Initial weight w0 is usually set to N1 . To identify the elastoplastic material properties of geomaterials, the PF without resampling, ie, SIS, is adopted as the preferable tool for analyzing numerical examples as seen in Figure 3. This paper focuses on the mechanical behavior of geomaterials, which can be described by soil‐water coupled finite element techniques. By means of the formulation of the soil‐water coupled FEM, Equation 7a can be rewritten in the following form42:
ut pt
¼
ut−1 pt−1
"
þ
½K
½K v T
½K v −θΔt ½K h
#−1 (
) F_ þ ½K v T pt vut þ vpt Q_ þ ð1−θÞ½K h pt
(17)
where the following notations are employed: [K] is the tangent stiffness matrix of the soil skeleton that is modeled by constitutive models for soils; [Kv] is the rectangular matrix that transforms the increment in nodal displacements to the volume change of each element; [Kv]T is the transpose of [Kv] that transforms the pore pressure of an element to the seepage force of element; [Kh] is the fluid stiffness matrix, a vector that represents the increment in applied force; {ut} is the nodal displacement vector at time t; {pt} is the nodal pore pressure vector at time t; θ is the time‐varying coefficient; and (0 < θ < 1) is the vector that represents the increment in the volume rate of the water flow, the system error vector for {ut}, and the system error vector for {pt}.
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Thus, state vector xt is constructed as
xt ¼
ut
(18)
pt
If state variables u1, u2, and u3 in {ut} are directly observed, for example, the observation matrix Ht in Equation 9 can be written in matrix form as 2
1 0
6 Ht ¼ 4 0 1 0 0
3
0
⋯
0
⋯
7 05
1
⋯
0
0
(19)
4 | N U M E R I CA L A NA LYS E S A N D D I S C U S S I O N S The proposed procedure, described in the previous sections, is first applied to hypothetical and actual soil tests to demonstrate its applicability. After the primary examination, the deformation of the ground observed during the construction of Kobe Airport on reclaimed land is investigated to confirm how helpful it is to actual construction work.
4.1 | Identification of yield function and corresponding parameters for hypothetical and actual soil tests To examine the numerical accuracy of the computational procedure described in the previous sections, hypothetical drained soil tests and actual undrained soil tests are analyzed under axisymmetric conditions.
4.1.1 | Data assimilation for hypothetical triaxial consolidated drained tests In the first application, synthetic soil tests were numerically carried out by a soil‐water coupled FEM under axisymmetric conditions using the set of parameters listed in Table 2 and assuming a value of 1.2 for nE. Three different cases for identification in Table 3 are dealt with based on the measured horizontal/vertical displacements in the top right corner of Figure 4 under a constant loading of 14 kPa per minute to examine the identifiability of nE and the possible set of unknowns among k, Λ, and M to be identified simultaneously. Optimizing the nE parameter really is not identifying a soil model. Rather, it is the parameter that controls the shape of the yield surface, of an otherwise unique constitutive model.where Λ = 1 − κ/λ is the irreversibility ratio, ν′ is the effective Poisson ratio, k is the coefficient of permeability, σ′v0 is the preconsolidation pressure, K0 is the coefficient of earth pressure at rest, σ′vi is the effective overburden pressure in situ, and Ki is the coefficient of in situ earth pressure at rest. TABLE 2 Parameters for hypothetical consolidated drained (CD) tests nE
D
Λ
M
ν′
k (m/day)
0.0502
0.8617
1.508
0.333
2.03 × 10−4
σ′v0 (kPa)
K0
σ′vi (kPa)
Ki
λ
e0
300
1.0
300
1.0
0.195
1.224
1.20
TABLE 3 Cases of identification Target Parameter Case
k (m/day)
Λ
Μ
nE
1
TBI
–
–
TBI
2
–
TBI
–
TBI
3
–
–
TBI
TBI
TBI = To be identified
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FIGURE 4
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Description of hypothetical consolidated drained (CD) tests [Colour figure can be viewed at wileyonlinelibrary.com]
