Selecting parameters to optimize in model calibration by inverse analysis Michele Calvello, Richard J. Finno * Department of Civil and Environmental Engineering, Northwestern University, Evanston, IL 60208, USA Received 28 May 2003; received in revised form 11 August 2003; accepted 26 March 2004 Available online 25 May 2004 Abstract A study evaluating the benefits of using inverse analysis techniques to select the appropriate parameters to optimize when calibrating a soil constitutive model is presented. The factors that affect proper calibration are discussed with reference to the op- timization of the elasto-plastic Hardening-Soil model for four layers of Chicago glacial clays. The models are initially calibrated using results from triaxial compression tests performed on specimens from four clay layers and subsequently re-calibrated using incli- nometer data that recorded the displacements of a supported excavation in these clays. Finite element simulations of both the triaxial tests and the supported excavation are performed. A parameter optimization algorithm is used to fit the computed results and ob- served data, expressed in the form of stress–strain curves and inclinometer readings, respectively. A procedure is presented which uses the results of sensitivity analyses conducted on the soil model parameters for the identification of the relevant and uncorrelated parameters to calibrate. In both cases the inverse analysis methodology effectively calibrates the soil parameters considered, which numerically converge to realistic values that minimize the errors between computed responses and experimental observations. Ó 2004 Elsevier Ltd. All rights reserved. 1. Introduction In a finite element simulation of a geotechnical problem, calibrations of the models used to reproduce soil behavior often pose significant challenges. Real soil is a highly nonlinear material, with both strength and stiffness depending on stress and strain levels. Numerous constitutive models have been developed that can cap- ture many of the important features of soil behavior. However, developing soil parameters for use in consti- tutive models is a procedure that involves much judg- ment and usually is best accomplished by experienced users of a particular model. An effective and more ob- jective way to calibrate a soil model employs inverse analysis techniques to minimize the difference between experimental data (laboratory or field tests) and nu- merically computed results [1,2]. For some large geotechnical engineering projects, for example, deep supported excavations in urban envi- ronments, it is usual to record ground movements developed during construction to evaluate the perfor- mance of the designed system. In some cases the data are used to control the construction process and update predictions of movements given the measured defor- mations at early stages of constructions. This procedure is referred to as the ‘‘observational method’’ [3–5]. This approach usually entails the use of pre-construction analysis and parametric studies coupled with much en- gineering judgment. Inverse analysis techniques con- ceptually can be used to enhance the conventional observational method practice by using the monitoring data to optimize automatically a numerical model of a geotechnical project. Recent work in related civil engi- neering fields (e.g. [6–9]) demonstrate that inverse modeling provides capabilities that help modelers sig- nificantly, even when the simulated systems are very complex. However, there are a number of issues that affect proper calibration, including the number of parameters to be optimized, which depends on both the site stra- tigraphy and number of parameters in the selected * Corresponding author. E-mail address: r-fi[email protected](R.J. Finno). 0266-352X/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compgeo.2004.03.004 Computers and Geotechnics 31 (2004) 411–425 www.elsevier.com/locate/compgeo
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Computers and Geotechnics 31 (2004) 411–425
www.elsevier.com/locate/compgeo
Selecting parameters to optimize in model calibrationby inverse analysis
Michele Calvello, Richard J. Finno *
Department of Civil and Environmental Engineering, Northwestern University, Evanston, IL 60208, USA
Received 28 May 2003; received in revised form 11 August 2003; accepted 26 March 2004
Available online 25 May 2004
Abstract
A study evaluating the benefits of using inverse analysis techniques to select the appropriate parameters to optimize when
calibrating a soil constitutive model is presented. The factors that affect proper calibration are discussed with reference to the op-
timization of the elasto-plastic Hardening-Soil model for four layers of Chicago glacial clays. The models are initially calibrated using
results from triaxial compression tests performed on specimens from four clay layers and subsequently re-calibrated using incli-
nometer data that recorded the displacements of a supported excavation in these clays. Finite element simulations of both the triaxial
tests and the supported excavation are performed. A parameter optimization algorithm is used to fit the computed results and ob-
served data, expressed in the form of stress–strain curves and inclinometer readings, respectively. A procedure is presented which uses
the results of sensitivity analyses conducted on the soil model parameters for the identification of the relevant and uncorrelated
parameters to calibrate. In both cases the inverse analysis methodology effectively calibrates the soil parameters considered, which
numerically converge to realistic values that minimize the errors between computed responses and experimental observations.