For the data assimilation, 200 particles are produced by means of uniform random numbers within the ranges given in Table 4. A sufficient number of particles are required so as to cover the space of parameters to be identified. Two parameters, such as k and nE, are identified in the CD (consolidated drained) test, while 1 parameter of nE is identified in the subsequent example for CU (consolidated undrained) test. The CD test has the 2‐dimensional parameter space and needs more particles than the CU test, which has the 1‐dimensional parameter space. And we adopted the uniform random numbers to generate particles within the search range for each parameter to be identified. Initial value for each parameter to be identified is evaluated as the average of its corresponding search range. Possible ranges for searching are dependent on the modeling for nE (see Figure 3) and site investigation for material parameters, eg, k, Λ, and M. Initial guesses of the diagonal terms for the error covariance matrix are assumed as the following term with parameter α: Rij ¼ ðαSÞ2 δij
(20)
where Rij is the error covariance matrix for the observation, S is the presumed maximum deformation, δij is Kronecker's delta, and α is assumed to be 0.1 in all cases. We considered only diagonal terms for error covariance matrix because each observation does not have correlation for each other. Each term is assumed to be 10% of the maximum deformation
TABLE 4 Range of particles Parameter
Range in Values for Particle Generation
nE
1.0 ≦ nE ≦ 2.0
k (m/day)
1.0 × 10−5 ≦ k ≦ 1.0 × 10−3
Λ
0.79 ≦ Λ ≦ 0.95
M
1.369 ≦ M ≦ 1.635
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based on the prediction at the design stage.18,19 System noise, Q, is assumed to be 0 because adopted system model is deterministic for this problem, and an initial guess of 1.5 for nE, corresponding to the modified Cam‐Clay model, is given for the data assimilation. Figures 5, 6, and 7 depict the time histories of the identification of nE and k, nE and Λ, and nE and Μ, respectively, along with the distribution of weight for the parameters obtained by Equation 20 10 minutes after the start of the tests. The identified values for the parameters are computed based on the mean estimate through the weighted average of 200 particles for the corresponding parameters using the weight distribution in Figures 5B, 6B, and 7B. As can be seen in Figures 5B, 6B, and 7B, the shape of the weight distribution for parameter nE is close to the normal distribution approximating its PDF around the prescribed value of 1.2 for the synthetic soil‐water coupled FEM computation, and the shape of this weight distribution becomes sharper with the progress of time. It is revealed that the identified parameters for all cases converge exactly to the value of 1.2 for nE, which is primarily assumed for the soil‐water coupled FEM computation in the synthetic soil tests, and the data assimilation for the soil tests has been successfully carried out for a set of nE and k or nE and Λ except for M.
4.1.2 | Data assimilation for actual triaxial consolidated undrained tests In the second application, triaxial CU tests were carried out in the following manner43: (i) A specimen with a diameter of 25 mm and a height of 100 mm was saturated, and a back pressure of 100 kPa was applied. (ii) The specimen was isotropically consolidated under a pressure of 9.8 kPa for 1 minute and kept under the same pressure for 60 minutes. (iii) The confining pressure was doubled for several stages under the same procedure. (iv) Procedures (ii) and (iii) were repeated up to confining pressures of 314 and 628 kPa. (v) A displacement corresponding to an axial strain rate of 0.05%/min and an axial strain of 15% were uniformly imposed on the upper surface of the specimen. A numerical computation was performed in axisymmetric conditions under the boundary conditions given in Figure 8 in which the lateral boundary was drained by filter paper during the isotropic consolidation, and horizontal movements of the specimen are fixed on the top and bottom pedestal. Table 5 lists the set of parameters for the computation; K0 is evaluated as K0 = ν′/(1 − ν′) for cohesive soil. nE is identified by the measurement data of excess pore pressure during the axial compression with the variance of σ2 = (0.2S)2 to estimate the optimum shape of the yield surface. The shape of the yield surface is governed by the dilatancy
FIGURE 5
Identification of parameters: nE and k [Colour figure can be viewed at wileyonlinelibrary.com]
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FIGURE 6
Identification of parameters: nE and Λ [Colour figure can be viewed at wileyonlinelibrary.com]
FIGURE 7
Identification of parameters: nE and M [Colour figure can be viewed at wileyonlinelibrary.com]
ET AL.
of soils, ie, volumetric change during shearing, and controlled by the parameter nE as explained in section 2. Because the dilatancy of the soil specimen generates the pore water pressure in the undrained test, the pore water pressure is the most important for the determination of the shape of the yield surface. Therefore, the data assimilation of the undrained test was focused on the reproduction of the pore water pressure and the identification of the parameter nE.
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FIGURE 8
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FE mesh and boundary conditions [Colour figure can be viewed at wileyonlinelibrary.com]
Fifty particles were generated within the range of 1.0 ≤ nE ≤ 2.0 by using the Halton sequences with low discrepancy, whose scatter uniformly distributed random numbers (b = 2; see Appendix A). Figures 9A, 9B, and 9C depict the distribution of weight at axial strain rates of 5, 10, and 15%, respectively. From these figures, it is revealed that the shape of the weight distribution becomes sharper and has a higher peak around nE = 1.3 along with the time lapse. This means that a constitutive model close to the modified Cam‐Clay model is calibrated because the specimen used in the laboratory testing is under a higher rate of shear than the actual ground. It is known that the modified Cam‐ Clay model can describe the shearing behavior of soils under the higher shear rate better than the original Cam‐Clay model.44 As is shown in the second section, nE = 1.0 corresponds to the original Cam‐Clay model and nE = 1.5 can approximate the modified Cam‐Clay model. In the CU test, the identified value of nE is 1.3. This means that the shear behavior of the test is similar to the one described by the modified Cam‐Clay model and to the one under the higher shear rate. A comparison between the calculated relation of the excess pore pressure and the axial strain, using the identified parameter and the observed one, is demonstrated in Figure 10 for the red line, under the confining pressure of 314 kPa, and for the blue line, under the confining pressure of 628 kPa. It can be seen that the results obtained are consistent with those obtained through the synthetic soil tests in the previous section.
4.2 | Identification of yield function and corresponding parameters of foundation ground for Kobe Airport project The above‐mentioned technique for data assimilation is applied to the deformation of the ground observed during the construction of Kobe Airport on reclaimed land45 to investigate how helpful it is to actual construction work, how much improved the permeability of the clay layer is with sand drains, and which constitutive model is selected to properly explain the entire ground behavior including lateral deformation.
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TABLE 5 Set of parameters for computation D 0.0576 K0 0.5127
FIGURE 9
Λ
Μ
ν′
k (m/day)
0.88
1.33
0.34
0.0019
σ′v0 (kPa)
λ
e0
157
0.15
1.18
Distribution of weight (left) 314 kPa and (right) 628 kPa
Kobe Airport was constructed on an artificially reclaimed island just off the coast of Kobe City, Japan. Figure 11 shows the cross section and the geological profile of Kobe Airport on the reclaimed land. The soft alluvial clay layers of Ac1, Ac2, Ac3, and Asc range up to 33 m in depth from the seabed. Vertical sand drains were installed to accelerate the settlement and to increase the strength and stiffness in the soft clay layers beneath the landfill and the embankment. This region is shown as SD in the figure. The diluvial sand layers of Ds1, Ds2, and Ds3 are exiting, which covers more than 30 m beneath the alluvial clay layers. These sand layers are sufficiently rigid because primary wave and secondary wave logging tests that estimated the N values of the layers were placed in the range of 20 to 100. The primary wave and secondary wave logging can estimate the shear
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Comparison between computation and experiment by using identified model [Colour figure can be viewed at wileyonlinelibrary.com]
FIGURE 11 Cross section of Kobe Airport on reclaimed land45
FIGURE 12
Location of measurement
instruments45
wave velocity of the ground, and the N value is calculated from the shear wave velocity. Below the sand layers, the diluvial clay layer called Ma13 and another diluvial sand layer of Ds4 can be seen. Ds4 is also a rigid sand layer, so that Ds4 as well as Ds1 to Ds3 were assumed to be elastic materials in the following numerical analysis. For better prediction of the ground deformation occurring in Kobe Airport, the material parameters of the clay layers improved by the sand drains, ie, SD layer in Figure 11, were intended for data assimilation because SD layer is soft and has greater influence on the deformation of the ground. For simplicity, the physical and mechanical properties in the layer were assumed to be homogeneous and the permeability k and the model parameter nE were adopted as the unknown parameters to be identified.