� 2004 Elsevier Ltd. All rights reserved.
1. Introduction
In a finite element simulation of a geotechnical
problem, calibrations of the models used to reproduce
soil behavior often pose significant challenges. Real soil
is a highly nonlinear material, with both strength and
stiffness depending on stress and strain levels. Numerousconstitutive models have been developed that can cap-
ture many of the important features of soil behavior.
However, developing soil parameters for use in consti-
tutive models is a procedure that involves much judg-
ment and usually is best accomplished by experienced
users of a particular model. An effective and more ob-
jective way to calibrate a soil model employs inverse
analysis techniques to minimize the difference betweenexperimental data (laboratory or field tests) and nu-
merically computed results [1,2].
For some large geotechnical engineering projects, for
example, deep supported excavations in urban envi-
412 M. Calvello, R.J. Finno / Computers and Geotechnics 31 (2004) 411–425
constitutive model, the interdependence of the model
parameters within the framework of the constitutive
model, the number of observations, and the type of
system under consideration.
In this paper, these factors are discussed and illus-trated by presenting results of inverse analyses used to
optimize the calibration of the Hardening-Soil (H-S)
model [10] for four layers of Chicago glacial clays. The
models are initially calibrated using results from triaxial
compression tests performed on specimens from the four
clay layers and subsequently re-calibrated using incli-
nometer data that recorded the displacements of a
supported excavation in these clays [11]. This paperdescribes the concepts of model calibration by inverse
analysis, summarizes the soil model used to define the
behavior of the clay considered, discusses the factors
that affect proper calibration, presents the results of the
model calibration from triaxial test data and from field
monitoring data and draws conclusions.
2. Model calibration by inverse analysis
In inverse analysis, a given model is calibrated by
iteratively changing input values until the simulated
output values match the observed data (i.e., observa-
tions). Fig. 1 shows a schematic of an inverse analysis
procedure. The input parameters are initially estimated
by conventional means. Much literature exists on thissubject, for example, a number of papers in the McGill
conference (i.e. [12–14]) describe how this first step is
done for a number of constitutive models for soils, i.e.,
a hyperbolic stress–strain model [12,13], and a
bounding surface model [14]. A numerical simulation
Numerical Model
Input parameters
Updated input parameters
Regression(objective function minimization)
Modeloptimized?
Initial input parameters
NO
YES
Computed results
ENDSTART
Optimized input parameters
Observations
Iter
ativ
e pr
oces
s
Fig. 1. Schematic of inverse analysis procedure.
of the problem is conducted and the simulated results
are compared to the available observations. A regres-
sion analysis is performed to minimize an objective
function, which quantifies the fit between computed
results and observations. Its minimization is attainedby the optimization of the input parameters needed to
perform the numerical simulation. If the model fit is
not ‘‘optimal’’, the procedure is repeated until the
model is optimized.
Inverse analysis algorithms allow the simultaneous
calibration of multiple input parameters. However,
identifying the important parameters to include in the
inverse analysis can be problematic. Indeed, in mostpractical problems it is not possible to use the regres-
sion analysis to estimate every input parameter of a
given simulation. The number and type of input pa-
rameters that one can expect to estimate simultaneously
depend upon many factors, including the characteristics
of the selected soil model, how the model parameters
are combined within the element stiffness matrix in a
finite element formulation, the site stratigraphy, thenumber and type of observations available, the char-
acteristics of the simulated system, and computational
time issues.
Fig. 2 shows a procedural flowchart used for the
identification of the parameters to optimize in the finite
element simulations of geotechnical problems. Note that
the first step of the procedure refers to the selection of
the model parameters that are relevant to the problemunder study. The last two steps, necessary if multiple soil
layers are calibrated simultaneously, refer to the selec-
tion of the total number of parameters that are opti-
mized in the simulation of the field scale problem. In this
paper, the first step is illustrated with results of simu-
lations of triaxial compression tests and the latter two
steps are illustrated with the simulation of a deep, sup-
ported excavation.In the work described herein, model calibration by
inverse analysis is conducted using UCODE [15], a
computer code designed to allow inverse modeling
posed as a parameter estimation problem. UCODE was
developed for ground-water models, but it can be ef-
fectively used in geotechnical modeling because it works
with any application software that can be executed in a
batch mode. Its model-independency allows the chosennumerical code to be used as a ‘‘closed box’’ in which
modifications only involve model input values. This is
an important feature of UCODE in that it allows one to
develop a procedure that can be easily employed in
practice and in which the engineer will not be asked to
use a particular finite element code or inversion algo-
rithm. Rather, macros can be written in a windows en-
vironment to couple UCODE with any finite elementsoftware. The commercial software PLAXIS 7.11 [16]
was used herein to simulate the soil behavior with the
H-S model [10].
Input.txtPLAXIS Input
PLAXIS Calculation
PLAXIS Output Output
ASCII I/OPLAXIS
Modeloptimized?
START
Initial parameters
NO
YES
Observations
Macro
Regression(multiple PLAXIS runs)
Best-fit parameters
END
Updated parameters
Fig. 3. Inverse analysis with UCODE and PLAXIS.
Fig. 2. Identification of soil parameters to optimize by inverse analysis.
M. Calvello, R.J. Finno / Computers and Geotechnics 31 (2004) 411–425 413
Fig. 3 shows a schematic of the interaction between
UCODE and PLAXIS during the inverse analysis.
PLAXIS, a Windows-based program, does not have any
option to save input or output files in ASCII format.
Therefore, windows-macros were written to convert thePLAXIS I/O into text files. The macros are needed to
produce model changes in PLAXIS-Input from an input
text file, switch between PLAXIS modules, calculate the
simulated model in PLAXIS-Calculation, and generate
an output text file from the PLAXIS-Output. Note that
the procedure needs no user intervention once the
analysis has been started. For more details, see [17].
3. Chicago glacial clays
Much of the subsoil in the Chicago area consists of
fairly distinct strata deposited during the advances and
retreats of a glacier during the Wisconsin Stage. Theadvance and retreat process, marked by terminal mo-
raines, created easily identifiable clay strata. In order of
deposition they are the Valparaiso, Tinley, Park Ridge,
Deerfield, Blodgett, and Highland Park tills [18]. Fig. 4
shows the soil profile at the site of the excavation con-
sidered herein, typical for the downtown area of Chi-
cago. All elevations refer to the Chicago City Datum
-200
-100
0
100
200
0 100 200 300 400
p=(σ1+2σ3)/3
q=σ 1
- σ3
Mohr-Coulombfailure line Shearing
yieldsurfaces
Yield CapSurface
Fig. 5. Hardening-Soil yield surfaces.
-20
-15
-10
-5
0
5
z (m)
Elevation(m CCD)
4.3
0.6-0.3
-4.6
-7.1
-10.8
-15.4
-18.5
Sand / Fill
Clay crust
Soft clay (Layer 1)
Soft-Medium clay (Layer 2)
Medium clay (Layer 3)
Stiff clay (Layer 4)
Very stiff clay
Hard Pan
STRATIGRAPHICUNIT
Blo
dget
tst
ratu
mD
eerf
ield
stra
tum
Park
Rid
gest
ratu
mT
inle
yst
ratu
m
Fig. 4. Subsurface profile.
414 M. Calvello, R.J. Finno / Computers and Geotechnics 31 (2004) 411–425
(CCD), the zero value of which corresponds to the av-
erage level of adjacent Lake Michigan. For the purposes
of the laboratory experimental program, four main
layers were identified in order of increasing depth: the
Upper Blodgett, the Lower Blodgett, the Deerfield, and
the Park Ridge. The laboratory experiments concen-trated on these four normally to lightly overconsolidated
clay layers because they are far more compressible than
the lower Tinley stratum clays, and thus they have the
largest effect on the soil mass response to the excavation.
Note that the Blodgett layer is a supraglacial till and
typically exhibits more variability than the underlying
Table 1
H-S input parameters
Parameter Explanation
Basic parameters
/ Friction angle
c Cohesion
w Dilatancy angle
Eref50 Secant stiffness in standard drained tria
Erefoed Tangent stiffness for primary oedomete
m Power for stress-level dependency of sti
Advanced parameters
Erefur Unloading–reloading stiffness
mur Poisson’s ratio
Rf Failure ratio qf=qak0 k0 value for normally consolidated soil
stratum [18]. In the inverse analysis described herein,
these soil strata are referred to as layers 1, 2, 3, and 4.