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FIGURE 13
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Finite element meshes, observational points, and boundary conditions used in the analysis
TABLE 6 Cases to be analyzed Parameters to be Identified
Lateral Displacement
nE
k (m/day)
Case 1
TBI
TBI
–
Case 2
TBI
TBI
○
TBI = To be identified
FIGURE 14
Loading condition considering the construction stages
The locations, at which the measurement instruments were installed and the observation data for this data assimilation were obtained, are presented in Figures 12 and 13. The settlement plate (3BC‐1) and the inclinometer (3K‐1) were installed 58.75 m south of the reclamation normal line, while the other settlement plates (3BC‐2, 3BC‐4, and KC‐5) were installed 10, 48, and 165 m north of the reclamation normal line, respectively. The 2 cases shown in Table 6 are analyzed to examine the applicability and the flexibility of the EC model in explaining properly the mechanical features over the entire foundation, namely, the lateral deformation in accordance with the settlement
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TABLE 7 Parameters of exponential contractancy (EC) model for foundation ground (modified from Ohta44) Soil Layer SD
ν
λ
κ
ei
Μ
k (m/day)
0.300
0.335
0.113
2.078
1.16
TBI −4
Ki
OCR
1.0
1.0
K0 1.0
Ac1
0.300
0.304
0.098
2.473
1.187
5.17 × 10
0.868
1.46
0.733
Ac2
0.300
0.313
0.113
2.150
1.117
2.00 × 10−4
0.790
1.41
0.683
0.773
1.36
0.675
0.750
1.33
0.658
0.729
1.20
0.671
0.746
1.19
0.696
0.719
1.19
0.665
Ac3‐1(1) Ac3‐1(2) Ac3‐2(1) Ac3‐2(2) Ac3‐2(3) Asc
0.300 0.300 0.300 0.300 0.300 0.300
Ds1
0.339 0.321 0.365 0.378 0.387 0.239
0.115
2.037
0.094
1.851
0.124
1.924
0.152
1.975
0.123
1.876
0.077
1.350
1.157 1.239 1.157 1.044 1.191 1.183
2
0.300
E = 14,000 kN/m
−4
1.43 × 10
−4
1.06 × 10
−5
9.42 × 10
−5
7.98 × 10
−5
7.42 × 10
−5
3.48 × 10 8.64 × 10
2
0.780
1.59
0.634
−1
–
–
–
−1
–
–
–
–
–
–
0.813
1.48
0.690
0.737
1.28
0.660
0.816
1.50
0.690
–
–
–
Ds2
0.300
E = 63,000 kN/m
8.64 × 10
Ds3
0.300
E = 28,000 kN/m2
8.64 × 10−1
Ma12(U) Ma12(M) Ma12(L) Ds4
0.300 0.300 0.300 0.300
0.300 0.256 0.295
0.113
1.359
0.085
1.158
0.113
1.251
E = 100,000 kN/m
2
1.091 1.170 1.083
−5
2.42 × 10
−5
2.91 × 10
−5
1.64 × 10 4.32 × 10
−1
behavior. The difference between Cases 1 and 2 is regarding with and without the consideration of lateral deformation as a part of measurement. Case 2 reflects the effect of both settlement and lateral deformation on the selection of yield surface, whereas Case 1 deals with only settlement measurement. We are trying to improve the numerical accuracy for the predicted behavior of embankment foundation by selecting an appropriate yield function through the identification of the EC model parameter, nE. Figure 13 presents a description of the 2D finite element meshes, the observational points, and the boundary conditions used for the analysis; the loading condition is depicted in Figure 14 considering the construction stages. As hydraulic boundary conditions, sand/gravel layers provided the imposed drained conditions, whereas clay layers provided the undrained conditions. The analysis was performed under plane strain conditions, and the material parameters, listed in Table 7, were used for representing the existing soil layers. In the table, ν, ei, Ki, overconsolidation ratio (OCR), and K0 denote the Poisson ratio, the initial void ratio, the ratio of the initial lateral to vertical pressure, the OCR, and the coefficient of earth pressure at rest, respectively. The parameter other than k, K0, Ki, and OCR in the improved ground is defined as the mass parameter, β, and is determined by Equation 21.46 The improved ground originally consists of the soft clay layers Ac1, Ac2, and Ac3 possessing slightly different material parameters. Hence, the averaged values of the material parameters weighted by the thickness of the layers were employed as the mass parameters of the improved ground. n
∑ bi hi ¼ β ¼ i¼1n ∑ hi
b1 h1 þ b2 h2 þ ⋯ þ bn hn h1 þ h2 þ ⋯ þ hn
(21)
i¼1
where bi means the parameter of the soil layer and hi is the thickness of each layer. We adopted the linear elasticity model for the sand/gravel layers and the embankment and the EC model for the clay layers. In the data assimilation, the authors attempted to identify nE for all the layers by using the EC model and k for the improved ground. TABLE 8 Range in values for particle generation Parameter
Range in Values for Particle Generation
nE
1.0 ≤ nE ≤ 5.0
k (m/day)
1.0 × 10−5 ≤ k ≤ 1.0 × 101
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FIGURE 15
Time history of identified parameters for Case 1
FIGURE 16
Time history of identified parameters for Case 2
ET AL.
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TABLE 9 Identified parameters at 565th days Identified Parameter nE
k (m/day)
Case 1
1.06
0.00243
Case 2
1.57
0.00305
Based on the above conditions, we conducted Monte Carlo simulations with 500 particles. The produced particles fell in the range shown in Table 8 and were efficiently scattered by the Halton sequence47; see the Appendix. The diagonal term for the error covariance matrix, α, was assumed to be 20% in all cases. Figures 15A and 16A show the time history of the identified parameters that are evaluated as the mean weighted average of all 500 particles by using the distribution of weight at each time; the weight in Figure 16B is in the lapse of 565 days for Cases 1 and 2, respectively. As seen in Figure 16A, the identified parameters, both nE and k, tend to converge after 400 days in both cases. The parameters identified by using the data measured up to 565th days are listed in Table 9. It is found that a significant difference exists for the identified nE between the 2 cases, whereas the permeability is identified at a constant value for both cases. As stated in the previous section, when nE = 1.0 is substituted into the yield function of the EC model, the yield function results in the original Cam‐Clay model and nE = 1.5 gives the closest shape to the yield function of the modified Cam‐Clay model. Thus, the results reveal that in Case 2, the identified model is closer to the modified Cam‐Clay model rather than that of Case 1 by taking the measurement of the lateral behavior into consideration. The recomputed results for settlement and lateral displacement by using the identified parameters at 565th days are depicted for each measured point in Figures 17 and 18, respectively. In this analysis, the measured data up to 565th days were used for the data assimilation, and then the simulation was conducted at the design stage by using the identified parameters. In these figures, the dashed lines represent the prediction by the direct analysis based on the Sekiguchi‐Ohta's constitutive model by using the parameters listed in Table 7. The results of Figure 18 show that the computed results by the proposed strategy are in better agreement with the measured data than those of the direct analysis, except several times when there is a slight underestimation in the predictions. Between the 2 cases, Case 1 has higher accuracy than Case 2 for predicting the settlement. In
FIGURE 17
Results of prediction for settlement [Colour figure can be viewed at wileyonlinelibrary.com]
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FIGURE 18