4. The H-S model
The soil model used to simulate the clay behavior is
the H-S model as implemented in PLAXIS 7.11. The H-
S model is an elasto-plastic, multi-yield surface, effective
stress soil model. Failure is defined by the Mohr–Cou-
lomb failure criterion. Two families of yield surfaces are
incorporated in the model to account for both volu-
metric and shear plastic strains. Fig. 5 shows the yieldsurfaces of the model in p–q stress space. A yield cap
surface controls the volumetric plastic strains. On this
cap, the flow rule is associative. On the shearing yield
surfaces, increments of plastic strain are nonassociative
and the plastic potential is defined to assure a hyperbolic
stress–strain response for a triaxial compression loading.
The basic characteristics of the model are a Mohr–
Coulomb failure with input parameters c, / and dilat-ancy angle, w, stress-dependent stiffness according to a
Initial estimates
Slope of faillure line in rn � s stress space
y-axis intercept in rn � s stress space
Function of /peak and /failure
xial test y-axis intercept in logðr3=pref Þ � logðE50Þ spacer loading y-axis intercept in logðrv=pref Þ � logðE50Þ spaceffness Slope of trendline in logðr3=prefÞ � logðE50Þ space
default¼ 3Eref50
default¼ 0.2
default¼ 0.9
conditions default¼ 1� sin/
M. Calvello, R.J. Finno / Computers and Geotechnics 31 (2004) 411–425 415
power law defined by input parameter, m, plastic
straining resulting from primary deviatoric loading with
an input parameter, Eref50 , and plastic straining from
primary compression with an input parameter Erefoed.
Elastic unloading–reloading is defined by input param-eters Eref
ur and mur.Table 1 shows the 10 H-S model input parameters,
their meaning and the conventional way of estimating
them. All parameters can be derived from the results of
drained triaxial compression and standard consolidation
tests. The failure parameters / and c are estimated as-
suming a Mohr–Coulomb failure criterion. The dilat-
ancy angle, w, depends on the volume changecharacteristics of the soil and is equal to zero for nor-
mally consolidated clays. The stiffness parameters Eref50 ,
Erefoed, and m are estimated by assuming the values the
50% secant and oedometric stiffnesses (E50 and Eoed,
respectively) are related to a reference pressure, pref ,usually set equal to 100 stress units. The advanced pa-
rameters are generally set equal to their default values.
Hence to begin a problem, one must estimate six pa-rameters for each soil type.
5. Optimization of laboratory data
Triaxial compression tests were performed on sam-
ples from the four most compressible clay layers on the
site of the Chicago Avenue and State Street subwayrenovation project. The triaxial testing program was
part of an experimental laboratory program conducted
to define the soil properties at the excavation site
[19,20]). The H-S model, used to simulate the behavior
of the clay specimens, was calibrated for the four clay
Table 2
Triaxial experimental program
Layer Depth of
samples (m)
Test type Test name Consolidation
stress (kPa)
1 5–6 CID TXC D1 107
D2 200
D3 400
CIU TXC U1 100
2 10.5–11.5 CID TXC D1 134
D2 220
D3 400
CIU TXC U1 130
3 12–13 CID TXC D1 175
D2 350
D3 450
CIU TXC U1 168
4 16.5–17.5 CID TXC D1 200
D2 350
D3 450
CIU TXC U1 204
layers based on these triaxial results using the inverse
analysis procedure presented in [2].
5.1. Experimental program
Table 2 summarizes the isotropically consolidated,
drained, and undrained triaxial compression tests per-
formed on soil samples from each of the four clay layers
considered. Fig. 6 shows the results of these tests. At
every layer, three drained (CID TXC) and one undrained
(CIU TXC) triaxial compression tests were conducted.
The test samples were first consolidated to different iso-
tropic effective stresses, the in situ vertical effective stress(tests D1 and U1) and two significantly higher stresses
(tests D2 and D3), and then sheared until failure. The
principal stress difference, the axial and volumetric
strains, and/or the excess pore pressures were recorded.
As expected for normally to lightly overconsolidated
clays, results of the drained tests show that the devia-
toric stresses and volumetric strains at failure generally
increase with depth at this site where the upper-strataclay samples are ‘‘weaker’’ than lower-strata ones and,
for a given layer, with increasing consolidation pressure.
Note that trends are not as clear for layers 1 and 2 de-
rived from the Blodgett stratum which is typically more
variable than the lower strata. For the undrained tests,
both the deviatoric stress and the excess pore pressures
at failure increase with increasing depth and, thus, in-
creasing effective consolidation pressure.