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Results of prediction for lateral displacement [Colour figure can be viewed at wileyonlinelibrary.com]
particular, Case 1 provides higher accuracy at 3BC‐4 than Case 2, and this means that a model close to the original Cam‐Clay model is applicable to settlement predictions. In addition, the results of the direct analysis lead to predictions with lower accuracy using inappropriate parameters, even if the model is appropriately selected. It can also be seen from Figure 18 that the identified model for Case 1, which is close to the original Cam‐Clay model, tends to considerably overestimate the lateral displacement. On the other hand, when the yield surface is selected to be closer to the modified Cam‐Clay model, such as Case 2, the computed values are in relatively good agreement with the overall measurements. These results clearly show that the entire ground behavior can be suitably represented by adjusting model parameter nE.
5 | CONCLUSIONS We have herein proposed a numerical strategy to simultaneously identify the elastoplastic constitutive model and its parameters by combining the PF, one of the nonlinear Kalman filters, with the EC model. A proper yield function is selected in‐between the original and the modified Cam‐Clay models by identifying the parameter, nE. To examine the validity of the proposed
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procedure, SIS is applied with a soil‐water coupled FEM to the measured data for the hypothetical CU test and the actual CD test for clay specimen. After examination, the proposed strategy is applied to the settlement and the lateral behavior of the foundation beneath the Kobe Airport Island constructed on reclaimed land. As a result, the identified model becomes closer to the modified Cam‐Clay model in the case of monitoring both the settlement and the lateral displacement. In addition, the results of recomputation, using the identified parameters, are in good agreement with the observation data. Thus, it can be concluded that a suitably identified parameter nE of the EC model can lead to a proper model for predicting the settlement and the lateral behavior with higher accuracy in applications to actual construction projects. However, it should be noted that the proposed technique was applied to the case where large deformations occurred and the material parameters were calibrated by the observed deformations. The computed responses were dominated by plastic deformations, and thus, elastic parameters are of little importance compared with the shape of the yield surface and the hardening employed. When the same model is applied to other cases in which small deformations are dominant, different conclusions may be derived regarding what parameters need to be optimized. AC KN OWL ED GE M EN TS This work has been supported by the Grant‐in‐Aid for Scientific Research (A), # 23248040, of the Japanese Society for the Promotion of Science (JSPS). This support is gratefully acknowledged. R E F E RENC E S 1. Bui HD. Inverse Problems in the Mechanics of Materials, An Introduction. USA: CRC Press; 1993. 2. Tarantola A. Inverse Problem Theory and Model Parameter Estimation. USA: SIAM; 2005. 3. Gioda G, Sakurai S. Back analysis procedures for the interpretation of field measurements in geomechanics. Int J Numer Anal Meth Geomech. 1987;11(6):555‐583. 4. Arai K, Ohta H, Kojima K, Wakasugi M. Application of back‐analysis to several test embankments on soft clay deposits. Soils Found. 1986;26 (2):60‐72. 5. Calvello M, Finno RJ. Selecting parameters to optimize in model calibration by inverse analysis. Comput Geotech. 2004;31:411‐425. 6. Tang Y‐G, Kung Gordon T‐C. Investigating the effect of soil models on deformations caused by braced excavations through an inverse‐analysis technique. Comput Geotech. 2010;37:769‐780. 7. Ledesma A, Gens A, Alonso EE. Estimation of parameters in geotechnical back analysis—I. Maximum likelihood approach. Comput Geotech. 1996;18(1):1‐27. 8. Gens A, Ledesma A, Alonso EE. Estimation of parameters in geotechnical back analysis—II. Application to a tunnel excavation problem. Comput Geotech. 1996;18(1):29‐46. 9. Honjo Y, Tsung LW, Guha S. Inverse analysis of an embankment on soft clay by extended Bayesian method. Int J Numer Anal Meth Geomech. 1994a;18:709‐734. 10. Honjo Y, Tsung LW, Sakajo S. Application of Akaike information criterion statistics to geotechnical inverse analysis: the extended Bayesian method. Struct Saf. 1994b;14:5‐29. 11. Bittanti S, Maier G, Nappi A. Inverse problems in structural elasto‐plasticity: a Kalman filter approach. In: Sawczuk A, Bianchi G, eds. Plasticity Today. London: Elsevier Applied Science Publishers; 1984:311‐329. 12. Stefano M, Corigliano A. Impact induced composite delamination: state and parameter identification via joint and dual extended Kalman filters. Comput Methods Appl Mech Eng. 2005;194:5242‐5272. 13. Nguyen LT, Datcheva M, Nestorovic T. Identification of a fault zone ahead of the tunnel excavation face using the extended Kalman filter. Mech Res Commun. 2013;53:47‐52. 14. Murakami A, Hasegawa T. Back analysis by Kalman filter finite elements and optimal location of observed points. In: Swoboda G, ed. Numerical Methods in Geomechanics. Rotterdam: Balkema; 1988:2051‐2058. 15. Murakami A. Application of Kalman Filtering to Some Geomechanical Problems Related to Safety Assessment. Doctoral Dissertation, Kyoto University; 1991. 16. Murakami A. The role of Kalman filtering in an inverse analysis of elasto‐plastic material. Proc Jpn Acad Ser B. 2002;78(8):250‐255. 17. Hommels A, Murakami A, Nishimura S. A comparison of the ensemble Kalman filter with the unscented Kalman filter: application to the construction of a road embankment. GEO Int. 2009;52‐54. 18. Murakami A, Shuku T, Nishimura S, Fujisawa K, Nakamura K. Data assimilation using the particle filter for identifying the elasto‐plastic material properties of geomaterials. Int J Numer Anal Methods Geomech. 2013;37:1642‐1669.
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How to cite this article: Murakami A, Shinmura H, Ohno S, Fujisawa K. Model identification and parameter estimation of elastoplastic constitutive model by data assimilation using the particle filter. Int J Numer Anal Methods Geomech. 2018;42:110–131. https://doi.org/10.1002/nag.2717
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A P P E N D IX A. Halton sequence This appendix presents the process of deriving the Halton sequence.44 It is a generalization of a Van der Corupt sequence. When defining a base as integer b ≥ 2, the base b expansion of integer n can be written in the following equation: n ¼ a0 þ a1 b þ a2 b2 þ ⋯ þ am bm 0 ≤ ai < b
(A:1)
where n is larger than 0. Then, the radical‐inverse function of n is defined by ϕ b ðn Þ ¼
a0 a1 a2 am þ 2 þ 3 þ ⋯ þ mþ1 b b b b
(A:2)
The Van der Corput sequence in base b is defined by the following formula: x n ¼ ðϕ b ðn ÞÞ A Halton sequence can be simply expressed by the k‐dimensional generalization of a Van der Corput sequence. X n ¼ ϕb1 ðnÞ; ⋯; ϕbk ðnÞ n ¼ 0; 1; 2; …
(A:3)
(A:4)
where b1, …, bk is pairwise prime and larger than 1. Figure A1 shows an example of comparing a uniform random number with a low‐discrepancy sequence.
FIGURE A1
Comparison of uniform random number and low‐discrepancy sequence