5.2. Optimization scheme
The observation points used for the inverse analysis
were selected from the stress–strain curves presented in
Fig. 6. The stress–strain curves of the drained test were
discretized by considering one observation point every
2% axial strain up to a maximum of ea ¼ 12%. Curves forthe undrained test were discretized using eight observa-
tion points per curve; the observation points were se-
lected more frequently at small strains, every 0.15% axial
strain up to ea ¼ 0:9%, so that the pore pressure varia-
tion would be adequately defined. Calvello and Finno [2]
showed that this number of observations points was
sufficient to define the stress–strain responses.
The initial values of the H-S input parameters of thefour clay layers were computed according to conven-
tional calibration procedures briefly summarized in
Table 3. The parameters optimized by inverse analysis
were chosen, according to the procedure shown in
Fig. 2, among the six H-S basic parameters. The four
advanced parameters, as suggested by the PLAXIS
manual, were always set equal to their default values.
The four clay layers were calibrated independently.To evaluate the relative importance of each input
parameter, a sensitivity analysis was conducted for the
six parameters at every layer, using the stress–strain
Fig. 6. Experimental results of triaxial tests.
416 M. Calvello, R.J. Finno / Computers and Geotechnics 31 (2004) 411–425
results of the triaxial tests as observations. The results of
these analyses are used to determine the number ofrelevant and uncorrelated parameters in each layer. The
characteristics of the H-S model as implemented in
PLAXIS, the type of observations, and the stress con-
ditions in the soil samples all influence the sensitivity of
the observations to changes in parameter values.
The relevant parameters are discerned based on the
composite scaled sensitivity, cssj:
cssj ¼XNDj¼1
oy 0iobj
� �bjx
1=2ii
� �2�����b
,ND
" #1=2
; ð1Þ
where y 0i is the ith simulated value; bj is the jth esti-
mated parameter; oy0i=obj is the sensitivity of the ithsimulated value with respect to the jth parameter; xjj
is the weight of the ith observation wherein the weight
of every observation is taken as the inverse of its error
variance, and ND is the number of observations. See
[2] for more details concerning x for the laboratory
data and [7] for the inclinometer data. The composite
scaled sensitivities indicate the total amount of infor-
mation provided by the observations for the estima-tion of parameter j and measure the relative
importance of the input parameters being simulta-
neously estimated.
Table 3
Initial and best-fit values of input parameters: laboratory results
The reference pressure is pref ¼ 100 kPa.aOptimized based on regression analysis.b Parameters do not affect predicted results since negligible; values not changed.
M. Calvello, R.J. Finno / Computers and Geotechnics 31 (2004) 411–425 417
The correlated parameters are discerned using the
correlation coefficients, corði; jÞ:
corði; jÞ ¼ covði; jÞvarðiÞ1=2varðjÞ1=2
; ð2Þ
where covði; jÞ equal the off-diagonal elements of the
variance–covariance matrix V ðb0Þ ð¼ s2ðX TxX Þ�1Þ, s2 isthe model error variance, and varðiÞ and varðjÞ refer tothe diagonal elements of V ðb0Þ. The values of corði; jÞindicate the correlation between the ith and jth pa-
rameters. Highly correlated parameters should not be
optimized simultaneously because many combinationscan lead to the same optimized result, but with unre-
alistic values of the optimized parameters. Values close
to )1.0 or 1.0 are indicative of parameters that cannot
be uniquely estimated with the observations used in the
regression.
Fig. 7. H-S input parameters: (a) composite scale
The estimates of parameter Erefoed could not be opti-
mized by inverse analysis because the H-S model im-
plemented in PLAXIS has an internal algorithm that
runs every time a new set of input parameters is speci-
fied. This algorithm considers the deviatoric stress re-sponse of an internally modeled compression test and
‘‘adjusts’’ the values of parameter Erefoed to produce a
hyperbolic curve in a triaxial stress–strain space. The
iterative nonlinear regression method used herein is
based on the value of a sensitivity matrix X ; Xij ¼oyi=obj. Therefore, parameter Eref
oed cannot be included in
the optimization because the regression algorithm would
compute wrong sensitivities every time its value is‘‘corrected’’ by PLAXIS.
Fig. 7 shows that the main parameters that affect the
observations, i.e., those with the higher values of cssj,
are /, Eref50 , and m for all four soil layers. The values of